Keywords

1 Introduction

Generally, the assumption of constant speed is made in rotordynamic studies for the sake of simplicity. This assumption is however very constraining since it allows the understanding of the behavior of rotors only when the permanent regime is achieved. Since the understunding of the rotors behavior in non-stationary working conditions is important, especially at start-up and shut-down periods, when the rotor goes through its critical speeds. Many studies have been carried out to understand the behavior of the rotating machineries in such conditions.

On the one hand, some of those studies were made by imposing an angular speed following a function of time which is often taken linear or exponential. Lalanne et al. [1] considered a dynamic model accounting only for the flexural behavior of the rotor and showed that the more important the acceleration through a critical speed is, the lower is the lateral displacement amplitude. Yamamoto [2] focused more on the gyroscopic term present in a rotor, whose magnitude is proportional to the angular speed. He considered a rotating speed varying periodically resulting on a rotor governed by differnetial equations having variable coefficients with time. He showed that contrary to all expectaions, there accurs no unstable vibrations in such working conditions. Al-Bedoor [3] studied the coupled torsional and lateral vibrations of unbalanced Jeffcot-rotor in the case of rotor-to-stator rubbing during start-up period. But he only dealt with angular speeds below the second critical one in order to simplify the gyroscopic effect term.

On the other hand, some other studies were carried with an imposed driving torque and not with an imposed angular speed. In this case a problem called ’unsufficient torque acceleration’ [4] may be encountered. If not enough torque is induced to the rotating machinery, the rotor stalls at the first critical speed and the angular speed gets trapped at resonance. Mastuura [5] was interested in this kind of working conditions. He considered a rigid rotor with a mass unbalance and investigated conditions under which the rigid rotor easily passes through critical speeds without stalling. Li et al. [6] proposed a new method in the instantaneous frequency domain to find the closed-form solutions of displacement, velocity and acceleration envelopes for passage through resonance. Their studies were however made under the assumption of constant speed acceleration rates for run-up and run-down processes in order to ensure tractable solutions. Srinivasan [7] focused on modeling the phenomenon of limited-torque acceleration through critical speed of a Jeffcott rotor, with a non-linear torque profile.

In this work, a most complete as possible dynamic model based on the finite element method is developed to describe the rotor dynamic behavior at non-stationary working conditions. It accounts for tracion-compression, bending and torsion. An imposed driving torque is induced to the system and no assumption is made on the angular speed of rotation. This requires the introduction of degree of freedom combining both the torsional deformation and the rigid body angular displacement as an unknown of the dynamic problem. The angular speed will then be deduced, not only fom the imposed time-varying torque, but also from the working conditions, namely, the passing through the critical speeds.

2 A New Dynamic Model Formulation for Rotors Working at Non Stationary Conditions

2.1 System Description

The fixed reference frame is denoted (XYZ) while the inertial reference frame is denoted (UVW). To describe the general orientation of the cross-section of the shaft element one first rotates by an angle \(\varPhi \) about the \(Z-\)axis then by an angle \(\theta \) about the new X’-axis and finally by an angle \(\varPsi \) about the final W’-axis. The Finite Element Method is used to model the shaft. The relatioships between Euler angles and the degrees of freedom of rotation at any point of the shaft in the global coordinate system are the following:

$$\begin{aligned} \theta _x = \theta .cos(\phi ) \end{aligned}$$
(1a)
$$\begin{aligned} \theta _y = \theta .sin(\phi ) \end{aligned}$$
(1b)
$$\begin{aligned} \theta _z = \phi +\psi \end{aligned}$$
(1c)

2.2 Governing Equations of the Rotor

The governing equations are obtained by the application of Lagrange equations on the expression of the kinetic energy of the different components of the rotor.

The Disk, it is assumed to be rigid, and it is fully characterized by its kinetic energy associated to the displacement of its center of mass C \((u_c,v_c,w_c) \) and to the rotational motion of its section. The associated displacement vector is then:

$$\begin{aligned} \left\{ \delta _c \right\} ^t = \lbrace u_c,v_c,w_c, \theta _{x_c}, \theta _{y_c},\theta _{z_c}\rbrace _{\lbrace 1,6\rbrace } \end{aligned}$$
(2)

The application of the Lagrange equations on the kinetic energy related to the disk \(T_D\) results on the following matrix equation:

$$\begin{aligned} \frac{d}{dt}\left( \frac{\partial T_D}{\partial \dot{\delta }}\right) -\frac{\partial T_D}{\partial \delta }= \left[ M_D ( \lbrace \delta _c\rbrace )\right] \left\{ \ddot{\delta }\right\} + \left[ C_D(\dot{\theta }_{z_c})\right] \left\{ \dot{\delta }\right\} \end{aligned}$$
(3)

such as:

$$[M_D(\left\{ \delta _c\right\} )]_{(6,6)}=\left[ \begin{matrix} m_D &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} m_D &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} m_D &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} I_d &{} 0 &{} \frac{I_p}{2}\theta _{y_c}\\ 0 &{} 0 &{} 0 &{} 0 &{} I_d &{} -\frac{I_p}{2}\theta _{x_c}\\ 0 &{} 0 &{} 0 &{} \frac{I_p}{2}\theta _{y_c} &{} -\frac{I_p}{2}\theta _{x_c} &{} I_p\\ \end{matrix}\right] $$

and \( \left[ C_D(\dot{\theta }_{z_c})\right] \) is the classical skew-matrix related to the gyroscopic effects.

The matrix \([M_D]\) is no longer a diagonal matrix as it is the case when studying the stationary regime, it has extra-diagonal terms variable with time due to the assumption of non-stationary regime inducing coupling between flexural and torsional behavior. This matrix is dissociated to a constant diagonal matrix and a time varying matrix with only time-varying extra-diagonal terms as follows:

$$\begin{aligned}{}[M_D(\left\{ \delta _c\right\} )]_{\lbrace 6,6\rbrace } = [{M_D}_{const}]_{\lbrace 6,6\rbrace }+[{M_D}_{var}(\left\{ \delta _c\right\} )]_{\lbrace 6,6\rbrace } \end{aligned}$$
(4)

The Mass Unbalance. The mass unbalance is defined by its mass \(m_u\) situated at a distance d from the geometric center of the shaft \(C (u_c,v_c)\). The application of the Lagrange equations on its kinetik energy \(T_u\) results on the following matrix equation:

$$\begin{aligned} \frac{d}{dt}\left( \frac{\partial T_u}{\partial \dot{\delta }}\right) -\frac{\partial T_u}{\partial {\delta }} = \left[ M_u(\theta _{z_c})\right] \left\{ \ddot{\delta }\right\} +\left\{ F_{nl_u}(\theta _{z_c})\right\} \end{aligned}$$
(5)

where

$$ \left[ M_u(\theta _{z_c})\right] =d\,m_u\left[ \begin{matrix} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} cos(\theta _{z_c})\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} - sin(\theta _{z_c})\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ cos(\theta _{z_c}) &{} - sin(\theta _{z_c}) &{} 0 &{} 0 &{} 0 &{} d\\ \end{matrix}\right] _{\lbrace 6,6\rbrace }$$
$$\left\{ F_{NL_u}(\theta _{z_c})\right\} ^t = -d\,m_u\,\dot{\theta }_{z_c}^2\left\{ \begin{matrix} sin(\theta _{z_c})\,\,\,\, cos(\theta _{z_c})\,\,\,\, 0\,\,\,\, 0\,\,\,\, 0\,\,\,\, 0\,\,\,\, \end{matrix}\right\} ^t_{\lbrace 6,1\rbrace }$$

In addition to the classical centrifugal force, due to the non-stationary regime working condition, an additional mass matrix with time varying components is also obtained. Again, this matrix reflects a coupling between the flexural behavior and the torsional one.

The Shaft. Using the finite element method, we obtain the following expression for the kinetic energy over a shaft element:

$$\begin{aligned} T_{s_e} = \frac{1}{2} \dot{\left\{ \delta _e \right\} }^t[M_{s_e}]\dot{\left\{ \delta _e \right\} } +T_{{gyr}_e} \end{aligned}$$
(6)

where \([M_{s_e}]\) is the elementary mass matrix including both the classical mass matrix and the secondary effects of the rotatory inertia matrix. \(T_{{gyr}_e}\) is the gyroscopic effect expression such as:

$$\begin{aligned} \begin{aligned} T_{{gyr}_e}&= \rho I_p \int _0^l \left\langle N_3\right\rangle \left\{ \delta \dot{\theta }_z \right\} -\left\langle \dfrac{\partial N_2}{\partial z} \right\rangle \left\{ \delta \dot{v} \right\} \, \left\langle \dfrac{\partial N_1}{\partial z} \right\rangle \left\{ \delta u \right\} \, \mathrm {d}z\\&+\rho I_p \int _0^l \left\langle N_3\right\rangle \left\{ \delta \dot{\theta }_z \right\} \left\langle \dfrac{\partial N_1}{\partial z} \right\rangle \left\{ \delta \dot{u} \right\} \, \left\langle \dfrac{\partial N_2}{\partial z} \right\rangle \left\{ \delta v \right\} \, \mathrm {d}z \end{aligned} \end{aligned}$$
(7)

where \(\lbrace \delta u\rbrace ^t=\langle u_1,\theta _{y1},\,u_2,\theta _{y2}\rangle \) and \(\lbrace \delta v\rbrace ^t=\langle v_1,\theta _{x1},\,v_2,\theta _{x2}\rangle \) are vectors used to describe the shaft element flexural behavior in both of the lateral directions. Whereas \(\lbrace \delta w\rbrace ^t=\langle w_1,\,w_2\rangle \) and \(\lbrace \delta \theta _z\rbrace ^t=\langle \theta _{z1},\theta _{z2} \rangle \) are used to describe the axial and torsional dispalements.

\( N_1\), \( N_2\) and \( N_3\) are shape functions vectors such as:

$$\begin{aligned} \langle N_1(z)\rangle = \langle 1-\frac{3z^2}{l^2}+\frac{2z^3}{l^3},\,0,\,0,\,0,\,z-\frac{2z^2}{l}+\frac{z^3}{l^2},\,0 ,\,\frac{3z^2}{l^2}-\frac{2z^3}{l^3},\,0,\,0,\,0,\, -\frac{z^2}{l}+\frac{z^3}{l^2},0\rangle \end{aligned}$$
(8a)
$$\begin{aligned} \langle N_2(z)\rangle = \langle 0, \, 1-\frac{3z^2}{l^2}+\frac{2z^3}{l^3},\,0,\,-z+\frac{2z^2}{l}-\frac{z^3}{l^2}, \,0,\,0,\,0,\,\frac{3z^2}{l^2}-\frac{2z^3}{l^3},\,0,\, \frac{z^2}{l}-\frac{z^3}{l^2}\,0,\,0,\rangle \end{aligned}$$
(8b)
$$\begin{aligned} \langle N_3(z)\rangle = \langle 0,\,0,\,0,\,0,\,0,\, 1-\frac{z}{l},\,0,\,0,\,0,\,0,\,0,\,\frac{z}{l} \rangle \end{aligned}$$
(8c)

In order to be able to write the kinetic energy related to the gyrosopic effect under a matrix form, we proceed by an integration by parts, which leads to the following formulation for the gyroscopic effects:

$$\begin{aligned} T_{{gyr}_e}= [S_{gyr_e}\left( \left\{ {\delta _e} \right\} \right) ]\lbrace \ddot{\delta _e} \rbrace + \lbrace Fnl_{gyr_e}(\dot{\left\{ \delta _e\right\} }) \rbrace \end{aligned}$$
(9)

where:

$$\begin{aligned} \begin{aligned} \,\![S_{gyr_e}\left( \left\{ {\delta _e} \right\} \right) ]&= -\left\langle N_3(l)\right\rangle ^{t} \left\{ \delta _e \right\} ^t [M_{67}^g(l)]^t - [M_{67}^g(l)] \left\{ \delta _e \right\} \left\langle N_3(l)\right\rangle \\&+\left\langle \dfrac{\partial N_3}{\partial z} \right\rangle ^{t} \left\{ \delta _e \right\} ^t [M_{67}^{g*}(l)]^t +[M_{67}^{g*}(l)] \left\{ \delta _e \right\} \left\langle \dfrac{\partial N_3}{\partial z} \right\rangle \end{aligned} \end{aligned}$$
(10)
$$\begin{aligned} \begin{aligned} \lbrace Fnl_{gyr_e}(\dot{\left\{ \delta _e\right\} },\dot{\left\{ \delta _e\right\} }) \rbrace&= -\left\langle N_3(l)\right\rangle ^{t} \left( \dot{\left\{ \delta _e \right\} }^t [M_{67}^g(l)] \dot{\left\{ \delta _e \right\} }\right) \\&- 2\left( \left\langle N_3(l)\right\rangle \dot{\left\{ \delta _e \right\} } \right) \left( [M_{67}^g(l)] \dot{\left\{ \delta _e \right\} } \right) \\&+\left\langle \dfrac{\partial N_3}{\partial z} \right\rangle ^{t} \left( \dot{\left\{ \delta _e \right\} }^t [M_{67}^{g*}(l)] \dot{\left\{ \delta _e \right\} }\right) \\&+2\left( \left\langle \dfrac{\partial N_3}{\partial z} \right\rangle \dot{\left\{ \delta _e \right\} } \right) \left( [M_{67}^{g*}(l)] \dot{\left\{ \delta _e \right\} } \right) \\ \end{aligned} \end{aligned}$$
(11)

such as

$$\begin{aligned}{}[M_{67}^g(l)]_e = \rho I_p \int _0^l \left( \left\langle \dfrac{\partial N_2}{\partial z} \right\rangle ^t \, \left\langle \dfrac{\partial N_1}{\partial z} \right\rangle - \left\langle \dfrac{\partial N_1}{\partial z} \right\rangle ^t \, \left\langle \dfrac{\partial N_2}{\partial z} \right\rangle \right) \mathrm {d}z \end{aligned}$$
(12a)
$$\begin{aligned}{}[M_{67}^{g*}(l)]_e =\rho I_p \int _0^l \left( \int _0^z \left( \left\langle \dfrac{\partial N_2}{\partial z} \right\rangle ^t \, \left\langle \dfrac{\partial N_1}{\partial z} \right\rangle - \left\langle \dfrac{\partial N_1}{\partial z} \right\rangle ^t \, \left\langle \dfrac{\partial N_2}{\partial z} \right\rangle \right) \mathrm {d}z\right) \mathrm {d}z \end{aligned}$$
(12b)

The application of the Lagrange equations on the kinetic energy of the shaft leads to:

$$\begin{aligned} \dfrac{d}{dt}\left( \frac{\partial T_{s_e}}{\partial \dot{\delta _e}}\right) -\frac{\partial T_{s_e}}{\partial \delta _e} =[M_{s_e}] \ddot{\left\{ \delta _e \right\} } + [S_{gyr_e}\left( \left\{ {\delta _e} \right\} \right) ]\lbrace \ddot{\delta _e} \rbrace + \lbrace Fnl_{gyr_e}(\dot{\left\{ \delta _e\right\} }) \rbrace \end{aligned}$$
(13)

\([S_{gyr_e}\left( \left\{ {\delta _e} \right\} \right) ]\) and \(\lbrace Fnl_{gyr_e}(\dot{\left\{ \delta _e\right\} })\) induce a strong non linearity to the system and couplig between the lateral and torsional degrees of freedom.

The calculation of the strain energy of the shaft and the virtual works of the bearings, assumed to be with linear stiffness and damping, leads to the classical stiffness matrix and we finally write the equation of the dynamics of rotating shaft element such as:

$$\begin{aligned} \begin{aligned} \,\![M_{s_e}]\ddot{\left\{ \delta _e \right\} }+[C_{s_e}]\dot{\left\{ \delta _e \right\} }+[K_{c_e}]{\left\{ \delta _e\right\} }&= -[S_{gyr_e}(\left\{ \delta _e \right\} )]\lbrace \ddot{\delta _e} \rbrace - \lbrace Fnl_{gyr_e}(\dot{\left\{ \delta _e \right\} }) \rbrace \end{aligned} \end{aligned}$$
(14)

2.3 Equations of the Rotor

Once the contribution of each of the constitutive elements of the rotor calculated, we consider them thouroughly to write the equation of the dynamic behavior of the rotor. As we have seen previously, the vector and matrix related to the gyroscopic effect depends explicitly on the displacement and velocity vectors \(\left\{ \delta _e \right\} \) and \(\dot{\left\{ \delta _e \right\} }\). We need then, to evaluate those vectors at each time step before doing the assembly over all the shaft elements. The choice of an explicit integration scheme is made. At each time step \(t_{i+1}\), vectors \(\left\{ \delta \right\} _{t_i}\) and \(\dot{\left\{ \delta \right\} }_{t_i}\) are assumed to be known and then the gyroscopic effect terms over an element, namely \([S_{gyr_e}(\left\{ \delta _e \right\} )]\) and \(\lbrace Fnl_{gyr_e}(\dot{\left\{ \delta _e \right\} }) \rbrace \) are calculated and assembled over the whole shaft.

Finally, the equation of motion, in the absence of axial efforts and in the presence of external ones can be written as follows:

$$\begin{aligned} \begin{aligned} ([M_{s}]&+[M_{D_{const}}]) \ddot{\left\{ \delta \right\} }_{i+1}+[C_s]\dot{\left\{ \delta \right\} }_i+ [K_c]{\left\{ \delta \right\} }_i =\\&-\left( \left[ S_{gyr}(\left\{ \delta \right\} _i)\right] +\left[ M_{D_{var}} ( \left\{ \delta \right\} _i )\right] +\left[ M_u(\theta _{z_c})\right] _i\right) \ddot{\left\{ \delta \right\} }_{i+1} \\ {}&- \left[ C_D(\dot{\theta }_{z_c})\right] _i\dot{\left\{ {\delta }\right\} }_i \\ {}&- \left( \lbrace Fnl_{gyr}(\dot{\lbrace \delta \rbrace }_i) \rbrace + \lbrace F_{nl_u}(\theta _{z_c})\rbrace _i \right) \\ {}&+ \lbrace F_{ext}\rbrace _i \end{aligned} \end{aligned}$$
(15)

where in the left side of the equation, the terms which are constant at each time step and, in the right side, the ones which need to be apdated at every time step.

Fig. 1.
figure 1

Rotor model

Full size image

3 Results and Discussions

Flexible Rotor with Mass Unbalance

We consider a rotor made of a shaft and a disk with a mass unbalance (Fig. 1) with the following proporties:

\(Shaft\,\, properties\)

length \(L = 0.6\) m; cross-sectional radius \(R_1 = 0.02\) m; \(\rho = 7800\)  kg.m\(^{-3}\); \(E = 2.10^11\)  N.m\(^{-2}\)

The shaft is descritized into 10 finite elemets.

\(Disk \,\, properties\)

Inner radius \(R_1 = 0.02\) m; Outer radius \(R_2 = 0.16\) m; Thikness \(h = 0.03\) m; \(\rho = 7800\)  kg.m\(^{-3}\)

Position of the disk: node number 7 of the shaft.

Fig. 2.
figure 2

Plot of the angular velocity for successful and unsuccessful acceleration through critical speeds

Full size image

\(Mass\,\, unbalance \)

Mass \(m_u = 1\%\) de la masse du disque; Eccentricity \( d = 0.1\) m

Bearings

\(k_{xx1} = k_{xx2} = k_{yy1} = k_{yy2} = 10^8\) N.m\(^{-1}\)

\(c_{xx1} = c_{xx2} = c_{yy1} = c_{yy2} = 0.01\) N.s.m\(^{-1}\)

In this example, the torque is constant at first, it brings the rotor to a speed of 1000 rpm and afterwards it follows a linear law of variation as a function of time. The problem of ’limited torque acceleration’ introduded by Gluse [4] for rigid rotors was encountered in our case. The first critical speed of the rotor is at \(\omega _{c1} \simeq 2513\) rpm.

Below some torque acceleration rate or damping values the rotor stalls in the vicinity of the first critical speed. In this case, the rotor generates large deflection displacements which results in important resistive torque. Which explains the dependency of the success or failure of the passing through the critical speeds on the applied driving torque. Figure 2 presents the passing through the first critical speed of the same rotor for different torque acceleration rates \(\alpha _{i, i\in \lbrace 1,2,3,4\rbrace }\). It confirms that the more the torque acceleration rate is important, the easier the passing through critical speeds is.

Fig. 3.
figure 3

Plot of the lateral diplacement amplitude for successful and unsuccessful acceleration through critical speeds

Full size image

It is also shown that, in accordance to the previous works, the higher the torque acceleration rate is, the lower is the lateral diplacement amplitude (Fig. 3).

4 Conclusion

The rotor dynamic model presented in this paper based on the finite element methods is a complete model taking into account the shear deformation, the secondary effects of rotatary inertia and the gyroscopic effects. It accounts for traction-compression, deflexion and torsion which makes it strongly non-linear with different coupling between lateral and torsional degrees of freedom. The new proposed formulation for the rotor dynamics makes no assumption on the instantaneous angular speed which implies the possibilty to simulate the rotor response for any non-stationary working condition with linear or non-linear driving torques.

If the maximum of the lateral vibrations when going through critical speeds was investigated in many previous works, no focus was made on the IAS considered in our case as unknown of the dynamic problem. The next step of the present work will be the analysis, in non staionary working conditons, of the IAS which is believed to provide a rich signal containing information not only about the torsional behavior but also about the lateral one due to the coupling between the different degrees of freedom as it is shown in the dynamic model. The rotor model will also be used as an identification tool at non-stationary working conditions.