Influence of a nonlinear asymmetric shock absorber on vibration of a bus subjected to harmonic excitation

. The main focus of paper is to get full understanding of four different types of shock absorbers characteristics and their effects on the vertical evaluation indexes of a bus subjected to harmonic excitation. The quarter car with two degree of freedom (2DOF) model was employed to calculate the vertical evaluation indexes. The bus is assumed to travel at a constant velocity on a road surface with a profile following a sinusoidal function. The four types of the shock absorber are Linear Symmetric (LS), Nonlinear Symmetric (NS), Linear Asymmetric (LA), and Nonlinear Asymmetric (NA). As for the LS type, the damping force is a linear function of the relative velocity, and the damping force is symmetric for the conditions of the positive and negative relative velocities. Same as for the LA type, it means that the damping force is linear, however it is asymmetric and differentiated according to the suspension state of stroke (compression or extension). As for the NS type, the damping force function is symmetric, nonlinear, differentiated according to the magnitude of the relative velocity. As for the last NA type, the damping force function is also nonlinear and differentiated according to the magnitude of the relative velocity, but it is also asymmetric. The obtained evaluation indexes of the relative displacement of the bus suspension, acceleration of the bus body, and the dynamic tire load within the frequency range of 0 and 25 (Hz), the common frequency range of the bus in operations. The results suggest that the NA shock absorber type is more effective in reducing the suspension dynamic deflection stroke, improving the road holding and maintaining the ride comfort. The systematic assessments of the shock absorber characteristics should guide interested readers in selecting the most appropriate damping coefficient.


Introduction
With the increasing requirements, especially for the ride comfort of buses, which are manufactured and assembled in Vietnam.Bus ride comfort was carried out by the multibody dynamic model to help the bus designers and manufacturers in order to improve and harmonize oscillatory comfort on the whole vehicle platform [1].The oscillatory model with 10DOF was employed to investigate the effects of bus vibrations on the comfort for the space of a driver, passenger in the middle part of the bus and passenger in the rear overhang [2].The determination of suspension parameters is very important, because it is directly related to ride comfort, safety and road holding during the operation of vehicles.The evaluation indexes of the ride comfort, workability, and operational safety of the bus suspension system were studied under the random excitation, with the change of the stiffness and damping values in the investigated domain [3,4].It shows that shock absorbers represent a critical element of a suspension, in order to better understand the influence of damping parameters on ride comfort of vehicles, many studies have been carried out [5,6].The multibody model of a larger bus was employed, in which a dynamic model for the MR (magnetic-rheological) damper was developed, a comparison to the traditional passive damper model showed the improvement [5].The 2DOF quarter car model with LA shock absorber, in which the damping ratio value changing with the stroke was used to analyze the vertical dynamics of an electric vehicle.The road profile was in type of triangular shape according to GB/T 4970-2009 standard, the vehicle moves over the bump with different speeds [6].The influence of LA shock absorber on the vehicle's ride comfort, was studied with the single bump effect, by using the 2DOF quarter car and the 4DOF half car models [7].The influence of LA shock absorber under the effect of the harmonic excitation was studied to improve the vehicle's ride comfort, by the 2 DOF quarter car model [8].Recently, an investigation of a bus's ride comfort with the two types of LS and LA shock absorbers was carried out, in which the quarter car model was employed, subjected under transient and random road excitations [9].
In Vietnam's practice, the types of shock absorbers have been recently used in the design and manufacture of buses are of symmetric/asymmetric and linear/nonlinear ones.This paper is conducted to get full understanding of the influences of the 4 types of common shock absorbers' characteristics on the bus vibration.The vibration evaluation indexes for a typical bus have been specifically identified corresponding to each type of shock absorber characteristic, in which the initial damping coefficient  is chosen as the same for all 4 types of shock absorbers.The evaluation indexes in terms of ride comfort, suspension working space, and road holding are determined under the excitation of the harmonic function.The obtained results are presented and analyzed systematically to help understand the nature of the influences of 4 common types of shock absorbers characteristics on the typical bus vertical dynamic evaluation indexes in the frequency domain.Especially, the peak values of the obtained gain responses at the corresponding resonance values of the natural frequencies of the suspension system and the tire, respectively.The type of NA shock absorber gives the best results in improving suspension performance, road holding, and maintaining ride comfort for the buses, which are being designed and manufactured in Vietnam.

Simulation model
This research intends to evaluate four types of shock absorbers in the suspension system and their effects on ride comfort, suspension working space and road holding of the bus.We develop a bus suspension model, compute the responses of acceleration, relative displacement and tire dynamic load (TDL), and establish its characteristics to understand the four types of shock absorbers that affect the bus vertical dynamic evaluation indexes.The bus model is simplified as a system consisting of masses, springs, and a damper as shown in Fig. 1.

Fig. 1.
The quarter car 2DOF model [10] The symbols, and the meanings of the quarter car 2DOF model parameters, are given in Table 1.In the Fig. 1, the bus body is denoted by the sprung mass  , and the suspension system connects the un-sprung mass  .The displacements of the sprung and un-sprung masses are denoted by  and  , respectively.The suspension consists of a spring  and an equivalent damping coefficient  , provides support to the bus body.The bus is assumed to travel at a constant speed  on a road surface with a road profile .The stiffness between the un-sprung mass and the road is the tire stiffness denoted by  .The sprung mass is 2000 (kg), and the un-sprung mass is 250 (kg).The spring stiffness  is 8.10 4 (N/m), corresponding to the natural frequency of the bus suspension system of about 1 (Hz) [1].The tire stiffness  is 1.10 6 (N/m).
The set of differential equations describing the 2DOF vertical dynamic model, is written in matrix form as shown in Eq. (1) as, [10]: where, the matrices included in the general dynamic equation are presented as: The variables  and  denote the velocity and acceleration.The subscripts  and  represent the tire and suspension, respectively.In addition, we introduce the shock absorber force  , which is linearly related to the relative velocity of the suspension system.The force is expressed by: where  is the relative velocity of the suspension, defined by: The equivalent damping coefficient  is related to the damping ratio  by  = 2   where the damping ratio value is in range of 0.2-0.4,[11].In the present study, we focus on four different damping characteristics.The initial value of the damping coefficient is  = 7500 (Ns/m), corresponding to  = 0.3.

Harmonic excitation
We assume the bus is moving at a constant velocity v on the surface of a road whose profile follows a sinusoidal harmonic function of time , as in Eq. ( 4): The considered road profile is characterized by the road bump length  , the road bump height  , and the velocity , see Fig. 2. The road bump is 1 (m) in length and 0.005 (m) in height.The velocity  is chosen to be in range of 1-100 (km/h) corresponding to the frequency of excitation in the range of 0.28-27.78(Hz), determined by Eq. ( 4).

The four types of shock absorbers
Four types of shock absorbers are investigated.The parameters of shock absorber models are given in Table 2.The first case is the Linear Symmetric (LS) shock absorber.In the LS type, the generated damping force is proportional to the relative velocity  of the suspension system.The damping coefficient is fixed at  = 7500 (N.s/m).
The second case is called Nonlinear Symmetric (NS) shock absorber.This type is characterized by a knee point  = 0.05 (m/s), which separates the shock absorber response into two domains: low and high speeds.The damping force is governed by  parameter at the low speed and by  parameter at high speed.For the NS type, the shock absorber force follows the following formulas: We choose  = 2 and  = 1, suggesting the damping strength at the low speed is twice as big as at high speed.The damping coefficient in the high-speed region is  .
The third case is Linear Asymmetric (LA) shock absorber.The shock absorber force follows the formula: The asymmetric coefficient  is taken as 0.4 [12], corresponding to the ratio of the compression and extension strokes of 30 to 70.Initially, the equivalent damping coefficient  is taken to be equal to the initial one c 0 , when the relative velocity,  = 0.
Finally, the fourth case is Nonlinear Asymmetric (NA) shock absorber.To some extent, the case is a combination of Case 2 and Case 3. The damping force changes proportionally but asymmetrically with the coefficient  .The slope of the damping force is altered at the knee-point value  with the coefficients  and .Similarly, initially, when  = 0, the equivalent damping coefficient  equals to  .Analytically, the damping force follows the following composite functions: In Fig. 3, a comparison is graphically made for the four types of shock absorbers.The relative velocity  is plotted against the shock absorber force  .For the first case, Linear Symmetric shock absorber, the damping force increases linearly with the velocity.The linear relationship is also present for the third model but with a knee-point at  = 0, resulting in a bilinear curve.The second and fourth models have three knee points at  = − ,  = 0, and  =  , breaking down the velocity range into four domains, namely,  =< − , − ≤  < 0, 0 <  ≤  , and  >  .In each domain, the damping force is proportional to the relative velocity of the suspension.

Evaluation indexes
Calculations are performed for each value of applied excitation frequency.The parameters of Suspension Dynamic Deflection (SDD), acceleration of the sprung mass namely Body Vibration Acceleration (BVA), and Tire Dynamic Load (TDL) are determined in the time domain, when in steady-state of the harmonic responses with the same frequency as the applied excitation frequency.The amplitude values of the obtained results in the time domain are determined at steady-state.Then, the evaluation indexes are determined in the frequency domain with the frequency range of 0-25 (Hz), a normal range for the vehicle operations [10].Especially at the resonance values corresponding to each type of damper, can be characterized most meaningfully and systematically.

Gain response of SDD
The relative displacement of the suspension to evaluate the Suspension Dynamic Deflection (SDD).The gain response of maximum and minimum relative displacement are defined as the corresponding ratio of the maximum and minimum relative displacements to the amplitude of the road excitation,  , [10], following the Eq. ( 8), and Eq. ( 9), respectively: It is easy to see that the working stroke of the suspension system with asymmetric shock absorber is asymmetric around the initial equilibrium position.To be able to compare the Suspension Dynamic Deflection Stroke (SDDS) parameter of different types of shock absorbers, the gain response of SDDS is determined by Eq. ( 10):

Gain response of BVA
The gain response of the Body Vibration Acceleration (BVA),  is defined as the ratio of the value of the sprung mass acceleration amplitude variation over time to the applied excitation acceleration amplitude.Mathematically, the  is calculated by Eq. ( 11):

Tire dynamic load -TDL
The maximum tire dynamic load  is defined as the ratio of the maximum dynamic force applied to the tire to the static reaction load [6].The  is computed by Eq. ( 12):

Calculation flowchart
For each type of the shock absorber, calculations are performed for each frequency of the harmonic excitation, the corresponding harmonic responses of SDD, BVA and TDL in the time domain are determined.And then, the corresponding amplitudes of these obtained steady-state harmonic responses are determined.Therefore, for each frequency of applied excitation, the evaluation indexes of SDD, BVA and TDL are determined by the Eqs.( 8)- (12).Next, do the same calculation for the next remaining excitation frequency values in the observed domain.The evaluation indexes are identified and analyzed in the observed frequency domain.The calculation flowchart is shown in Fig. 4.

Gain response of SDD
The gain responses of maximum relative displacement  are calculated by Eq. ( 8) in the frequency domain, for the two cases of LS and NS, Fig. 5.Both of the cases, the obtained  are symmetrical to the zero frequency in the domain.The responses have two peaks associated with the two resonances of the system.The first is the suspension's natural frequency  of about 1 (Hz), and the second is the tire's natural frequency  of about 10 (Hz).It can be seen that the first peak of the NS shock absorber, which occurs at a low frequency, is lower than that of the LS shock absorber.The phenomenon happens because the NS case has a more significant damping coefficient when || <  , corresponding to the low-frequency domain.In addition, the NS shock absorber also has a lower  at the second peak.The gains are lower about 33 % at the first peak and 17 % at the second peak.On average, the  of the NS shock absorber is lower by about 12 %.As for the cases of asymmetric shock absorbers, the obtained  and  are calculated by the Eqs.(8)(9), in the frequency domain, Fig. 6.Unlike the previous cases, the gain responses are not symmetric to the zero-frequency axis.In this case, the damping coefficient value depends on the shock absorber states, namely, tension or compression.The value of the damping coefficient is smaller in compression than in tension.The  of the NA case is smaller about 133 % on an average than that of the LA case in the entire observed frequency domain.And, the  are lower about 56 % at the first peak and 40 % at the second peak.With the obtained  , at the 2 peaks of the LA case are lower than the NA case.However, in the frequency domain outside the 2 resonant frequencies, the  of the LA case is larger than the NA case.In the entire observed frequency range, the  of the NA case is smaller about 15 % on an average than that of the LA case.With both types of asymmetric shock absorbers, the obtained  and  all have negative values, with the applied excitation greater than 5 (Hz), which indicates that the suspension system is working mostly in the state under compression during the harmonic applied excitation.Due to the asymmetrical characteristic of the shock absorbers, the suspensions move up and down around a position lower than their initial equilibrium position.Therefore, the asymmetry around the zero axis (initial equilibrium position) of the suspension relative displacement corresponds to two types of asymmetric shock absorbers, LA and NA, showing the influence of asymmetric properties of displacement.

Gain response of SDDS
The obtained results of  for all four cases of damping characteristics are calculated by Eq. ( 10) and shown in the frequency domain in Fig. 7.It shows that the obtained  of the LS type is similar to the LA type.The obtained  of the NS type is identical to the NA type.Therefore, the  depends only on the linear or nonlinear properties of the shock absorbers, but completely independent of the asymmetrical properties.At first and second natural frequencies, the peak values of  with the NS type are smaller, respectively, by 33 % and 17 % compared to the LS type.With the asymmetric types of shock absorbers, the peak values of  with the NA type are smaller by 31 % and 18 % compared to the LA type, at the first and second natural frequencies, respectively.On average, the  of the NS and NA shock absorbers are lower about 12 % and 13 % than the LS and LA shock absorbers, respectively.
The obtained  clearly shows that the working stroke of the suspension system does not depend on the asymmetrical characteristic but only on the nonlinear characteristic of the shock absorbers.In addition, the asymmetrical characteristic of the shock absorbers only affects the equilibrium position of the suspension working stroke.

Gain response of BVA
The obtained  are calculated by Eq. ( 11), in the frequency domain for the four types of shock absorbers, in Fig. 8.The  for the four types of the shock absorbers peak at the first natural frequency  .From the peak, the  drop quickly with the frequency.At the first natural frequency, where the peaks of the  occur, the maximum of  are usually in the range of 1.0-2.5, and is the range of interest in the design of the automotive suspension systems [11].Within the range, the figure shows that the NS and NA shock absorber types have significantly lower peaks than those of LS and LA shock absorber types.However,  of the NS and NA shock absorber types are higher in the frequency range of 3-7 (Hz).Similar to the  , the  depends only on the linear or nonlinear properties of the shock absorbers, but not almost on the asymmetrical properties.
In Table 3, the characteristics of the  of the four types of shock absorbers are presented.The table shows the maximum and the mean of the  .The relative deviation in the case of LS shock absorber type is also explained.Compared to the LS case, the NS case has a 33 % lower the maximum of  .The NA case has a 27 % the maximum of  lower.However, the mean of  is higher for the NS case (13 %) and the NA case (7 %), respectively.The nonlinear shock absorbers significantly reduce the peak of the  .Therefore, the combination of the nonlinear and asymmetric characteristics, as in the NA case, simultaneously reduces the peak of the  at the low frequency.And the mean of the  in the type of NA shock absorber is not also much increased in the observed frequency range.

Tire dynamic load -TDL
The obtained  are calculated by Eq. ( 12) for the four types of shock absorbers in the frequency domain, Fig. 9.The  reaches its peak at the second resonant frequency or the natural frequency of the tire.It is easy to see that the  is similar for symmetric and asymmetric shock absorbers, but it is different for linear and nonlinear shock absorbers.In the other words, the  depends only on the nonlinearity and not on the asymmetry of the shock absorbers.In the frequency range of 7-15 (Hz), the  of linear shock absorbers are higher than those of nonlinear shock absorbers.Therefore, the nonlinear shock absorbers increase road holding because the  has a lower value in this range of frequency domain.However, in the lower frequency range, the  are vice versa.The peaks of the  of the nonlinear shock absorbers are lower, about 15 %, than those of the linear shock absorbers.On average, the  of the NS and NA shock absorbers are slightly smaller about 3 % than the LS and LA shock absorbers.

Conclusions
In this research paper, we study the effects of the characteristics of four types of shock absorbers on the vertical dynamic behaviors of a bus.The dynamic quarter car 2DOF model is employed, subjected to harmonic excitation.The characteristics of four types of the shock absorbers, namely, Linear Symmetric, Nonlinear Symmetric, Linear Asymmetric, and Nonlinear Asymmetric are investigated.In particular, we investigate the effects shock absorber types on the evaluation indexes of the relative displacement, acceleration, and tire dynamic force.The ratio of output and input amplitudes namely the 'Gain response or Transmissibility' represents the inputoutput relationships of evaluation indexes are studied and presented in the frequency domain, in the normal common range for the vehicle operations (0-20 (Hz)).The vertical dynamic behaviors, especially at the resonance region, of the bus could be characterized most meaningfully.The influences of shock absorber characteristics on the bus vibration are systematically full understanding, and the following conclusions are obtained: 1) The gain responses of the Suspension Dynamic Deflection Stroke ( ) only depend on the linear or nonlinear properties of the shock absorbers, but completely independent of the asymmetrical properties.The average values of  of the NS and NA shock absorbers are lower about 10 % than the LS and LA shock absorbers.
2) The gain responses of the Body Vibration Acceleration ( ) are significantly reduced at the peaks with the types of nonlinear shock absorbers, namely NS and NA.The obtained  also only depend on the linear or nonlinear properties and almost independent on the asymmetrical properties of the shock absorbers.In addition, the peak of the  significantly decreases at the low frequency with the combination of the nonlinear and asymmetric characteristics of the shock absorber, namely NA, and the mean value of the  is not also much increased in the entire vehicles' common frequency range.
3) The maximum tire dynamic loads ( ) depend only on the nonlinearity and not on the asymmetry of the shock absorbers.At the peak performance, the nonlinear shock absorber types have 15 % lower  than those of the linear shock absorber types.

Fig. 3 .
Fig. 3.The relationship between the damping force  and the relative velocity  for the four types of shock absorbers

Fig. 5 .Fig. 6 .
Fig. 5.The  of the LS and NS shock absorbers in the frequency domain

Fig. 7 .
Fig. 7.The  of the four shock absorbers in the frequency domain

Fig. 8 .
Fig. 8.The  of the four types of the shock absorbers in the frequency domain

Fig. 9 .
Fig. 9.The  of the four types of the shock absorbers in the frequency domain

Table 1 .
The

Table 2 .
Shock absorber model parameters Symbol Description Value  Initial damping coefficient [Ns/m] 7500  Knee-point [m/s] 0.05  Slope of low-speed damping coefficient 2  Slope of high-speed damping coefficient 1  Asymmetric coefficient 0.4

Table 3 .
The maximum and mean values of  for the four types of the shock absorbers