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

2.1. 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. 1The 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 $m_s$, and the suspension system connects the un-sprung mass $m_u$. The displacements of the sprung and un-sprung masses are denoted by $x_s$ and $x_u$, respectively. The suspension consists of a spring $k_s$ and an equivalent damping coefficient $c_{eq}$, provides support to the bus body. The bus is assumed to travel at a constant speed $v$ on a road surface with a road profile $y$. The stiffness between the un-sprung mass and the road is the tire stiffness denoted by $k_t$. The sprung mass is 2000 (kg), and the un-sprung mass is 250 (kg). The spring stiffness $k_s$ is 8.10⁴ (N/m), corresponding to the natural frequency of the bus suspension system of about 1 (Hz) [1]. The tire stiffness $k_t$ is 1.10⁶ (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 $\dot{x}$ and $\ddot{x}$ denote the velocity and acceleration. The subscripts $t$ and $s$ represent the tire and suspension, respectively. In addition, we introduce the shock absorber force $F_c$, which is linearly related to the relative velocity of the suspension system. The force is expressed by:

where $dz$ is the relative velocity of the suspension, defined by:

The equivalent damping coefficient $c_{eq}$ is related to the damping ratio $\xi$ by $c_{eq}=2\xi \sqrt{m_s{k}_s}$ 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 $c_0=$7500 (Ns/m), corresponding to $\xi =$0.3.

2.2. 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 $t$, as in Eq. (4):

The considered road profile is characterized by the road bump length $d_1$, the road bump height $d_2$, and the velocity $v$, see Fig. 2. The road bump is 1 (m) in length and 0.005 (m) in height. The velocity $v$ 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).

Fig. 2The road profile described by the sinusoidal harmonic function [10]

2.3. 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 $dz$ of the suspension system. The damping coefficient is fixed at $c_0=$7500 (N.s/m).

The second case is called Nonlinear Symmetric (NS) shock absorber. This type is characterized by a knee point $d{z}_c=$0.05 (m/s), which separates the shock absorber response into two domains: low and high speeds. The damping force is governed by $\kappa$ parameter at the low speed and by $\lambda$ parameter at high speed. For the NS type, the shock absorber force follows the following formulas:

We choose $\kappa =\text{2}$ and $\lambda =\text{1}$, 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 $c_0$.

The third case is Linear Asymmetric (LA) shock absorber. The shock absorber force follows the formula:

The asymmetric coefficient $e_D$ 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 $c_{eq}$ is taken to be equal to the initial one c₀, when the relative velocity, $dz=$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 $e_D$. The slope of the damping force is altered at the knee-point value $d{z}_c$ with the coefficients $\kappa$ and $\lambda$. Similarly, initially, when $dz=$0, the equivalent damping coefficient $c_{eq}$ equals to $c_0$. 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 $dz$ is plotted against the shock absorber force $F_c$. 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 $dz=$0, resulting in a bilinear curve. The second and fourth models have three knee points at $dz=-d{z}_c$, $dz=$0, and $dz=d{z}_c$, breaking down the velocity range into four domains, namely, $dz=<-d{z}_c$, $-d{z}_c\le dz<$0, $0<dz\le d{z}_c$, and $dz>d{z}_c$. In each domain, the damping force is proportional to the relative velocity of the suspension.

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

2.4. 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.

3.1. Gain response of SDD

The gain responses of maximum relative displacement $G_{SDDmax}$ 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 $G_{SDDmax}$ 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 $f_{n1}$ of about 1 (Hz), and the second is the tire’s natural frequency $f_{n2}$ 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 $\left|dz\right|<d{z}_c$, corresponding to the low-frequency domain. In addition, the NS shock absorber also has a lower $G_{SDDmax}$ at the second peak. The gains are lower about 33 % at the first peak and 17 % at the second peak. On average, the $G_{SDDmax}$ of the NS shock absorber is lower by about 12 %.

Fig. 4The calculation flowchart

As for the cases of asymmetric shock absorbers, the obtained $G_{SDDmax}$ and $G_{SDDmin}$ 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 $G_{SDDmax}$ 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 $G_{SDDmax}$ are lower about 56 % at the first peak and 40 % at the second peak. With the obtained $G_{SDDmin}$, 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 $G_{SDDmin}$ of the LA case is larger than the NA case. In the entire observed frequency range, the $G_{SDDmin}$ 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 $G_{SDDmax}$ and $G_{SDDmin}$ 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.

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

Fig. 6The GSDDmax and GSDDmin of the LA and NA shock absorbers in the frequency domain

3.2. Gain response of SDDS

The obtained results of $G_{SDDS}$ 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 $G_{SDDS}$ of the LS type is similar to the LA type. The obtained $G_{SDDS}$ of the NS type is identical to the NA type. Therefore, the $G_{SDDS}$ 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 $G_{SDDS}$ 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 $G_{SDDS}$ 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 $G_{SDDS}$ of the NS and NA shock absorbers are lower about 12 % and 13 % than the LS and LA shock absorbers, respectively.

The obtained $G_{SDDD}$ 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.

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

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

3.3. Gain response of BVA

The obtained $G_{BVA}$ are calculated by Eq. (11), in the frequency domain for the four types of shock absorbers, in Fig. 8. The $G_{BVA}$ for the four types of the shock absorbers peak at the first natural frequency $f_{n1}$. From the peak, the $G_{BVA}$ drop quickly with the frequency. At the first natural frequency, where the peaks of the $G_{BVA}$ occur, the maximum of $G_{BVA}$ 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, $G_{BVA}$ of the NS and NA shock absorber types are higher in the frequency range of 3-7 (Hz). Similar to the $G_{SDDS}$, the $G_{BVA}$ 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 $G_{BVA}$ of the four types of shock absorbers are presented. The table shows the maximum and the mean of the $G_{BVA}$. 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 $G_{BVA}$. The NA case has a 27 % the maximum of $G_{BVA}$ lower. However, the mean of $G_{BVA}$ is higher for the NS case (13 %) and the NA case (7 %), respectively. The nonlinear shock absorbers significantly reduce the peak of the $G_{BVA}$. Therefore, the combination of the nonlinear and asymmetric characteristics, as in the NA case, simultaneously reduces the peak of the $G_{BVA}$ at the low frequency. And the mean of the $G_{BVA}$ in the type of NA shock absorber is not also much increased in the observed frequency range.

3.4. Tire dynamic load – TDL

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

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