Vehicle Vibration Analysis in Changeable Speeds Solved by Pseudoexcitation Method

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 halfcar, 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.


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 2 Mathematical Problems in Engineering 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.

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 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  The last formula in the above 3 equations is the conventional expressions of self-spectrum density or mutual spectrum density.
If the pseudoresponse values y 1 and y 2 are considered in the above system, the following equations can be validated: Then the matrixes of power spectrum density are as follows:

2.5
If the pseudoexcitation of a random process is x t S xx e iwt , then it can give That is the power spectrum density of accelerations, and in the same way, we can obtain the following equations: 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.

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.
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.l 1 is the distance between chair and vehicle mass center.l 2 , l 3 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: where 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 by where S q n 0 is the unevenness coefficient of road profile.n 0 is the referenced spatial frequency, n 0 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 n 0 is set as S q n 0 64 × 10 −6 m 2 /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 h 0 e jwt , 3.4 where h 0 is the amplitude of unevenness degree of roads.The expression of unevenness degree of roads in space domain is as follows: q h 0 e jΩs , 3.5 wt Ωv, 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 where v 0 is the initial velocity of the automobile and a is its acceleration.Then 3.6 can be rewritten as wdt Ωds,

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

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
Mathematical Problems in Engineering 7 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: where τ l/v, l is the distance between two wheel axles, then it has q 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 1 e −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.

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

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 }, then ˙ z t h g w ˙ q t iw h g w q t , 3.15 ¨ z t −w 2 h g w q t .

3.16
Substitute 3.15 and 3.16 into the system equation, then the system frequency response function can be obtained as follows: where The frequency response functions of the relative displacement of suspension and the dynamic loads of tires are as follows, respectively:

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:

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 { ¨ z 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 is q n e iwt w 4 h g w h g w T S q n w 4 H w H q w H q w T H w T S q n .

3.21
By substituting w 2nπ 2as v 0 2 1/2 into 3.21 , the spatial acceleration power spectrum density of system responses can be obtained as follows: H s, n H q s, n H q s, n T H s, n T S q n , 3.22  where

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 v 0 2 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 z z s, n }, S rd s, n , and S tf s, n , we can obtain the dynamic characteristics of vehicles and driving comfort of vehicles.

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 1.0 ms and the running distance as s 150 m, respectively.The other mechanical model parameters of the five-DOF automobile system are shown in Table 1.
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.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.

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.

Figure 2 :
Figure 2: The mechanical model of a half-car, five-DOF automobile system.

Figure 5 :
Figure 5: 3D spectrum of relative displacement of vehicle suspension.

Figure 6 :
Figure 6: 2D spectrum of relative displacement of vehicle suspension.
, it can be seen that if it multiplies a constant S xx before the excitation e iwt , it can create a pseudoexcitation,

Table 1 :
Mechanical model parameters of the five-DOF automobile system.