Nonlinear Oscillator and a Non-ideal Energy Source

Abstract

In this chapter the motion of the non-ideal system which contains a nonlinear one degree-of-freedom oscillator and a non-ideal energy source is considered. In such a non-ideal oscillator-motor system there is an interaction between motions of the oscillator and of the motor as it was already explained in the previous chapter. However, due to nonlinear properties of the oscillator in the non-ideal system beside the Sommerfeld effect some additional phenomena are evident. Depending on the parameter properties of the oscillator the motion is regular or irregular.

In this chapter the motion of the non-ideal system which contains a nonlinear one degree-of-freedom oscillator and a non-ideal energy source is considered. In such a non-ideal oscillator-motor system there is an interaction between motions of the oscillator and of the motor as it was already explained in the previous chapter. However, due to nonlinear properties of the oscillator in the non-ideal system beside the Sommerfeld effect some additional phenomena are evident. Depending on the parameter properties of the oscillator the motion is regular or irregular. Results on motion of the non-ideal systems with nonlinear oscillators are published in Dimentberg et al. (1997), Warminski et al. (2001), Warminski and Kecik (2006) Dantas and Balthazar (2007), Felix et al. (2009a), Zukovic and Cveticanin (2007, 2009), Nbendjo et al. (2012), Cveticanin and Zukovic (2015a, b) etc.

This chapter is divided into five sections. In Sect. 3.1, a generalization of the model of the non-ideal oscillator-motor is done: a strong nonlinear oscillator is coupled with a motor with nonlinear torque property. The model of the structure-motor system is generalized by assuming that the driving torque is a nonlinear function of the angular velocity and the oscillator is with strong nonlinearity. The oscillator-motor system is assumed as a non-ideal one where not only the motor affects the motion of the oscillator but also vibrations of the oscillator have an influence on the motor motion. The model of the motor-structure system is described with two coupled strong nonlinear differential equations. An improved asymptotic analytic method based on the averaging procedure is developed for solving such a system of strong nonlinear differential equations. The steady state motion and its stability is studied. Results available the discussion of the Sommerfeld effect. A new procedure for determination of parameters of the non-ideal system for which the Sommerfeld effect does not exist is developed. For these critical values of the parameter the Sommerfeld effect is suppressed. In Sect. 3.2, the suggested theoretical consideration is applied for pure nonlinear oscillator driven by a motor with nonlinear torque characteristics. As a special case the pure nonlinear oscillator where the order of the nonlinearity is a positive rational number is investigated. The influence of the order of nonlinearity on the dynamic properties of the system is also analyzed. The numerical simulation is done for the motor oscillator system, where the motor torque is a cubic function of the angular velocity and the oscillator is with pure nonlinearity. The obtained results are compared with those obtained analytically (Cveticanin and Zukovic 2015b). In Sect. 3.3, a pure strong nonlinear oscillator which is coupled to a non-ideal source whose torque is a linear function of the angular velocity is considered. An analytical solving procedure based on averaging is developed. The approximate solution has the form of the Ateb function. The resonant case is considered. Steady-state solution and characteristic points are determined. Special attention is given to suppression of the Sommerfeld effect. The section ends with some numerical examples (Cveticanin and Zukovic 2015a). In Sect. 3.4 the non-ideal system with the Duffing oscillator of cubic type is analyzed. The hardening Duffing oscillator with one stable fixed point which is coupled to a non-ideal energy source is mathematically modelled and analytically solved. Approximate solution of the problem is calculated. Condition for the steady-state motion are obtained. Stability of motion is investigated and phenomena of jump in the amplitude-frequency diagram is treated. Using the numerical simulation the chaotic motion is detected. A procedure for controlling chaos is introduced (Zukovic and Cveticanin 2007). Finally, in Sect. 3.5, the non-ideal system with bistable Duffing oscillator, which has three fixed points, is considered. The semi-trivial and non-trivial solutions are determined. Based on the semi-trivial solutions the conditions for quenching of the amplitude of the mechanical system are obtained. In this chapter the stability of non-trivial solutions is investigated. Based on the signs of the Lyapunov exponents (Lyapunov 1893) regions of chaos and hyperchaos are determined (Nbendjo et al. 2012).

3.1 Nonlinear Oscillator Coupled with a Non-ideal Motor with Nonlinear Torque

In the previous chapter the systems which have the following limitations are considered:

• the elastic force of the structure is assumed to be linear (Zukovic and Cveticanin 2009), or with small nonlinearity (Dimentberg et al. 1997; Warminski et al. 2001; Dantas and Balthazar 2006; Felix et al. 2009b),

• the torque property of the motor is assumed to be a linear function of the angular velocity (Dantas and Balthazar 2003; Tsushida et al. 2003 and 2005; Souza et al. 2005a and 2005b; Castao et al. 2010).

In this section a generalization to the model of motor-structure system is done. It is assumed that the elastic property of the system need not be linear or with a small nonlinearity but with a strong nonlinearity of any order, described with any positive rational exponent of the displacement. The motor torque is assumed to be a nonlinear function of angular velocity. No limitation to the form of the forcing torque is introduced. Such a generalization gives us an opportunity to give a more realistic view of the dynamics of the system.

In general, the model of the non-ideal system is an oscillator-motor one which has two degrees of freedom. The motion is described with a system of two coupled differential equations (see Felix et al. 2009b)

$$\begin{aligned} m \ddot{x}+\hat{f}_{1}(x)=\hat{f}_{2}(x,\dot{x})+\hat{F}(\varphi ,\dot{\varphi } ,\ddot{\varphi },\hat{q}),\qquad I\ddot{\varphi }=\mathcal {M(}\dot{\varphi })+ \hat{R}(\varphi ,\dot{\varphi },\ddot{x},\hat{q}), \end{aligned}$$

(3.1)

where x and $\varphi$ are generalized coordinates of the system (displacement and angular position), $\hat{f}_{1}$ is the deflection function of the oscillator, $\hat{f}_{2}$ is the function which describes other properties of the oscillator (damping, relaxation, hysteresis...), $\hat{F}$ and $\hat{R}$ are coupling functions of the oscillator and the motor, m is the mass of the oscillator, I is the moment of inertia of the rotating part of the motor and $\hat{q}$ is the measure of the unbalance of the rotor. The deflection function of the oscillator $\hat{f}_{1}(x)$ is usually assumed to be linear or weakly nonlinear, while the function $\hat{f} _{2}$ is supposed to be a small one. In general, the torque of the electro-motor is

$$\begin{aligned} \mathcal {M(}\dot{\varphi })=L(\dot{\varphi })-H(\dot{\varphi }), \end{aligned}$$

(3.2)

where $\dot{\varphi }$ is the angular velocity of the motor and $L(\dot{ \varphi })$ and $H(\dot{\varphi })$ are driving and resisting torques. The torque property of the motor is usually assumed to be a linear function of the angular velocity

$$\begin{aligned} \mathcal {M(}\dot{\varphi })=V_{m}^{*}-C_{m}^{*}\dot{\varphi }, \end{aligned}$$

(3.3)

where $C_{m}^{*}$ and $V_{m}^{*}$ are characteristics of the motor (Felix et al. 2009b). Comparing the real motor torque with the (3.3) it is evident that the assumed model represents the first approximation of the real one.

The aim is to make the generalization of the problem on non-ideal systems considering the whatever any nonlinear oscillator and the improved version of the motor torque model. The function $\hat{f}_{1}(x)$ in (3.1) need not to be a small nonlinear function, but may be a strong nonlinear one . The elastic force in the oscillator is the function of any rational order of the displacement (integer or non-integer). Besides, the motor torque (3.2) need not to be a linear velocity function (3.3), but it may be a nonlinear one.

3.1.1 Nonlinear Motor Torque Property

As it is stated by Nayfeh and Mook (1979), for determination of the influence of the motion on the motor properties it is necessary to know the characteristics of the motor. Kononenko and Korablev (1959) plotted experimentally obtained torques as a function of the frequency or angular velocity of rotor for various types of DC motors, for an asynchronous and a synchronous motor. For the most of the mentioned characteristics it is common that they are nonlinear .

The DC series wound motor , which is considered in this section, has a net of nonlinear characteristics for various constant control or regulator parameters (see Kononenko 1969). Mathematical model of the motor characteristics is assumed in the form

$$\begin{aligned} \mathcal {M(} \dot{\varphi })=(V_{m}^{*}-C_{m}^{*}\dot{\varphi })^{n}, \end{aligned}$$

(3.4)

where $V_{m}^{*}$ and $C_{m}^{*}$ are constant parameters and $n=2,3,4,\ldots$ is a positive integer. It should be mentioned that the relation (3.4) includes the linear model (3.3) for $n=1$. The DC series wound motor develops a large torque and can be operated at low speed. It is a motor that is well suited for starting heavy loads. Because of that it is often used for industrial cranes and winches, where very heavy loads must be moved slowly and lighter loads moved very rapidly. Introducing the dimensionless time parameter $\tau$, the relation (3.4) transforms into

$$\begin{aligned} \Gamma (\Omega )=(V_{m}-C_{m}\Omega )^{n}, \end{aligned}$$

(3.5)

where $\Gamma (\Omega )$ is the dimensionless driving torque with dimensionless parameters $V_{m}$ and $C_{m}$ and angular velocity $\Omega$.

Fig. 3.1

Torque curves for various values of control parameter $V_{m}$ and constant parameter n: a linear property ($n=1$), b cubic property ($n=3$)

In Fig. 3.1 the torque curves for various values of control parameter $V_{m}$ and constant parameter n are plotted: (a) linear property ($n=1$), (b) cubic property ($n=3$). It can be concluded that for increasing of the control parameter $V_{m}$ the curves move to right in the $\Gamma (\Omega )-\Omega$ plane. For the arbitrary value of the motor frequency, the higher the control parameter $V_{m}$, the higher the value of the torque $\Gamma (\Omega )$. In Fig. 3.2, the torque curves for constant value of parameter $V_{m}$ and various values of the order of nonlinearity is plotted. Increasing the order of nonlinearity n the curves move to the left in the $\Gamma (\Omega )-\Omega$ plane.

Fig. 3.2

Torque curves for various values of the nonlinearity order n and constant control parameter $V_{m}$

Namely, for a certain constant frequency, the motor torque is higher if the order of nonlinearity n is smaller. Nevertheless, the driving torque is zero and independent on the order n for the angular velocity $\Omega =V_{m}/C_{m}$. The higher the control parameter $V_{m}$, the higher the value of the angular velocity $\Omega$ for which the driving torque is zero.

3.1.2 Solution Procedure in General

Introducing the motor characteristics (3.5) into (3.1) and after some modification dimensionless differential equations follow as

$$\begin{aligned} y\text {''}+f_{1}(y)= & {} f_{2}(y,y^{\prime })+F(\varphi ,\varphi ^{\prime },\varphi \text {''},q), \end{aligned}$$

(3.6)

$$\begin{aligned} \varphi \text {''}= & {} \Gamma \mathcal {(}\varphi ^{\prime })+R(\varphi ,\varphi ^{\prime },y\text {''},q), \end{aligned}$$

(3.7)

where y and $\varphi$ are the dimensionless generalized coordinates, q is the dimensionless unbalance measure, $f_{1}$, $f_{2}$, F, R and $\Gamma$ are the functions of the dimensionless coordinates and parameters and $(^{\prime })=$d / d$\tau$, $(\text {''})=$d$^{2}/$d$\tau ^{2}$ with dimensionless time $\tau$.

To classify the ‘small’ and ‘arge’ values in the system (3.6), (3.7), we introduce a small parameter $\varepsilon<<1$. Due to the physical sense of the problem the functions $f_{2}$, F, R and $\Gamma$ are small and we have

$$\begin{aligned} y\text {''}+f_{1}(y)= & {} \varepsilon f_{2}(y,y^{\prime })+\varepsilon F(\varphi ,\varphi ^{\prime },\varphi \text {''},q), \end{aligned}$$

(3.8)

$$\begin{aligned} \varphi \text {''}= & {} \varepsilon ^{2}\Gamma \mathcal {(}\varphi ^{\prime })+\varepsilon ^{2}R(\varphi ,\varphi ^{\prime },y\text {''},q). \end{aligned}$$

(3.9)

The terms on the right side of Eqs. (3.8) and (3.9) are small values of the first and second order, respectively, but different from zero. It must be mentioned that, in this paper, we analyze the system with small foundation damping and in the Eq. (3.8) the damping term is of order $\varepsilon$.

In the previous investigation it was assumed that the function $f_{1}(y)$ is a linear one, and the Eq. (3.8) is with small perturbed terms on the right-hand side of the equation. In this paper the generalization of the problem is done, as the function $f_{1}(y)$ need not be linear. The suggested mathematical procedure is based on the method described for the perturbed linear differential equation.

If $\varepsilon =0$ the Eqs. (3.8) and (3.9) simplify into two uncoupled differential equations

$$\begin{aligned} y^{\prime \prime }+f_{1}(y)=0,\qquad \varphi ^{\prime \prime }=0. \end{aligned}$$

(3.10)

For the case when $f_{1}(y)$ is the linear deflection function, the differential equation (3.10) is a linear one and has the exact solution in the form of the trigonometric function. Otherwise, the trigonometric function represents only the approximate solution of the nonlinear differential equation (3.10). In the papers (Cveticanin 2009, 2009; Cveticanin and Pogany 2012) it is already shown that the approximate solution of trigonometric type is very close to the numerical solution of (3.10) and represents a satisfactory asymptotic solution. It gives as the opportunity to assume the asymptotic solution to (3.10) in the form

$$\begin{aligned} y=a\cos (\omega (a)t+\psi ),\qquad \varphi ^{\prime }=\Omega , \end{aligned}$$

(3.11)

where a, $\psi$ and $\Omega$ are arbitrary constants which satisfy the initial conditions. It is worth to say that the frequency of vibration $\omega$ of the nonlinear differential equation (3.10) depends on the amplitude a and has to satisfy exactly or approximately the relation

$$\begin{aligned} -a\omega ^{2}\cos (\omega t+\psi )+f_{1}(a\cos (\omega t+\psi ))\approx 0. \end{aligned}$$

(3.12)

It is of special interest to consider the resonant case (see Cveticanin 1995), when the difference between the frequency of vibration of the structure $\omega (a)$ and of the driving frequency $\Omega$ is small. Due to the fact that $\omega$ depends on a, there is a trace of frequencies which have to satisfy the relation

$$\begin{aligned} \Omega -\omega (a)=(\varepsilon \sigma )^{2}, \end{aligned}$$

(3.13)

where $\varepsilon \sigma<<1$. The solution of (3.11) and its first time derivative for the resonant case are

$$\begin{aligned} y=a\cos (\varphi +\psi ),\qquad y^{\prime }=-a\Omega \sin (\varphi +\psi ). \end{aligned}$$

(3.14)

The method suggested in this paper requires the solution of (3.8) and (3.9) to be close to (3.14) which is the solution of (3.11). Namely, the solution of (3.8) is the perturbed version of (3.14), where the parameters are time variable. Using the procedure given by Kononenko (1969) and Cveticanin (1992) the solution of (3.8) is suggested in the form

$$\begin{aligned} y=a(\tau )\cos (\varphi (\tau )+\psi (\tau ))\equiv a\cos (\varphi +\psi ), \end{aligned}$$

(3.15)

and

$$\begin{aligned} y^{\prime }=-a(\tau )\Omega (\tau )\sin (\varphi (\tau )+\psi (\tau ))\equiv -a\Omega \sin (\varphi +\psi ). \end{aligned}$$

(3.16)

The first time derivative of (3.15) is

$$\begin{aligned} y^{\prime }=a^{\prime }\cos (\varphi +\psi )-a(\Omega ^{\prime }+\psi ^{\prime })\sin (\varphi +\psi ). \end{aligned}$$

(3.17)

Comparing (3.17) with (3.16), it follows

$$\begin{aligned} a^{\prime }\cos (\varphi +\psi )-a\psi ^{\prime }\sin (\varphi +\psi )=0. \end{aligned}$$

(3.18)

Substituting the solution (3.15) and the corresponding first (3.16) and second time derivative of (3.16) into (3.8), we obtain

$$\begin{aligned}&-a^{\prime }\Omega \sin (\varphi +\psi )-a\Omega (\Omega +\psi ^{\prime })\cos (\varphi +\psi )-a\Omega ^{\prime }\sin (\varphi +\psi ) \nonumber \\&+f_{1}(a\cos (\varphi +\psi )) \nonumber \\= & {} \varepsilon f_{2}(a\cos (\varphi +\psi ),-a\omega \sin (\varphi +\psi ))+\varepsilon F(\psi ,\Omega ,\Omega ^{\prime },q), \end{aligned}$$

(3.19)

where according to (3.11) the differential equation (3.9) transforms into two first order differential equations

$$\begin{aligned} \Omega ^{\prime }= & {} \varepsilon ^{2}\Gamma \mathcal {(}\Omega )+\varepsilon ^{2}R, \end{aligned}$$

(3.20)

$$\begin{aligned} \varphi ^{\prime }= & {} \Omega , \end{aligned}$$

(3.21)

with

$$\begin{aligned} R=R(\varphi ,\Omega ,-a\Omega ^{2}\cos (\varphi +\psi ),q). \end{aligned}$$

Using the relation (3.12) and neglecting terms with the second order small parameter $O(\varepsilon ^{2})$, the relation (3.18) with (3.19) gives two first order differential equations

$$\begin{aligned} a^{\prime }= & {} - \frac{\varepsilon }{\Omega }(F+f_{2})\sin (\varphi +\psi )+a(\Omega ^{2}-\omega ^{2})\sin (\varphi +\psi )\cos (\varphi +\psi ) \end{aligned}$$

(3.22)

$$\begin{aligned} a\psi ^{\prime }= & {} -a\frac{\Omega ^{2}-\omega ^{2}}{\Omega }\cos ^{2}(\varphi +\psi )-\frac{\varepsilon (F+f_{2})}{\Omega }\cos (\varphi +\psi ) \end{aligned}$$

(3.23)

where

$$\begin{aligned} F=F(\varphi ,\Omega ,\Omega ^{\prime },q),\qquad f_{2}=f_{2}(a\cos (\varphi +\psi ),-a\Omega \sin (\varphi +\psi )). \end{aligned}$$

Equations (3.20)–(3.23) are four first order differential equations which correspond to two second order differential equations (3.8) and (3.9). Our task is to solve and analyze these equations.

Averaging procedure

Due to complexity of Eqs. (3.20)–(3.23) it is a heavy task to solve them. This is the reason that the approximate solution procedure for the system of differential equations (3.20)–(3.23) is introduced. In order to eliminate all resonances for the dynamic system described with (3.22) and (3.23), we define the resonant surface by rewriting the relation (3.13) into

$$\begin{aligned} \Omega (\tau )-\omega (a)=(\varepsilon \sigma )^{2}, \end{aligned}$$

with $a=a(\tau )$. Now, we perform the averaging over the slow varying variables and apply the standard averaging procedure (see Zhuravlev and Klimov 1988; Cveticanin 1993, 2003). Averaging Eqs. (3.20)–(3.23) over the period of vibration gives

$$\begin{aligned} a^{\prime }= & {} - \frac{1}{\Omega }(\bar{F}(\psi ,\Omega ,q)+\bar{f}_{2}(a)), \end{aligned}$$

(3.24)

$$\begin{aligned} a\psi ^{\prime }= & {} -a\frac{\Omega ^{2}-\omega ^{2}}{2\Omega }-(\bar{F} ^{*}(\psi ,\Omega ,q)+\bar{f}_{2}^{*}(a)), \end{aligned}$$

(3.25)

$$\begin{aligned} \Omega ^{\prime }= & {} \Gamma \mathcal {(}\Omega )+\bar{R}(\psi ,\Omega ,a,q), \end{aligned}$$

(3.26)

where $\bar{F}$, $\bar{F}^{*}$, $\bar{f}_{2},$ $\bar{f}_{2}^{*}$ and $\bar{R}$ are averaged functions F, $f_{2}$ and R, respectively. Equations (3.24) and (3.25) give variations of the amplitude and initial phase of vibration of the oscillator, while (3.26) describes the variation of the averaged angular velocity of the motor. Solving these equations we obtain $a-t$, $\psi -t$ and $\Omega -t$ relations for various values of parameter. Equations describe the non-stationary motion of the system and give us very objective qualitative analysis of the problem.

It is of special interest to study the influence of the motion of the motor on the oscillator, but also of the oscillator on the motor. It requires the analysis of the coupled system of differential equations (3.24)–(3.26). Solutions $a-t$ and $\Omega -t$ have to be compared with corresponding relations for the case when there is not an interaction between the support and the motor. Then, $\bar{F}=\bar{R}=0$ and the Eqs. (3.24)–(3.26) simplify into

$$\begin{aligned} a^{\prime }= & {} -\frac{\bar{f}_{2}(a)}{\omega (a)},\qquad \xi ^{\prime }= \frac{\bar{f}_{2}^{*}(a)}{a}, \end{aligned}$$

(3.27)

$$\begin{aligned} \Omega ^{\prime }= & {} \Gamma \mathcal {(}\Omega ). \end{aligned}$$

(3.28)

Separating variables in Eqs. (3.27)$_{1}$ and (3.28) and after some calculation, we have

$$\begin{aligned} s(a_{0})-s(a)= & {} \tau , \end{aligned}$$

(3.29)

$$\begin{aligned} \Gamma _{1}\mathcal {(}\Omega )-\Gamma _{1}\mathcal {(}\Omega _{0})= & {} \tau , \end{aligned}$$

(3.30)

where

$$\begin{aligned} s(a)=\int \frac{\bar{f}_{2}(a)}{\omega (a)}da,\qquad \Gamma _{1}\mathcal {(} \Omega )=\int \frac{d\Omega }{\Gamma \mathcal {(}\Omega )}, \end{aligned}$$

$a_{0}$ is the initial amplitude of the oscillator vibration and $\Omega _{0}$ is the initial angular velocity of the motor. Based on (3.27)$_{2}$ and (3.29) the $\psi -a$ and $\psi -\tau$ relations, for the initial phase of vibration of the oscillator, are calculated.

Comparing Eqs. (3.24)–(3.26) and (3.27), (3.28), it is evident that the most significant difference in amplitude and angular velocity is for the resonant case when the difference between the angular velocity of the motor $\Omega$ and the eigenfrequency of the oscillator $\omega$ is quite small.

3.1.3 Steady-State Motion and Its Stability

The steady state motion of the coupled oscillator-motor system is described with the system of three algebraic equations

$$\begin{aligned} \bar{F}(\psi ,\Omega ,q)+\bar{f}_{2}(a)= & {} 0, \end{aligned}$$

(3.31)

$$\begin{aligned} a\frac{\Omega ^{2}-\omega ^{2}}{2\Omega }+\bar{F}^{*}(\psi ,\Omega ,q)+ \bar{f}_{2}^{*}(a)= & {} 0, \end{aligned}$$

(3.32)

$$\begin{aligned} \Gamma \mathcal {(}\Omega )+\bar{R}(\psi ,\Omega ,a,q)= & {} 0, \end{aligned}$$

(3.33)

which represent right-hand sides of Eqs. (3.24)–(3.26). Eliminating the parameter $\xi$ by combining Eqs. (3.31) and (3.32), and also (3.31) and (3.33), a system of two algebraic equations is obtained

$$\begin{aligned} Q_{1}(a,\Omega ,q)=0,\qquad Q_{2}(a,\Omega ,q,V_{m})=0, \end{aligned}$$

(3.34)

where equations give $a-\Omega$ relations for various values of q and parameter $V_{m}$ of the driving torque (see Eq. (3.3)). Algebraic equations (3.34) are nonlinear and very complex. The solution of (3.34) gives the amplitude of oscillator vibration and the angular velocity of motor for the steady state motion. Very often, the solution of (3.34) is analyzed graphically by plotting of the frequency-response curves $a-\Omega$ for the oscillator (relation (3.34)$_{1}$) and the motor (relation (3.34)$_{2}$). The intersection of the curves give us the steady-state parameters of the system.

To analyze the stability of the steady-state solution the Jacobi determinant is formed

$$\begin{aligned} J=\left[ \begin{array}{lll} \frac{{d{\bar{f}}_{2}(a)}}{da} &{} \frac{\partial \bar{F}}{\partial \psi } &{} \frac{\partial \bar{F}}{\partial \Omega } \\ \frac{\Omega ^{2}-\omega ^{2}}{2\Omega }-\frac{a\omega }{\Omega }\frac{ d\omega }{da}+\frac{\bar{f}_{2}^{*}(a)}{da} &{} \frac{\partial \bar{F} ^{*}}{\partial \psi } &{} \frac{\partial \bar{F}^{*}}{\partial \Omega }-a \\ \frac{\partial \bar{R}}{\partial a} &{} \frac{\partial \bar{R}}{\partial \psi } &{} \frac{\partial \bar{R}}{\partial \Omega }+\frac{d\Gamma }{d\Omega } \end{array} \right] . \end{aligned}$$

(3.35)

The characteristic equation is

$$\begin{aligned} J_{3}\lambda ^{3}+J_{2}\lambda ^{2}+J_{1}\lambda +J_{0}=0, \end{aligned}$$

(3.36)

where

$$\begin{aligned} J_{3}= & {} 1,\qquad J_{2}=-\left( \frac{d\bar{f}_{2}(a)}{da}+\frac{\partial \bar{F}^{*}}{\partial \psi }+\frac{\partial \bar{R}}{\partial \Omega }+ \frac{d\Gamma }{d\Omega }\right) ,\qquad J_{0}=-\det (J), \nonumber \\ J_{1}= & {} \left( \frac{d\bar{f}_{2}(a)}{da}+\frac{\partial \bar{F}^{*}}{ \partial \psi }\right) \left( \frac{\partial \bar{R}}{\partial \Omega }+ \frac{d\Gamma }{d\Omega }\right) +\frac{d\bar{f}_{2}(a)}{da}\frac{\partial \bar{F}^{*}}{\partial \psi } \nonumber \\&+\frac{\partial \bar{F}}{\partial \psi }\left( \frac{\Omega ^{2}-\omega ^{2}}{2\Omega }-\frac{a\omega }{\Omega }\frac{d\omega }{da}+\frac{\bar{f} _{2}^{*}(a)}{da}\right) \nonumber \\&-\frac{\partial \bar{F}}{\partial \Omega }\frac{\partial \bar{R}}{\partial a}-\frac{\partial \bar{R}}{\partial \psi }\left( \frac{\partial \bar{F} ^{*}}{\partial \Omega }-a\right) . \end{aligned}$$

(3.37)

Using the Routh–Hurwitz criteria it can be concluded that the stability for the steady-state solution (3.31)–(3.33) is satisfied if

$$\begin{aligned} J_{0}>0,\qquad J_{1}>0,\qquad J_{2}>0,\qquad J_{1}J_{2}-J_{0}J_{3}>0. \end{aligned}$$

(3.38)

Fig. 3.3

Characteristic points

3.1.4 Characteristic Points on the Steady State Curves

Let us determine the locus of characteristic points and characteristic control parameter in the frequency-response curve (3.34)$_{1}$ where the jump phenomena appears (see Fig. 3.3). The criteria is that at these points P and R the both curves given with (3.34) have the same direction of the joint tangent. Namely, for

$$\begin{aligned} dQ_{1}= \frac{\partial Q_{1}}{\partial \Omega }d\Omega +\frac{\partial Q_{1}}{ \partial a}da=0,\qquad dQ_{2}=\frac{\partial Q_{2}}{\partial \Omega }d\Omega +\frac{\partial Q_{2}}{\partial a}da=0, \end{aligned}$$

(3.39)

the equality of the direction of the tangents of the curves (3.34)$_{1}$ and (3.34)$_{2}$ in P is

$$\begin{aligned} \frac{da}{d\Omega }\equiv -\left( \frac{\frac{\partial Q_{1}}{\partial \Omega }}{\frac{\partial Q_{1}}{\partial a}}\right) _{\Omega _{P},a_{P},V_{mP}}=-\left( \frac{\frac{\partial Q_{2}}{\partial \Omega }}{ \frac{\partial Q_{2}}{\partial a}}\right) _{\Omega _{P},a_{p},V_{mp}}. \end{aligned}$$

(3.40)

Solving algebraic equations (3.40) and also (3.34), values $\Omega _{P},a_{P},V_{mP}$ for a peak point are obtained. The frequency $\Omega _{P}$ corresponds to the amplitude $a_{P}$ and gives the control parameter for motor torque $V_{mP}$. Using the value of the control parameter $V_{mP}$, Eq. (3.34) give us the additional pair of ($\Omega _{Q},a_{Q}$) values due to the fact that system of algebraic equations is nonlinear. Points P and Q correspond to the same value of control parameter $V_{mP}$.

The same procedure is applied for determining the $\Omega _{R},a_{R},V_{mR}$ and also ($\Omega _{S},a_{S}$) which corresponds to $V_{mR}$. The critical frequency $\Omega _{R}$ with correspondent amplitude $a_{R}$ gives the value of the control parameter $V_{mR}$ for which the jump phenomena to the point S appears during decreasing of the control parameter $V_{m}$. In the region between $V_{mP}$ and $V_{mR}$ in the amplitude-frequency diagram a gap exists.

To eliminate the Sommerfeld effect the control parameter $V_{m}$ has to be beyond the interval ($V_{mP},V_{mR})$. The number of stable steady-state solutions outside this interval is only one.

3.1.5 Suppression of the Sommerfeld Effect

The Sommerfeld effect does not appear if, for all of values of the driving torque, only one steady-state response of the oscillator exists. Then, the intersection between the amplitude-frequency curves of the oscillator and of the motor has only one unique solution. Using this criteria the parameters of the system have to be calculated. For technical reasons we suggest an approximate analytical method for determination of the parameters of the non-ideal system where Sommerfeld effect does not exist. The basic requirement of the method is that the bone curve $Q_{3}(a,\Omega )$ of the amplitude-frequency characteristic of the oscillator (3.34)$_{1}$ and the amplitude-frequency curve of the motor (3.34)$_{2}$ have the equal gradient for the extreme steady-state position ($a^{*},\Omega ^{*}$) . Namely, the following relations have to be satisfied

$$\begin{aligned} Q_{1}(a^{*},\Omega ^{*},q^{*})= & {} 0,\qquad Q_{2}(a^{*},\Omega ^{*},q^{*},V_{m}^{*})=0,\qquad Q_{3}(a^{*},\Omega ^{*})=0, \nonumber \\ \frac{da}{d\Omega }\equiv & {} -\left( \frac{\frac{\partial Q_{1}}{\partial \Omega }}{\frac{\partial Q_{1}}{\partial a}}\right) _{\Omega ^{*},a^{*},V_{m}^{*},q^{*}}=-\left( \frac{\frac{\partial Q_{3}}{ \partial \Omega }}{\frac{\partial Q_{3}}{\partial a}}\right) _{\Omega ^{*},a^{*},V_{m}^{*},q^{*}}. \end{aligned}$$

(3.41)

Solving the system of algebraic equations (3.41) the parameter $q^{*}$ is obtained, for which only one solution for $\Omega ^{*},a^{*}$ and $V_{m}^{*}$ exists. Due to the fact that for either value of the control parameter $V_{m}$ the amplitude-frequency curve (3.34)$_{2}$ remains parallel to $Q_{2}(a,\Omega ,q^{*},V_{m}^{*})=0$ and also to the bone curve $Q_{3}(a,\Omega )=0$ it can be concluded that there is only one intersection between any $Q_{2}(a,\Omega ,q^{*},V_{m})=0$ and the amplitude-frequency curve $Q_{1}(a,\Omega ,q^{*})=0$.

Remark 1

Using the same relations (3.41) instead of $q^{*}$ another critical parameter of the system can be calculated (for example $C_{m}^{*})$.

3.1.6 Conclusion

Analyzing the results the following is concluded:

• 1. The generalized non-ideal mechanical system contains the nonlinear oscillator of any order and a motor with the driving torque which need not to be the linear function of the angular velocity.

• 2. The approximate solution procedure of the problem suggested in the text is suitable for the near resonant case and gives the results which are possible to be used for the stability analysis and discussion of the characteristic properties of the system.

• 3. The approximate value of the control parameter for the non-ideal source is analytically calculated applying the method of equating the gradient of the both amplitude-frequency curves (of the oscillator and of the motor) in the intersection points. The criteria for the Sommerfeld effect is obtained.

• 4. It can be concluded that the method developed in the text gives the parameter values for which the Sommerfeld effect is suppressed. For these parameters there exists only one steady-state response of the oscillator for all values of the driving torque.

• 5. The suitable choice of non-ideal system parameters available the motion without jumps.

• 6. The analytically obtained results show a good agreement with numerically obtained ones. It proves the correctness of the analytic procedures.

3.2 Pure Nonlinear Oscillator and the Motor with Nonlinear Torque

Let us consider a motor-structure system shown in Fig. 3.4.

Fig. 3.4

Model of the motor-structure non-ideal system

A motor with an unbalance is connected to a viscoelastic structure with nonlinear properties. The motion occurs in a horizontal plane and is constrained so that the motor executes a rectilinear motion along the x -axis. The elastic force of the structure is assumed as a pure nonlinear displacement function

$$\begin{aligned} F_{e}=kx\left| x\right| ^{\alpha -1}, \end{aligned}$$

(3.42)

where $\alpha \ge 1$ and $\alpha \in \mathbb {R}$ is a positive rational number (integer or non-integer) which represents the order of nonlinearity and k is the coefficient of rigidity. Experimental investigation on a significant number of materials, for example: aluminum, titanium and other aircraft materials (Prathap and Varadan 1976), copper and copper alloys (Lo and Gupta 1978), aluminum alloys and annealed copper (Lewis and Monasa 1982), wood (Haslach 1985), ceramic materials (Colm and Clark 1988), hydrophilic polymers (Haslach 1992; Pilipchuk 2010), composites (Chen and Gibson 1998), polyurethane foam (Patten et al. 1998), felt (Russell and Rossing 1998), etc., show that the stress-strain properties of the material are nonlinear. The nonlinear dependence of the restoring force on the deflection is a polynomial whose exponent is of positive integer or non-integer order. For most of these materials the damping properties are also nonlinear. However, for the mentioned metallic materials and their alloys the order of nonlinearity in the damping force is small and the linear damping force model gives a good approximation. Thus, the damping force-velocity function is

$$\begin{aligned} F_{d}=c \dot{x}, \end{aligned}$$

(3.43)

where c is the damping constant. Mass of the system is M, the moment of inertia of the motor rotor is J, mass of the rotor unbalance is m and the length of the rotor unbalance is d. The considered non-ideal system has two degrees-of-freedom, represented by the generalized coordinates x and $\varphi$ and the motion is described with two Lagrange differential equations

$$\begin{aligned} \frac{d}{dt}\frac{\partial T}{\partial \dot{x}}-\frac{\partial T}{\partial x} +\frac{\partial U}{\partial x}+\frac{\partial \Phi }{\partial \dot{x}}= & {} Q_{x}, \nonumber \\ \frac{d}{dt}\frac{\partial T}{\partial \dot{\varphi }}-\frac{\partial T}{ \partial \varphi }+\frac{\partial U}{\partial \varphi }+\frac{\partial \Phi }{\partial \dot{\varphi }}= & {} Q_{\varphi }, \end{aligned}$$

(3.44)

where T is the kinetic energy, U is the potential energy, $\Phi$ is the dissipative function and $Q_{x}$ and $Q_{\varphi }$ are the generalized forces. The kinetic energy , potential energy and the dissipation function are expressed by

$$\begin{aligned} T= \frac{1}{2}M\dot{x}^{2}+\frac{1}{2}m(\dot{x}-d\dot{\varphi }\cos \varphi )^{2}+\frac{1}{2}m(d\dot{\varphi }\sin \varphi )^{2}+\frac{1}{2}J\dot{\varphi } ^{2}, \end{aligned}$$

(3.45)

$$\begin{aligned} U=\frac{k}{\alpha +1}x^{\alpha +1},\qquad \Phi =\frac{1}{2}c\dot{x}^{2}, \end{aligned}$$

(3.46)

where $\varepsilon<<1$ and a dot denotes differentiation with respect to time t. For the driving torque (3.2) the generalized forces are

$$\begin{aligned} Q_{x}=0,\qquad Q_{\varphi }=\mathcal {M(}\dot{\varphi }). \end{aligned}$$

(3.47)

Equations of motion (3.44) have the form

$$\begin{aligned} \ddot{x}(M+m)+kx\left| x\right| ^{\alpha -1}+c\dot{x}-md(\ddot{ \varphi }\cos \varphi -\dot{\varphi }^{2}\sin \varphi )= & {} 0, \nonumber \\ (J+md^{2})\ddot{\varphi }-md\ddot{x}\cos \varphi= & {} \mathcal {M}(\dot{\varphi } ), \end{aligned}$$

(3.48)

and the initial conditions are

$$\begin{aligned} x(0)=x_{0},\;\dot{x}(0)=0,\;\varphi (0)=0,\;\dot{\varphi }(0)=\omega _{0}. \end{aligned}$$

(3.49)

It is convenient to normalize the coordinates and time according to

$$\begin{aligned} x\longrightarrow y=x/l,\qquad t\longrightarrow \tau =\Omega ^{*}t, \end{aligned}$$

(3.50)

where l is the initial length of the non-deformed spring and $\Omega ^{*}$ is the synchronous angular velocity of the rotor. By introducing (3.50) the differential equations (3.48) transform into

$$\begin{aligned} y\text {''}+\varepsilon \zeta _{1}y^{\prime }+p^{2}y\left| y\right| ^{\alpha -1}= & {} \varepsilon \mu _{1}(\varphi \text {''}\cos \varphi -\varphi ^{\prime 2}\sin \varphi ), \nonumber \\ \varphi \text {''}= & {} \varepsilon ^{2}[\eta _{2}y\text {''}\cos \varphi +\mathcal {M}(\varphi ^{\prime })], \end{aligned}$$

(3.51)

with non-dimensional initial conditions

$$\begin{aligned} y(0)=A,\;y^{\prime }(0)=0,\;\varphi (0)=0,\;\varphi ^{\prime }(0)=\frac{ \omega _{0}}{\Omega ^{*}}=\omega , \end{aligned}$$

(3.52)

where

$$\begin{aligned} p= & {} \frac{\omega ^{*}}{\Omega ^{*}},\qquad \omega ^{*2}=\frac{ kl^{\alpha -1}}{M+m},\qquad A=\frac{x_{0}}{l},\qquad \varepsilon =\frac{m}{ M+m}, \nonumber \\ \varepsilon \mu _{1}= & {} \left( \frac{m}{M+m}\right) \left( \frac{d}{l} \right) ,\qquad \varepsilon ^{2}\eta _{2}=\left( \frac{d}{l}\right) \frac{ ml^{2}}{(J+md^{2})}, \nonumber \\ \varepsilon \zeta= & {} \left( \frac{m}{M+m}\right) \frac{c}{\Omega ^{*}m} ,\qquad \qquad \varepsilon ^{2}\mathcal {M}(\varphi ^{\prime })=\frac{ \mathcal {M}(\Omega ^{*}\varphi ^{\prime })}{(J+md^{2})\Omega ^{*2}}, \end{aligned}$$

(3.53)

$\Omega ^{*}$ is the synchronous angular velocity (Dimentberg et al. 1997) and prime denotes differentiation with respect to $\tau$. It is worth to say that $\varepsilon<<1$ is a small positive parameter. Using the expression (3.51) we have

$$\begin{aligned} y\text {''}+p^{2}y\left| y\right| ^{\alpha -1}= & {} -\varepsilon (\mu _{1}\varphi ^{\prime 2}\sin \varphi +\zeta _{1}y^{\prime }), \end{aligned}$$

(3.54)

$$\begin{aligned} \varphi \text {''}= & {} \varepsilon ^{2}[\mathcal {M}(\varphi ^{\prime })-\eta _{2}p^{2}y\left| y\right| ^{\alpha -1}\cos \varphi ]. \end{aligned}$$

(3.55)

The Eqs. (3.54) and (3.55) represent the system of two coupled differential equations which describe the motion of the non-ideal system given in Fig. 3.4. Comparing (3.54) and (3.55) with (3.8) and (3.9) we have

$$\begin{aligned} f_{1}=p^{2}y\left| y\right| ^{\alpha -1},\quad f_{2}=-\zeta _{1}y^{\prime },\quad F=-\mu _{1}\varphi ^{\prime 2}\sin \varphi ,\quad R=-\eta _{2}p^{2}y\left| y\right| ^{\alpha -1}\cos \varphi . \end{aligned}$$

(3.56)

Now, the differential equations (3.54) and (3.55) have to be solved.

3.2.1 Approximate Solution Procedure

For the case when the small parameter $\varepsilon$ tends to zero, the differential equations (3.54) and (3.55) transform into

$$\begin{aligned} y\text {''}+p^{2}y\left| y\right| ^{\alpha -1}=0,\qquad \varphi \text {''}=0. \end{aligned}$$

(3.57)

The two differential equations (3.57) are uncoupled and can be solved independently but according to the initial conditions (3.52). Equation (3.57)$_{1}$ is a second order pure nonlinear differential equation with rational order of nonlinearity. The approximate analytical solution of the (3.57)$_{1}$ is assumed in the form of a harmonic function (3.11)$_{1}$ with the frequency (see Cveticanin 2009; Cveticanin and Pogany 2012)

$$\begin{aligned} \omega =\omega _{\alpha }\sqrt{p^{2}}a^{(\alpha -1)/2}, \end{aligned}$$

(3.58)

where

$$\begin{aligned} \omega _{\alpha }=\sqrt{\frac{\alpha +1}{2}}\frac{\sqrt{\pi }\Gamma (\frac{ 3+\alpha }{2(\alpha +1)})}{\Gamma (\frac{1}{\alpha +1})}, \end{aligned}$$

(3.59)

and $\Gamma$ is the gamma function (Gradstein and Rjizhik 1971). The relation (3.58) is the exact analytically obtained frequency of vibration of the nonlinear elastic structure (3.57)$_{1}$. The asymptotic solution (3.11)$_{1}$ approximately satisfies the differential equation (3.57)$_{1}$, i.e.,

$$\begin{aligned} -a\omega ^{2}\cos (\omega t+\psi )+p^{2}a^{\alpha }\cos ^{\alpha }(\omega t+\psi )\approx 0. \end{aligned}$$

(3.60)

Using the generalized procedure given in Sect. 3.3 and substituting the approximate solution with time variable parameters (3.15) and the corresponding derivatives into (3.54) it follows

$$\begin{aligned}&-a^{\prime }\Omega \sin (\varphi +\psi )-a\Omega (\Omega +\psi ^{\prime })\cos (\varphi +\psi ) \\&+p^{2}a^{\alpha }\cos ^{\alpha }(\varphi +\psi ) \nonumber \\= & {} -\varepsilon \mu _{1}\Omega ^{2}\sin \varphi +\varepsilon \zeta _{1}a\Omega \sin (\varphi +\psi ), \nonumber \end{aligned}$$

(3.61)

where for the resonant condition (3.13) and using the relation (3.60), the differential equation (3.61) simplifies into

$$\begin{aligned} -a^{\prime }\Omega \sin (\varphi +\psi )-a(\Omega ^{2}-\omega ^{2}+\Omega \varphi ^{\prime })\cos (\varphi +\psi )=\varepsilon \mu _{1}\Omega ^{2}\sin \phi +\varepsilon \zeta _{1}a\Omega \sin (\varphi +\psi ). \end{aligned}$$

(3.62)

The relations (3.18) and (3.62) are the two first order differential equations which correspond to the second order differential equation (3.54). Solving the Eqs. (3.18) and (3.62) for $a^{\prime }$ and $\psi ^{\prime }$ the relations (3.22) and (3.23) for (3.116) follow

$$\begin{aligned} a^{\prime }= & {} \varepsilon \mu _{1}\Omega \sin \varphi \sin (\varphi +\psi )-\varepsilon \zeta _{1}a\sin ^{2}(\varphi +\psi ) \nonumber \\&-a\frac{\Omega ^{2}-\omega ^{2}}{\Omega }\sin (\varphi +\psi )\cos (\varphi +\psi ), \end{aligned}$$

(3.63)

$$\begin{aligned} a\psi ^{\prime }= & {} -a\frac{\Omega ^{2}-\omega ^{2}}{\Omega }\cos ^{2}(\varphi +\psi )-\varepsilon \mu _{1}\Omega \sin \varphi \cos (\varphi +\psi ) \nonumber \\&+\varepsilon \zeta _{1}a\sin (\varphi +\psi )\cos (\varphi +\psi ). \end{aligned}$$

(3.64)

Substituting the solution (3.15) into the differential equation (3.55) and using the relations (3.56) and (3.60) we obtain (3.20) with (3.21), i.e.,

$$\begin{aligned} \Omega ^{\prime }=\varepsilon ^{2}\mathcal {M}(\Omega )-\varepsilon ^{2}\eta _{2}a\omega ^{2}\cos (\varphi +\psi )\cos \varphi ,\qquad \varphi ^{\prime }=\Omega . \end{aligned}$$

(3.65)

Equations (3.63)–(3.65) represent the four first order differential equations which correspond to the second order differential equations (3.54) and (3.55). After averaging equations transform into

$$\begin{aligned} a^{\prime }= & {} -\frac{\zeta _{1}a}{2}+\frac{\mu _{1}\Omega }{2}\cos \psi , \end{aligned}$$

(3.66)

$$\begin{aligned} \psi ^{\prime }= & {} -\frac{\Omega ^{2}-\omega ^{2}}{\Omega }+\frac{\mu _{1}\Omega }{2a}\sin \psi , \end{aligned}$$

(3.67)

$$\begin{aligned} \Omega ^{\prime }= & {} \mathcal {M}(\Omega )-\frac{\eta _{2}a\Omega ^{2}}{2} \cos \psi . \end{aligned}$$

(3.68)

Equations (3.66)–(3.68) describe the non-stationary motion of the system.

3.2.2 Steady-State Motion and Its Properties

If there is no interaction between the oscillator and the motor the amplitude of vibration of the oscillator decreases exponentially from the initial amplitude A (due to viscous damping) as $a=A\exp (-\gamma \tau /2)$, and the variation of the angular velocity of the motor satisfies the relation

$$\begin{aligned} \Omega =\frac{V_{m}}{C_{m}}-\frac{V_{m}}{C_{m}}\frac{1}{ (1+C_{m}V_{m}^{n-1}(n-1)\tau )^{1/(n-1)}}. \end{aligned}$$

(3.69)

In Fig. 3.5, $\Omega -\tau$ curves for various values of parameter n are plotted.

Fig. 3.5

$\Omega -\tau$ curves for various values of the nonlinearity order n: $n=2$ (full line), $n=3$ (dotted line)

Namely, for $V_{m}=1$ and $C_{m}=1$ the $\Omega -\tau$ relations according to (3.69) are:

for $n=2$

$$\begin{aligned} \Omega =1-\frac{1}{1+\tau }, \end{aligned}$$

(3.70)

for $n=3$

$$\begin{aligned} \Omega =1-\frac{1}{(1+2\tau )^{1/2}}. \end{aligned}$$

(3.71)

Analyzing the relations (3.69) i.e., (3.70) and (3.71), it is obvious that for any value of parameter n, the angular velocity is zero for $\tau =0$, and tends to the constant steady-state value $V_{m}/C_{m}$. The smaller the value of the parameter n, the steady-state value is achieved in a shorter time.

For the steady-state respons , Eqs. (3.66)–(3.68) have the form

$$\begin{aligned} \frac{\zeta _{1}a}{2}= & {} \frac{\mu _{1}\Omega }{2}\cos \psi , \end{aligned}$$

(3.72)

$$\begin{aligned} \frac{\Omega ^{2}-\omega ^{2}}{\Omega }a= & {} \frac{\mu _{1}\Omega }{2}\sin \psi , \end{aligned}$$

(3.73)

$$\begin{aligned} \mathcal {M}(\Omega )= & {} \frac{\eta _{2}a\Omega ^{2}}{2}\cos \psi . \end{aligned}$$

(3.74)

Solving the algebraic equations (3.72)–(3.74), the steady-state properties of the system are determined.

Eliminating ${\xi }$ from (3.72) and (3.73) and also from (3.72) and (3.74) following two $a-\Omega$ relations are obtained

$$\begin{aligned} \left( \mu _{1}\Omega \right) ^{2}=a^{2}\left( \zeta _{1}^{2}+4\left( \frac{\Omega ^{2}-\omega ^{2}}{\Omega }\right) ^{2}\right) , \end{aligned}$$

(3.75)

$$\begin{aligned} \zeta _{1}a^{2}\Omega =2\frac{\mu _{1}}{\eta _{2}}\mathcal {M}(\Omega ), \end{aligned}$$

(3.76)

i.e., after substituting (3.58) and (3.4) it follows

$$\begin{aligned} \left( \mu _{1}\Omega \right) ^{2}= & {} a^{2}\left( \zeta _{1}^{2}+4\left( \frac{\Omega ^{2}-\omega _{\alpha }^{2}p^{2}a_{{}}^{(\alpha -1)}}{\Omega } \right) ^{2}\right) , \end{aligned}$$

(3.77)

$$\begin{aligned} \zeta _{1}a^{2}\Omega= & {} 2\frac{\mu _{1}}{\eta _{2}}(V_{m}-C_{m}\Omega )^{n}. \end{aligned}$$

(3.78)

The solution of the system of algebraic equations (3.75) and (3.76) depends on the motor torque function but also on the order of nonlinearity of the oscillator. Solving algebraic equations (3.77) and (3.78) for a and $\Omega$, the steady-state phase angle $\psi$ is calculated. Namely, the relation for phase is

$$\begin{aligned} 2 \frac{\Omega ^{2}-\omega ^{2}}{\Omega \zeta _{1}}=\tan \psi , \end{aligned}$$

(3.79)

and is obtained by dividing Eqs. (3.72) and (3.73).

3.2.3 Characteristic Points

According to the procedure given in Sect. 3.3, characteristic points in amplitude-frequency curves can be calculated. Solving the system of algebraic equations (3.77) and (3.78) and also the relation

$$\begin{aligned} \frac{da}{d\Omega }\equiv & {} \frac{2\mu _{1}^{2}\Omega ^{3}-\Omega a^{2}\zeta _{1}^{2}-8a^{2}\left( \Omega ^{2}-\omega ^{2}\right) \Omega }{ a\zeta _{1}^{2}\Omega ^{2}+4a\left( \Omega ^{2}-\omega ^{2}\right) ^{2}-4(\alpha -1)a\left( \Omega ^{2}-\omega ^{2}\right) \omega ^{2}} \\= & {} -\frac{\zeta _{1}a^{2}+2nC_{m}\frac{\mu _{1}}{\eta _{2}} (V_{m}-C_{m}\Omega )^{n-1}}{2\zeta _{1}a\Omega }, \end{aligned}$$

where $\omega$ is given as (3.58), the parameters $a_{P},$ $\Omega _{P}$ and $V_{mP}$ of the characteristic point P are obtained.

Due to complexity to the suggested calculation an approximate solution procedure is recommended. Using the fact that the locus of the point P is near the position of the point P’, where solutions of the Eq. (3.77) bifurcate from one to two, i.e., from three to two real solutions, it is suggested to consider the characteristics of P’ instead of P. Thus, for

$$\begin{aligned} \Omega _{P^{\prime }}=\omega _{p^{\prime }}=\omega _{\alpha }a_{p^{\prime }}^{(\alpha -1)/2}\sqrt{p^{2}}, \end{aligned}$$

(3.80)

the relation (3.77) gives the peak amplitude

$$\begin{aligned} a_{P^{\prime }}=\left( \frac{\zeta _{1}}{\omega _{\alpha }\mu _{1}\sqrt{p^{2} }}\right) ^{\frac{2}{\alpha -3}}. \end{aligned}$$

(3.81)

Substituting (3.81) into (3.80), the locus $\Omega _{P^{\prime }}$ for $a_{P^{\prime }}$ is obtained

$$\begin{aligned} \Omega _{P^{\prime }}=\left( \frac{\zeta _{1}}{\mu _{1}}\right) ^{\frac{ \alpha -1}{\alpha -3}}\left( \omega _{\alpha }\sqrt{p^{2}}\right) ^{\frac{2}{ 3-\alpha }}. \end{aligned}$$

(3.82)

Equations (3.81) and (3.82) with (3.78) give the value of a control parameter $V_{mP^{\prime }}$ of the motor torque

$$\begin{aligned} V_{mP^{\prime }}=C_{m}\left( \frac{\zeta _{1}}{\mu _{1}}\right) ^{\frac{ \alpha -1}{\alpha -3}}\left( \omega _{\alpha }\sqrt{p^{2}}\right) ^{\frac{2}{ 3-\alpha }}+\left( \frac{\eta _{2}}{2}\right) ^{\frac{1}{n}}\left( \frac{ \zeta _{1}}{\mu _{1}}\right) ^{\frac{2\alpha }{n(\alpha -3)}}\left( \omega _{\alpha }\sqrt{p^{2}}\right) ^{\frac{6}{n(3-\alpha )}}. \end{aligned}$$

(3.83)

For this approximate value of the control parameter $V_{mP^{\prime }}$ the Sommerfeld effect has to appear. Analyzing the relation (3.77) it is obvious that the position of the extreme point P’ is on a line

$$\begin{aligned} a= \frac{\mu _{1}}{\zeta _{1}}\Omega . \end{aligned}$$

The gradient of the line does not depend on the order of nonlinearity $\alpha$, but only on the parameters $\mu _{1}$ and $\zeta _{1}$.

3.2.4 Suppression of the Sommerfeld Effect

As it is previously shown Eqs. (3.81)–(3.83) give us values $\Omega _{P^{\prime }}$ and $a_{P^{\prime }}$ of the point P’ and also the corresponding control parameter $V_{mP^{\prime }},$ which forces the amplitude-frequency torque curve through the point P’. If the amplitude-frequency torque curve and the backbone curve (3.80) have the equal gradients in the P’ i.e.,

$$\begin{aligned} \frac{da}{d\Omega }\equiv \frac{2}{\omega _{\alpha }(\alpha -1)\sqrt{p^{2}} a^{\frac{\alpha -3}{2}}}=-\frac{\zeta _{1}a^{2}+2nC_{m}\frac{\mu _{1}}{\eta _{2}}(V_{m}-C_{m}\Omega )^{n-1}}{2\zeta _{1}a\Omega }, \end{aligned}$$

(3.84)

the additional condition for suppression of the Sommerfeld effect in the system is obtained. Namely, solving the system of four algebraic equations (3.81)–(3.83) and (3.84) four unknown values are obtained: $a^{*}$, $\Omega ^{*}$, $V_{m}^{*}$ and also $\alpha ^{*}$, $\zeta _{1}^{*}$, $\mu _{1}^{*}$, $\eta _{2}^{*}$ or $C_{m}^{*}$. The forth mentioned parameter is the control parameter for elimination of the Sommerfeld effect.

The left side of the Eq. (3.84) is the gradient of the backbone curve in the point P’. Using (3.81) the gradient is

$$\begin{aligned} \left( \frac{da}{d\Omega }\right) _{P^{\prime }}=\frac{2\mu _{1}}{\zeta _{1}(\alpha -1)}. \end{aligned}$$

It depends on the order of nonlinearity $\alpha$: for $\alpha <1$ it is negative, for $\alpha >1$ it is positive, while for $\alpha =1$ it represents an orthogonal direction. The bending is higher for $\alpha$ significantly higher or smaller than 1.

Using relations (3.81)–(3.83), Eq. (3.84) is rewritten as

$$\begin{aligned} \frac{4\zeta _{1}}{1-\alpha }=\gamma +C_{m}n\mu _{1}\left( \frac{2}{\eta _{2} }\right) ^{\frac{1}{n}}\left( \frac{\zeta _{1}}{\mu _{1}}\right) ^{\frac{2}{n }\left( (n-1)+\frac{n-3}{\alpha -3}\right) }\left( \omega _{\alpha }\sqrt{ p^{2}}\right) ^{\frac{2(n-3)}{(3-\alpha )n}}. \end{aligned}$$

(3.85)

If the order of nonlinearity $\alpha$ and parameters n, $\gamma$, $\mu _{1}$ and $\eta _{2}$ are known, solving the relation (3.85), for example, for the parameter $C_{m}^{*}$, we have

$$\begin{aligned} C_{m}^{*}=\frac{3+\alpha }{1-\alpha }\frac{1}{n}\left( \frac{\eta _{2}}{2 }\right) ^{\frac{1}{n}}\left( \frac{\zeta _{1}}{\mu _{1}}\right) ^{1-\frac{2 }{n}\left( (n-1)+\frac{n-3}{\alpha -3}\right) }\left( \omega _{\alpha }\sqrt{ p^{2}}\right) ^{\frac{2(n-3)}{(\alpha -3)n}}. \end{aligned}$$

(3.86)

Substituting (3.86) into (3.87) the control parameter $V_{m}^{*}$ is

$$\begin{aligned} V_{m}^{*}=\left( \frac{3+\alpha }{1-\alpha }\frac{1}{n}+1\right) \left( \frac{\eta _{2}}{2}\right) ^{\frac{1}{n}}\left( \frac{\zeta _{1}}{\mu _{1}}\right) ^{ \frac{2\alpha }{n(\alpha -3)}}\left( \omega _{\alpha }\sqrt{p^{2}}\right) ^{ \frac{6}{(3-\alpha )n}}. \end{aligned}$$

(3.87)

It can be concluded that for parameter values (3.86) and (3.87) in the non-ideal system with known order of nonlinearity $\alpha$ the jump phenomena is excluded .

3.2.5 Numerical Examples

In this section, numerical examples of electro-motors connected with pure nonlinear elastic structures are considered. As is shown in the previous section, there are numerous materials whose elastic properties are strong nonlinear. We choose two of them: the copper alloy with $\alpha =4/3$ and aluminium alloy with $\alpha =5/3$ (Jutte 2008). The corresponding elastic forces are

$$\begin{aligned} F_{e1}=kx\left| x\right| ^{1/3},\qquad F_{e2}=kx\left| x\right| ^{2/3}. \end{aligned}$$

(3.88)

If the motor-structure system is driven with the motor torque of cubic type

$$\begin{aligned} \mathcal {M}( \dot{\varphi })=(V_{m}^{*}-C_{m}^{*}\dot{\varphi })^{3}. \end{aligned}$$

(3.89)

differential equations of motion are:

for $\alpha =4/3$

$$\begin{aligned} y\text {''}+p^{2}y\left| y\right| ^{1/3}= & {} -\varepsilon (\mu _{1}\varphi ^{\prime 2}\sin \varphi +\zeta _{1}y^{\prime }), \end{aligned}$$

(3.90)

$$\begin{aligned} \varphi \text {''}= & {} \varepsilon ^{2}[(V_{m}-C_{m}\varphi ^{\prime })^{3}-\eta _{2}p^{2}y\left| y\right| ^{1/3}\cos \varphi ], \end{aligned}$$

(3.91)

for $\alpha =5/3$

$$\begin{aligned} y\text {''}+p^{2}y\left| y\right| ^{2/3}= & {} -\varepsilon (\mu _{1}\varphi ^{\prime 2}\sin \varphi +\zeta _{1}y^{\prime }), \end{aligned}$$

(3.92)

$$\begin{aligned} \varphi \text {''}= & {} \varepsilon ^{2}[(V_{m}-C_{m}\varphi ^{\prime })^{3}-\eta _{2}p^{2}y\left| y\right| ^{2/3}\cos \varphi ]. \end{aligned}$$

(3.93)

Corresponding averaged differential equations of motion obtained analytically are

$$\begin{aligned} a^{\prime }= & {} -\frac{\zeta _{1}a}{2}+\frac{\mu _{1}\Omega }{2}\cos \psi , \end{aligned}$$

(3.94)

$$\begin{aligned} \xi ^{\prime }= & {} -\frac{\Omega ^{2}-\omega _{\alpha }^{2}p^{2}a^{(\alpha -1)}}{\Omega }+\frac{\mu _{1}\Omega }{2a}\sin \psi , \end{aligned}$$

(3.95)

$$\begin{aligned} \Omega ^{\prime }= & {} (V_{m}-C_{m}\Omega )^{3}-\frac{\eta _{2}a\Omega ^{2}}{2} \cos \psi , \end{aligned}$$

(3.96)

where dependently on the order $\alpha$ the frequency constants are $\omega _{4/3}=0.96916$ and $\omega _{5/3}=0.940\,81$ (see Cveticanin 2009). For $\mu _{1}=0.15$, $\eta _{2}=0.05$, $p^{2}=1$, $C_{m}=1$ and $\zeta _{1}=0.1$ the steady-state amplitude-frequency relations (3.77) for these systems are:

for $\alpha =4/3$

$$\begin{aligned} 0.0225\Omega ^{2}=a^{2}\left( 0.01+4\left( \frac{\Omega ^{2}-0.93927a_{{}}^{1/3}}{\Omega }\right) ^{2}\right) , \end{aligned}$$

(3.97)

for $\alpha =5/3$

$$\begin{aligned} 0.0225\Omega ^{2}=a^{2}\left( 0.01+4\left( \frac{\Omega ^{2}-0.885\,12a_{{}}^{2/3}}{\Omega }\right) ^{2}\right) , \end{aligned}$$

(3.98)

while the relation (3.78) for the motor is

$$\begin{aligned} 0.1a^{2}\Omega =6(V_{m}-\Omega )^{3}. \end{aligned}$$

(3.99)

In Fig. 3.6, amplitude-frequency steady-state diagrams for various values of control parameter $V_{m}$ and order of nonlinearity $\alpha$ are plotted.

Fig. 3.6

Frequency-response curves for various $\alpha$ obtained: a analytically (full line), b numerically (circles - for the increasing of the $V_{m}$, squares - for decreasing of the $V_{m})$

Analytically obtained (full line) curves (3.97)–(3.99) are compared with numerically (3.90)–(3.93) ones, obtained by increasing (circle) and decreasing (squares) of the control parameter $V_{m}$. Numerical solutions are obtained applying the Runge–Kutta procedure. We note that there is a hysteresis in diagrams in the region P-Q-R-S (see Fig. 3.6), where two stable steady-state responses exist. This phenomena, called Sommerfeld effect , depends on the orders of nonlinearity $\alpha$.

Using relations (3.81)–(3.83) coordinates of peaks ($a_{P^{\prime }},\Omega _{P^{\prime }})$ in the diagram (3.77) and corresponding control parameters $V_{mP^{\prime }}$ are calculated and shown in Table 3.1. Approximate values $V_{mP^{\prime }}$ are compared with the exact numerically obtained value $V_{mP}$. It can be seen that they are in good agreement.

Table 3.1 Coordinates of peak in the $a_{p}$-$\Omega _{p}$ diagram for control parameter $V_{mP}$ and certain value of $\alpha$

For the value of the control parameter $V_{mP}$ the amplitude-frequency curve (3.78) represents the boundary for which the Sommerfeld effect exists.

Fig. 3.7

Time history diagrams for $\alpha =5/3$: a non-stationary diagram for $V_{m}=1.2$; b, c Steady-state diagrams for $V_{m}=1.2$; d non-stationary motion for $V_{m}=1.4$

The question is whether the Sommerfeld effect in this non-ideal system can be suppressed . Using the procedure given in this section, the value of the parameter $\alpha$ for which the Sommerfeld effect is suppressed can be calculated. For the driving torque with cubic nonlinearity ($n=3$) the relation (3.85) simplifies to

$$\begin{aligned} \frac{4\zeta _{1}}{1-\alpha }=\zeta _{1}+3C_{m}\mu _{1}\left( \frac{2}{\eta _{1}}\right) ^{\frac{1}{3}}\left( \frac{\zeta _{1}}{\mu _{1}}\right) ^{\frac{ 4}{3}}. \end{aligned}$$

(3.100)

Substituting the previously mentioned parameter values and $C_{m}=1$ into (3.100) we obtain the critical value of the order of nonlinearity

$$\begin{aligned} \alpha ^{*}=0.59851. \end{aligned}$$

(3.101)

For $\alpha =\alpha ^{*}$ the Sommerfeld effect is suppressed. It means, that for all values of $\alpha \ge 1$ the Sommerfeld effect exists.

To prove the correctness of analytically and numerically obtained results, let us plot curves for the steady-state and of the non-stationary motion. In Fig. 3.7, amplitude-frequency diagrams of the oscillator with nonlinearity order $\alpha =5/3$ for the control parameter $V_{m}=1.2$ (non-stationary motion in Fig. 3.7a and two steady-state motions in Fig. 3.7b, c) and for the control parameter $V_{m}=1.4$ (non-stationary motion in Fig. 3.7d) are shown. Comparing these diagrams with values shown in Fig. 3.7, it can be seen that they are in good agreement.

Based on the obtained results, it can be concluded that the methods developed for the general type of the non-ideal system (mentioned in the previous sections) are applicable for the pure nonlinear oscillator coupled with a non-ideal source excited with the nonlinear torque.

3.3 Pure Strong Nonlinear Oscillator and a Non-ideal Energy Source

Let us consider a non-ideal energy source which is settled on a foundation which is a nonlinear oscillator (Fig. 3.4) The elastic characteristic of the oscillator is pure nonlinear and described with an elastic force which is proportional to the displacement x with the positive rational exponent $\alpha \in \mathbb {R}$ (integer or noninteger)

$$\begin{aligned} F_{e}=kx\left| x\right| ^{\alpha -1}, \end{aligned}$$

(3.102)

where k is the rigidity constant. The damping property of the system is supposed to be a linear velocity function and is given with the damping force

$$\begin{aligned} F_{d}=c \dot{x}, \end{aligned}$$

(3.103)

where c is the damping coefficient.

The oscillator is driven by a motor which has an unbalance m which is on the distance d to the rotor shaft. Position of the unbalance is varying in time and is described with the angle $\varphi$. As it is suggested in Dantas and Balthazar (2003), Tsuchida et al. (2003, 2005), Souza et al. (2005a, b), Felix et al. (2009a), Castao et al. (2010), the motor torque is a linear function of angular velocity

$$\begin{aligned} \mathcal {M(} \dot{\varphi })=V_{\varphi m}-C_{\varphi m}\dot{\varphi }, \end{aligned}$$

(3.104)

where $C_{\varphi m}$ and $V_{\varphi m}$ are constant values and $\dot{ \varphi }$ is the angular velocity of the motor. The system executes a rectilinear motion and the displacement is given with the variable x. This motion has an effect on the rotation of the rotor of the motor.

The suggested oscillator-motor system has two-degrees-of-freedom. The two generalized coordinates are x and $\varphi$. In general, the Lagrange differential equations of motion for the system are

$$\begin{aligned} \frac{d}{dt}\frac{\partial T}{\partial \dot{x}}-\frac{\partial T}{\partial x} +\frac{\partial U}{\partial x}+\frac{\partial \Phi }{\partial \dot{x}}= & {} Q_{x}, \nonumber \\ \frac{d}{dt}\frac{\partial T}{\partial \dot{\varphi }}-\frac{\partial T}{ \partial \varphi }+\frac{\partial U}{\partial \varphi }+\frac{\partial \Phi }{\partial \dot{\varphi }}= & {} Q_{\varphi }, \end{aligned}$$

(3.105)

where T is the kinetic energy, U is the potential energy, $\Phi$ is the dissipation function and $Q_{x}$ and $Q_{\varphi }$ are the corresponding generalized forces.

The kinetic energy, potential energy and the dissipation function are, respectively,

$$\begin{aligned} T=\frac{1}{2}M\dot{x}^{2}+\frac{1}{2}m(\dot{x}-d\dot{\varphi }\cos \varphi )^{2}+\frac{1}{2}m(d\dot{\varphi }\sin \varphi )^{2}+\frac{1}{2}J\dot{\varphi } ^{2}, \end{aligned}$$

(3.106)

$$\begin{aligned} U=\frac{k}{\alpha +1}x^{\alpha +1},\qquad \Phi =\frac{1}{2}c\dot{x}^{2}, \end{aligned}$$

(3.107)

where a dot denotes differentiation with respect to time t, M is mass of the oscillator, J is the moment of inertia of the motor rotor, m is mass of the rotor unbalance and d is the length of the rotor unbalance. The virtual work of the motor torque is

$$\begin{aligned} \delta A=\mathcal {M(} \dot{\varphi })\delta \varphi , \end{aligned}$$

(3.108)

and the generalized forces

$$\begin{aligned} Q_{x}=0,\quad Q_{\varphi }=\mathcal {M(}\dot{\varphi }). \end{aligned}$$

(3.109)

Substituting (3.106)–(3.109) into (3.105), the mathematical model of the system is obtained (Cveticanin 2009)

$$\begin{aligned} \ddot{x}(M+m)+kx\left| x\right| ^{\alpha -1}+c\dot{x}-md(\ddot{ \varphi }\cos \varphi -\dot{\varphi }^{2}\sin \varphi )= & {} 0, \nonumber \\ (J+md^{2})\ddot{\varphi }-md\ddot{x}\cos \varphi= & {} \mathcal {M}(\dot{\varphi } ). \end{aligned}$$

(3.110)

In this case the effect of gravitational force is omitted. The model (3.110) represents a system of two coupled strong nonlinear second order differential equations. To find the exact (closed form) solution of (3.110) is even impossible. An approximate solving procedure for the resonant motion of the system is suggested. It is based on the averaging procedure adopted for the system (3.110). The influence of the nonlinearity order on the Sommerfeld effect is discussed. A method for suppressing Sommerfeld effect is developed. The critical parameters for the Sommerfeld phenomena are approximately determined. As examples the steady-state resonant motions of the oscillators with non-integer order driven by a non-ideal force are considered.

3.3.1 Model of the System

It is convenient to normalize the coordinates and time in (3.110) according to

$$\begin{aligned} x\longrightarrow y=x/l,\qquad t\longrightarrow \tau =\Omega ^{*}t, \end{aligned}$$

(3.111)

where l is the initial length of the non-deformed spring and $\Omega ^{*}$ is the synchronous angular velocity of the rotor (see Dimentberg et al. 1997). By introducing (3.111) and the notation $\phi \rightarrow \psi _{1}$, the differential equations (3.110)) transform into

$$\begin{aligned} y\text {''}+\zeta y^{\prime }+p^{2}y\left| y\right| ^{\alpha -1}= & {} \mu (\varphi \text {''}\cos \varphi -\varphi ^{\prime 2}\sin \varphi ), \nonumber \\ \varphi \text {''}= & {} \eta y\text {''}\cos \varphi +(V_{m}^{*}-C_{m}^{*}\varphi ^{\prime }), \end{aligned}$$

(3.112)

where

$$\begin{aligned} p^{2}= & {} \frac{kl^{\alpha -1}}{\Omega ^{*2}(M+m)},\qquad \qquad \zeta =\left( \frac{c}{M+m}\right) \frac{1}{\Omega ^{*}},\qquad \mu =\frac{dm}{l(M+m)} ,\qquad \nonumber \\ \eta= & {} \frac{dml}{J+md^{2}},\qquad V_{m}^{*}=\frac{V_{\varphi m}}{ (J+md^{2})\Omega ^{*2}},\qquad C_{m}^{*}=\frac{C_{\varphi m}}{ (J+md^{2})\Omega ^{*}}, \end{aligned}$$

(3.113)

and $(^{\prime })$ ${\equiv }(\mathrm{d}/\mathrm{d}\tau$), $(\text {''})$ ${\equiv }(\mathrm{d}^{2}/\mathrm{d}\tau ^{2})$. After some modification, we have

$$\begin{aligned} y\text {''}(1-\mu \eta \cos ^{2}\varphi )+p^{2}y\left| y\right| ^{\alpha -1}+\zeta y^{\prime }= & {} \mu \left[ (V_{m}^{*}-C_{m}^{*}\varphi ^{\prime })\cos \varphi -\varphi ^{\prime 2}\sin \varphi \right] , \nonumber \\ \varphi \text {''}(1-\mu \eta \cos ^{2}\varphi )+\mu \eta \varphi ^{\prime 2}\cos \varphi \sin \varphi= & {} -\eta (\zeta y^{\prime }+p^{2}y\left| y\right| ^{\alpha -1})\cos \varphi \nonumber \\&+(V_{m}^{*}-C_{m}^{*}\varphi ^{\prime }). \end{aligned}$$

(3.114)

Due to the physical properties of the system, it can be concluded that the parameters $\mu$ and $\eta$ are small in comparison to 1 and can be treated as the product of a small parameter $\varepsilon$ and constants $\mu _{1}$ and $\eta _{1}$, i.e., $\mu =\varepsilon \mu _{1}$ and $\eta =\varepsilon \eta _{1}$. The same is valid for $V_{m}^{*}$ and $C_{m}^{*}$, and also the damping parameter $\zeta$, i.e., we have $V_{m}^{*}=\varepsilon V_{m}$, $C_{m}^{*}=\varepsilon C_{m}$ and $\zeta =\varepsilon \zeta _{1}$. Then, the system of differential equations (3.114) is simplified into

$$\begin{aligned} y\text {''}+p^{2}y\left| y\right| ^{\alpha -1}= & {} -\varepsilon (\mu _{1}\varphi ^{\prime 2}\sin \varphi +\zeta _{1}y^{\prime }), \nonumber \\ \varphi \text {''}= & {} -\varepsilon \eta _{1}p^{2}y\left| y\right| ^{\alpha -1}\cos \varphi +\varepsilon V_{m}(1-K_{m}\varphi ^{\prime }), \end{aligned}$$

(3.115)

where $\varepsilon<<1$ is a small positive parameter and $K_{m}=C_{m}/V_{m}$. In (3.115) the small terms of the second order are neglected. The Eq. (3.115) represent the system of two coupled differential equations which describe the motion of the non-ideal system given in Cveticanin and Zukovic (2015a).

3.3.2 Analytical Solving Procedure

Let us rewrite the differential equations (3.115) into a system of four first order differential equations

$$\begin{aligned} y^{\prime }= & {} z, \nonumber \\ z^{\prime }= & {} -p^{2}y\left| y\right| ^{\alpha -1}+\varepsilon F_{1}, \nonumber \\ \varphi ^{\prime }= & {} \Omega , \nonumber \\ \Omega ^{\prime }= & {} \varepsilon F_{2}, \end{aligned}$$

(3.116)

where

$$\begin{aligned} F_{1}=-(\mu _{1}\Omega ^{2}\sin \varphi +\zeta z),\qquad F_{2}=V_{m}(1-K_{m}\Omega )-\eta _{1}p^{2}y\left| y\right| ^{\alpha -1}\cos \varphi . \end{aligned}$$

(3.117)

For $\varepsilon =0$, the Eq. (3.116) transform into

$$\begin{aligned} y^{\prime }=z,\qquad z^{\prime }=-p^{2}y\left| y\right| ^{\alpha -1},\qquad \varphi ^{\prime }=\Omega ,\qquad \Omega ^{\prime }=0. \end{aligned}$$

(3.118)

The first two differential equations describe the motion of a pure integer or noninteger order nonlinear oscillator (see Cveticanin et al. 2012; Cveticanin and Pogany 2012). The exact analytical solution of (3.118) is

$$\begin{aligned} y= & {} a \text {ca}(\alpha ,1,\nu t),\qquad z=-\frac{2ha^{(\alpha +1)/2}}{\alpha +1} \text {sa}(1,\alpha ,\nu t), \nonumber \\ \varphi= & {} C_{1}t+C_{2},\qquad \Omega =C_{1}, \end{aligned}$$

(3.119)

where

$$\begin{aligned} \nu =ha^{(\alpha -1)/2},\qquad h=\left| p\right| \sqrt{\frac{\alpha +1}{2}}, \end{aligned}$$

(3.120)

$C_{1}$ and $C_{2}$ are constants of integration, A is an arbitrary constant and ca$(\alpha ,1,\nu t)=$ ca and sa$(1,\alpha ,\nu t)=$ sa are the cosine and sine Ateb-functions given by Droniuk et al. (1997, 2010), Droniuk and Nazarkevich (2010) (see Appendix). Namely, (3.119) is the generating solution of the generating Eq. (3.118). Based on that solution, the trial solution for (3.116) is introduced.

Let us express y, z, $\varphi$ and $\Omega$ as functions of new variables a, $\psi$, $\varphi$ and $\Omega$, i.e.

$$\begin{aligned} y=a\text {ca}(\alpha ,1,\psi )\equiv a\text {ca}(\psi ), \end{aligned}$$

$$\begin{aligned} z=-\frac{2h}{\alpha +1}a^{(\alpha +1)/2}\text {sa}(1,\alpha ,\psi )\equiv - \frac{2h}{\alpha +1}a^{(\alpha +1)/2}\text {sa}(\psi ), \end{aligned}$$

(3.121)

and $\psi _{2}\rightarrow \Theta$. According to the expressions for the derivatives of Ateb functions (A22) and (A23) (see Appendix), the first time derivatives of (3.121) follow

$$\begin{aligned} y^{\prime }= & {} a^{\prime }\text {ca}(\psi )-\frac{2\psi ^{\prime }}{\alpha +1} a\text {sa}(\psi ), \nonumber \\ z^{\prime }= & {} -ha^{(\alpha -1)/2}A^{\prime }\text {sa}(\psi )-\frac{2h\psi ^{\prime }}{\alpha +1}a^{(\alpha +1)/2}\text {ca}^{\alpha }(\psi ). \end{aligned}$$

(3.122)

Substituting (3.122) into (3.116) and using the relations (A21) (see Appendix) and (3.120), the modified equations of motion are

$$\begin{aligned} \psi ^{\prime }= & {} ha^{(\alpha -1)/2}-\varepsilon F_{1}\frac{\alpha +1}{2h} a^{-(\alpha +1)/2}\text {ca}(\psi ), \nonumber \\ a^{\prime }= & {} -\frac{\varepsilon F_{1}}{h}a^{(1-\alpha )/2}\text {sa}(\psi ), \nonumber \\ \varphi ^{\prime }= & {} \Omega , \nonumber \\ \Omega ^{\prime }= & {} \varepsilon F_{2}, \end{aligned}$$

(3.123)

where

$$\begin{aligned} F_{1}= & {} -\mu _{1}\Omega ^{2}\sin \varphi +\zeta _{1}\frac{2h}{\alpha +1} a^{(\alpha +1)/2}\text {sa}(\psi ), \\ F_{2}= & {} V_{m}(1-K_{m}\Omega )-\eta _{1}p^{2}a\text {ca}(\psi )\left| a \text {ca}(\psi )\right| ^{\alpha -1}\cos \varphi . \end{aligned}$$

Equation (3.123) are the four first order differential equations which in the first approximation correspond to (3.166). In these equations the trigonometric and Ateb periodic functions exist. It is well known that the period of trigonometric functions sin$\varphi$ and cos$\varphi$ is 2$\pi$, while of the Ateb functions sa($\psi$) and ca($\psi$) is 2$\Pi _{\alpha }$ , where the expression for $\Pi _{\alpha }$ is given in Appendix (see Eq. (A14))

$$\begin{aligned} \Pi _{\alpha }= \text {B}\left( \frac{1}{\alpha +1},\frac{1}{2}\right) , \end{aligned}$$

(3.124)

and B is the beta function (Abramowitz and Stegun 1964). Introducing the new variable

$$\begin{aligned} \psi = \frac{\Pi _{\alpha }}{2\pi }\bar{\psi }, \end{aligned}$$

(3.125)

we obtain the Ateb functions sa($\frac{\Pi _{\alpha }}{2\pi }\bar{\psi }$) and ca($\frac{\Pi _{\alpha }}{2\pi }\bar{\psi }$) whose period is also 2$\pi$ as is for the trigonometric functions sin$\varphi$ and cos$\varphi$. Substituting (3.125) into (3.123), it follows

$$\begin{aligned} \bar{\psi }^{\prime }= & {} \left( \frac{2\pi }{\Pi _{\alpha }}\right) ha^{(\alpha -1)/2}-\varepsilon F_{1}\frac{\alpha +1}{2h}\left( \frac{2\pi }{ \Pi _{\alpha }}\right) a^{-(\alpha +1)/2}\text {ca}\left( \frac{\Pi _{\alpha }}{ 2\pi }\bar{\psi }\right) , \nonumber \\ a^{\prime }= & {} -\frac{\varepsilon F_{1}}{h}a^{(1-\alpha )/2}\text {sa}\left( \frac{ \Pi _{\alpha }}{2\pi }\bar{\psi }\right) , \nonumber \\ \varphi ^{\prime }= & {} \Omega , \nonumber \\ \Omega ^{\prime }= & {} \varepsilon F_{2}, \end{aligned}$$

(3.126)

where

$$\begin{aligned} F_{1}=-\mu _{1}\Omega ^{2}\sin \varphi +\zeta _{1}\frac{2h}{\alpha +1} a^{(\alpha +1)/2}\text {sa}\left( \frac{\Pi _{\alpha }}{2\pi }\bar{\psi }\right) , \end{aligned}$$

$$\begin{aligned} F_{2}=V_{m}\left( 1-K_{m}\Omega \right) -\eta _{1}p^{2}A\text {ca}\left( \frac{\Pi _{\alpha }}{ 2\pi }\bar{\psi }\right) \left| a\text {ca}\left( \frac{\Pi _{\alpha }}{2\pi }\bar{\psi } \right) \right| ^{\alpha -1}\cos \psi . \end{aligned}$$

(3.127)

Differential equations (3.126) represent the mathematical model of the non-ideal system for the non-resonant case which is not of a significant interest. Much more important case is the resonant one.

3.3.3 Resonant Case and the Averaging Solution Procedure

Let us introduce the new variable $\theta$ which satisfies the relation

$$\begin{aligned} \bar{\psi }=\theta +\varphi , \end{aligned}$$

(3.128)

with time derivative

$$\begin{aligned} \bar{\psi }^{\prime }=\theta ^{\prime }+\Omega . \end{aligned}$$

(3.129)

Substituting (3.129) and (3.128) into (3.126), differential equations with variables A, $\theta$, $\varphi$ and $\Omega$ follow as

$$\begin{aligned} \theta ^{\prime }= & {} \left[ \left( \frac{2\pi }{\Pi _{\alpha }}\right) ha^{(\alpha -1)/2}-\Omega \right] -\varepsilon F_{1}\frac{\alpha +1}{2h} \left( \frac{2\pi }{\Pi _{\alpha }}\right) a^{-(\alpha +1)/2}\text {ca}\left( \frac{ \Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) , \nonumber \\ a^{\prime }= & {} -\frac{\varepsilon F_{1}}{h}a^{(1-\alpha )/2}\text {sa}\left( \frac{ \Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) , \nonumber \\ \varphi ^{\prime }= & {} \Omega , \nonumber \\ \Omega ^{\prime }= & {} \varepsilon F_{2}, \end{aligned}$$

(3.130)

where

$$\begin{aligned} F_{1}= & {} -\mu _{1}\Omega ^{2}\sin \varphi +\gamma \frac{2h}{\alpha +1} a^{(\alpha +1)/2}\text {sa}\left( \frac{\Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) ,\qquad \nonumber \\ F_{2}= & {} V_{m}(1-K_{m}\Omega )-\eta _{1}p^{2}a\text {ca}\left( \frac{\Pi _{\alpha } }{2\pi }(\theta +\psi _{1})\right) \left| a\text {ca}\left( \frac{\Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) \right| ^{\alpha -1}\cos \varphi . \nonumber \\&\end{aligned}$$

(3.131)

For the case when

$$\begin{aligned} \left( \frac{2\pi }{\Pi _{\alpha }}\right) ha^{(\alpha -1)/2}-\Omega =\varepsilon \sigma , \end{aligned}$$

(3.132)

the condition of nonlinear resonance is satisfied. Thus, according to (3.130)$_{1}$, $\theta ^{\prime }$ is of the order $\varepsilon$. For $\theta ^{\prime }$ of the order $O(\varepsilon )$, the relation (3.129) yields the difference between the frequencies ($\bar{\psi }^{\prime }-\varphi ^{\prime })\equiv$ ($\bar{\psi }^{\prime }-\Omega )$ to be also of the $\varepsilon$ order.

To solve the system of differential equations (3.130) is not an easy task. It is the reason the averaging procedure suggested by Zhuravlev and Klimov (1988) is adopted for this special case. The averaging is done over the period 2$\pi$ of the variable $\varphi$. The averaged differential equations are

$$\begin{aligned} \theta ^{\prime }= & {} \left[ \left( \frac{2\pi }{\Pi _{\alpha }}\right) ha^{(\alpha -1)/2}-\Omega \right] -\varepsilon \bar{F}_{1\theta }\frac{ \alpha +1}{2h}\left( \frac{2\pi }{\Pi _{\alpha }}\right) a^{-(\alpha +1)/2}, \nonumber \\ a^{\prime }= & {} -\frac{\varepsilon \bar{F}_{1A}}{h}a^{(1-\alpha )/2}, \nonumber \\ \varphi ^{\prime }= & {} \Omega ,\qquad \Omega =\varepsilon \bar{F}_{2}, \end{aligned}$$

(3.133)

where

$$\begin{aligned} \bar{F}_{1A}= & {} -\mu _{1}\Omega ^{2}\bar{f}_{1}+\zeta _{1}\frac{2h}{\alpha +1 }a^{(\alpha +1)/2}\bar{f}_{2}, \nonumber \\ \bar{F}_{1\theta }= & {} -\mu _{1}\Omega ^{2}\bar{f}_{3}+\zeta _{1}\frac{2h}{ \alpha +1}a^{(\alpha +1)/2}\bar{f}_{4}, \nonumber \\ \bar{F}_{2}= & {} V_{m}(1-K_{m}\Omega )-\eta _{1}p^{2}A^{\alpha }\bar{f}_{5}, \end{aligned}$$

(3.134)

and

$$\begin{aligned} \bar{f}_{1}= & {} \frac{1}{2\pi }\mathop {\displaystyle \int }\limits _{0}^{2\pi }\sin \varphi \text {sa}\left( \frac{\Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) d\varphi ,\quad \bar{f}_{2}= \frac{1}{2\pi }\mathop {\displaystyle \int }\limits _{0}^{2\pi }\text {sa}^{2}\left( \frac{\Pi _{\alpha }}{ 2\pi }(\theta +\varphi )\right) d\varphi , \nonumber \\ \bar{f}_{3}= & {} \frac{1}{2\pi }\mathop {\displaystyle \int }\limits _{0}^{2\pi }\sin \varphi \text {ca}\left( \frac{\Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) d\varphi ,\quad \nonumber \\ \bar{f}_{4}= & {} \frac{1}{2\pi }\mathop {\displaystyle \int }\limits _{0}^{2\pi }\text {sa}\left( \frac{\Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) \text {ca}\left( \frac{\Pi _{\alpha }}{2\pi } (\theta +\varphi )\right) d\varphi , \nonumber \\ \bar{f}_{5}= & {} \frac{1}{2\pi }\mathop {\displaystyle \int }\limits _{0}^{2\pi }\cos \varphi \text {ca}\left( \frac{\Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) \left| \text {ca}^{\alpha -1}\left( \frac{\Pi _{\alpha }}{2\pi }(\theta +\varphi )\right) \right| d\varphi . \end{aligned}$$

(3.135)

Being the sine and cosine Ateb periodic functions, they are suitable for Fourier series expansion . The finite Fourier approximation of the functions is according to Droniuk et al. (2010, 2010)

$$\begin{aligned} \text {sa}(1,\alpha ,\psi )= & {} \frac{a_{0}}{2}+\mathop {\displaystyle \sum }\limits _{n=1}^{\infty }a_{n}\sin \frac{\pi n\psi }{\Pi _{\alpha }}, \nonumber \\ \text {ca}(\alpha ,1,\psi )= & {} \mathop {\displaystyle \sum }\limits _{n=1}^{\infty }b_{n}\cos \frac{ \pi n\psi }{\Pi _{\alpha }}, \end{aligned}$$

(3.136)

where the coefficient in the series are

$$\begin{aligned} a_{0}= & {} \frac{2}{\Pi _{\alpha }}\mathop {\displaystyle \int }\limits _{0}^{\Pi _{\alpha }}sa(1,\alpha ,\psi )d\psi ,\qquad a_{n}=\frac{2}{\Pi _{\alpha }} \mathop {\displaystyle \int }\limits _{0}^{\Pi _{\alpha }}sa(1,\alpha ,\psi )\sin \frac{\pi n\psi }{ \Pi _{\alpha }}d\psi , \nonumber \\ b_{n}= & {} \frac{2}{\Pi _{\alpha }}\mathop {\displaystyle \int }\limits _{0}^{\Pi _{\alpha }}ca(\alpha ,1,\psi )\cos \frac{\pi n\psi }{\Pi _{\alpha }}d\psi \end{aligned}$$

(3.137)

and $\psi \equiv \frac{\Pi _{\alpha }}{2\pi }(\theta +\varphi )$. In this calculation the Fourier series expansion of the function ca$(\psi )\left| \text {ca}^{\alpha -1}(\psi )\right|$ is also introduced as (see Mickens 2004; Cveticanin 2008)

$$\begin{aligned} ca(\psi )\left| \text {ca}^{\alpha -1}(\psi )\right| =\mathop {\displaystyle \sum }\limits _{n=1}^{\infty }c_{n}\cos (n\psi ), \end{aligned}$$

(3.138)

with

$$\begin{aligned} c_{n}=\frac{4}{\pi }\mathop {\displaystyle \int }\limits _{0}^{\pi /2}ca(\psi )\left| \text {ca} ^{\alpha -1}(\psi )\right| \cos (n\psi )d\psi . \end{aligned}$$

(3.139)

For practical reasons, it is suitable to determine the solution in the first approximation. Then, using the first terms of the Fourier series (3.136), the expressions (3.134) and (3.135) are transformed into

$$\begin{aligned} \bar{f}_{1}=\frac{1}{2}a_{1}\cos \theta ,\quad \bar{f}_{2}=\frac{1}{2} a_{1}^{2},\quad \bar{f}_{3}=-\frac{1}{2}b_{1}\sin \theta ,\quad \bar{f} _{4}=0,\quad \bar{f}_{5}=\frac{1}{2}c_{1}\cos \theta , \end{aligned}$$

(3.140)

and

$$\begin{aligned} \bar{F}_{1A}= & {} -\frac{1}{2}\mu _{1}a_{1}\Omega ^{2}\cos \theta +\zeta _{1} \frac{h}{\alpha +1}a^{(\alpha +1)/2}a_{1}^{2}, \nonumber \\ \bar{F}_{1\theta }= & {} \frac{1}{2}\mu _{1}b_{1}\Omega ^{2}\sin \theta ,\quad \bar{F}_{2}=V_{m}(1-K_{m}\Omega )-\frac{1}{2}\eta _{1}p^{2}a^{\alpha }c_{1}\cos \theta \end{aligned}$$

(3.141)

where $a_{1}$, $b_{1}$ and $c_{1}$ are the coefficients calculated according to (3.137) and (3.139) for $n=1$. Substituting (3.141) into (3.133) the simplified averaged differential equations are

$$\begin{aligned} \theta ^{\prime }= & {} \left[ \left( \frac{2\pi }{\Pi _{\alpha }}\right) ha^{(\alpha -1)/2}-\Omega \right] -\frac{\varepsilon \mu _{1}}{2}b_{1}\Omega ^{2}\frac{\alpha +1}{2h}\left( \frac{2\pi }{\Pi _{\alpha }}\right) a^{-(\alpha +1)/2}\sin \theta , \nonumber \\ A^{\prime }= & {} -\frac{1}{h}a^{(1-\alpha )/2}\left( -\frac{\varepsilon \mu _{1}}{2} a_{1}\Omega ^{2}\cos \theta +\varepsilon \zeta _{1}a_{1}^{2}\frac{h}{\alpha +1}a^{(\alpha +1)/2}\right) , \nonumber \\ \Omega ^{\prime }= & {} \varepsilon \left( V_{m}(1-K_{m}\Omega )-\frac{1}{2} \eta _{1}c_{1}p^{2}a^{\alpha }\cos \theta \right) . \end{aligned}$$

(3.142)

The equations describe the transient motion in the resonant case .

Steady - state solution

Equating the right side of the Eq. (3.142) to zero and after some modification the steady-state equations up to the first order approximation are

$$\begin{aligned} \varepsilon \mu _{1}\Omega ^{2}\sin \theta= & {} \frac{1}{b_{1}}\frac{4h}{\alpha +1}\left[ ha^{(\alpha -1)/2}-\left( \frac{ \Pi _{\alpha }}{2\pi }\right) \Omega \right] a^{(\alpha +1)/2}, \end{aligned}$$

(3.143)

$$\begin{aligned} \varepsilon \mu _{1}\Omega ^{2}\cos \theta= & {} \frac{2\varepsilon \zeta _{1}a_{1}h}{\alpha +1}a^{(\alpha +1)/2}, \end{aligned}$$

(3.144)

$$\begin{aligned} \eta _{1}c_{1}p^{2}a^{\alpha }\cos \theta= & {} 2V_{m}(1-K_{m}\Omega ). \end{aligned}$$

(3.145)

Eliminating the variable $\theta$ in the Eqs. (3.143) and (3.144) the frequency - response relation is as follows

$$\begin{aligned} 1=\frac{a^{(\alpha +1)}}{\Omega ^{4}\mu _{1}^{2}}\left( \left( \frac{2\zeta _{1}a_{1}h}{\alpha +1}\right) ^{2}+\left( \frac{4h}{\varepsilon b_{1}}\frac{ ha^{(\alpha -1)/2}-\left( \frac{\Pi _{\alpha }}{2\pi }\right) \Omega }{ \alpha +1}\right) ^{2}\right) . \end{aligned}$$

(3.146)

Dividing Eqs. (3.144) and (3.145) the $a-\theta$ expression as a function of the control parameter $V_{m}$ is obtained

$$\begin{aligned} V_{m}(1-K_{m}\Omega )=\frac{\eta _{1}c_{1}a_{1}p^{2}}{\mu _{1}(\alpha +1)} \frac{\zeta _{1}h}{\Omega ^{2}}a^{(3\alpha +1)/2}. \end{aligned}$$

(3.147)

Characteristic points

The characteristic point P for the curves (3.72) and (3.147) corresponds to the peak amplitude and exists if the condition of the equality of the gradients $da/d\Omega$ for the both curves is satisfied. Due to complexity of the expressions, it is suggested to use the approximate procedure for obtaining of this point. Instead of the exact characteristic point P, the point P’ is determined, in which the bifurcation of the solutions of the Eq. (3.72) appears. It is known that the locus of these two points (P and P’) are quite close to each other.

Let us assume that in (3.66) the left hand side of the equation is zero, i.e.,

$$\begin{aligned} ha^{(\alpha -1)/2}-\left( \frac{\Pi _{\alpha }}{2\pi }\right) \Omega =0. \end{aligned}$$

(3.148)

Substituting (3.148) into (3.72) the peak amplitude is obtained

$$\begin{aligned} a_{P}=\left( \frac{1}{\mu _{1}}\left( \frac{\Pi _{\alpha }}{2\pi h}\right) ^{2}\left( \frac{2\zeta _{1}a_{1}h}{\alpha +1}\right) \right) ^{2/(\alpha -3)}. \end{aligned}$$

(3.149)

Substituting (3.149) into (3.148), the locus $\Theta _{P}$ for $a_{P}$ is obtained

$$\begin{aligned} \Omega _{P}=\left( \left( \frac{\Pi _{\alpha }}{2\pi h}\right) ^{(\alpha +1)}\left( \frac{2\zeta _{1}a_{1}h}{\mu _{1}(\alpha +1)}\right) ^{(\alpha -1)}\right) ^{1/(\alpha -3)}. \end{aligned}$$

(3.150)

Equations (3.149) and (3.150) with (3.147) give the value of a control parameter $V_{mP^{\prime }}$ of the motor torque

$$\begin{aligned} V_{mP}=\frac{q_{2}c_{1}a_{1}p^{2}}{\mu _{1}(\alpha +1)(1-K_{m}\Omega _{P})} \frac{\zeta _{1}h}{\Omega _{P}^{2}}a_{P}^{(3\alpha +1)/2}. \end{aligned}$$

(3.151)

For this approximate value of the control parameter $V_{mP}$ the Sommerfeld effect has to appear.

Analyzing the relations (3.149) and (3.150) it can be seen that the relations are independent on the value of $\eta _{1}$.

3.3.4 Suppression of the Sommerfeld Effect

As it is previously shown the Eqs. (3.149)–(3.151) give us the values $a_{P^{\prime }}$, $\Omega _{P^{\prime }}$ of the point P’ and also the corresponding control parameter $V_{mP}$ for which the Sommerfeld effect exists and the amplitude-frequency torque curve is forced through the point P’. Our intention is to suppress the Sommerfeld effect which is evident for P’.

Let us calculate the gradient of the backbone curve (3.148) in the bifurcation point P. Substituting $a_{P}$ given with the relation (3.149) into the first derivative $(da/d\Omega )$ of the relation (3.148), the required gradient is obtained

$$\begin{aligned} \left( \frac{da}{d\Omega }\right) _{P}=\frac{\alpha +1}{\alpha -1}\frac{\mu _{1}}{ \zeta _{1}a_{1}}\left( \frac{2\pi }{\Pi _{\alpha }}\right) . \end{aligned}$$

(3.152)

The gradient is the function of the order of the nonlinearity $\alpha$. For $\alpha <1$ the direction of the tangent is such to give an obtuse angle and for $\alpha >1$ a sharp angle. For $\alpha =1$ the tangent is orthogonal.

The gradient of the amplitude-frequency torque curve (3.147) for the bifurcation point is also worth to be determined. The first derivative $(dA/d\Omega )$ for (3.147) is calculated and the coordinates $\Omega _{P}$ for $a_{P}$ given with relations (3.149) and (3.150) have to be substituted

$$\begin{aligned} \left( \frac{da}{d\Omega }\right) _{P}=V_{mP}\frac{2(2-3K_{m}\Omega _{P})\mu _{1}}{\eta _{1}c_{1}a_{1}p^{2}\zeta _{1}h}\frac{\alpha +1}{3\alpha +1}\frac{ \Omega _{P}}{(a_{P})^{(3\alpha -1)/2}}. \end{aligned}$$

(3.153)

Equating the gradients (3.152) and (3.153) in the P’ and assuming the condition (3.151), we have

$$\begin{aligned} \frac{3\alpha +1}{\alpha -1}\frac{\mu _{1}}{\zeta _{1}a_{1}}\left( \frac{ 2\pi }{\Pi _{\alpha }}\right) =\frac{2(2-3K_{m}\Omega _{P})}{(\alpha +1)(1-K_{m}\Omega _{P})}\frac{a_{P}}{\Omega _{P}}. \end{aligned}$$

(3.154)

The relation (3.154) depends on the parameter of the torque $K_{m}$, parameters of eccentricity $\mu _{1}$, and also on the order of nonlinearity $\alpha$. Due to the motor properties (3.3) and the sign of the gradient of the $V_{m}$ curve, it can be concluded that the relation (3.154) is valid only for $\alpha <1$.

Let us solve the relation (3.154) for the driving parameter

$$\begin{aligned} \tilde{K}_{m}=\frac{2-p_{1}}{(3-p_{1})\Omega _{P}}, \end{aligned}$$

(3.155)

where

$$\begin{aligned} p_{1}=\frac{(3\alpha +1)(\alpha +1)}{\alpha -1}\frac{\mu _{1}}{\zeta _{1}a_{1}}\left( \frac{\pi }{\Pi _{\alpha }}\right) \frac{\Omega _{P}}{a_{P}} . \end{aligned}$$

(3.156)

Then the value $\tilde{V}_{m}$ is the necessary one for elimination of the Sommerfeld effect. Substituting the value of $\tilde{K}_{m}$ (3.155) into the relation for $V_{mP}$ (3.151), the corresponding control parameter $\tilde{V}_{m}$ is obtained

$$\begin{aligned} \tilde{V}_{m}=\frac{(3-p_{1})\eta _{1}c_{1}a_{1}p^{2}}{\mu _{1}(\alpha +1)} \frac{\zeta _{1}h}{\Omega _{P}^{2}}a_{P}^{(3\alpha +1)/2}. \end{aligned}$$

(3.157)

Finally, the parameter $\tilde{C}_{m}$ is

$$\begin{aligned} \tilde{C}_{m}=\frac{\tilde{V}_{m}}{\tilde{K}_{m}}=\frac{\eta _{1}c_{1}a_{1}p^{2}(3-p_{1})^{2}}{\mu _{1}(\alpha +1)(2-p_{1})}\frac{\zeta _{1}h}{\Omega _{p}}a_{P}^{(3\alpha +1)/2}. \end{aligned}$$

(3.158)

For the parameter value (3.158) i.e., (3.155) and (3.157) the jump phenomena in the non-ideal system with known order of nonlinearity $\alpha$ is excluded. For all values of the control parameter $V_{m}$ there is always only one steady point in the $a-\Omega$ curve, i.e., only one intersection point between $V_{m}$ and $a-\Omega$ curve . Then, the Sommerfeld effect does not exist.

3.3.5 Numerical Examples of Non-ideal Driven Pure Nonlinear Oscillators

For physical interpretation, a motor-support system mounted in a soya extraction plant is considered: mass of the system is ($M+m$) of 5 kg, radius of the motor rotor is 0.120 m, measure of the unbalance is $md=0.0125$ kgm, length l is 0.05 m, moment of inertia of the motor rotor J $=0.698\times 10^{-3}$ kgm$^{2}$, the synchronous speed of the rotor is $\Omega ^{*}=1450$ rpm and the damping coefficient is $c=37.\,961$ kg/s. The three types of nonlinearity are considered: one, with the order of nonlinearity $\alpha =2/3$ (smaller than 1) and coefficient of rigidity $k=0.42470\times 10^{5}$ N/m$^{2/3}$, second, with $\alpha =1$ (linear case) and coefficient of rigidity $k=1.\,152\,8\times 10^{5}$ N/m and with the order of nonlinearity $\alpha =5/3$ (higher than 1) and coefficient of rigidity $k=8.\,494\,1\times 10^{5}$ N/m$^{5/3}$. Two values of the motor parameters are considered: $C_{\varphi m}=1.898$ Nm/s and $C_{\varphi m}=18.98$ Nm/s. The corresponding non-dimensional parameters are according to (3.53): $\varepsilon =0.1$, $\mu _{1}=0.5$, $\eta _{1}=0.5$, $p^{2}=1$ and $\zeta _{1}=0.5$. Two values for $C_{m}$ are 1 and 10. Varying the control parameter $V_{m}$ of the motor, the steady-state properties of the system are analyzed. For $C_{m}=1$ the initial control parameter for the increasing case is $V_{m}=0.3$ and for the decreasing case it is $V_{m}=1.2$, while for $C_{m}=10$ it is $V_{m}=3$ and $V_{m}=12$, respectively. The analytical results are obtained by solving the relations (3.72) and (3.147), and numerical ones by integrating of the original differential equations of motion (3.48). For numerical calculation the Runge–Kutta procedure is applied.

Fig. 3.8

Frequency-response steady state curve obtained analytically for $\alpha =5/3$ and $C_{m}=10$ with characteristic points

The numerical procedure is as follows: for a certain value of the control parameter $V_{m}$ the relations (3.48) are numerically solved and the amplitude and frequency are computed. The amplitude of vibration is computed for the maximal disposition of the oscillator from the equilibrium position, while the frequency of vibration is the averaged value for $\dot{\psi }_{1}$ during the steady state motion. These values are the initial values for the further numerical calculation. Increasing the control parameter $V_{m}$ for a small value, the new steady-state parameters are computed. The numerical calculation is repeated for the new higher value of the control parameter and with initial conditions which are the previous steady-state parameters. After the significant repetition of the numerical process for increasing of the control parameter $V_{m}$, an amplitude-frequency curve for increasing of the control parameter is obtained. In contrary, if the same procedure is applied but the control parameter is decreased the another amplitude-frequency curve is obtained which partly differs from the first one.

Fig. 3.9

Frequency-response curves for $\alpha =5/3$ and $C_{m}=10$ obtained analytically (full line) and numerically (circles - for the increasing of $V_{m}$, squares - for decreasing of $V_{m})$

If during the numerical solution of (3.48) the obtained amplitudes significantly differ for the two infinitesimal close values of the control parameter $V_{m}$, it is evident that the jump phenomena occurs. It gives us the chance to “recognized” the Sommerfeld effect during the numerical process.

In Figs. 3.8 and 3.9 the amplitude - frequency diagrams obtained analytically by solving (3.72) and (3.147) and numerically by solving (3.48) for $C_{m}=10$ and $\alpha =5/3$ are plotted. The control parameter $V_{m}$ is varied and the steady-state amplitude - frequency values are obtained. The computation time was 200 periods of vibration T. The steady-state numerical solution is reached after approximately 30T. The period T is computed numerical. For the certain value of control parameter $V_{m}$ the steady state amplitude and frequency are calculated. These values were used as the initial conditions for computation of the steady-state values for the new value of the control parameter $V_{m}$.

Fig. 3.10

Frequency-response steady state curve obtained analytically for $\alpha =5/3$ and $C_{m}=1$ with characteristic points

From the both figures it is obvious that for slow increasing of $\Omega$ and variation of the motor parameter $V_{m}$, the amplitude of vibration increases up to the point P. Using the relations (3.149)–(3.151) the parameters of the characteristic point are obtained: $a_{P}=0.9$, $\Omega _{P}=0.92$ and $V_{mP}=9.42$. Along the curve $V_{mP}$ the jump phenomena from P to Q ($a_{Q}=0.098$, $\Omega _{Q}=0.94$) occurs. For that position $\Omega _{Q}>\Omega _{P}$, but the amplitude $a_{Q}$ is significantly smaller than $a_{P}$. Further increase of the control parameter $V_{m}$ causes the further decrease of the amplitude of vibration for increase of the frequency $\Omega$. If the value of the control parameter $V_{m}$ is decreased, the steady-state locus in $a-\Omega$ plane moves to left along the $a-\Omega$ curve to the point R. The frequency $\Omega$ decreases and the amplitude a increases slowly up to the value of $a_{R}=0.1$ at the frequency $\Omega _{R}=0.6$ (the values are obtained numerically). For $V_{mR}=6.01$ a sudden change in the amplitude of vibration appears. The amplitude jumps along the curve $V_{mR}$ to the value $a_{S}=0.3$ with $\Omega _{S}=0.6$. The further decrease of the control parameter $V_{m}$ produces amplitude and frequency decrease. The region in the $a-\Omega$ plane bounded with the part SP and RQ of the $a-\Omega$ curves and also RS and PQ parts of the $V_{m}-\Omega$ curves represent the hysteresis and the jump from P to Q and R to S in the system is the so called Sommerfeld effect well known in the non-ideal mechanical systems .

Fig. 3.11

Frequency-response curves for $\alpha =5/3$ and $C_{m}=1$ obtained analytically (full line) and numerically (circles - for the increasing of $V_{m}$, squares - for decreasing of $V_{m})$

In Fig. 3.9 the values obtained numerically by solving of the original differential equations of motion (3.48) for increasing of the control parameter $V_{m}$ is plotted with circles (o) and for decreasing of the control parameter with a filled squares ($\blacksquare$). Comparing the analytically obtained solutions (full line) with the numerically obtained values it can be seen that the results are in a good agreement.

Fig. 3.12

Frequency-response steady state curve obtained analytically for $\alpha =2/3$ and $C_{m}=10$ with characteristic points

In Figs. 3.10 and 3.11 the frequency-response curve for the oscillator with order of nonlinearity $\alpha =2/3$ and motor parameter $C_{m}=1$ is plotted. The curves in Fig. 3.10 are obtained by solving the approximate relations (3.72) and (3.147), while in Fig. 3.11 numerically solved differential equations (3.48) are plotted. Both figures show that for slow increasing of $\Omega$ and variation of the motor parameter $V_{m}$, the amplitude of vibration increases up to the point P whose properties are $a_{P}=0.9$, $\Omega _{P}=0.9$ and $V_{mP}=1.12$.

Fig. 3.13

Frequency-response curves for $\alpha =2/3$ and $C_{m}=10$ obtained analytically (full line), numerically (circles - for the increasing of $V_{m}$, squares - for decreasing of $V_{m})$

Along the curve $V_{mP}$ the jump phenomena from P to Q ($a_{Q}=0.055$, $\Omega _{Q}=1.115$) occurs: the frequency $\Omega _{Q}$ is higher than $\Omega _{P}$, and the amplitude $a_{Q}$ is smaller than $a_{P}$. During further increase of the control parameter $V_{m}$, the amplitude of vibration decreases while the frequency $\Omega$ increases. If the value of the control parameter $V_{m}$ is decreased, the steady-state locus in $a-\Omega$ plane moves to left along the $a-\Omega$ curve to the point R. The frequency decreases to $\Omega _{R}=0.6$ and the amplitude increases to $a_{R}=0.125$ when $V_{mR}=0.6$. Then a sudden change in the amplitude of vibration appears. The amplitude jumps along the curve $V_{mR}$ to the amplitude $a_{S}=0.3$ and frequency $\Omega _{S}=0.585$. The further decrease of the control parameter $V_{m}$ produces amplitude and frequency decrease. The bounded region SPQR represents the hysteresis plane. Comparing Figs. 3.8 and 3.10 it is seen that the Sommerfeld effect occurs for lower values of the motor parameter $V_{m}$ if $C_{m}=1$ than if it is $C_{m}=10$.

Fig. 3.14

Frequency-response steady state curve obtained analytically for $\alpha =2/3$ and $C_{m}=1:$ a with characteristic points, b for increasing of $V_{m}$, c for decreasing of $V_{m}$

In Fig. 3.11 the numerically obtained $a-\Omega$ diagrams are plotted: $V_{m}$ is increased (o), and $V_{m}$ is decreased ($\blacksquare$). These numerically obtained results by solving (3.48) are compared with analytically obtained results (full line curves). The difference between the analytical and numerical solutions is negligible.

In Figs. 3.12 and 3.13 the analytically and numerically obtained amplitude - frequency curves for the oscillator with the order of nonlinearity $\alpha =2/3$ and motor parameter $C_{m}=10$ are plotted. The analytical solutions are obtained by solving (3.72) and (3.147), and the numerical solutions by solving (3.48). The computation procedure was the same to that made in the previous calculation.

First, the control parameter $V_{m}$ was slowly increased. The steady-state frequency and amplitude increase, too, up to P, when the control parameter has the value $V_{mP}=12.45$ and the amplitude and frequency are $a_{P}=0.075$, $\Omega _{P}=1.24$. These values are obtained analytically by solving the relations (3.149)–(3.151). For that value of control parameter ($V_{mP}=12.45)$ a sudden change in amplitude occurs for the almost the same frequency. Namely, there is the jump to the point Q with the following values of the amplitude and frequency: $a_{Q}=0.475,$ $\Omega _{Q}=1.22$. Increasing the value of $V_{m}$ gives decreasing of the amplitude a and increasing of $\Omega$. If the control parameter is $V_{m}$ and the frequency $\Omega$ are decreased, the amplitude of vibration increases up to $a_{R}=1.0$ for $V_{mR}=10.55$ and $\Omega _{R}=1.03$. At that value of the control parameter the jump to another steady state position occurs with amplitude $a_{S}=0.02$ and frequency $\Omega _{S}=1.04$. Further decrease of the control parameter gives also the decrease of the amplitude and frequency. Finally, it can be concluded that the Sommerfeld effect and the jump phenomena occur for $V_{mP}=12.45$ and $V_{mR}=10.55$.

Comparing the analytically obtained curve (full line) and the numerically obtained curves for the case when the control parameter increases (o) and when it decreases ($\blacksquare$) it can be seen that the difference is negligible (see Fig. 3.13).

Fig. 3.15

Frequency-response curves for $\alpha =2/3$ and $C_{m}=1$ obtained analytically (full line) and numerically (circles - for the increasing of $V_{m}$, squares - for decreasing of $V_{m})$

In Figs. 3.14 and 3.15 the frequency-response curve for the oscillator with order of nonlinearity $\alpha =2/3$ and motor parameter $C_{m}=1$ is plotted. The curves in Fig. 3.14 are obtained by solving the approximate relations (3.72) and (3.147), while in Fig. 3.15 numerically solved differential equations (3.48) are plotted. In Fig. 3.14b the steady-state motion for increasing value of the control parameter $V_{m}$ is plotted. The amplitude of vibration increases slowly with the frequency up to the point $A_{1}$ with coordinates $a_{A1}=0.1$, $\Omega _{A1}=1.24$ for the control parameter $V_{mA1}=1.249$. For that value of the control parameter the other steady state motion is with the amplitude $a_{A2}=0.875$, and frequency $\Omega _{A2}=1.04$. For this value of the control parameter the amplitude increases significantly, and the frequency decreases. For higher values of the control parameter the tendency of increase but also decreasing of the frequency is evident. For the control parameter $V_{mA3}=1.305$ the peak amplitude $a_{A3}=1.025$ for $\Omega _{A3}=1.03$ is reached. For this value of control parameter the another steady state motion is with amplitude and frequency $a_{A4}=0.455$, $\Omega _{A4}=1.25$, respectively. Further increase of the control parameter $V_{m}$ gives the decrease of the amplitude and increase of the frequency.

In Fig. 3.14c the procedure is repeated but in the opposite direction: the control parameter $V_{m}$ is decreased. Decreasing $V_{m}$, the amplitude is increasing up to $a_{B1}=$ 0.7and the frequency is decreased to $\Omega _{B1}=1.13$ for $V_{mB1}=1.261$. At that value of the control parameter a jump to the amplitude $a_{B2}=0.9$ and frequency $\Omega _{B2}=1.04$ occurs. Further decrease of the control parameter causes decrease of the amplitude but increase of the frequency to the boundary values $a_{B3}=0.475,$ $\Omega _{B3}=1.12$ for $V_{mB3}=1.190$. For that value of control parameter the other steady state motion is with parameters $a_{B4}=0.025$, $\Omega _{B4}=1.19$. Decreasing $V_{m}$ the amplitude and frequency decrease. In Fig. 3.14a the four values of the control parameter $V_{m}$ for with the amplitude - frequency curve of the motor is the tangent of the steady state curve are shown. It means, that for $\Omega$ slowly increased two times the jump phenomena appear: from $A_{1}$ to $A_{2}$ (the amplitude jumps to a higher value) and from $A_{3}$ to $A_{4}$ (the amplitude jumps to a smaller value) as is shown in Fig. 3.14b. For slow decreasing of $\Omega$ the amplitude jumps two times in the steady-state curve, too (see Fig. 3.14c) from $B_{1}$ to $B_{2}$ (the amplitude jumps to a higher value) and from $B_{3}$ to $B_{4}$ (the amplitude jumps to a smaller value).

Fig. 3.16

Frequency-response steady state curve obtained analytically for $\alpha =1$ and $C_{m}=1$ with characteristic points

Fig. 3.17

Frequency-response steady state curve obtained analytically for $\alpha =2/3$ and: a $C_{m}=0.85$, and b $C_{m}=5$

In Fig. 3.15 the numerically obtained $a-\Omega$ diagrams are plotted: $V_{m}$ is increased (o), and $V_{m}$ is decreased ($\blacksquare$). These numerically obtained results by solving (3.48) are compared with analytically obtained results (full line curves). The difference between the analytical and numerical solutions is negligible.

In Fig. 3.16 the influence of the control parameter $V_{m}$ on amplitude-frequency curve obtained analytically by solving (3.72) and (3.147) for the linear oscillator $\alpha =1$ and motor parameter $C_{m}=1$ is analyzed. In the diagram two values of the control parameter for which the Sommerfeld effect occurs is obtained. For the case of increasing frequency, the control parameter for which two steady state motions P($a_{P}=1,$ $\Omega _{P}=1)$ and Q($a_{Q}=0.14$, $\Omega _{Q}=1.24)$ exist is $V_{mP}=1.251$. For the case of decreasing frequency, for the control parameter $V_{mR}=1.101$ the jump is from R($a_{R}=425$, $\Omega _{R}=1.06)$ to S($a_{S}=0.075$, $\Omega _{S}=0.7)$.

In Fig. 3.17 we plot the case where the Sommerfeld effect is suppressed. The parameters of the system are $\alpha =2/3,$ $\varepsilon =0.1$, $\mu _{1}=0.5$, $\eta _{1}=0.5$, $p^{2}=1$ and $\zeta _{1}=1.2$. Two values of $C_{m}$ are considered: $C_{m}=0.85$ (Fig. 3.17a) and $C_{m}=5$ (Fig. 3.17b) while $V_{m}$ is varied. The first value for $C_{m}$ is computed according to (3.158) and the second value is an arbitrary one. It can be seen that for $C_{m}=5$ the $V_{m}$ intersects the A-$\Omega$ curve in one, two or three points giving the Sommerfeld effect. For $C_{m}=0.85$ only one intersection point of $V_{m}$ with the amplitude frequency exists and no amplitude jump exists. Thus, the Sommerfeld effect is eliminated.

Comparing the curves and the results plotted in Figs. 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15 and 3.16 the following is concluded:

• 1. The amplitude-frequency curve bends on right for $\alpha >1$ (Figs. 3.8 and 3.9), on the left for $\alpha <1$ (Figs. 3.12, 3.13, 3.14 and 3.15) and is straight for $\alpha =1$ (Fig. 3.16).

• 2. For $C_{m}=10$ the lines for $V_{m}$ curves in Figs. 3.8, 3.9, 3.12 and 3.13 are closed to the vertical direction and this case is close to the ideal system. For the ideal system, the oscillator has no influence on the motion of the motor and there is no coupling between these two motions. The ideal system has one-degree-of-freedom and the motion of the oscillator is forced with a time periodical excitation force. The mathematical model of the system is a second order differential equation. For $C_{m}=10$ (see Figs. 3.14, 3.15 and 3.16) the $V_{m}$ curves are bent and the properties of non-ideal system are highly significant. For $C_{m}=1$ and $\alpha =1$ two characteristic values of the control parameter $V_{m}$ are evident (see Fig. 3.16), while for $C_{m}=10$ and $\alpha =2/3$ even four (Figs. 3.14 and 3.15).

• 3. In the system with order of nonlinearity $\alpha =2/3$ the Sommerfeld effect can be eliminated by suitable assumption of the value of the parameter $C_{m}$ (see Fig. 3.17). Namely, for $C_{m}=5$ the effect of jump occurs (Fig. 3.14b), while for the critical value $C_{m}=0.85$ and any value of $V_{m}$, the Sommerfeld effect is eliminated.

3.3.6 Conclusion

Analyzing the results the following is concluded:

• 1. In the non-ideal system which contains a pure nonlinear oscillator (order of the nonlinearity is integer or non-integer) and a motor with linear torque properties (a non-ideal source) for certain parameter values the resonant phenomena appears.

• 2. The approximate averaging procedure, based on the introduction of additional slow variables, as it is assumed that the angular velocity and the frequency of the structure are functions of these variables, is appropriate for analytical analysis of the problem. The suggested averaging procedure gives the equations which are suitable for analysis of the near resonant case. Results which are obtained are applicable for the analysis of the characteristic properties of the system:

  • (a) The amplitude-frequency curve bends on right for the nonlinearity of the order higher than 1, i.e., the order is a positive rational number higher than 1. The amplitude-frequency curve bends on the left, if the nonlinearity is a positive rational number smaller than 1. For the linear case,the backbone curve of the amplitude-frequency steady state curve is straight.

  • (b) The approximate value of the control parameter for the non-ideal source is analytically calculated applying the method of equating the gradient of the both amplitude-frequency curves (of the oscillator and of the motor) in the intersection points. The criteria for the Sommerfeld effect is analytically obtained.

  • (c) The Sommerfeld effect occurs not only for the linear case and when the nonlinearity is higher than 1 (as it is previously published), but also if the nonlinearity is a positive rational number smaller than 1.

  • (d) The parameters of the energy source effect the steady-state properties of the non-ideal mechanical system. The both parameters of the motor torque ($V_{m\phi }$ and $C_{m\phi })$ and also their rate determine the motion properties. Hence, independently on the order of nonlinearity, the higher the value of $C_{m\phi }$ the system tends to the ideal one.

  • (e) For the case when the nonlinearity is of the order smaller than 1, the amplitude jump effect may occur two times during increase of the motor property $V_{m\phi }$ and two times during its decrease. For certain motor property the characteristic curve may be the tangent of the amplitude-frequency curve in two points during increase of $V_{m\phi }$ and also during its decrease, and for these values the jump effect occur. In the linear non-ideal oscillator and in the oscillator with a nonlinearity higher than 1 the jump phenomena is evident once during increasing and once during decreasing of $V_{m\phi }$. Namely, the characteristic of the motor is the tangent of the amplitude-frequency curve for one $V_{m\phi }$ during increase, and for other value during decrease.

• 4. The method given in this text obtains motor torque parameters which can suppress the Sommerfeld effect in the non-ideal mechanical system with the positive nonlinearity of order smaller than 1. The Sommerfeld effect and the jump phenomena in the amplitude for non-ideal system may be suppressed by these critical parameter values.

• 5. Comparing the analytical and numerical solutions it is evident that they are in a good agreement independently on the order of nonlinearity.

3.4 Stable Duffing Oscillator and a Non-ideal Energy Source

Let us consider a motor operating on a structure with strong cubic nonlinearity (Fig. 3.18). Oscillator is of Duffing type and is connected with the motor with limited power supply. The motor has an eccentric mass which is the part of the non-ideal perturbation source. The driving of the system comes from the unbalanced rotor linked to the oscillator fed by an electric motor. The driven system is taken as a consequence of the dynamics of the whole system (oscillator plus rotor). Duffing oscillator , connected with non-ideal energy source, is of hardening type with strong cubic nonlinearity which was widely investigated by Krylov and Bogolubov (1943), Ueda (1985), Fang and Dowell (1987), Pezeshki and Dowell (1988), Yuste and Bejarano (1986, 1990), Cheng et al. (1991), Chen and Cheung (1996), Chen et al. (1998), Gendelman and Vakakis (2000), Mickens (2001, 2006), Andrianov (2002), He (2002), Hu and Xiong (2003), Andrianov and Awrejcewicz (2003a, b), Amore and Aranda (2005), Cveticanin (2004, 2006, 2011), Ozis and Yildirm (2007), Belendez et al. (2007), Cveticanin et al. (2010), Kovacic et al. (2010) and others. The structure of mass M is connected to a fixed basement by a non-linear spring and a linear viscous damper (damping coefficient c). The nonlinear spring stiffness is given by $k_{1}x+k_{2}x^{3}$, where x denotes the structure displacement with respect to some equilibrium position in the absolute reference frame. The motion of the structure is due to an in-board non-ideal motor driving an unbalanced rotor. We denote by $\varphi$ the angular displacement of the rotor unbalance, and model it as a particle of mass m and radial distance d from the rotating axis. The moment of inertia of the rotating part is J. For the resonant case the structure has an influence on the motor input or output. The forcing function is dependent of the system it acts on and the source is of non-ideal type.

Fig. 3.18

Model of the motor-structure non-ideal system

The non-ideal problem has two - degrees of freedom, represented by the generalized coordinates x and $\varphi$. The kinetic energy T , potential energy U and the dissipative function $\Phi$ are expressed by

$$\begin{aligned} T= \frac{1}{2}M\dot{x}^{2}+\frac{1}{2}m(\dot{x}^{2}+d^{2}\dot{\varphi }^{2}-2d \dot{x}\dot{\varphi }\sin \varphi )+\frac{1}{2}J\dot{\varphi }^{2}, \end{aligned}$$

(3.159)

$$\begin{aligned} U=\frac{1}{2}k_{1}x^{2}+\frac{1}{4}k_{2}x^{4}-(M+m)gx-mgd\cos \varphi ,\qquad \Phi =\frac{1}{2}c\dot{x}^{2}. \end{aligned}$$

(3.160)

A dot denotes differentiation with respect to time t. Lagrange’s equations of motion for the system are in general

$$\begin{aligned} \frac{d}{dt}\left( \frac{\partial T}{\partial \dot{x}}\right) -\frac{ \partial T}{\partial x}+\frac{\partial U}{\partial x}+\frac{\partial \Phi }{ \partial \dot{x}}= & {} Q_{x}, \\ \frac{d}{dt}\left( \frac{\partial T}{\partial \dot{\varphi }}\right) -\frac{ \partial T}{\partial \varphi }+\frac{\partial U}{\partial \varphi }+\frac{ \partial \Phi }{\partial \dot{\varphi }}= & {} Q_{\dot{\varphi }}, \end{aligned}$$

where $Q_{x}$ and $Q_{\dot{\varphi }}$ are generalized forces.

The differential equations of motion have the form

$$\begin{aligned} \ddot{x}(M+m)+c\dot{x}-md(\ddot{\varphi }\sin \varphi +\dot{\varphi }^{2}\cos \varphi )+k_{1}x+k_{2}x^{3}= & {} (M+m)g, \nonumber \\ (J+md^{2})\ddot{\varphi }-md\ddot{x}\sin \varphi +mgd\sin \varphi= & {} \mathcal { M}(\dot{\varphi }), \end{aligned}$$

(3.161)

where $\mathcal {M}(\dot{\varphi })$ is the motor torque. The most often used model of the torque is the linear moment-speed relation (see Dimentberg et al. 1997)

$$\begin{aligned} \mathcal {M}(\dot{\varphi })=M_{0}\left( 1-\frac{\dot{\varphi }}{\Omega }\right) , \end{aligned}$$

(3.162)

where $M_{0}$ and $\Omega$ are constant values.

It is convenient to normalize the coordinates and time according to

$$\begin{aligned} x\longrightarrow y=\frac{\Omega ^{2}}{g}x,\qquad t\longrightarrow \tau =\Omega t, \end{aligned}$$

(3.163)

where g is the gravity constant. By introducing (3.163) the differential equations (3.161) transform into

$$\begin{aligned} y\text {''}+\zeta y^{\prime }+p^{2}y+\gamma y^{3}= & {} 1+\mu (\varphi \text {''}\sin \varphi +\varphi ^{\prime 2}\cos \varphi ), \nonumber \\ \varphi \text {''}= & {} \eta y\text {''}\sin \varphi -\eta \sin \varphi +F(1-\varphi ^{\prime }), \end{aligned}$$

(3.164)

where

$$\begin{aligned} p= & {} \frac{\omega ^{*}}{\Omega },\qquad \omega ^{*2}=\frac{k_{1}}{M+m },\qquad \gamma =\frac{k_{3}g^{2}}{(M+m)\Omega ^{6}},\qquad \zeta =\frac{c}{ \Omega (M+m)}, \nonumber \\ \mu= & {} \frac{m}{M+m}\frac{d\Omega ^{2}}{g},\qquad \eta =\frac{gmd}{\Omega ^{2}(J+md^{2})},\qquad F=\frac{M_{0}}{\Omega ^{2}(J+md^{2})}, \end{aligned}$$

(3.165)

and prime denotes differentiation with respect to $\tau$. The differential equations (3.164) are non-linear and coupled.

3.4.1 Asymptotic Solving Method

In the regime near resonant the difference between the excitation frequency is close to the natural frequency. For the near resonant case one can write

$$\begin{aligned} \varphi ^{\prime }-p=\varepsilon \sigma , \end{aligned}$$

(3.166)

where $\varepsilon \sigma$ is the detuning parameter with the small parameter $\varepsilon<<1$. Expressing parameters of equations (3.165) by

$$\begin{aligned} \zeta =\varepsilon \zeta _{1},\qquad \gamma =\varepsilon \gamma _{1},\qquad \mu =\varepsilon \mu _{1},\qquad \eta =\varepsilon \eta _{1},\qquad F=\varepsilon F_{1}, \end{aligned}$$

(3.167)

the differential equations of motions have the form

$$\begin{aligned} z^{\prime \prime }+p^{2}z= & {} \varepsilon \mu _{1}\left. \left( \varphi ^{\prime \prime }\sin \varphi +\varphi ^{\prime 2}\cos \varphi \right) -\varepsilon \zeta _{1}z^{\prime }-\varepsilon \gamma _{1}\left( z+ \frac{1}{p^{2}}\right) ^{3}\right] , \nonumber \\ \varphi ^{\prime \prime }= & {} \varepsilon \eta _{1}z^{\prime \prime }\sin \varphi -\varepsilon \eta _{1}\sin \varphi +\varepsilon F_{1}(1-\varphi ^{\prime }), \end{aligned}$$

(3.168)

where the new variable is

$$\begin{aligned} z=y-\frac{1}{p^{2}}. \end{aligned}$$

(3.169)

Expressing $z\text {''}$ and $\varphi \text {''}$ from the Eq. (3.168) and assuming only the terms to $O(\varepsilon ^{2})$ we obtain

$$\begin{aligned} z\text {''}+p^{2}z= & {} \varepsilon \left[ \mu _{1}\varphi ^{\prime 2}\cos \varphi -\zeta _{1}z^{\prime }-\gamma _{1}(z+\frac{1}{p})^{3}\right] +\varepsilon ^{2}\ldots , \nonumber \\ \varphi \text {''}= & {} \varepsilon \left[ F_{1}(1-\varphi ^{\prime })-\eta _{1}(1+p^{2}z)\sin \varphi \right] +\varepsilon ^{2}.... \end{aligned}$$

(3.170)

Following the reference (Warminski et al. 2001)

$$\begin{aligned} z=a\cos (\varphi +\psi ), \end{aligned}$$

(3.171)

where a and $\psi$ are the new coordinates. The first derivative is

$$\begin{aligned} z^{\prime }=-ap\sin (\varphi +\psi ), \end{aligned}$$

(3.172)

when

$$\begin{aligned} a^{\prime }\cos (\varphi +\psi )-a(\omega +\psi ^{\prime }-p)\sin (\varphi +\psi )=0. \end{aligned}$$

(3.173)

Determining the second derivative of z and substituting (3.171) and (3.172) into (3.170) we obtain

$$\begin{aligned}&-a^{\prime }p\sin (\varphi +\psi )-ap(\omega +\psi ^{\prime })\cos (\varphi +\psi )+p^{2}a\cos (\varphi +\psi ) \nonumber \\= & {} \varepsilon \mu _{1}\omega ^{2}\cos \varphi +ap\varepsilon \zeta _{1}\sin (\varphi +\psi )-\varepsilon \gamma _{1}\left[ a\cos (\varphi +\psi )+\frac{1}{p} \right] ^{3}, \nonumber \\ \omega ^{\prime }= & {} \varepsilon F_{1}(1-\omega )-\varepsilon \eta _{1}[1+ap^{2}\cos (\varphi +\psi )]\sin \varphi . \end{aligned}$$

(3.174)

Equations (3.173) and (3.174) lead to the derivatives $a^{\prime }$, $\psi ^{\prime },\varphi ^{\prime }$ and $\omega ^{\prime }$

$$\begin{aligned} a^{\prime }= & {} -\sin (\varphi +\psi )\left\{ \frac{\varepsilon \mu _{1}}{p}\omega ^{2}\cos \varphi +a\varepsilon \zeta _{1}\sin (\varphi +\psi ) \right. \nonumber \\&\left. -\frac{1}{p}\varepsilon \gamma _{1}\left[ a\cos (\varphi +\psi )+\frac{1}{p} \right] ^{3}\right\} , \nonumber \\ a\psi ^{\prime }= & {} \varepsilon \sigma a-\cos (\varphi +\psi )\left\{ \frac{ \varepsilon \mu _{1}}{p}\omega ^{2}\cos \varphi +a\varepsilon \zeta _{1}\sin (\varphi +\psi ) \right. \nonumber \\&\left. -\frac{1}{p}\varepsilon \gamma _{1}\left[ a\cos (\varphi +\psi )+\frac{1}{p} \right] ^{3}\right\} , \nonumber \\ \omega ^{\prime }= & {} \varepsilon F_{1}(1-\omega )-\varepsilon \eta _{1}[1+ap^{2}\cos (\varphi +\psi )]\sin \varphi ,\qquad \varphi ^{\prime }=\omega . \end{aligned}$$

(3.175)

Unfortunately, the closed form analytical solution for (3.175) is complicate to be obtained and the numerical methods are convenient.

3.4.2 Stability of the Steady State Solution and Sommerfeld Effect

Introducing the averaging procedure (see Bogolyubov and Mitropolskij 1974)

$$\begin{aligned} a^{\prime }=\frac{\varepsilon }{2\pi }\mathop {\displaystyle \int }\limits _{0}^{2\pi }f_{a}d\varphi ,\qquad a\psi ^{\prime }=\frac{\varepsilon }{2\pi }\mathop {\displaystyle \int }\limits _{0}^{2\pi }f_{\psi }d\varphi ,\qquad \omega ^{\prime }=\mathop {\displaystyle \int }\limits _{0}^{2\pi }f_{\omega }d\varphi , \end{aligned}$$

(3.176)

where

$$\begin{aligned} f_{a}= & {} -\frac{1}{p}\sin (\varphi +\psi )\left\{ \mu _{1}\omega ^{2}\cos \varphi +ap\zeta _{1}\sin (\varphi +\psi )-\gamma _{1}\left[ a\cos (\varphi +\psi )+\frac{1 }{p}\right] ^{3}\right\} , \nonumber \\ f_{\psi }= & {} a\sigma -\frac{1}{p}\cos (\varphi +\psi )\{\mu _{1}\omega ^{2}\cos \varphi +ap\zeta _{1}\sin (\varphi +\psi ) \nonumber \\&\left. -\gamma _{1}\left[ a\cos (\varphi +\psi )+\frac{1}{p}\right] ^{3}\right\} , \nonumber \\ f_{\omega }= & {} F_{1}(1-\omega )-\eta _{1}[1+ap^{2}\cos (\varphi +\psi )]\sin \varphi , \end{aligned}$$

(3.177)

the differential equations (3.175) are transformed into

$$\begin{aligned} \omega ^{\prime }= & {} \varepsilon \left[ F_{1}(1-\omega )+\frac{1}{2}\eta _{1}ap^{2}\sin \psi \right] , \nonumber \\ a^{\prime }= & {} -\frac{\varepsilon }{2}\left( a\zeta _{1}+\frac{\mu _{1}\omega ^{2} }{p}\sin \psi \right) , \nonumber \\ \psi ^{\prime }= & {} \varepsilon \sigma -\frac{\varepsilon }{2p}\left( \frac{\mu _{1}\omega ^{2}}{a}\cos \psi -\frac{3\gamma _{1}a^{2}}{4}-\frac{3\gamma _{1} }{p^{2}}\right) . \end{aligned}$$

(3.178)

For the steady-state response, Eq. (3.178) have the form

$$\begin{aligned} F_{1}(1-\omega _{S})+ \frac{1}{2}\eta _{1}a_{S}p^{2}\sin \psi _{S}= & {} 0, \nonumber \\ a_{S}+\frac{\mu _{1}\omega _{S}^{2}}{p}\sin \psi _{S}= & {} 0, \nonumber \\ \sigma -\frac{1}{2p}\left( \frac{\mu _{1}\omega _{S}^{2}}{a_{S}}\cos \psi _{S}- \frac{3\gamma _{1}a_{S}^{2}}{4}-\frac{3\gamma _{1}}{p^{2}}\right)= & {} 0, \end{aligned}$$

(3.179)

where S denotes the steady state values. Combining the second and the third Eq. (3.179) we obtain

$$\begin{aligned} a_{S}^{2}\left\{ \frac{\zeta _{1}^{2}}{4}+\left[ \sigma +\frac{3\gamma _{1}}{2p}\left( \frac{ a_{S}^{2}}{4}+\frac{1}{p^{2}}\right) \right] ^{2}\right\} =\left( \frac{\mu _{1}\omega _{S}^{2}}{2p} \right) ^{2}, \end{aligned}$$

(3.180)

while combining the first and the second Eq. (3.179) yields

$$\begin{aligned} F_{1}\mu _{1}\omega _{S}^{2}(1-\omega _{S})=\frac{1}{2}\zeta _{1}\eta _{1}a_{S}^{2}p^{3}. \end{aligned}$$

(3.181)

Expressing $a_{S}$ in (3.181) and substituting into (3.180) the $\omega _{S}(F_{1})$ function is obtained

$$\begin{aligned} 8F_{1}(1-\omega _{S})\left\{ \frac{\zeta _{1}^{2}}{4}+\left[ \omega _{S}-p+\frac{3\gamma _{1}}{2p^{5}}\left( \frac{\mu _{1}\omega _{S}^{2}F_{1}(1-\omega _{S})}{2\zeta _{1}\eta _{1}p}+1\right) \right] ^{2}\right\} -\mu _{1}\zeta _{1}\eta _{1}p\omega _{S}^{2}=0. \end{aligned}$$

(3.182)

The number of real solution is one, two or three and it depends on the control parameter $F_{1}$. To determine which of the solutions actually correspond to a realizable motion, we need to consider the stability of the solutions. We determine the stability by determining the nature of the singular points of (3.179). To accomplish this, we let

$$\begin{aligned} a=a_{S}+a_{1},\qquad \psi =\psi _{S}+\psi _{1},\qquad \omega =\omega _{S}+\omega _{1}. \end{aligned}$$

(3.183)

Substituting (3.183) into (3.179) and neglecting all but the linear terms in $a_{1},\psi _{1},\omega _{1}$ we obtain

$$\begin{aligned} \omega _{1}^{\prime }= & {} \varepsilon \left[ -F_{1}\omega _{1}+ \frac{1}{2}\eta _{1}a_{1}p^{2}\sin \psi _{S}+\frac{\psi _{1}}{2}\eta _{1}a_{S}p^{2}\cos \psi _{S}\right] , \nonumber \\ a_{1}^{\prime }= & {} -\frac{\varepsilon }{2}\left( a_{1}\zeta _{1}+\psi _{1}\frac{ \mu _{1}\omega _{S}^{2}}{p}\cos \psi _{S}+2\frac{\mu _{1}\omega _{S}\omega _{1}}{p}\sin \psi _{S}\right) , \nonumber \\ \psi _{1}^{\prime }= & {} -\omega _{1}-\frac{\varepsilon }{2p}\left( 2\frac{\mu _{1}\omega _{S}\omega _{1}}{a_{S}}\cos \psi _{S}-a_{1}\frac{\mu _{1}\omega _{S}^{2}}{a_{S}^{2}}\cos \psi _{S} \right. \\&\left. -\psi _{1}\frac{\mu _{1}\omega _{S}^{2}}{a_{S}}\sin \psi _{S}-\frac{ 3\gamma _{1}a_{S}a_{1}}{2}\right) . \nonumber \end{aligned}$$

(3.184)

Equations (3.184) are linear and have the solution in the form

$$\begin{aligned} (a_{1},\psi _{1},\omega _{1})=(a_{10},\psi _{10},\omega _{10})\exp (\lambda \tau ), \end{aligned}$$

where $\lambda$ is an eigenvalue of the coefficient of matrix. The solutions are stable and hence corresponding motions reliable, if the real part of each eigenvalue is negative or zero.

For the parameter values

$$\begin{aligned} \varepsilon =0.1,\quad \zeta _{1}=0.2,\quad \gamma _{1}=0.1,\quad \eta _{1}=1.0,\quad \mu _{1}=0.5, \end{aligned}$$

(3.185)

using the relations (3.180) and (3.181), i.e., (3.182) the response-control parameter $F_{1}$ and the frequency of vibration-control parameter $F_{1}$ diagrams are plotted (Fig. 3.19).

Fig. 3.19

Jump effect for: a amplitude-control parameter curve, b frequency-control parameter curve

Fig. 3.20

Frequency-response curve obtained: a analytically, b numerically

In Fig. 3.20 the frequency-response curves obtained numerically and analytically by solving the Eqs. (3.180) and (3.181) are plotted. Comparing the solutions it can be concluded that the difference between “exact” numerical solution and approximate analytical solution is negligible. Analyzing the obtained curves and the relation (3.181) it is evident that the curve depends on the control parameter $F_{1}$. For $\omega _{S}=2/3$ the maximal response

$$\begin{aligned} a_{S\max }=\frac{2}{3p}\sqrt{\frac{2\mu _{1}}{3\zeta _{1}\eta _{1}p}F_{1}}, \end{aligned}$$

is a function of $F_{1}$, but is not dependent on the parameter of non-linearity $\gamma _{1}$.

Figures 3.19 and 3.20 show the characteristic points in the diagrams: point R and point T. Between these points the solution is unstable while all those outside this region are stable. Point R is the peak in $a_{S}-\omega _{S}$, $a_{S}-F_{1}$ and $F_{1}-\omega _{S}$ diagrams. Analyzing the relation (3.182) and equating the first derivative to zero ($dF_{1}/d\omega _{S}=0)$

$$\begin{aligned} 0= & {} -8F_{1}\left\{ \frac{\zeta _{1}^{2}}{4}+\left[ \omega _{S}-p+\frac{3\gamma _{1}}{2p^{5}}\left( \frac{ \mu _{1}\omega _{S}^{2}F_{1}(1-\omega _{S})}{2\zeta _{1}\eta _{1}p}+1\right) \right] ^{2}\right\} \\&+2F_{1}(1-\omega _{S})\left[ \omega _{S}-p+\frac{3\gamma _{1}}{2p^{5}}\left( \frac{\mu _{1}\omega _{S}^{2}F_{1}(1-\omega _{S})}{2\zeta _{1}\eta _{1}p}+1\right) \right] \\&\left[ 1+\frac{3\gamma _{1}}{2p^{5}}\frac{\mu _{1}F_{1}(2\omega _{S}-3\omega _{S}^{2})}{2\zeta _{1}\eta _{1}p}\right] -2\mu _{1}\zeta _{1}\eta _{1}p\omega _{S}. \end{aligned}$$

the peak is obtained. The locus of the peak (point R) is according to (3.180)

$$\begin{aligned} \omega _{SR}=p-\frac{3\varepsilon \gamma _{1}}{2p}\left( \frac{a_{SR}^{2}}{4}+ \frac{1}{p^{2}}\right) , \end{aligned}$$

(3.186)

and the amplitude is $a_{SR}=\mu _{1}\omega _{SR}^{2}/\tilde{\alpha }p$. Substituting (3.186) into the relation (3.181) we obtain the value of the control parameter for the peak amplitude

$$\begin{aligned} F_{1R}=\frac{\eta _{1}p\mu _{1}\omega _{SR}^{2}}{2\zeta _{1}(1-\omega _{SR})} . \end{aligned}$$

(3.187)

Substituting the parameter values (3.185) into (3.187) the numeric value of the control parameter is $F_{1R}=18.246$. For that calculated value of the control parameter and the known value of the parameter of nonlinearity ($\gamma _{1}=0.1$) the amplitude and frequency of Q are obtained by solving the relations (3.180) and (3.181). For the control parameter $F_{1T}=4.57$ two stable solutions are obtained: T with $a_{ST}$ and $\omega _{ST}$ and P with $a_{SP}$ and $\omega _{SP}$. We note that there are gaps in the diagrams (see Figs. 3.19 and 3.20) where no steady state response exists. The gaps are not the same for increasing and decreasing the control parameter $F_{1}$. Increasing the control parameter causes the increase of the amplitude and frequency of vibrations to R. Then the effect of jump to smaller amplitude and higher frequency in Q appears. Decreasing the control parameter $\xi _{1}$ decreases the frequency and increases the amplitude to T and then jump into P with higher amplitude and smaller frequency occurs. The same hysteresis is seen in Fig. 3.20. This phenomena of jump is called the Sommerfeld effect in non-ideal systems.

To eliminate the Sommerfeld effect for the certain value of the control parameter $F_{1R}$ the parameter of nonlinearity has to be calculated. The Eq. (3.180) indicates the peak amplitude

$$\begin{aligned} a_{SR}= \frac{\mu _{1}}{\zeta _{1}p}\omega _{SR}^{2}. \end{aligned}$$

(3.188)

The function $a_{SR}(\omega _{SR})$ is independent on the parameter of non-linearity $\gamma _{1}$. The relation (3.181) gives the dependence of amplitude $a_{S}$ on $\omega _{S}$ for various values of control parameter $\xi _{1}$. Using (3.188) and (3.181) we obtain

$$\begin{aligned} F_{1R}(1-\omega _{SR})-\frac{1}{2}\eta _{1}\frac{\mu _{1}}{\zeta _{1}}\omega _{SR}^{2}p=0. \end{aligned}$$

(3.189a)

$\omega _{SR}$ is the solution of (3.189a) for the control parameter $F_{1R}$. Substituting $\omega _{SR}$ into (3.188) the amplitude $a_{SR}$ is calculated. Introducing the so obtained values $\omega _{SR}$ and $a_{SR}$ into (3.180) the value of nonlinear parameter $\gamma _{1R}$ is determined

$$\begin{aligned} \gamma _{1R}=\frac{8}{3}\frac{p^{3}}{a_{SR}^{2}p^{2}+4}(p-\omega _{SR}). \end{aligned}$$

(3.190)

For $\gamma _{1}<\gamma _{1R}$ only one stable solution exists for the control parameter $F_{1R}$ and the motion is without jump.

Fig. 3.21

Frequency-control parameter curve for various values of parameter of non-linearity

In Fig. 3.21 the $a_{S}-F_{1}$ curve for two values of parameter of nonlinearity $\gamma _{1}$ is shown. It is seen that for $\gamma _{1}=0.1$ the bending of the curve is smaller than for $\gamma _{1}=0.15$. For $F_{1R}=42.1$ and $\gamma _{1R}=0.15$ two real values of amplitude exist and for $\gamma _{1}=0.1$ only one. It can be concluded that for the real system with parameters (3.185) which works with control parameter $F_{1R}=42.1$ the parameter of non-linearity of the structure has to be $\gamma _{1}<0.15$. Then, the Sommerfeld effect is eliminated.

Fig. 3.22

Trajectories in the phase space for various values of the control parameter: a periodic solution with period 1, b periodic solution with period 2, c periodic solution with period 3, d periodic solution with period 4, e periodic solution with period 5, f periodic solution with period 9, g, h chaotic solution

3.4.3 Numerical Simulation and Chaotic Behavior

To prove the analytically obtained results the numerical experiment is done. The system (3.51) is rewritten in the form

$$\begin{aligned} y_{1}^{\prime }= & {} y_{2}, \nonumber \\ y_{2}^{\prime }= & {} \frac{1}{1-\mu \eta \sin ^{2}y_{3}}(-\zeta y_{2}-py_{1}-\gamma y_{1}^{3}+1 \nonumber \\&+\mu \sin y_{3}(\mu y_{4}^{2}\cos y_{3}-\eta \sin y_{3}+F(1-y_{4}))), \nonumber \\ y_{3}^{\prime }= & {} y_{4,} \\ y_{4}^{\prime }= & {} \frac{\eta \sin y_{3}}{1-\mu \eta \sin ^{2}y_{3}}(-\zeta y_{2}-py_{1}-\gamma y_{1}^{3}+1+\mu y_{4}^{2}\cos y_{3} \nonumber \\&+\mu \sin y_{3}\left( -\eta \sin y_{3}+F(1-y_{4})\right) ) \nonumber \\&-\eta \sin y_{3}+F(1-y_{4}). \nonumber \end{aligned}$$

(3.191)

A number of numerical simulations are done for various control parameter F. Applying the Runge–Kutta numerical procedure with the fixed step length the system of four first order differential equations (3.191) is solved. The results are plotted in Fig. 3.22. The phase space of the system has out of four dimensions, but we were chiefly interested in position of the oscillator itself. We also plotted the Poincaré map which represents the surface of section ($y_{1}$($\tau _{n}$),$y_{2}$($\tau _{n}$)). The points ($y_{1}$($\tau _{n}$),$y_{2}$($\tau _{n}$)) are captured for $\tau _{n}=nT$, where $n=1,2,\ldots$, with period $T=2\pi$/$\bar{\Omega }$. The angular velocity $\overline{\Omega }$ is obtained numerically

$$\begin{aligned} \overline{\Omega }=\frac{\varphi \left( \tau _{1}\right) -\varphi \left( 0\right) }{\tau _{1}}=\frac{y_{3}\left( \tau _{1}\right) -y_{3}\left( 0\right) }{\tau _{1}}, \end{aligned}$$

(3.192)

where $\tau _{1}$ is a long time period for numerical calculation.

For the parameter values of the system, mentioned in the previous section, and the control parameter $F=7.0$ the trajectory in the phase space and the Poincaré map are plotted (Fig. 3.22a). The motion is periodic with period 1 and frequency $\overline{\Omega }=0.999$. For $F=15$ the periodic motion is with period 2 (Fig. 3.22b) and for $F=21$ with period 3 (Fig. 3.22c). Increasing the value of the control parameter the motion is periodic but the period number increases, too (see Fig. 3.22d–f). For high values of control parameter ($F=80$ and $F=10000)$ the strange attractor is obtained (Fig. 3.22g, h). The existence of strange attractor signifies chaos which is evident only if the certain criteria for the maximal (local) Lyapunov exponent are satisfied. For computational reasons the vector notation for the system of Eq. (3.191) is introduced

$$\begin{aligned} Y^{\prime }=G(Y,P), \end{aligned}$$

(3.193)

where $Y=[y_{1},y_{2},y_{3},y_{4}]^{T}$ is the state space vector, $G=[ g_{1},g_{2},g_{3},g_{4}]$, P($p,\alpha ,\eta ,\mu ,F$) is set of parameters and [...]$^{T}$ is denoting transpose. The equations for small deviations $\delta Y$ from the trajectory Y(t) are

$$\begin{aligned} \delta Y^{\prime }=L_{ij}(Y(t))\delta Y,\qquad i,j=1,2,\ldots 4, \end{aligned}$$

(3.194)

where $L_{ij}=\partial g_{i}/\partial y_{j}$ is the Jacobian matrix of derivatives

$$\begin{aligned} L_{11}= & {} L_{13}=L_{14}=0,\qquad L_{12}=1, \nonumber \\ L_{21}= & {} -\frac{p+3\alpha y_{1}^{2}}{1-\mu \eta \sin ^{2}y_{3}},\qquad L_{22}=-\frac{\zeta }{1-\mu \eta \sin ^{2}y_{3}},\qquad \nonumber \\ L_{23}= & {} \frac{-\mu y_{4}^{2}\sin y_{3}+\mu \eta (1-y_{4})\cos y_{3}}{1-\mu \eta \sin ^{2}y_{3}}-\frac{Q_{1}}{(1-\mu \eta \sin ^{2}y_{3})^{2}}, \nonumber \\ L_{24}= & {} \frac{2\mu y_{4}\cos x_{3}-\mu \eta \sin y_{3}}{1-\mu \eta \sin ^{2}y_{3}},\quad L_{44}=\frac{\mu \eta y_{4}\sin (2y_{3})-F}{1-\mu \eta \sin ^{2}y_{3}}, \nonumber \\ L_{31}= & {} L_{32}=L_{33}=0,\qquad L_{34}=1, \nonumber \\ L_{41}= & {} -\frac{\eta p\sin y_{3}+3\eta \gamma y_{1}^{2}\sin y_{3}}{1-\mu \eta \sin ^{2}y_{3}},\qquad L_{42}=-\frac{\zeta \eta \sin y_{3}}{1-\mu \eta \sin ^{2}y_{3}}, \nonumber \\ L_{43}= & {} \frac{\mu \eta y_{4}^{2}\cos (2y_{3})-\eta \cos x_{3}(\zeta y_{2}+py_{1}+\gamma y_{1}^{3})}{1-\mu \eta \sin ^{2}y_{3}} \nonumber \\&+\frac{Q_{2}}{(1-\mu \eta \sin ^{2}y_{3})^{2}}, \end{aligned}$$

(3.195)

where

$$\begin{aligned} Q_{1}= & {} -\mu \eta \sin (2y_{3})[\zeta y_{2}+py_{1}+\gamma y_{1}^{3}-\mu y_{4}^{2}\cos y_{3}-\mu F(1-y_{4})\sin y_{3}], \nonumber \\ Q_{2}= & {} \mu \eta \sin (2y_{3})[F(1-y_{4})+0.4\mu \eta y_{4}^{2}\sin (2y_{3})-\eta \sin y_{3}(\zeta y_{2}+py_{1}+\gamma y_{1}^{3})]. \nonumber \\&\end{aligned}$$

(3.196)

The maximal Lyapunov exponent of the system is then as defined by Wolf et al. (1985)

$$\begin{aligned} \lambda =\lim \frac{1}{t}\log \frac{\left\| \delta Y(t)\right\| }{ \left\| \delta Y(0)\right\| }. \end{aligned}$$

(3.197)

In Fig. 3.23 we present the local Lyapunov exponent for the control parameters $F=80$ and $F=10000$. For the both parameter values the positive Lyapunov exponent is calculated. As it is discussed by Wolf et al. (1985), if the system contains at least one positive Lyapunov exponent the motion is chaotic . The two initially nearby orbits (or trajectories) diverge from each other and the separation of two nearby trajectories increases exponentially with time due to sensitive dependence on initial conditions.

Fig. 3.23

Positive Lyapunov exponent for the control parameters: a $F=80$ and b $F=10000$

The transition from periodic motion to chaos is by periodic doubling. In Fig. 3.24 the $y_{1}-F$ bifurcation diagram is plotted. The bifurcation diagram is obtained for the long time integration of the differential equations (3.191) of the motion. After decay of transient motion a steady state motion is established. The fixed values of the parameters are $\alpha =0.1,$ $\gamma =9$, $p=1$, $\mu =8.373$, $\eta =0.05$ and the control parameter we choose to work with is F. By increasing parameter F we found sequences of period doubling bifurcations (Fig. 3.24). Diagram starts from $F=6$ where periodic motion with period $n=1$ exists. This periodic solution bifurcates on $F\approx 8$ onto period 2T. Further period doubling bifurcation (period 4, 8, 16,...) leads to chaotic motion for $F\approx 20$. After this interval of chaos the periodic motion with period 3 exists. Further increase of parameter F leads to the new period doubling bifurcation with periods 5, 9, ... and finally to chaos for $F\in [28,33]$. The next regions of chaotic motion are for $F\in [40,56]$ and $F>62$.

Fig. 3.24

Period doubling bifurcation diagram

Comparing the bifurcation parameter values in Fig. 3.24 with the results of numerical experiment (see Fig. 3.22) it can be concluded that for the control parameter $F=80$ chaotic motion exists.

3.4.4 Chaos Control

It is of interest to control the chaos in the motion of the system (3.191) and specially of oscillator. There are many methods for chaos control. Dantas and Balthazar (2003) show that if we use an appropriate damping coefficient the chaotic behavior is avoided. The method of Pyragas (1992, 1995) is based on the addition of a special kind of time-continuous perturbation (external force control), which does not change the form of the desired unstable periodic solution, but under certain conditions can stabilize it. The method of Pyragas was applied in the paper. For the external force $\Phi (t)$ the model (3.191) becomes

$$\begin{aligned} y_{1}^{\prime }= & {} y_{2}+\Phi (t), \nonumber \\ y_{2}^{\prime }= & {} \frac{1}{1-\mu \eta \sin ^{2}y_{3}}\{-\zeta y_{2}-py_{1}-\gamma y_{1}^{3}+1+\mu \sin y_{3}[\mu y_{4}^{2}\cos y_{3} \nonumber \\&-\eta \sin y_{3}+F(1-y_{4})\}, \nonumber \\ y_{3}^{\prime }= & {} y_{4}, \nonumber \\ y_{4}^{\prime }= & {} \frac{\eta \sin y_{3}}{1-\mu \eta \sin ^{2}y_{3}}(-\zeta y_{2}-py_{1}-\gamma y_{1}^{3}+1+\mu y_{4}^{2}\cos y_{3} \nonumber \\&+\mu \sin y_{3}\left( -\eta \sin y_{3}+F(1-y_{4})\right) )-\eta \sin y_{3}+F(1-y_{4}). \end{aligned}$$

(3.198)

The external force $\Phi (t)$ is defined as

$$\begin{aligned} \Phi (t)=K[y_{up}(t)-y_{1}(t)], \end{aligned}$$

(3.199)

where K is an adjustable weight of the perturbation $\Phi (t)$ and $y_{up}(t)$ is a component of the unstable periodic solution of (3.191) which we wish to stabilize. The function $y_{up}(t)$ is time periodical with period T. For $\Phi (t)$ zero the system has a strange attractor.

Fig. 3.25

The two-periodic solution: before stabilization (−) and after stabilization (–)

However, by selecting the constant K one can achieve the desired stabilization. Using the shooting method suggested by Van Dooren and Janssen (1996) the unstable two periodic unstable solution is detected (Fig. 3.25). Varying the value of the constant K in the interval $\left[ 0.1,2\right]$ it is concluded that for $K\in [0.3,2]$ the stabilization is achieved (Fig. 3.25).

Fig. 3.26

External force-time history diagram

For $K=2$ the function $\Phi (t)$ tends to a very small value (Fig. 3.26) and the component $y_{1}$, which is the solution after control, is very close to $y_{up}(t)$.

3.4.5 Conclusion

During passage through resonance of the motor-structure system which is modeled as a stable Duffing oscillator with non-ideal excitation severe vibrations appear. The energy of the system is not used for increasing of the rotation velocity, but is spent for vibrations which are harmful. Very often in the motion of the system near resonance the jump phenomena occurs: at the same value of the control parameter of the motor the amplitude of vibration skips to a higher value with lower frequency or to smaller amplitude with higher frequency. The manifestation depends on the direction of variation of the control setting. The jump phenomena and the increase in power required by a source operating near resonance are manifestations of a non-ideal energy source and are referred to as Sommerfeld effect. The Sommerfeld effect contributes to transform a regular vibration to an irregular chaotic one. It is concluded that in spite of the fact that the structure is modeled as the stable Duffing oscillator chaos appears. In the system the chaos is achieved by period doubling bifurcation. From engineering point of view it is necessary to eliminate the jump effect and the chaotic motion. The elimination of the jump phenomena for the certain control parameter is achieved by using the structure with coefficient of nonlinearity which is smaller than the critical value (3.190). Chaos is controlled using the external force control procedure where the added force does not change the form of the desired unstable periodic solution, but under certain conditions can stabilize it.

Comparing the results obtained applying the approximate analytic methods with those obtained numerically it is concluded that the difference is negligible. It proves the correctness of the used analytic procedure.

3.5 Bistable Duffing Oscillator Coupled with a Non-ideal Source

The model, shown in Fig. 3.27, represents an one degree-of-freedom cart connected to a fixed frame by a nonlinear spring and a dashpot (Warminski et al. 2001) . Motion of the cart is due to a non-ideal motor with unbalanced rotor. In the absolute reference frame x denotes the cart displacement and $\varphi$ denotes the angular displacement of the rotor. Elastic force of the spring is a cubic function of cart position x.

Fig. 3.27

Model of the system

Motion of the system is described with the following equations (Kononenko 1976)

$$\begin{aligned} \ddot{x}(M+m)+c\dot{x}-md(\ddot{\varphi }\sin \varphi +\dot{\varphi }^{2}\cos \varphi )-k_{1}x+k_{2}x^{3}= & {} 0, \nonumber \\ (J+md^{2})\ddot{\varphi }-md\ddot{x}\sin \varphi= & {} \mathcal {M}(\dot{\varphi } ), \end{aligned}$$

(3.200)

where $\mathcal {M}(\dot{\varphi })$ is the motor torque. For further consideration, let us introduce the dimensionless length and dimensionless time

$$\begin{aligned} y=\frac{x}{d},\quad \tau =\omega t, \end{aligned}$$

(3.201)

where d is the distance of the unbalanced mass to the shaft center and

$$\begin{aligned} \omega =\sqrt{\frac{k_{1}}{m+M}}. \end{aligned}$$

(3.202)

Using (3.201) and (3.202), the dimensionless equations of motion are

$$\begin{aligned} y\text {''}+\zeta y^{\prime }-y+\gamma y^{3}= & {} \frac{m}{M+m}(\varphi \text {''}\sin \varphi +\varphi ^{\prime 2}\cos \varphi ), \nonumber \\ \varphi \text {''}= & {} \eta y\text {''}\sin \varphi +\mathcal {M}(\varphi ^{\prime }), \end{aligned}$$

(3.203)

where

$$\begin{aligned} \zeta= & {} \frac{c}{\omega (M+m)},\quad \gamma =\frac{d^{2}k_{2}}{\omega ^{2}(M+m)},\quad \mu =\frac{m}{M+m}, \nonumber \\ \eta= & {} \frac{md^{2}}{(J+md^{2})},\quad \mathcal {M}\left( \varphi ^{\prime }\right) =\frac{\mathcal {M}\left( \dot{\varphi }\right) }{\omega ^{2}\left( J+m_{2}d^{2}\right) }, \end{aligned}$$

(3.204)

and $(^{\prime })=d/d\tau$, $(\text {''})=d^{2}/d\tau ^{2}$, $\zeta$ is the damping coefficient, $\gamma$ is the nonlinear parameter of the potential, while $\mu$ and $\eta$ are physical characteristics of the system. The torque is assumed to be linear , i.e.,

$$\begin{aligned} \mathcal {M}(\varphi ^{\prime })=E_{1}-E_{2}\varphi ^{\prime }, \end{aligned}$$

(3.205)

with voltage or the strength of the motor $E_{1}$ and with characteristic parameter of the motor $E_{2}$. For (3.205) the equations of motion are

$$\begin{aligned} y\text {''}+\zeta y^{\prime }-y+\gamma y^{3}= & {} \mu (\varphi \text {''}\sin \varphi +\varphi ^{\prime 2}\cos \varphi ), \nonumber \\ \varphi \text {''}= & {} \eta y\text {''}\sin \varphi +E_{1}-E_{2}\varphi ^{\prime }. \end{aligned}$$

(3.206)

If this system has a principal parametric resonance 2:1, we assume that the oscillator vibrates with frequency $\Omega$, but the frequency of non-ideal force is equal to $\Omega /2$. As the voltage source is alternated, $E_{1}$ is periodic and has the form

$$\begin{aligned} E_{1}=u_{0}\cos \left( \frac{\upsilon _{0}}{2}\tau \right) , \end{aligned}$$

(3.207)

where $u_{0}$ is the amplitude of the voltage source and $\upsilon _{0}=\Omega /\omega$. Substituting (3.207) into (3.206) we obtain

$$\begin{aligned} y\text {''}+\zeta y^{\prime }-y+\gamma y^{3}= & {} \mu (\varphi \text {''}\sin \varphi +\varphi ^{\prime 2}\cos \varphi ), \nonumber \\ \varphi \text {''}= & {} \eta y\text {''}\sin \varphi +u_{0}\cos \left( \frac{\upsilon _{0}}{2}\tau \right) -E_{2}\varphi ^{\prime }. \end{aligned}$$

(3.208)

Due to the strong nonlinearity there is no exact analytical solution for (3.208). Let us assume the approximate solution in the form

$$\begin{aligned} y= & {} A_{0}+A_{1}\cos (\upsilon _{0}\tau )+A_{2}\sin (\upsilon _{0}\tau ), \nonumber \\ \varphi= & {} B_{0}+B_{1}\cos \frac{(\upsilon _{0}\tau )}{2}+B_{2}\sin \frac{ (\upsilon _{0}\tau )}{2}, \end{aligned}$$

(3.209)

where $A_{0}$ is the amplitude of the structure, $B_{0}$ is the amplitude of the rotor at rest, $A=\sqrt{A_{1}^{2}+A_{2}^{2}}$ is the amplitude of oscillator and $B=\sqrt{B_{1}^{2}+B_{2}^{2}}$ is the rotor amplitude. To obtain approximate solutions we expand the nonlinear function sin$\varphi$ and cos$\varphi$ in the Taylor series until third order around the lower steady state for $\varphi =0$. Substituting (3.209) into (3.208), balancing the harmonics and neglecting the derivatives of the second order and terms having derivatives in a power higher than one, the set of first order approximate differential equations follows as

$$\begin{aligned} 0= & {} \zeta A_{1}^{\prime }+2\omega A_{2}^{\prime }-\upsilon _{0}\mu (B_{2}B_{1}^{\prime }+B_{1}B_{2}^{\prime })\left( 1- \frac{B_{1}^{2}}{4}-\frac{B_{2}^{2}}{12}+\frac{5}{4}B_{0}^{2}\right) +\zeta \upsilon _{0}A_{2} \\&+\left( -1-\upsilon _{0}^{2}+\frac{3}{4}\gamma (A_{1}^{2}+A_{2}^{2})+3\gamma A_{0}^{2}\right) A_{1}+\frac{\upsilon _{0}^{2} }{4}\mu \left( 1-\frac{B_{0}^{2}}{2}\right) (B_{1}^{2}-B_{2}^{2}), \end{aligned}$$

$$\begin{aligned} 0= & {} \gamma A_{2}^{\prime }+2\omega A_{1}^{\prime }+\upsilon _{0}\mu (B_{1}B_{1}^{\prime }-B_{2}B_{2}^{\prime })\left( 1-\frac{B_{1}^{2}}{6}- \frac{B_{0}^{2}}{2}\right) -\zeta \upsilon _{0}A_{1} \\&+\left( -1-\upsilon _{0}^{2}+\frac{3}{4}\gamma (A_{1}^{2}+A_{2}^{2})+3\gamma A_{0}^{2}\right) A_{2}+\frac{\upsilon _{0}^{2} }{2}\mu B_{1}B_{2}\left( 1-\frac{B_{0}^{2}}{2}\right) , \end{aligned}$$

$$\begin{aligned} 0= & {} \omega \varepsilon _{2}B_{2}(A_{1}^{\prime }+\frac{\upsilon _{0}}{2} A_{2})\left( 1-\frac{B_{1}^{2}}{4}-\frac{B_{2}^{2}}{12}-\frac{B_{0}^{2}}{2} \right) +E_{2}B_{1}^{\prime } \\&+\upsilon _{0}B_{2}^{\prime }+\frac{\upsilon _{0}}{2}E_{2}B_{1}-\frac{ \upsilon _{0}^{2}}{4}B_{1}-u_{0} \\&+\upsilon _{0}\eta (B_{1}A_{2}^{\prime }+\frac{\upsilon _{0}}{2} A_{1}B_{1})\left( 1-\frac{B_{1}^{2}}{6}-\frac{B_{0}^{2}}{2}\right) , \end{aligned}$$

$$\begin{aligned} 0= & {} \upsilon _{0}\eta B_{1}(A_{1}^{\prime }+\frac{\upsilon _{0}}{2} A_{2})\left( 1-\frac{B_{2}^{2}}{4}-\frac{B_{1}^{2}}{12}-\frac{B_{0}^{2}}{2} \right) +E_{2}B_{2}^{\prime } \\&-\upsilon _{0}B_{1}^{\prime }-\frac{\upsilon _{0}}{2}E_{2}B_{1}-\frac{ \upsilon _{0}^{2}}{4}B_{2} \\&-\upsilon _{0}\eta (B_{2}A_{2}^{\prime }+\frac{\upsilon _{0}}{2} A_{1}B_{2})\left( 1+\frac{B_{2}^{2}}{6}-\frac{B_{0}^{2}}{2}\right) , \end{aligned}$$

$$\begin{aligned} 0= & {} A_{0}\left( \frac{3}{2}\gamma (A_{1}^{2}+A_{2}^{2})+\gamma A_{0}^{2}-1\right) , \nonumber \\ 0= & {} \frac{\upsilon _{0}^{2}}{8}\eta B_{0}\left( A_{1}(B_{2}^{2}-B_{1}^{2}\right) -2A_{2}B_{1}B_{2}). \end{aligned}$$

(3.210)

Relations (3.210)$_{5}$ and (3.210)$_{6}$ are satisfied for

$$\begin{aligned} \gamma A_{0}^{2}=1-\frac{3}{2}\gamma A,\quad B_{0}=0. \end{aligned}$$

(3.211)

Besides, let us assume for small oscillations that $B^{2}=0$.Using this assumption and (3.211) the Eqs. (3.210)$_{1}$–(3.210)$_{4}$ give the steady states when

$$\begin{aligned} A_{1}^{\prime }=0,\quad A_{2}^{\prime }=0,\quad B_{1}^{\prime }=0,\quad B_{2}^{\prime }=0, \end{aligned}$$

as

$$\begin{aligned} 0= & {} \zeta \upsilon _{0}A_{2}+\left( -1-\upsilon _{0}^{2}+\frac{3}{4}\gamma (A_{1}^{2}+A_{2}^{2})+3\gamma A_{0}^{2}\right) A_{1}+\frac{\upsilon _{0}^{2} }{4}\mu (B_{1}^{2}-B_{2}^{2}), \nonumber \\ 0= & {} -\zeta \upsilon _{0}A_{1}+\left( -1-\upsilon _{0}^{2}+\frac{3}{4}\gamma (A_{1}^{2}+A_{2}^{2})+3\gamma A_{0}^{2}\right) A_{2}+\frac{\upsilon _{0}^{2} }{2}\mu B_{1}B_{2}, \nonumber \\ u_{0}= & {} \eta \frac{\upsilon _{0}^{2}}{2}(A_{1}B_{1}+A_{2}B_{2})+\frac{ \upsilon _{0}}{2}E_{2}B_{2}-\frac{\upsilon _{0}^{2}}{4}B_{1}-\eta \frac{ \upsilon _{0}^{2}}{8}A_{2}B_{2}B_{1}^{2}, \nonumber \\ 0= & {} -\eta B_{1}\frac{\upsilon _{0}^{2}}{8}A_{2}B_{2}^{2}-\frac{\upsilon _{0} }{2}E_{2}B_{1}-\frac{\upsilon _{0}^{2}}{4}B_{2}-\eta \frac{\upsilon _{0}^{2} }{2}(A_{1}B_{2}-A_{2}B_{1}). \end{aligned}$$

(3.212)

Equation (3.212) represent the set of four coupled equations. According to some specific considerations the system of Eq. (3.212) have semi-trivial and non-trivial solutions.

3.5.1 Semi-trivial Solutions and Quenching of Amplitude

The semi-trivial solution means physically that one part of the system is oscillating while the other one is at rest. This gives the condition for quenching of amplitude of oscillation of one part of the system: of the mechanical part and of the non-ideal force.

Fig. 3.28

Amplitude-frequency curve of non-ideal system for semi-trivial case. (Nbendjo et al. 2012)

Quenching of amplitude in the mechanical part satisfies the following requirements

$$\begin{aligned} A_{1}=A_{2}=0,\quad B_{1}\ne 0,\quad B_{2}\ne 0. \end{aligned}$$

(3.213)

For these values the mechanical part of the system does not vibrate. This case can be used as a technique of control whose objective is to cancel vibration of the mechanical system. According to (3.210) the condition of quenching phenomenon in the space of the parameters of the system is derived and given as

$$\begin{aligned} B=\frac{2u_{0}}{\upsilon _{0}\sqrt{E_{2}^{2}+(\frac{\upsilon _{0}}{2})^{2}}}. \end{aligned}$$

(3.214)

In Fig. 3.28, the amplitude of non-ideal force B as function of the frequency $\upsilon _{0}$, given with (3.211), is plotted. For numerical purposes, the following set of parameters (Souza et al. 2005a, b) is considered

$$\begin{aligned} \zeta =0.02,\quad \gamma =0.1,\quad \mu =0.1,\quad \eta =0.25,\quad E_{2}=1.5. \end{aligned}$$

Analytical solution is compared with numerical solution of (3.210) using the fourth-order Runge Kutta algorithm (see Fig. 3.28).

It appears that the amplitude of the non-ideal force decreases as the frequency increases. To analyze the stability of this semi-trivial solution, the amplitude modulation equations are given by

$$\begin{aligned} \dot{A}_{1}= & {} f_{1}(A_{1},A_{2},B_{1},B_{2}), \nonumber \\ \dot{A}_{2}= & {} f_{2}(A_{1},A_{2},B_{1},B_{2}), \nonumber \\ \dot{B}_{1}= & {} f_{3}(A_{1},A_{2},B_{1},B_{2}), \nonumber \\ \dot{B}_{2}= & {} f_{4}(A_{1},A_{2},B_{1},B_{2}). \end{aligned}$$

(3.215)

Perturbing the solution $A_{1},A_{2},B_{1}$ and $B_{2}$ with $\delta A_{1},\delta A_{2},\delta B_{1}$ and $\delta B_{2}$ and substituting into (3.215), after linearization a set of differential equations is obtained. Stability conditions are based on the eigenvalues of the Jacobian (Warminski and Kecik 2006). If the solution is complex, and the real part of the eigenvalue is always negative, the system is stable.

Quenching of amplitude of the non-ideal force occurs for

$$\begin{aligned} B_{1}=0,\quad B_{2}=0,\quad A_{1}\ne 0,\quad A_{2}\ne 0. \end{aligned}$$

(3.216)

This situation represents the case where the non-ideal forces does not swing and the structure vibrates. Using the set of differential equations for steady-state motion (3.212) and assuming the stationarity of solutions leads after some calculations to the following non-linear equation of the amplitude

$$\begin{aligned} A= \sqrt{\frac{(\frac{15}{4}\gamma )^{2}+\zeta ^{2}\upsilon _{0}^{2}+(2-\upsilon _{0}^{2})^{2}}{\frac{15}{2}\gamma (2-\upsilon _{0}^{2})} }. \end{aligned}$$

(3.217)

Analyzing of this equation shows the evidence of mechanical part which is at equilibrium and thus the system remains stable.

3.5.2 Non-trivial Solutions and Their Stability

Non-trivial solutions represent the case where both systems vibrate and

$$\begin{aligned} A_{1}\ne 0,\quad A_{2}\ne 0,\quad B_{1}\ne 0,\quad B_{2}\ne 0. \end{aligned}$$

(3.218)

Analytically, we were supposed to use the set of Eq. (3.210) and to derive the amplitude of both systems. Unfortunately, the system is strongly nonlinear and it is quite impossible to obtain an analytical expression of amplitude.

Fig. 3.29

a Amplitude of the mechanical part as function of the frequency; b Amplitude of the non-ideal source as function of the frequency (Nbendjo et al. 2012)

Moreover, to deal with such a question, we solved directly the base equations (3.208) numerically using the fourth-order Runge–Kutta algorithm and discuss the amplitude resonance curves. Afterwards, we explore the stability of the system using Lyapunov spectrum and bifurcation sequences . The frequency response curve is thus obtained from (3.208) and presented in Fig. 3.29a for the evaluation x and in Fig. 3.29b for evaluation of $\varphi$ for $u_{0}=1$.

Fig. 3.30

a Lyapunov spectrum and b bifurcation diagram as function of $\upsilon _{0}$ for $u_{0}=1$ (Nbendjo et al. 2012)

Figure 3.29a reveals a set of subharmonic resonances instead of the internal resonance 2:1 as expected. Concerning the amplitude of non-ideal forces, it decreases as frequency increases and the effect of internal resonance is visible by their appearance.

Fig. 3.31

Lyapunov spectrum and corresponding bifurcation diagram as function of $u_{0}$ for $\upsilon _{0}=1.5$ (Nbendjo et al. 2012)

Focusing on the stability of the system allows one to display the Lyapunov spectrum as function of frequency for the specify value of $u_{0}=1$.

Fig. 3.32

Region in parameter space of ($u_{0},\upsilon _{0}$) where hyperchaos is detected (Nbendjo et al. 2012)

It appears, for example in Fig. 3.30a, that the Lyapunov spectrum is positive for certain values of frequency which is necessary for the presence of chaos in the system. On the other hand, for another set of frequencies, Fig. 3.30a shows more than one positive Lyapunov exponent , indicating hyperchaos on the system. These effects are confirmed via the corresponding bifurcation diagram (see Fig. 3.30b). Setting $\upsilon _{0}=1.5$ allows one to observe the stability of the system as $u_{0}$ increases.

It appears in Fig. 3.31a that the system shows a transient from periodicity to quasiperiodicity and later to chaos. To view how these transient arise, the corresponding bifurcation diagram is presented in Fig. 3.31b, which confirms the results obtained from the Lyapunov spectrum. It shows that for $u_{0}=0.36$ there is a crisis in which sudden change in chaotic attractors occurs. Summarizing, it is concluded that there are regions in which all Lyapunov exponents are less or equal to zero for periodic or quasiperiodic orbits. If one of the Lyapunov exponents is positive, chaos is evident. If more than one positive Lyapunov spectrum exists, the presence of the hyperchaos for a specific set of parameters is indicated (Fig. 3.32).

Based on these consideration it can be concluded that the stability of the systems deeply depends on the voltage of the energy source.

3.5.3 Conclusion

Alternating strength of the voltage of the source may be the reason for the limited power supply for the bistable Duffing oscillator. Thus, in the oscillator - non-ideal source system for certain parameters quenching phenomena occurs: quenching of amplitude in the mechanical part or quenching of amplitude of the non-ideal force. Besides, in the system an explosion of resonances 2:1 to a set of subharmonic resonances is revealed. Based on stability analysis of the non-trivial solutions we concluded that the periodicity of the voltage source perturbs the limited power supply and increases the possibility of the appearance of chaos and hyperchaos of the systems. Consequently, when the voltage source is alternated, if the choice of the characteristic is bad, the system can become unstable.

Appendix: Ateb Functions

3.1.1 Sine Ateb Function

The incomplete Beta function is defined as (see Abramowitz and Stegun 1964, 1979)

$$\begin{aligned} B_{x}(p,q)=\mathop {\displaystyle \int }\limits _{0}^{0\le x\le 1}t^{p-1}(1-t)^{q-1}dt, \end{aligned}$$

(A1)

while for the case $x=1$, the complete Beta function is obtained

$$\begin{aligned} B(p,q)=\mathop {\displaystyle \int }\limits _{0}^{1}t^{p-1}(1-t)^{q-1}dt, \end{aligned}$$

(A2)

where p and q are real numbers. For the positive integrands in (A1) and (A2), the interval for the incomplete beta function is

$$\begin{aligned} 0\le B_{x}(p,q)\le B(p,q). \end{aligned}$$

(A3)

For the functions B(p, q) and $B_{x}(p,q)$ the following identities are evident

$$\begin{aligned} B(p,q)=B(q,p),\quad B_{x}(p,q)=B(p,q)-B_{y}(p,q), \end{aligned}$$

(A4)

where $x+y=1$.

Let us determine the inverse of the half of the incomplete beta function (A1)

$$\begin{aligned} x\rightarrow \frac{1}{2}B_{x}(p,q)=\frac{1}{2}\mathop {\displaystyle \int }\limits _{0}^{0\le x\le 1}t^{p-1}(1-t)^{q-1}dt. \end{aligned}$$

(A5)

Introducing the notation (Cveticanin 2014)

$$\begin{aligned} \frac{1}{2}B_{x}(p,q)=w, \end{aligned}$$

(A6)

the half of the incomplete beta function is

$$\begin{aligned} \frac{1}{2}\mathop {\displaystyle \int }\limits _{0}^{0\le x\le 1}t^{p-1}(1-t)^{q-1}dt=w. \end{aligned}$$

(A7)

For the the new variables

$$\begin{aligned} t=\bar{v}^{1/p},\quad x=v^{1/p}, \end{aligned}$$

(A8)

the boundary of integration $0\le x\le 1$ transforms into $0\le \left| v\right| \le 1$ and the integral (A7) has the forms

$$\begin{aligned} \frac{1}{2p}\mathop {\displaystyle \int }\limits _{0}^{0\le v\le 1}(1-\bar{v}^{1/p})^{q-1}d\bar{v} =w,\quad \frac{1}{2p}\mathop {\displaystyle \int }\limits _{0}^{-1\le v\le 0}(1-\bar{v} ^{1/p})^{q-1}d\bar{v}=w. \end{aligned}$$

(A9)

Let us consider the first integral in (A9). For $v=0$ the integral is zero, and for $v=1$ it is according to (A5)

$$\begin{aligned} \frac{1}{2p}\mathop {\displaystyle \int }\limits _{0}^{1}(1-\bar{v}^{1/p})^{q-1}d\bar{v}=\frac{1}{2} B(p,q). \end{aligned}$$

(A10)

Thus, the value of the function w (A9) is bounded

$$\begin{aligned} 0\le w\le \frac{1}{2}B(p,q), \end{aligned}$$

(A11)

as B(p, q) is finite. Now, the inverse for the integral (A9) is constructed. This inverse depends on the three parameters p, q and w. With notation given by Rosenberg (1963, 1966) we have

$$\begin{aligned} v=sa\left( \frac{1-p}{p},\frac{1-q}{q},w \right) , \end{aligned}$$

(A12)

The second integral in (A9) gives

$$\begin{aligned} -v=sa\left( \frac{1-p}{p},\frac{1-q}{q},-w\right) . \end{aligned}$$

(A13)

It follows that sa$(\frac{1-p}{p},\frac{1-q}{q},w)$ is the inverse of (A9) on the interval $-\frac{1}{2}B(p,q)\le w\le \frac{1}{2}B(p,q)$. Formula (A9) defines w uniquely as a function of v in this interval. Using the odd property (A13) of inverse of (A9), Rosenberg (1963) named the function ’sine Ateb function ’ and noted as sa.

The period of the function is

$$\begin{aligned} 2\Pi (p,q)=2B(p,q). \end{aligned}$$

(A14)

3.1.2 Cosine Ateb Function

The change of variables

$$\begin{aligned} t=1- \bar{u}^{1/q},\quad x=1-u^{1/q}, \end{aligned}$$

(A15)

transforms the interval $0\le x\le 1$ into $0\le \left| u\right| \le 1$. Substituting (A15) into (A7), for $1\ge u\ge 0$, it yields

$$\begin{aligned} -\frac{1}{2q}\mathop {\displaystyle \int }\limits _{1}^{0\le u\le 1}(1-\bar{u}^{1/q})^{p-1}d\bar{u} =w. \end{aligned}$$

(A16)

Due to the property of the beta function (A4), we have

$$\begin{aligned} -\frac{1}{2q}\mathop {\displaystyle \int }\limits _{0}^{0\le u\le 1}(1-\bar{u}^{1/q})^{p-1}d\bar{u}+ \frac{1}{2q}\mathop {\displaystyle \int }\limits _{0}^{1}(1-\bar{v}^{1/q})^{p-1}d\bar{u}=w, \end{aligned}$$

(A17)

i.e.,

$$\begin{aligned} -\frac{1}{2q}\mathop {\displaystyle \int }\limits _{0}^{0\le u\le 1}(1-\bar{u}^{1/q})^{p-1}d\bar{u} =-\frac{1}{2}B(p,q)+w. \end{aligned}$$

(A18)

Using the notation of the inverse function (A12) i.e. (A13) and the period $\Pi (p,q)=B(p,q)$ we obtain

$$\begin{aligned} u=\pm sa\left( \frac{1-q}{q},\frac{1-p}{p},\frac{1}{2}\Pi (p,q)\mp w\right) . \end{aligned}$$

(A19)

The inverse function u Rosenberg (1963) called ‘cosine Ateb function’ and noted it with ca:

$$\begin{aligned} ca\left( \frac{1-q}{q},\frac{1-p}{p},w\right) =sa\left( \frac{1-p}{p},\frac{1-q}{q},\frac{1}{2}\Pi (p,q)\pm w\right) . \end{aligned}$$

(A20)

The ca function is an even function (see Senik (1969a, b), Drogomirecka (1997)) with period $2\Pi (p,q)$ given with (A14).

3.1.3 Properties of Ateb Functions

The relation which satisfy the sa and the ca Ateb functions is (Gottlieb 2003, Gricik and Nazarkevich 2007)

$$\begin{aligned} \text {sa}^{2}\left( \frac{1-p}{p},\frac{1-q}{q},w\right) +\text {ca}^{\alpha +1}\left( \frac{1-q }{q},\frac{1-p}{p},w\right) =1. \end{aligned}$$

(A21)

The first derivatives of sa and ca functions are

$$\begin{aligned} \frac{d}{dw}\text {sa}\left( \frac{1-p}{p},\frac{1-q}{q},w\right)= & {} \text {ca}^{\alpha }\left( \frac{1-q}{q},\frac{1-p}{p},w\right) , \end{aligned}$$

(A22)

$$\begin{aligned} \frac{d}{dw}\text {ca}\left( \frac{1-q}{q},\frac{1-p}{p},w\right)= & {} -\frac{2}{\alpha +1} \text {sa}\left( \frac{1-p}{p},\frac{1-q}{q},w\right) . \end{aligned}$$

(A23)

The Fourier series expansion of the ca Ateb function (Gricik et al. 2009) is as follows

$$\begin{aligned} \text {ca}\left( \frac{1-q}{q},\frac{1-p}{p},w\right) =\mathop {\displaystyle \sum }\limits _{n=1}^{\infty }a_{n}\cos \frac{\pi nw}{\Pi (p,q)}, \end{aligned}$$

(A24)

where $a_{0}=0$ and

$$\begin{aligned} a_{n}=\frac{2}{\Pi (p,q)}\mathop {\displaystyle \int }\limits _{0}^{\Pi (p,q)}\text {ca}\left( \frac{1-q}{q}, \frac{1-p}{p},w\right) \cos \frac{\pi nw}{\Pi (p,q)}dw. \end{aligned}$$

(A25)

Namely, the coefficient is enough to be calculated for $w\in \left[ 0,\Pi (p,q)/2\right]$.