Mechanical Structures: Mathematical Modeling

Abstract

The organization and arrangement of irrelated or distributed elements in a system or an object is defined as mechanical structure. The elements in a mechanical structure can exhibit the characteristics of different parameters. In order to investigate the feature of a mechanical structure many factors should be considered and defined. This chapter aims to introduce a basic definition of mechanical structures with focusing on vehicle and its suspension system as the main mechanical structure target.

4.1 Overview

The organization and arrangement of irrelated or distributed elements in a system or an object is defined as mechanical structure. The elements in a mechanical structure can exhibit the characteristics of different parameters. The main parameters which mainly considerable for such structure are mass, elasticity and damping [1]. In order to investigate the feature of a mechanical structure about the mentioned parameters, many factors should be considered and defined. Some of the most important factors which are the basic definition of mechanical structures are described and focused in the following sections. In this chapter, car suspension system has been considered as the main mechanical structure target.

4.1.1 Degrees of Freedom

The number of the parameters which are defining the configuration independently is known as the degree of freedom (DOF). As the other general definition, in order to describe the motion of a system, the coordinate position which is independent is required to be identified. The number of these independent coordinate positions is showing the number of DOF [2].

Vehicle can be considered as a wellknown mechanical system. A vehicle can be defined as a rigid body which has highly stiff suspension and able to move on a two-dimensional space or on a flat plane. The actual degree of freedom for such vehicle is three as it has two components of translation and one angle of rotation. In the case of car suspension system, degree of freedom has been considered as one or two degrees. Many researches around the car suspension as a mechanical structure considered the DOF of suspension system as two. However, in order to performing some complex mathematical models about the mechanical structure, the DOF for suspension system has been assumed as one.

4.1.2 Periodical System Response

When a mechanical system is under excitation by an internal or external forces, it shows some of the properties of a vibratory response. This motion is called periodic motion which might be in regular interval and repetitive or irregular [3]. The mentioned system response is going to be considered in a period of time T. The time for a complete cycle of motion will be considered as T.

4.1.3 Harmonic Motion

Periodic motion in the simplest form of is actually the harmonic motion in which cosine and sine functions as oscillatory functions are utilized to represent the observed or actual motion. The motion described using a continuous sine or cosine function is referred to as steady state [4]. w(t) or actual displacement may be written in the form of following equation [4]:

$$w(t) = \left| A \right|\cos (\omega t + \phi )$$

(4.1)

here:

$w(t)$ :

is actual displacement

$\left| A \right|$ :

real amplitude of motion

$\omega$ :

is circular frequency in radians per second

$\phi$ :

is an arbitrary phase angle in radians.

(4.1) The superposition is the combination  of sine and cosine functions, and can be shown as follows [5]:

$$w(t) = A_{\text{R}} \cos \omega t - A_{\text{I}} \sin \omega t$$

(4.2)

where A_R and A_I are real numbers of amplitude of motion as follows [5]:

$$A_{\text{R}} = \left| A \right|\cos \phi ,$$

(4.3)

$$A_{\text{I}} = \left| A \right|\sin \phi ,$$

(4.4)

The expression of phase angle is the following equation [5]:

$$\phi = \tan^{ - 1} \left( {\frac{{A_{\text{I}} }}{{A_{\text{R}} }}} \right)$$

(4.5)

The constant $\left| A \right|$ in (4.1) is related to the constants A_I and A_R in (4.2) by following equation [5]:

$$\left| A \right| = \left( {A_{\text{R}}^{2} + A_{\text{I}}^{2} } \right)^{1/2}$$

(4.6)

4.1.4 Frequency

Frequency has been defined as the cycle’s amount which is known as hertz, in every second of the motion which represents the period’s reciprocal [3]. The frequency can be shown by following equation [6]:

$$f = \frac{1}{T}$$

(4.7)

Radians per second which is $\omega$ or circular frequency can be shown as following equation [3].

$$\omega = 2\pi f$$

(4.8)

4.1.5 Amplitude

The periodical amount of the response (maximum amount) of the system is amplitude. For instance, $\left| A \right|$ is the motion’s amplitude if the motion is stated by (4.1).

4.1.6 The Mean Square Amplitude

The average of the time of response’s square can be defined as the amplitude mean square [3]. So that, for instance, consider the following equation:

$$\overline{{w^{2} }} = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int\limits_{0}^{T} {w^{2} (t){\text{d}}t} .$$

(4.9)

The positive value of the square root of the mean square amplitude is the root mean square amplitude. In order to have the harmonic oscillation of (4.2), the root mean square amplitude is equal to $A/\sqrt 2$ and it is independent of the phase.

4.1.7 Free and Forced Vibrations

As the outcome of some initial conditions and without any external disturbances, the free vibrations can be defined as the system’s motion [7]. By presence of applying the continues and external disturbance the system’s motions will be the defined as Forced vibrations [7].

4.1.8 Phasor

When the system’s harmonic motion is represented by a rotating vector is a phasor [3]. The can be represented in the complex formula of the periodic motion of (4.1) and (4.2) which will be more appropriate for manipulating mathematically is shown as follows [8]:

$$w(t) = A{\text{e}}^{j\omega t} ,$$

(4.10)

here $w(t)$ and A are complex with the complex amplitude stated as follows [8]:

$$A = A_{\text{R}} + jA_{\text{I}}$$

(4.11)

Figure 4.1 illustrates the phasor form of (3.8). The real motion’s amplitude is $\left| A \right|$ and it is the length of the vector. Counter clockwise rotation of the vector with angular velocity, on the imaginary and real axis of $\omega$ plan causes projection, changes agreeably with time t changing. A cycle of motion is 360° rotation of the vector.

Fig. 4.1

Harmonic motion’s phasor diagram [10]

The actual measured or observed motion is the real phasor’s component or the complex description. So, the actual motion is shown in following [9]:

$$w(t) = \text{Re} \left[ {A{\text{e}}^{j\omega t} } \right].$$

(4.12)

Equation (4.2) yields by utilizing ${\text{e}}^{j\omega t} = \cos \omega t + \sin \omega t$ and substitution of $A = A_{\text{R}} + jA_{\text{I}}$ into the (4.12). As it is shown in (4.5), the real and imaginary ratio of phasor’s components will result the $\phi$ which is the phase of the motion.

The vibration is the main reason of utilizing the negative sign in (4.2). This selection guarantee that the $\phi$ phase has positive value and meanwhile A_t has a constant value which should be evaluated using boundary conditions making the negative sign choosing to be insignificant against the result.

As the phasor known as vector, same category of harmonic motions with same frequency can be sum up as vector. This is the reason behind complex notation for linear motions, as superposition’s principle remains valid, which means separate summation of the imaginary and real components that form motions but individually.

Many responses presented this chapter are written using complex notation because it’s a convenient way of analyzing systems having superimposed responses (i.e. like in the case of active controls simulations). The complex descriptions (the real parts) will recover the real motion. In the case that motion has direct description in the actual form, then will be indicated in the subsequent text.

4.2 Single Degree of Freedom Mechanical Systems

As shown in Fig. 4.2, a mass M being supported on a spring with neglectable mass. The w is a unique variable which illustrating the system’s displacement, so that the system possesses a single degree of freedom (SDOF) [3, 11]. Followed by appropriate.

Fig. 4.2

Undamped and damped single degree of freedom systems [11]

By the reason of the gravity, the constant force is neglectable if the resting point of the coordinate system is the origin of the system. The free body diagram of the Fig. 4.2 illustrates that the amount of the w (Positive) which can be achieved by equilibrium position and the mass displacement, force the spring to apply the Kw (restoring force) due to the elongation. Due to the mass releasing, mass acceleration will occur by the spring. The relation between the restoring force and acceleration are shown in the following formula which is based on Newton’s second law of motion (see Eq. 4.13).

$$M\frac{{{\text{d}}^{2} w}}{{{\text{d}}t^{2} }} = - Kw$$

(4.13)

The simple single degree of freedom system’s motion will be described by differential equation (see Eq. 4.14)

$$\frac{{{\text{d}}^{2} w}}{{{\text{d}}t^{2} }} + \left( {\frac{K}{M}} \right)w = 0$$

(4.14)

Equation (4.14) is a second-order ordinary differential equation furthermore, along these lines must have an answer which is indicated as far as two obscure constants or amplitudes of movement. Despite the fact that the above analysis is clear, it illustrates the fundamental procedure in which flexible systems are commonly examined. By dividing the into components (squares). For some underlying conditions the re-establishing and inertial powers are adjusted, subsequently giving the differential condition portraying the movement of the system.

4.3 SODF Free Motion

As the mechanical systems in free motion having harmonic response, (4.14) solution can be form by (4.2). Consequently, the following equation can describe the actual motion [11].

$$w(t) = A_{\text{R}} \cos \omega t - A_{\text{I}} \sin \omega t$$

(4.15)

where A_R and A_I are real amplitudes of motion.

In order to determine the condition to have natural vibration, the relation for frequency $\omega$ should be provided. By substitution of (4.14) into (4.15) we have can get the following equation [11]:

$$\omega_{n} = \sqrt {\frac{K}{M}}$$

(4.16)

and thus, the solution of (4.14) becomes as follows:

$$w(t) = A_{\text{R}} \cos \omega_{n} t - A_{\text{I}} \sin \omega_{n} t$$

(4.17)

With considering the (4.17), motion can be completely specified if A_R and A_I being defined as unknown constants. These unknown constants will be defined by utilizing initial or boundary conditions. One of the most essential factors of system’s characteristics is $\omega_{n}$ which is resonance or natural frequency. It is noticeable to mentioned that by increasing the K (stiffness) and M (mass) in SDOF systems, the natural frequency will increase. The mentioned states are generally correct if the system is elastic and linear.

To determine the total system’s motion, it is mandatory to utilize the introductory conditions first. For instance, if at t = 0 the system will have a real velocity w and an underlying genuine removal $\dot {w}$ then the unknown constants in (4.17) can be determined from the following conditions [11]:

$$w(0) = A_{\text{R}} ,$$

(4.18)

$$\dot {w} (0) = - \omega_{n} A_{\text{I}} ,$$

(4.19)

where the use of the overdot is a conservative documentation for separation as for time. The watched reaction will be achieved by fathoming for the A_R and A_I parameters from Eqs. (4.18) and (4.19) and utilizing these two equations into (4.17). The genuine reaction of the system to subjective starting conditions will be obtained by (4.20) to (4.23) [11].

$$w(t) = w(0)\cos \omega_{n} t + \frac{{\dot {w(0)}}}{{\omega_{n} }}\sin \omega_{n} t$$

(4.20)

It is also possible to write the motion as follows:

$$w(t) = \left| A \right|\cos (\omega_{n} t + \phi )$$

(4.21)

in which $\phi$ (phase angle) is obtained from (4.5) the following:

$$\phi = \tan^{ - 1} \left( { - \frac{{\dot {w (0)}}}{{\omega_{n} w(0)}}} \right)$$

(4.22)

Equation (4.6) will give and the motion’s amplitude, and then the following equation will be the result (see Eq. 4.23)

$$\left| A \right| = \left[ {[w(0)]^{2} + \left( {\frac{{\dot {w(0)}}}{{\omega_{n} }}} \right)^{2} } \right]^{1/2}$$

(4.23)

Thus the response of the single degree of freedom is observable at $\omega_{n}$ (Natural Frequency) as simple harmonic motion, with $\left| A \right|$ (amplitude) and $\phi$ (phase angle) obtained by (4.22) and (4.23) respectively.

4.4 Damped Motion of Single Degree of Freedom Systems

In all of the real systems, the source of vibration is a type of mechanism (damping). Also the vibration energy will be loosen amid the motion cycle. When there is proportional resistant force and the act is in opposite direction of the velocity there is the simples type of damping [3]. Thus, the damping force is specified by the following equation:

$$F_{d} = - C\frac{{{\text{d}}w}}{{{\text{d}}t}},$$

(4.24)

where C is the damping coefficient. Figure 4.2 shows a system with a single degree of freedom including the mentioned damping type. This type of damping is known as viscous damping. With considering the additional damping force of new system’s force equalization will drive the Eqs. (4.25)–(4.33) [12]. It has been also shown in Fig. 4.2 as the free body diagram.

$$M\frac{{{\text{d}}^{2} w}}{{{\text{d}}t^{2} }} + C\frac{{{\text{d}}w}}{{{\text{d}}t}} + Kw = 0$$

(4.25)

It is currently more advantageous to utilize a perplexing depiction of the motion. Accordingly, an answer is accepted of the form the following equation:

$$w(t) = A{\text{e}}^{\gamma t} ,$$

(4.26)

In this stage w(t) is a variable but complex. $\gamma$ will be obtained by (4.25) into (4.26) substitution (see Eq. 4.27).

$$\gamma = - \frac{C}{2M} \pm j\sqrt {\frac{K}{M} - \left( {\frac{C}{2M}} \right)^{2} } .$$

(4.27)

By $C_{c} = 2M\omega_{n}$, it is possible to show C in the form of circular damping. By utilizing $\zeta = C/C_{c}$ the ratio of damping will be achieved. (4.27) then reduces to the following:

$$\gamma = - \omega_{n} \zeta \pm j\omega_{n} \sqrt {1 - \zeta^{2} ,}$$

(4.28)

where, is the natural undamped frequency shown by (4.16) is $\omega_{n}$.

When $\zeta > 1$, In (4.28) both terms will be real, and it point toward a response which decaying steadily without oscillation. It is the definition of overdamped system. When $\zeta = 1$, it is a system which is named critically damped. Figure 4.3 illustrate that the value of $\zeta$ shows the minimum required damping for oscillatory motion prevention. Also this value shows that it is guaranteed in the short time the system will be in rest position again. There will be a real square root and $\gamma$ will have a negative genuine part if $\zeta < 1$. In addition the root in amplitude is with expanding the time. Therefore, the reaction will waver at a damped characteristic frequency (see 4.29).

Fig. 4.3

Different value of damping and their response

$$\omega_{d} = \omega_{n} \sqrt {1 - \zeta^{2} }$$

(4.29)

This type of system is known light damping system as auxiliary underdamped system. The watched reaction to the predetermined introductory conditions characterized in Sect. 4.3 is gained by utilizing the (4.25) real part and the initial conditions to solve for the constants A_I and A_R as described before. The actual displacement is shown as the following equation:

$$w(t) = {\text{e}}^{{ - \omega_{n} \zeta t}} \left[ {w(0)\cos \omega_{d} t + \frac{{\dot{w} (0) + \zeta \omega_{n} w(0)}}{{\omega_{d} }}\sin \omega_{d} t} \right]$$

(4.30)

The simple harmonic form of the equation is as follows:

$$w(t) = \left| A \right|{\text{e}}^{{ - \omega_{n} \zeta t}} \cos (\omega_{d} t + \phi ),$$

(4.31)

$\phi$ can be written as follows:

$$\phi = \tan^{ - 1} \left( { - \frac{{\dot {w}(0) + \zeta w(0)}}{{w(0)\omega_{d} }}} \right)$$

(4.32)

Then $\left| A \right|$ can be written as follows:

$$\left| A \right| = \left\{ {[w(0)]^{2} + [\dot{w}(0) + \zeta \omega_{n} w(0)]^{2} /\omega_{d}^{2} } \right\}^{{\frac{1}{2}}}$$

(4.33)

According to (4.31), harmonic motion which is in the frequency of rod and the amplitude as the outcome of the $\left| A \right|{\text{e}}^{{ - \omega_{n} \zeta t}}$ (which is decline by increasing the time) are the constituent of the response. Focusing on the (4.29) showing that the natural frequency is higher than the damped frequency.

In active control models, the damping characteristics and presence are essential. They are also important because they represent a process in which the system response can be decrease by passive means. The existence and feature of the damping is essential for both active and passive control. The SDOF system with different damping and its general response against and time cure has been shown in Fig. 4.3. As the result of the (4.33), a response with oscillating at the rod is occurred by a light damping and its amplitude is decreasing slowly with the time. The response goes near to equilibrium position by a critical damping, but it does not pass it. The respond is going to be constant about the oscillatory motion with heavy damping; the motion will be expressively slow down even near to the equilibrium position by the damping force and it takes time to return to the initial position.

4.5 Forced Response of SDOF Systems

Excitation of most of the systems is with continues disturbance which is going to be applied on the system. This type of excitation is more than the ones mentioned free motion initial excitation. Assume to write the complex form amplitude which is constant and with a harmonic force as the disturbance, shows as follows [13]:

$$f(t) = F{\text{e}}^{j\omega t}$$

(4.34)

Here, the applied force and its relative phase and the amplitude which will be defined by a complex number which is F. (4.14) has to be a homogeneous differential equation so that it should be modified with considering the disturbance force [13].

$$M\frac{{{\text{d}}^{2} w}}{{{\text{d}}t^{2} }} + Kw = F{\text{e}}^{j\omega t}$$

(4.35)

In order to state the (4.36), it is noticeable to remember that the disturbance has been assumed to be applied during the whole-time t and in this condition the component of transient response will be zero. So that the response will be considered as steady state and it will be harmonically beginning in a complex form [13].

$$w(t) = A{\text{e}}^{j\omega t}$$

(4.36)

here A and w(t) are complex generally. By substitution of this assumption in (3.33), (4.37) will be obtained as follows [13]:

$$\left( { - \omega^{2} + \frac{K}{M}} \right)A = \frac{F}{M}$$

(4.37)

A is an amplitude with complex response and unknown. It can be achieved by reorganizing the (4.37). So that we can obtain the following equation [13]:

$$A = \frac{F/K}{{1 - (\omega /\omega_{n} )^{2} }}$$

(4.38)

The response of the elastic systems can be shifted to the disturbance which is harmonic and steady state with the above Eq. (4.38). Theoretically, the displacement amplitude is infinite for the excitations which are steady state and continues, whenever the $\omega = \omega_{n}$. A large response will be the result of the frequency very close to the natural one during the system driving. From the control point of view, this condition is so essential because the system has been driven on resonance.

The above approach is being so difficult to extension for systems which are complex. For such systems, impedance method is an appropriate method. Here the mechanical input impedance is defined as the complex amplitude ratio of the drive point’s force input to velocity. So that, in (4.36) for harmonic displacement, $\dot{w}$ is the velocity such as following [8, 10]:

$$\dot{w}(t) = j\omega A{\text{e}}^{j\omega t}$$

(4.39)

The velocity thus has complex amplitude $j\omega A$, and input impedance is given as follows [13]:

$$Z_{i} = \frac{F}{j\omega A}$$

(4.40)

Figure 4.4 illustrates the system input impedance, and it is driven as following:

Fig. 4.4

Magnitude of forced response of a single degree of freedom system [13]

$$Z_{i} = \frac{{ - jK[1 - (\omega /\omega_{n} )^{2} ]}}{\omega }$$

(4.41)

By utilizing the (4.40), the system’s response to a disturbance force which is harmonic and steady state can be calculated if the input impedance is measured. The amplitude can be measured by (3.42) if we know the relation between the force response and damping (see Eq. (4.42)).

$$A = \frac{F/K}{{1 - (\omega /\omega_{n} )^{2} + j2\zeta (\omega /\omega_{n} )}}$$

(4.42)

Figure 4.5 presents the non-dimensional displacement response amplitude equal to $\left| {AK/F} \right|$ The relationship between the $\left| {AK/F} \right|$ and $\omega /\omega_{n}$ which are the non-dimensional displacement response amplitude and input frequency, for different $\zeta$ which is the ratio of damping is shown in fig. 4.5. This figure also indicates that the less value of damping will cause large value of response reduction near or on resonance and not that much different than the resonance condition. On the other hand, at resonance, system will be bounded by damping.

Fig. 4.5

Phases of forced response of a single degree of freedom system [13]

Figure 4.5 shows the phase response. The displacement response’s phase of the system’s excitation force has almost 180° flipping change of the phase for low value of light damping. The reason behind this fact is the incremental behavior of the frequency of the excitation over the resonance frequency. The phase sharpness transaction has the opposite relationship with damping amount. This relationship is important for observation of the system with many degrees of freedom.

4.6 Car Suspension Types as Mechanical Structures and Performance

As it has been mentioned before, vehicle is considered as the target mechanical structure. On vehicle, the suspension system is focused to be modeled mathematically. By considering the principles of control and control functions, the vehicle suspension system can be categorized into three main categories which are passive, semi-active and active suspensions.

4.6.1 Passive Suspension

While the suspension system has no other power and actuator and the coefficient of damping and stiffness is fixed, the system will be named as passive suspension. Figure 4.6 shows a typical type passive suspension system. As the figure shows, this type of system contains springs, dampers and sprung and unsprung masses. This type of suspension system is considered as traditional mechanical structures which have low cost, reliable performance, no additional energy and simple structure. Passive suspension system is widely utilizing in many types of vehicle. Based on random vibration theory, due to inability of adjusting stiffness and damping in passive suspensions, there is no adoptability for different road and it can only ensure the specific operating conditions to achieve optimal damping effect. Due to this feature, by utilizing this type of suspension good ride comfort and handling stability will be hard to be acquired [14].

Fig. 4.6

Passive suspension system [15]

Due to the mentioned characteristics of mechanical structures which have been mentioned in the beginning of the chapter, the passive suspension has the following two main defects:

• 1. It has large travel stroke while the frequency is reducing. The reason behind it was that the natural frequency of the system square is propositional and in reverse direction of the suspension travel stroke.

• 2. All of the mentioned parameters which are mentioned in the beginning of chapter are restricted due to the suspension components’ limitation about the stiffness and damping; in addition, there is no the possibility of meeting with different speed, load, road conditions and so on.

There are a few aspects which might improve the passive suspension systems’ performance.

• 1. Finding the optimal suspension and mechanical structures’ parameters by the aid of modeling and simulation;

• 2. Utilizing the variable-damping shock absorbers and gradient stiffness springs to make the suspension system to be adoptable for different conditions;

• 3. Utilization of the multi-link suspension including stabilizer bar [16].

4.6.2 Semi-active Suspension

Whenever there is the possibility of adjusting the shock absorber and elastic element stiffness based on the need, the suspension system will be named as semi-active damping. Semi-active suspension concept was proposed by Karnopp et al. in 1974 with the name of Skyhook semi-active suspension. Due to difficulty of adjusting the stiffness of spring, mainly regulating the shock absorber damping will be focused on semi-active suspensions. Figure 4.7 shows the semi-active suspension systems components.

Fig. 4.7

Semi-active suspension system [17]

There is no any specific component of dynamic control in semi-active suspension system. There are some sensors which measuring the velocity and sending the data as signal to car Engine Control Unit (ECU). ECU will measure the input signal and calculate the required control force. In order to compensate the vibration, the signal will be transferred to the shock absorbers to control the damping. Investigation of the actuators (shock absorbers) and the control architecture are two main aspects which have to be focused on semi-active suspensions. In addition, less size of the device and low cost are two advantages of the semi-active suspension in comparison with the active ones [18].

4.6.3 Active Suspension

Due to the need of more accurate and better performance vehicle, different technologies have been utilized to increase the performance of the suspension systems. In this order, different types of manufacturing techniques and technologies, minicomputers, microprocessors, electrohydraulic systems, and different types of controllers have been innovated to be utilized in the suspension systems. To make the vehicle to be smart enough to control itself about the vibration in different road condition and vehicle load, active suspension system will be installed. In order to achieve this goal, an effective control architecture is needed for active suspension system [19]. Figure 4.8 shows an active suspension system.

Fig. 4.8

¼ model of active suspension system [20]

As the active suspensions are the computer-controlled systems, they have the following features:

• (1) Force generation by a power source;

• (2) Utilized components to be able to be functioning continuously and passing the mentioned force;

• (3) Utilized different type of sensors and huge amount of data sets.

With considering the mentioned features, this type of suspension systems required mechanics, electronics, and control knowledge together. In braking and turning conditions, the springs will be deformed and the inertia force will be generated. The active suspension system will generate a different type of inertia force to compensate the one generated due to the braking and turning to reduce the care position changing [21].

In Table 4.1, a comparison of different type of car suspension system is illustrated.

Table 4.1 Three types of suspension systems’ technical comparison [22]

In this chapter, it has been focused to just model the active and passive suspension systems. No matter how and what kind, only the amount of created force is having been discussed.

4.6.4 Mathematical Modeling of Passive Suspension System

Mathematical modeling of the system and determining the requirements for its design are the first step in every system design. In order to evaluate the designed control system, several models will be tested by simulation techniques. This procedure shows the importance of the mathematical modeling of the system which is as the prerequisite for the control architecture and its design [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39].

Providing the model description by parameters determination is the key to create a mathematical model of a system.

There are three types of well-known models in the case of vibration control (here suspension control) which are weight function, the state space, and transfer function description. It is possible to categorize them  as the continues and discrete time mathematical models. In order to model the passive suspension system mathematically, it is considered to start the procedure with simplifying the system as it has been illustrated in Fig. 4.6. In addition, adding more detail for it to be capable of being utilized in mathematical model (Fig. 4.9).

Fig. 4.9

Vehicle ¼ passive suspension system mathematical model [15]

In Fig. 4.9 which illustrating a passive suspension in a quarter vehicle model, there are sprung mass M₁ and unsprung mass M₂ which are vehicle body and an assembly of the wheel and axle, respectively. The linear spring with stiffness k₂ represents the tire when is contacting the road during the car is traveling. The main components of the passive suspension system are damper and spring. D represent the linear damper with average coefficient of damping D and a linear actual spring with coefficient of stiffness k₁. x₀(t) shows the displacement of the vehicle body (sprung mass), and x₂(t) is representing the displacement of wheel and axle (unsprung Mass). x_i(t) is representing the vertical road profile.

Unlike the definition and mathematical modeling which are explained in the beginning of this chapter (SDOF), here there is a different mathematical modeling as this system has two degrees of freedom. So that the vehicle suspension system’s dynamics and two differential equations of degrees of freedom motion will be represented as follows [40]:

$$M_{1} \ddot{x}_{0} (t) + D \left[ {\dot{x}_{0} (t) - \dot{x}_{2} (t)} \right] + k_{1} \left[ {x_{0} (t) - x_{2} (t)} \right] = 0$$

(4.43)

$$M_{2} \ddot{x}_{2} (t) - D\left[ {\dot{x}_{0} (t) - \dot{x}_{2} (t)} \right] + k_{1} \left[ {x_{2} (t) - x_{0} (t)} \right] + k_{2} \left[ {x_{2} (t) - x_{i} (t)} \right] = 0$$

(4.44)

By applying the Laplace transformation on (4.44), we can get (4.45) and it is passive suspension system transfer function [40].

$$\frac{{X_{0} }}{{X_{1} }} = \frac{{k_{2} (Ds + k_{1} )}}{{M_{1} M_{2} s^{4} + (M_{1} + M_{2} )Ds^{3} + (M_{1} k_{1} + M_{1} k_{2} + M_{2} k_{1} )s^{2} + Dk_{2} s + k_{1} k_{2} }}$$

(4.45)

In order to do simulation by utilizing the mentioned mathematical model, the value of parameters which could be different for each car should be defined. Here is an example of these parameters’ definition which has been given as Table 4.2.

Table 4.2 Vehicle ¼ passive suspension model simulation input parameters [22, 41]

4.6.5 Mathematical Modeling Active Suspension System

In this chapter, it has been focused to just model the active and passive suspension systems. No matter how and what kind, only the amount of created force is having been discussed.

In order to utilize pavement method and its roughness as input for modeling of the body of vehicle in vertical vibration, ¼ vehicle dynamic vibration model has been chosen. This model dose not include all of the features such as geometrical information and pitching and rolling angle vibration, but it is still covering almost all of the features such as the load and suspension system’s stress change information. Here, the characteristics of the ¼ vehicle dynamic vibration model have been provided [42]:

• (1) Two sets of suspension system installed in the vehicle which are the suspensions for front side and back side are independently functioning from each other of each other.

• (2) The suspension itself and the tires have to be investigated together.

• (3) The components which are elastic can be simply can be assumed as damping and springs.

• (4) Decrease the system parametric description under precondition of keeping accuracy and effectiveness.

• (5) Mass of the actuator will not be considered, and force is their only output.

Figure 4.10 illustrates a vehicle ¼ model of active suspension system. Everything about the parameters in this figure is same as passive suspension’s assumptions. The only different is the u which is the active control force and has been generated by the actuator of active suspension.

Fig. 4.10

¼ model of active suspension system [20]

Form Fig. 4.10, dynamics of the vehicle suspension system can be analyzed. Also, it is possible to create two differential equations of degrees of freedom motion as follows [23]:

$$M_{1} \ddot{x}_{0} (t) + D \left[ {\dot{x}_{0}(t) - \dot {x}_{2} (t)} \right] + k_{1} \left[ {x_{0} (t) - x_{2} (t)} \right] = \text{u}$$

(4.46)

$$M_{2} \ddot{x}_{2} (t) - D\left[ {\dot{x}_{0} (t) - \dot{x}_{2} (t)} \right] + k_{1} \left[ {x_{2} (t) - x_{0} (t)} \right] + k_{2} \left[ {x_{2} (t) - x_{1} (t)} \right] = - u$$

(4.47)

Equations. (4.48) to (4.58) illustrate the mathematical modeling of an active suspension system with state space method [22, 23].

$$x_{1} = x_{2} (t),x_{2} = x_{0} (t),x_{3} = \dot{x}_{2} (t),x_{4} = \dot{x}_{0} (t)$$

(4.48)

Equation of the system state space can be shown as follows:

$$\frac{{{\text{d}}X}}{{{\text{d}}t}} = AX + BU$$

(4.49)

In (4.49), state variable matrixes can be shown as follows:

$$x = [\begin{array}{*{20}l} {x_{1} } \hfill & {x_{2} } \hfill & {x_{3} } \hfill & {x_{4} } \hfill \\ \end{array} ]$$

(4.50)

A and B are constant matrixes, and they can be shown as follows:

$$\left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ {\frac{{K_{1} + K_{2} }}{{m_{2} }}} & {\frac{{K_{1} }}{{m_{2} }}} & {\frac{D}{{m_{2} }}} & {\frac{D}{{m_{2} }}} \\ {\frac{{K_{1} }}{{m_{1} }}} & {\frac{{K_{1} }}{{m_{1} }}} & {\frac{D}{{m_{1} }}} & {\frac{D}{{m_{1} }}} \\ \end{array} } \right]$$

(4.51)

$$\left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ {\frac{{K_{2} }}{{m_{2} }}} & {\frac{1}{{m_{2} }}} \\ 0 & { - \frac{1}{{m_{1} }}} \\ \end{array} } \right]$$

(4.52)

The matrix belongs to the input variable of the system which is as follows:

$$U = [\begin{array}{*{20}l} {x_{i} (t)} \hfill & u \hfill \\ \end{array} ]^{\text{T}}$$

(4.53)

Output matrix of the suspension system is shown as follows:

$$Y = CX + DU$$

(4.54)

In (4.45), Y is the output variable matrix and it is shown as follows:

$$Y = \left\{ {k_{2} [x_{i} (t) - x_{1} ]\quad \ddot{x}_{0} (t)\quad x_{0} (t)} \right\}$$

(4.55)

The following equation is another representation of Y:

$$Y = \left\{ {k_{2} [x_{i} (t) - x_{1} ]\quad \ddot{x}_{2} \quad x_{2} } \right\}$$

(4.56)

C and D are constant matrixes, and they can be shown as follows:

$$\left[ {\begin{array}{*{20}c} { - k_{2} } & 0 & 0 & 0 \\ {\frac{{k_{1} }}{{m_{1} }}} & { - \frac{{k_{1} }}{{m_{1} }}} & {\frac{D}{{m_{1} }}} & {\frac{D}{{m_{1} }}} \\ 0 & 1 & 0 & 0 \\ \end{array} } \right]$$

(4.57)

$$\left[ {\begin{array}{*{20}c} {k_{2} } & 0 \\ 0 & { - \frac{1}{{m_{1} }}} \\ 0 & 0 \\ \end{array} } \right]$$

(4.58)

Eliminating the pavement condition by utilizing a controller was the main goal of this section. It means adjusting the force which is generated by active suspension system to compensate the road condition changes. It is worthy to mention that the road excitation and changes can be assumed as noise based on the definitions in previous chapters. To achieve the goal of this chapter, in the next chapter definition of noise cancelation and its common methods will be focused.