Vehicle Vibration Analysis in Changeable Speeds Solved by Pseudoexcitation Method

Abstract

The vehicle driving comfort has become one of the important factors of vehicle quality and receives increasing attention. In this paper, the mechanical and mathematical models of the half-car, five degrees of freedom (DOF) of a vehicle were established, as well as the pseudoexcitation model of road conditions for the front wheel and the rear wheel. By the pseudoexcitation method, the equations of transient response and power spectrum density were established. After numerical simulation to vehicle vibration response of changeable driving, the results show that the pseudoexcitation method is more convenient than the traditional method and effectively solves the smoothness computation problems of vehicles while the pseudoexcitation method is used to analyze vehicle vibration under nonstationary random vibration environments.

1. Introduction

The mechanical and mathematical model of vehicle systems is usually simplified as a multiple-mass, complicated vibration system.Due to road excitation, vehicles may come into complicated vibration, which is disadvantageous to passenger health and goods protection [1, 2]. Therefore, it is important and necessary to control the vehicle’s vibration within a limited and comfortable grade in order to ensure safety steering and physical health of drivers and passengers, as well as the operating stability of man-vehicle-road system. In the process of automobile moving, the random and changeable road surface is the main factor to induce vehicle vibration. Therefore, investigation of vehicle’s stochastic vibration [3, 4] induced by road excitation has been a significant problem of vehicle design and its performance simulation.

At present, for this kind of problems, the Fourier transform analysis is used to investigate the dynamic characteristics of constant driving problems of automobiles based on stationary random vibration theory. After finishing vibration model of vehicles, it is important to derive the frequency characteristic of vehicle vibration responses and to establish power spectrum density function of road excitation and vehicle vibration responses [5]. Then it can be used to analyze the influence of vehicle structural parameters and road excitation on vehicle random vibration [6]. Although this method was relatively simple, its derivation process is too complicated, that is, this method needs not only to derive the frequency response characteristics of vehicle vibration system, but also to derive the frequency response characteristics of vehicle vibration response values.

However, in some circumstances, vehicles are running in changeable speeds, such as in accelerating starting period and decelerating stopping period. In these cases, the road excitation and the vehicle dynamic response in time domain are nonstationary [7, 8]. The stochastic vibration analysis method based on Fourier transform and its inverse transform has been used to study the changeable speed response of vehicles in unevenness roads, but its computation work is enormous. The pseudoexcitation method was used to analyze the stochastic vibration of structural systems [9–11]. By pseudoexcitation method, stochastic vibration analysis was carried on a two-DOF system vibration of a quarter-vehicle model [12–14], in which the vibration response of a constant speed moving vehicle was investigated to a stationary random road excitation. The changeable speed vehicle vibration response was also conducted under one-point road excitation [15]. In addition, some last investigations [16, 17] have dealt with a quarter-car model with a harmonic excitation while the study [18] considers the additional stochastic component in the road surface roughness. In this study, the time-space frequency relationship of vehicle vibration under changeable speed moving was derived by pseudoexcitation method, and then the equation of transient power spectrum density of vehicle vibration response under nonstationary random road excitation input was obtained. At the end, we also conducted the numerical simulations to the vibration responses of a half-car, five-DOF vehicle system under changeable speed moving conditions.

2. Theory of Pseudoexcitation Method

When a linear system is randomly excited by self-spectrum density 𝑆_𝑥𝑥(𝑤), the self-power spectrum of its response 𝑦 is 𝑆_𝑦𝑦(𝑤)=|𝐻(𝑤)|²𝑆_𝑥𝑥(𝑤), (2.1) where 𝐻(𝑤) is the frequency response function, and its meaning is shown in Figure 1 [19], that is, the corresponding harmonic response is 𝑦=𝐻(𝑤)𝑒^𝑖𝑤𝑡 when the random excitation is replaced by harmonic excitation 𝑒^𝑖𝑤𝑡. From Figure 1, it can be seen that if it multiplies a constant 𝑆_𝑥𝑥 before the excitation 𝑒^𝑖𝑤𝑡, it can create a pseudoexcitation,𝑥(𝑡)=𝑆_𝑥𝑥𝑒^𝑖𝑤𝑡. (2.2)

Figure 1 

The principle of pseudoexcitation method.

After multiplying a same constant to its response value, it can give the following equations: 𝑦^∗𝑦=𝑦²𝑆_𝑥𝑥=𝑆_𝑦𝑦,𝑥^∗𝑦=𝑆_𝑥𝑥𝑒^{−𝑖𝑤𝑡}⋅𝑆_𝑥𝑥𝑒^𝑖𝑤𝑡=𝑆_𝑥𝑥𝐻=𝑆_𝑥𝑦,𝑦^∗𝑥=𝑆_𝑥𝑥𝑒^{−𝑖𝑤𝑡}⋅𝑆_𝑥𝑥𝐻𝑒^𝑖𝑤𝑡=𝑆_𝑥𝑥𝐻=𝑆_𝑠𝑦. (2.3) The last formula in the above 3 equations is the conventional expressions of self-spectrum density or mutual spectrum density.

If the pseudoresponse values 𝑦₁ and 𝑦₂ are considered in the above system, the following equations can be validated:𝑦₁^∗𝑦₂=𝐻₁^∗𝑆_𝑥𝑥𝑒^{−𝑖𝑤𝑡}⋅𝐻₂𝑆_𝑥𝑥𝑒^𝑖𝑤𝑡=𝐻₁^∗𝑆_𝑥𝑥𝐻₂=𝑆_𝑦1𝑦2,𝑦₂^∗𝑦₁=𝐻₂^∗𝑆_𝑥𝑥𝐻₁=𝑆_𝑦2𝑦1. (2.4)

Then the matrixes of power spectrum density are as follows:𝑆_𝑦𝑦=𝑦^∗⋅𝑦^𝑇,𝑆_𝑥𝑦={𝑥}^∗⋅𝑦^𝑇,𝑆_𝑦𝑥=𝑦^∗⋅{𝑥}^𝑇. (2.5)

If the pseudoexcitation of a random process is 𝑥(𝑡)=𝑆_𝑥𝑥𝑒^𝑖𝑤𝑡, then it can give 𝑥̇=𝑖𝑤𝑆_𝑥𝑥𝑒^𝑖𝑤𝑡,𝑥̈=−𝑤²𝑆_𝑥𝑥𝑒^𝑖𝑤𝑡,𝑆_̈𝑥̈𝑥=𝑤⁴𝑆_𝑥𝑥. (2.6)(2.7)

That is the power spectrum density of accelerations, and in the same way, we can obtain the following equations: 𝑆𝑦^̈‌𝑦^̈‌=𝑦̈^∗⋅𝑦̈^𝑇,𝑆𝑦^̈‌𝑦₁^̈₂=𝑦̈₁^∗⋅𝑦̈₂^𝑇, (2.8)(2.9) where ∗ is a complex conjugate, and 𝑇 is a matrix transfer.

After computing the vibration response of the system in the case of the pseudo harmonic excitation, all the power spectrum densities of them can be solved according to (2.5)–(2.9). Then the self-spectrum density and mutual spectrum density of them can also be obtained.

3. Vibration Analysis of the Half-Car, Five-DOF Vehicle System inChangeable Speeds

3.1. Mechanical and Mathematical Models of the System

For analyzing automobile vibration, it is important to establish its mechanical and mathematical model of the automobile structural system, so that the vibration characteristic response value of the mathematical model of vehicle vibration can be solved and obtained. For the mechanical modeling of automobiles, a seven-DOF mechanical model [20] has been developed to investigate the influence of active and semiactive suspension to automobile dynamic performance. In addition, the moving smartness and operating stability of automobiles were also investigated by spatial mechanical models of automobiles [21]. In this study, a half-car, five-DOF linear mechanical model [22] of an automobile system was developed as shown in Figure 2.

Figure 2 

The mechanical model of a half-car, five-DOF automobile system.

In Figure 2, 𝑚_𝑠 is the mass of driver and chair and 𝑚_𝑏 is the mass of automobile structure. 𝑚_𝑓,𝑚_𝑟 are the nonspring supported mass of front and rear suspensions, respectively. 𝑘_𝑠 and 𝑐_𝑠 are the rigidity coefficient and damping coefficient of the chair, respectively. 𝑘_𝑓,𝑘_𝑟 are the rigidity coefficient of front and rear suspensions, respectively. 𝑘_𝑡𝑓,𝑘_𝑡𝑟 are the rigidity coefficient of front and rear wheels, respectively. 𝑞_𝑓,𝑞_𝑟 are the road excitation forces at front and rear wheels, respectively. 𝑙₁ is the distance between chair and vehicle mass center. 𝑙₂,𝑙₃ are the distances from the vehicle mass center to front and rear wheel axles, respectively.

By Lagrange equations, the mathematical model of the vehicle mechanical model in Figure 2 is as follows:[𝑀]𝑍̈+[𝐶]𝑍̇+[𝐾]{𝑍}=[𝐹]{𝑄}, (3.1)

where 𝑚𝑀=^𝑠00000𝑚_𝑏00000𝑚_𝑝00000𝑚_𝑓00000𝑚_𝑟,𝑐𝐶=^𝑠−𝑐_𝑠𝑐_𝑠𝑙₁00−𝑐_𝑠𝑐_𝑠+𝑐_𝑓+𝑐_𝑟−𝑐_𝑠𝑙₁−𝑐_𝑓𝑙₂+𝑐_𝑟𝑙₃−𝑐_𝑓−𝑐_𝑟𝑐_𝑠𝑙₁−𝑐_𝑠𝑙₁−𝑐_𝑓𝑙₂+𝐶𝑟𝑙₃𝑐_𝑠𝑙²₁+𝑐_𝑓𝑙²₂+𝑐_𝑟𝑙²₃𝑐_𝑓𝑙₂−𝐶𝑟𝑙₃0−𝑐_𝑓𝑐_𝑓𝑙₁𝑐_𝑓00−𝑐_𝑟−𝑐_𝑟𝑙₃0𝑐_𝑟,𝑘𝐾=^𝑠−𝑘_𝑠𝑘_𝑠𝑙₁00−𝑘_𝑠𝑘_𝑠+𝑘_𝑓+𝑘_𝑟−𝑘_𝑠𝑙₁−𝑘_𝑓𝑙₂+𝑘_𝑟𝑙₃−𝑘_𝑓−𝑘_𝑟𝑘_𝑠−𝑘_𝑠𝑙₁−𝑘_𝑓𝑙₂+𝑘_𝑟𝑙₃𝑘_𝑠𝑙²₁+𝑘_𝑓𝑙²₂+𝑘_𝑟𝑙²₃𝑘_𝑓𝑙₂−𝑘_𝑟𝑙₃0−𝑘_𝑓𝑘_𝑓𝑙₂𝑘_𝑓+𝑘_𝑡𝑓00−𝑘_𝑟−𝑘_𝑟𝑙₃0𝑘_𝑟+𝑘_𝑡𝑟,𝑘𝐹=000000_𝑡𝑓00𝑘_𝑡𝑟,{𝑍}=𝑧_𝑠𝑧_𝑏𝑧_𝑝𝑧_𝑓𝑧_𝑟^𝑇,{𝑄}=𝑞_𝑓𝑞_𝑟^𝑇. (3.2)

3.2. Road Excitation

The unevenness degree of road profile can be generally described by power spectrum density. The international GB7031 recommends that the power spectrum density of road profile is described by𝑆_𝑞(𝑛)=𝑆_𝑞𝑛₀𝑛𝑛₀^−𝑤, (3.3) where 𝑆_𝑞(𝑛₀) is the unevenness coefficient of road profile. 𝑛₀ is the referenced spatial frequency, 𝑛₀=0.1(𝑚⁻¹). 𝑛 is the spatial frequency (𝑚⁻¹). 𝑤 is the frequency exponent of the graded road spectrum and generally chosen as 2. In this study, 𝑆_𝑞(𝑛₀) is set as 𝑆_𝑞(𝑛₀)=64×10⁻⁶(𝑚²/𝑚⁻¹), that is, Grade B road condition.

When automobiles move in changeable speeds, the excitations of automobile systems are different in time domain and space domain. It is not stationary in space domain but in time domain. However, the automobile’s mechanical responses are all nonstationary. By the inherent characteristics of frequency response function 𝐻(𝑤) of vehicle system in time domain and the relation of time frequency 𝑤 and space frequency 𝑛, the transient frequency response function 𝐻(𝑠,𝑛) can be obtained, then we can solve the stochastic vibration of the vehicle system in changeable speed moving [11, 23].

The unevenness degree of roads in time domain is shown as follows [24]: 𝑞(𝑡)=ℎ₀𝑒^𝑗𝑤𝑡, (3.4) where ℎ₀ is the amplitude of unevenness degree of roads. The expression of unevenness degree of roads in space domain is as follows: 𝑞=ℎ₀𝑒^𝑗Ω𝑠,𝑤𝑡=Ω𝑣, (3.5)(3.6) where Ω is the spatial angular frequency.

When the automobile is moving in a constant speed, 𝑠=𝑣𝑡, it has the following relation, 𝑤=Ω𝑠 or 𝑓=𝑛𝑣.

When the automobile is moving in a changeable speed, it has 𝑠=𝑣₀+𝑎𝑡²2, (3.7) where 𝑣₀ is the initial velocity of the automobile and 𝑎 is its acceleration. Then (3.6) can be rewritten as 𝑤𝑑𝑡=Ω𝑑𝑠,𝑤=Ω𝑑𝑠𝑣𝑑𝑡=2𝑛𝜋₀+𝑎𝑡=2𝑛𝜋2𝑎𝑠+𝑣₀^21/2. (3.8)

Equation (3.8) reflects the time-space frequency relation of automobiles in an accelerated moving.

3.3. Pseudoexcitation of Random Road at Front and Rear Wheels

By the time-frequency expression of unevenness degree of roads, a pseudoexcitation of road 𝑞(𝑡) can be built which is corresponding with the road excitation 𝑞(𝑡), as follows: 𝑞(𝑡)=𝑆_𝑞(𝑛)𝑒^𝑖𝑤𝑡. (3.9)

Hypothesize that the road excitations on four wheels of the automobile are the same and the delay relation [14] between front wheel excitation 𝑞_𝑓(𝑡) and rear wheel excitation 𝑞_𝑟(𝑡) is as follows: 𝑞_𝑟(𝑡)=𝑞_𝑓(𝑡−𝜏), (3.10) where 𝜏=𝑙/𝑣,𝑙 is the distance between two wheel axles, then it has𝑞_𝑓(𝑡)=𝑆_𝑞(𝑛)𝑒^𝑖𝑤𝑡,𝑞_𝑟(𝑡)=𝑞_𝑓(𝑡−𝜏)=𝑆_𝑞(𝑛)𝑒^{𝑖𝑤(𝑡−𝜏)}=𝑒^{−𝑖𝑤𝑡𝜏}𝑞(𝑡). (3.11) Therefore, the excitation input is written as follows:𝑞(𝑡=𝑞)^𝑓(𝑡)𝑞_𝑟(𝑡)=1𝑒^{−𝑖𝑤𝜏}𝑞(𝑡)=𝐻_𝑞(𝑤)𝑞(𝑡). (3.12)

The road excitations from front wheel and rear wheel can be simplified as an excitation input {𝑞(𝑡)}, and its frequency response characteristic is {𝐻_𝑞(𝑤)}. Thus, the two-point excitations are simplified as one-point excitation.

3.4. Formulation of System Response

For a multiple degrees-of-freedom system, its frequency response characteristic is the complex number ratio of response vector and excitation vector. For the half-car, five-DOF vehicle system in this study, if we supposed that its frequency response is [𝐻(𝑤)], then the relation of pseudoresponse and pseudoexcitation is {𝑧(𝑡)}=[𝐻(𝑤)]𝑞(𝑡). (3.13)

Substituting (3.12) into (3.13) gives{𝑧(𝑡)}=[𝐻(𝑤)]𝐻_𝑞(𝑤)𝑞(𝑡)=ℎ_𝑔(𝑤)𝑞(𝑡). (3.14)

Since {ℎ_𝑔(𝑤)}=[𝐻(𝑤)]{𝐻_𝑞(𝑤)}, then𝑧̇(𝑡)=ℎ_𝑔(𝑤)𝑞̇(𝑡)=𝑖𝑤ℎ_𝑔(𝑤𝑞(𝑡)𝑧̈(𝑡),)=−𝑤²ℎ_𝑔(𝑤)𝑞(𝑡). (3.15)(3.16)

Substitute (3.15) and (3.16) into the system equation, then the system frequency response function can be obtained as follows:[𝐻(𝑤)]=[𝐾]−𝑤²[𝑀]+𝑖𝑤[𝐶]⁻¹𝐹, (3.17) where [𝐻(𝑤)]_5×2=𝐻(𝑤)_𝑠𝐻(𝑤)_𝑏𝐻(𝑤)_𝑝𝐻(𝑤)_𝑓𝐻(𝑤)_𝑟. (3.18)

The frequency response functions of the relative displacement of suspension and the dynamic loads of tires are as follows, respectively:𝐻_𝑟𝑑(𝑤)=𝑧_𝑏(𝑤)−𝑙₂𝑧_𝑝(𝑤)−𝑧_𝑓(𝑤)𝑞_𝑓(𝑤),𝐻_𝑡𝑓(𝑤)=𝑧_𝑓(𝑤)−𝑧_𝑟(𝑤)𝑞_𝑓(𝑤)𝑘_𝑡𝑓. (3.19)

By substituting (3.13) into (3.19), it gives the frequency response functions of the relative displacement of suspension and the dynamic loads of tires as follows: 𝐻_𝑟𝑑(𝑤)=𝐻_𝑏(𝑤)−𝑙₂𝐻_𝑝(𝑤)−𝐻_𝑓(𝑤),𝐻_𝑡𝑓(𝑤)=𝐻_𝑓(𝑤)−𝐻_𝑟(𝑤)𝑘_𝑡𝑓. (3.20)

It can be found that the system response result by pseudoexcitation method is the same as the result obtained by Fourier transform analysis. After obtaining the automobile structural parameters and the road excitation parameters, the pseudoexcitation responses {𝑧(𝑡)} and {𝑧̈(𝑡)} can be solved in accordance with (3.14) and (3.16). The response power spectrum can be achieved according to (2.5) and (2.6).

The power spectrum matrix of vertical acceleration of the system is𝑆_̈𝑧̈𝑧(𝑤)=𝑧̈(𝑡)^∗⋅𝑧̈(𝑡)^𝑇=𝑤⁴ℎ_𝑔(𝑤)𝑆_𝑞(𝑛)𝑒^{−𝑖𝑤𝑡}ℎ_𝑔(𝑤)^𝑇𝑆_𝑞(𝑛)𝑒^𝑖𝑤𝑡=𝑤⁴ℎ_𝑔(𝑤)ℎ_𝑔(𝑤)^𝑇𝑆_𝑞(𝑛)=𝑤⁴[𝐻(𝑤)]𝐻_𝑞(𝑤)𝐻_𝑞(𝑤)^𝑇[𝐻(𝑤)]^𝑇𝑆_𝑞(𝑛). (3.21)

By substituting 𝑤=2𝑛𝜋(2𝑎𝑠+𝑣₀²)^1/2 into (3.21), the spatial acceleration power spectrum density of system responses can be obtained as follows:𝑆_̈𝑧̈𝑧(𝑠,𝑛)=(2𝜋𝑛)⁴2𝑎𝑠+𝑣₀²²[𝐻(𝑠,𝑛)]𝐻_𝑞(𝑠,𝑛)𝐻_𝑞(𝑠,𝑛)^𝑇[𝐻(𝑠,𝑛)]^𝑇𝑆_𝑞(𝑛), (3.22) where 𝐻(𝑠,𝑛)=[𝐾]−4𝑛²𝜋²2𝑎𝑠+𝑣₀²[𝑀]+𝑖2𝑛𝜋2𝑎𝑠+𝑣₀²[𝐶]⁻¹𝐻𝐹,_𝑞(=1𝑒𝑠,𝑛)_{−𝑖(2𝑛𝜋2𝑎𝑠+𝑣02𝜏)}𝑆_𝑞(𝑛)𝑒^{−𝑖(2𝑛𝜋2𝑎𝑠+𝑣02𝑡)}. (3.23)

In the same way, according to the frequency response function of relative displacement of suspensions and dynamic loads of tires, we can obtain the transient spatial power spectrum density functions. It is important to substitute 𝑤=2𝑛𝜋(2𝑎𝑠+𝑣₀²)^1/2 into them to get the corresponding power spectrum density as follows: 𝑆_𝑟𝑑(𝑠,𝑛)=𝐻_𝑟𝑑(𝑠,𝑛)𝑆_𝑞(𝑛)𝐻_𝑟𝑑(𝑠,𝑛)^𝑇,𝑆_𝑡𝑓(𝑠,𝑛)=𝐻_𝑡𝑓(𝑠,𝑛)𝑆_𝑞(𝑛)𝐻_𝑡𝑓(𝑠,𝑛)^𝑇. (3.24)

By means of computing {𝑆_̈𝑧̈𝑧(𝑠,𝑛)}, 𝑆_𝑟𝑑(𝑠,𝑛), and 𝑆_𝑡𝑓(𝑠,𝑛), we can obtain the dynamic characteristics of vehicles and driving comfort of vehicles.

4. Computing Case

In order to check the validation of the above mathematical models of transient response analysis of the half-car, five-DOF automobile system based on pseudoexcitation, we carried out the simulations as follows. To mimic an accelerating process of the automobile, set the acceleration as 𝑎=1.0m⁄s and the running distance as 𝑠=150m, respectively. The other mechanical model parameters of the five-DOF automobile system are shown in Table 1.

Table 1 

Mechanical model parameters of the five-DOF automobile system.

Figures 3 and 4 show the 3D (three-dimensional) and 2D (two-dimensional) spectrums of body acceleration, respectively. Figures 5 and 6 show the 3D and 2D spectrums of relative displacement of vehicle suspension, respectively. Figures 7 and 8 show the 3D and 2D spectrums of seat acceleration, respectively. Figures 9 and 10 show the 3D and 2D spectrums of tire acceleration, respectively. Figure 11 shows the 2D spectrum of tire dynamic load. The nonstationary response spectrum analysis of the vehicle system shows that the low-spatial frequency ingredient has a main role with vehicle speed increasing when the vehicle moves in a constant acceleration. With the speed increasing, the peak values of power spectrum of seat and body acceleration, as well as the low frequency values of power spectrum of dynamic tire loads, are not monotonously increasing, and some local values are decreasing. The changing of relative displacement of vehicle suspensions is not large. The nonlinearity of the vehicle suspension is not included in the current study, this is the limitation of this vehicle dynamic model, and it will be considered in the coming study.

Figure 3 

3D spectrum of body acceleration.

Figure 4 

2D spectrum of body acceleration.

Figure 5 

3D spectrum of relative displacement of vehicle suspension.

Figure 6 

2D spectrum of relative displacement of vehicle suspension.

Figure 7 

3D spectrum of seat acceleration.

Figure 8 

2D spectrum of seat acceleration.

Figure 9 

3D spectrum of tire acceleration.

Figure 10 

2D spectrum of tire acceleration.

Figure 11 

2D spectrum of tire dynamic load.

5. Conclusion

By pseudoexcitation method, the vibration response characteristics of the half-car, five-DOF automobile system were obtained. The results show that the pseudoexcitation method is more convenient than the traditional method and effectively solves the smoothness computation of vehicles while the pseudoexcitation method is used to analyze vehicle vibration under nonstationary random vibration.

Acknowledgments

The authors are grateful for the research Grants from the Program for New Century Excellent Talents in University (no. NCET-08-0103), the Scientific Research Foundation of China Postdoctor (nos. 20070420203 and 200801391), the Natural Science Foundation of China (no. 50875041), National 973 Program (2007CB210305-2), Basic Research Expense of National Universities (no. N090503001), Program for Changjiang Scholars and Innovative Research Team in University, and a research fund from Northeastern University, China.