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1.Automotive Engineering Research Institute，Jiangsu University，Zhenjiang 212013，P.R.China；2.Department of Mechanical Engineering，Wayne State University，Detroit 48202，USA

Abstract: The dynamic responses of suspension system of a vehicle travelling at varying speeds are generally nonstationary random processes，and the non-stationary random analysis has become an important and complex problem in vehicle ride dynamics in the past few years. This paper proposes a new concept，called dynamic frequency domain（DFD），based on the fact that the human body holds different sensitivities to vibrations at different frequencies，and applies this concept to the dynamic assessment on non-stationary vehicles.The study mainly includes two parts，the first is the input numerical calculation of the front and the rear wheels，and the second is the dynamical response analysis of suspension system subjected to non-stationary random excitations.Precise time integration method is used to obtain the vertical acceleration of suspension barycenter and the pitching angular acceleration，both root mean square（RMS）values of which are illustrated in different accelerating cases. The results show that RMS values of non-stationary random excitations are functions of time and increase as the speed increases at the same time.The DFD of vertical acceleration is finally analyzed using time-frequency analysis technique，and the conclusion is obviously that the DFD has a trend to the low frequency region，which would be significant reference for active suspension design under complex driving conditions.

Key words：non-stationary random process；suspension system；vehicle modeling；dynamical frequency domain（DFD）；ride comfort

0 Introduction

Road excitations to a vehicle may be non-sta?tionary random processes as a result of the vehicle’s varying speeds. It has been shown that non-station?ary characteristics can have significant effects on ve?hicle ride comfort，handling performance，and safe?ty［1-3］. With vehicle technologies moving towards in?tellectualization，electrification and integration，the detrimental consequences due to non-stationary driv?ing can be profound. For instance，several studies have reported that such non-stationary vibration may shorten the power battery life of new energy ve?hicles［4-5］，which is a research hotspot in automotive industry，bringing great challenges to ensure the sta?bility and reliability of the complex electromechani?cal hydraulic coupling systems in these vehicles［6-7］.Therefore，understanding the non-stationary vibra?tion characteristics is essential for the analysis and development of control systems to improve the qual?ity of vehicle driving. Traditional suspension control focused on the entire frequency range or some finite fixed frequency range of interest［8］. However，the frequency ranges of interest for suspension control are often time-varying since the vibration responses of a vehicle body subjected to non-stationary ran?dom excitations are non-stationary processes. More?over，the resonant frequency bands will change as well due to the varying speeds.

The dynamics of vehicles subjected to station?ary road excitations is well understood［9］，and re?search results have been extensively applied to the automotive industry. However，several key obsta?cles remain for the non-stationary problem. The first step that is essential for studying the non-sta?tionary random vibration problem is to obtain accu?rate road information. The frequency domain model is one of the most effective road excitation models and has been employed extensively in vehicle dy?namics analysis due to the well-known theories.Parkhilovskii［10］and Dodds and Robson［11］were among the early researchers to put forth frequency domain road models. They developed statistic char?acteristics analyses to profile the road roughness for different kinds of road surfaces，with excitations to the wheels modeled as ergodic stationary random processes. Such models are generally inadequate be?cause more complex dynamic analysis of vehicle structural systems is required in modern automotive technologies. Time domain models are more suit?able to describe the road profile and have widely been adopted for several decades［12-17］. However，the amount of computation time was huge. To over?come the obstacle，Marzbanrada and Ahmadib［18］proposed a linear filtering white noise model which resulted in a much less computation time and faster speed. Zhang et al.［19］used a novel method called Cholesky decomposition filtering white noise to gen?erate new random signals. However，the process was too complex and the simulation precision was not sufficient in practice.

From aforementioned research，it is clear that neither the frequency nor the time domain model can satisfactorily analyse the non-stationary random processes. Recent research efforts have focused on developing suitable models and analysis methods for vehicles subjected to non-stationary random excita?tions［20-22］. Lei et al.［23］modeled and simulated the power spectral density（PSD）of non-stationary ran?dom road excitations as a Wiener process. Zhang et al.［24］used the equivalent covariance method to es?tablish a non-stationary random input model for a single wheel and resolved the correlation between the front and the rear wheels with a variable time lag. However，the derivative process employed in these two papers was too tedious to be applied effi?ciently to a vehicle system analysis. In Refs.［25-26］，a non-geometric approach was applied to de?fine the spectral characteristics of non-stationary pro?cesses in the response analysis of a simple oscilla?tor，with only the first three non-stationary spectral characteristics of the response being considered.Marbato and Conte［27］pointed out that the approach to describe non-stationary random processes was not unique，and extended definitions of spectral characteristics of non-stationary processes from realto complex-valued functions to more adequately model the processes.

In summary，many studies have been devoted to understand the responses of suspension systems subjected to road excitations and most of these works were based on the assumption of stationary ergodic random vibration processes in which frequency do?main analysis is sufficient.However，for non-station?ary random vibration analysis，statistical characteris?tics must be examined in both the time and frequency domains.In this paper，the response of a half-car sus?pension system to non-stationary road excitations was examined for different acceleration cases. Linear time variable method was applied to transfer the road stationary random process in the spatial domain to the non-stationary inputs in the time domain. The vertical acceleration of suspension barycenter and the pitch angular acceleration were obtained by the pre?cise time integration method（PTIM）. The response vibration level with indicator as root mean square（RMS）values varying with time was illustrated and finally the dynamic frequency domain（DFD）charac?teristics were performed using time-frequency analy?sis method.

1 General Description of Responses of Vehicle Suspension System to Road Excitation

In this paper，we consider the vehicle suspen?sion system as a linear， multi-degree-of-freedom（MDOF）vibration system. Since the system is ex?cited by non-stationary random excitations，the re?sponses are also non-stationary processes. In order to describe the characteristics of the non-stationary responses more adequately，it is necessary to intro?duce additional concepts in the time-frequency do?main.

Road surface irregularity is the main excitation source that causes the suspension system to vi?brate. The road profiles are stationary random pro?cesses and are usually described by the PSD in spa?tial frequency domain. In general，the frequency band of road surface irregularity does not exceed 10 m-1，as shown in Fig.1. From Fig.1，we can see that the road surface can be divided into eight classes，represented by letters from A to H，with H class representing the worst case of road surface（i.e.，this road surface is the roughest）. For linear MODF suspension systems，the resonant frequen?cies are usually in the range of 1—10 Hz. For a ve?hicle traveling at a constant speed，the response PSD of the suspension system is the product of the PSD of road surface irregularity and the square of Fourier transform of frequency response function（FRF）of the vibration system. In this case，the resonant frequency band of responses is compara?tively stable. However，when the vehicle travels at varying speeds，the excitation spectrum will be rep?resented in both time and frequency domains. The responses are non-stationary signals and hence the peak band，called the dynamic frequency domain in this paper，is also varying，as shown in Fig.1. Ve?hicle ride comfort and handling performances can be improved efficiently by attenuating the vertical vibration.

Fig.1 Vibration responses of vehicle suspension system subjected to different excitations including both stationary and non-sta?tionary random processes

2 Modeling of Vehicle Suspension and Road Excitation

A half vehicle model is employed to describe the ride performance. It is easier to compute than a full car model，and also capable of modeling the pitch angular acceleration and the coherence charac?teristics between the front and the rear wheels.When a vehicle travels at varying speeds，its verti?cal acceleration，suspension displacement，and dy?namic tire loads are much more complex than those in the constant speed case. The vehicle mod?el is illustrated in Fig.2. Table 1 shows the values of the parameters for the system considered in this study.

Fig.2 Half vehicle model with 4-DOF

Table 1 Parameters and values of suspension system

2.1 Road surface excitation

The loads at the front and the rear wheels gen?erated by road surface excitations are the same in the spatial frequency domain，but there is a certain time difference in time domain.

The road surface irregularity is a stationary ran?dom process but the excitation to each wheel is a non-stationary random process when the vehicle speed is varying. It represents a linear time varying system with a stationary random process as input and the output is a non-stationary random process.Assume that the road surface is of class C and the ir?regularity coefficient is

wheren0=0.1 m-1is a reference spatial frequency.The front wheel inputqf（t）can be obtained by em?ploying a linear relationship between the input/out?put of a vibratory system as

wherev（t）is the vehicle speed expressed as a func?tion of timet. For a vehicle with constant accelera?tiona，the speed isv(t)=v0+at，Ωc=2πnc，nc=0.01 m-1is the cutoff spatial frequency.A（t）=is the non-stationary modulation function andW（t）the stationary process.

2.1.1 Input to the front wheel

In order to get the front wheel inputqf（t），PTIM is adopted because of its high computation precision and faster efficiency［28］. We can rewrite Eq.（2）in the standard form of PTIM as

whereH=-v(t)ΩcandR=A(t)W(t). The numerical solution to Eq.（3）includes two parts：The homogeneous and particular solutions. The homogeneous solution may be written in the form of

whereqf（0）is the initial condition. If time interval Δtwas determinated，the exponential matrixeH?iΔtin the equation may be easily obtained［29-30］. Next，we need to calculate the particular solution

The above integral can be evaluated using the 3-point Gauss-Legendre integral formula given by

whereξiis called the Gauss integration point，Xithe integration coefficient，ando(f) the residual item of error.Suppose that

Substituting Eqs.（6—7）into Eq.（5）gives

2.1.2 Input to the rear wheel

As mentioned above，the time lagτbetween the two wheel inputs is constant when the driving speed does not vary，butτis a function time when the vehicle speed varies. The time lagτcan be ob?tained by dividing the wheel baseLby the speedvas

Since the road surface irregularity is de?scribed in the spatial domain while the excitation to the wheels is expressed in time domain，the excitation at timetcan be expressed by the road roughness ats（t）［31］. Under the premise that the front input was abtained，the rear input could be expressed by

This equation cannot be transformed by FFT but can be expanded through Taylor series expan?sion. By omitting the higher order items，we can get（omiting the second order itme）

Taking the derivative of both sides of Eq.（11），we obtain

Given

we can get the rear wheel input with the front wheel input and instantaneous speed of

2.1.3 Validation of PTIM simulation on a cantilever beam

Compared with other methods，the responses of a cantilever beam subjected to different types of excitation are analyzed to verify the reliability and accuracy of PTIM. The geometry and input-output points setting of the cantilever beam are shown in Fig.3. The length of the beam is 1 000 mm，and the excitation point and measurement point are set at 300 mm and 700 mm from the fixed end，re?spectively. On one hand，the analytical responses at the measurement point subjected to the excita?tions including sinusoidal or random signals can be obtained exactly through vibration theory. On the other hand，the responses can also be calculated by finite element analysis method. In this section，sev?eral methods such as Newmark，Wilson-θand NPIM are selected to compare with the analysis re?sults.

Fig.3 Geometrical parameters of cantilever beam model

To avoid the resonance case，it is better to cal?culate the natural frequencies of the cantilever beam first，as shown in Table 2. The values are exactly according to analytical calculation.

Table 2 The first five natural frequencies of the beam

Fig.4 Responses at 4 Hz excitation and detail illustration

We use two different sinusoidal signals at 4 Hz and 23.9 Hz to excite the beam，and the responses are shown in Fig.4 and Fig.5，respectively. From the figures we can conclude that PTIM has an accu?rate calculation with the analytical results. No mat?ter in low-frequency or high-frequency domain，it has higher accuracy than the other two methods，which can meet the requirements of input and output calculation in vehicle dynamics.

Fig.5 Responses at 23.9 Hz excitation and detail illustration

2.2 Governing equations of motion

According to dynamic law，the equation of mo?tion of the vehicle system shown in Fig.2 is

The items’meanings in Eq.（15）are shown in Table 1. The relationships amongza，zb，zc，andφcare

Next，the state vector is set as

and the output vector is

PTIM is still to be used and the state equations may be written as

where the terms in the above equations are defined as

3 Simulation and Discussions

The suspension system responses of vehicle traveling at variable speeds may be calculated by the following steps.

Step 1Determine the excitation signals of the front and the rear wheels according to running condi?tions，road surface irregularity，and the structural parameters of vehicle.

Step 2Establish the half vehicle model and obtain the vibration performance.

Step 3Use the numerical integration ap?proach and PTIM to simulate the interest responses of suspension，mainly including the vertical accelera?tion of suspension barycenter and the proposed pitch?ing angular acceleration.

Step 4Analyse the response by time-frequen?cy method，and focus on the DFD’s varying trend with speeds，obtaining the DFD characteristics of suspension system under non-stationary random ex?citation.

The PSD of C class road surface is shown in Fig.6（a），and the altitude data in spatial amplitude domain may be obtained using trigonometric series method as shown in Fig.6（b）.

In this paper，the vehicle starts moving at ini?tial state of rest with an acceleration of 2 m/s2. With the traveling time of 14 s，the traveling distance is 196 m. The excitation signals of the front and the rear wheels in time domain are exhibited in Fig.7.Fig.7 shows that there is a time lag of the two in?puts，and the lag is a variable that has an absolute relationship with the vehicle velocity.

Fig.6 Illustration of road surface irregularity

Fig.7 Excitation of the front and the rear wheels in time do?main

Fig.8（a）shows the vertical acceleration of sus?pension barycenter，and Fig.8（b） represents the pitching angular acceleration.Xaxis represents the simulation time. On one hand，we can draw the con?clusion from the figures that the level of vibration in?creases with the increased speed，especially after 8 s. When the speed is 57.6 km/h，the amplitude in?creases obviously. On the other hand，if the vehicle rides at the varying speeds，no matter the vertical acceleration of suspension barycenter nor the pitch?ing angular acceleration would be stationary process as illustrated distinctly in Fig.8（c）and Fig.8（d）.The comparison between the two cases implies that the vibration response level of vehicles increases continuously due to the non-stationary random exci?tation，so that some new control algorithms should be considered to deal with this situation. This would be the follow-up work of this paper.

Fig.8 Responses of suspension systems

In order to compare the responses under differ?ent traveling conditions，three different acceleration cases are used for simulation and the RMS of re?sponses are shown in Fig.9. The results show that both the vertical acceleration of suspension barycen?ter and pitching angular acceleration have the in?creasing trend，and the RMS are different in the cas?es which implies that the responses of vehicle sus?pension system may be affected by the traveling con?ditions.

Fig.9 RMS of responses under different accelerating condi?tions

The former study only gives the response anal?ysis of suspension systems in time domain，but the analysis in frequency domain is not presented. How?ever，the latter attracts more interest. Take the ver?tical acceleration response for time-frequency analy?sis and the time-frequency spectrum diagram is giv?en in Fig.10. Fig.10 clearly presents the response varying tendency in both time and frequency do?mains. The yellow region in the red parallelogram represents the DFD of the responses. The initial central frequency is about 8 Hz and moves toward the lower frequency region continually with the in?creased speed. The result shows that the response of suspension in this case may not be stationary pro?cess，and the DFD has changed obviously.

Fig.10 Time-frequency analysis result of vertical accelera?tion of suspension barycenter

The frequencies corresponding to the peak val?ues of response spectrum is bound to shift from high to low frequency because of the variable-speed trav?eling. That is to say，in non-stationary situation，we have to use time-frequency analysis method to study the responses of suspension systems. According to Fig.10，at the starting moment，the frequency corre?sponding to the peak values of response spectrum is 8 Hz，and it changes as the vehicle speeds up. How?ever，the frequencies still belong to the sensitive band of human’s body. The results show that the concept of DFD is of great significance for the analy?sis of vehicle suspension response，and the suspen?sion control with ride comfort as the goal should be based on the finite frequency domain.

4 Conclusions

The dynamic response analysis of vehicles un?der non-stationary random excitation condition has become the research hotspot. Based on the human’s sensitive to vibration in different frequency range，some conclusions can be drawn from our study.

（1）The paper proposes a new concept named DFD analysis which would be a new innovation in non-stationary random vibration research field. It is the basement of non-stationary random vibration control for vehicles traveling in complex cases.More works should be conducted with this concept and advanced algorithms would be developed by our research team in the future.

（2）The excitations to the front and the rear wheels in time domain are derived and the digital process is acquired using PTIM. This method has more precise advantages compared with the tradi?tional numerical methods such as Newmark and Wil?son-θ.

（3）By analysing the responses of vertical ac?celeration at the barycenter of suspension and pitch?ing angular acceleration of suspension in three differ?ent accelerations，1 m/s2，2 m/s2，3 m/s2，the RMS tendency are obtained.

（4）In addition，the time-frequency spectrum diagram is obtained by using time-frequency analy?sis method，and the results show that the RMS in?creases with the increase of speed and the range of interest of DFD moves to the low frequency region，which can provide an important reference for the de?sign and application of active suspension.