Abstract
Automobilerobot (selfdriving automobile) is being researched and developed vigorously. When the automobilerobot is moving on the road surface, the low frequency vibration excitation not only influences the ride comfort of the automobilerobot but also strongly affects the durability of the vehicle’s structures. To research the automobilerobot’s vibration in the low frequency region, a dynamic model of the vehicle is established to calculate the vibration equations in the time region. Based on the theory of the Laplace transfer function, the automobilerobot’s vibration equations in the time region are transformed and converted to the vibration equations in the frequency region. Then, the effect of the design parameters and operation parameters on the characteristic of the automobilerobot’s accelerationfrequency is simulated and analyzed to evaluate the ride comfort as well as the durability of the automobilerobot’s structures in the frequency region. The research results show that the design parameters of the stiffness, mass, and road wavelength remarkably affect the characteristic of the automobilerobot’s accelerationfrequency. To reduce the resonant amplitude of the accelerationfrequency in the vertical and pitching direction of the automobilerobot, the stiffness parameters of the automobilerobot's and tires should be reduced while the mass of the automobilerobot’s body should be increased. Additionally, the road’s roughness also needs to be decreased or the road’s quality needs to be enhanced to reduce the resonant amplitude of the automobilerobot’s accelerationfrequency.
1. Introduction
To increase driver comfort, vehicle manufacturers are developing selfdriving car systems, also known as automobilerobot. The vibration isolation structure of automobilerobot is similar to the vibration isolation structure of the traditional automobile. The isolation systems of the automobilerobot have been used to reduce the vibration excitations from the road surface transmitted to the automobilerobot’s body. In the design process of the vehicle’s suspension systems, the structures of the suspension system were designed by the spring and damper with the stiffness parameter and damping parameter. The study showed that these parameters greatly affected the ride comfort of the vehicle [1]. In order to enhance the ride comfort of the vehicle or automobile, these design parameters were optimized by the genetic algorithm [23]. By searching for the best stiffness and damping parameters for the automobile’s suspension systems, the automobile’s ride comfort has been improved in comparison with the passive suspension systems. However, the automobile’s ride comfort was still low under the high speeds of the automobile’s moving or the automobile’s moving on the poor road surface roughness. Therefore, the automobile’s suspension systems were improved by using the control damping forces of semiactive suspension systems [45] or semiactive air suspension systems [6]. The research results showed that with the control damping forces of the semiactive suspension systems used, the automobile’s ride comfort was better than that of the automobile’s optimal suspension systems under different operation conditions. However, the research also indicated that the control performance of the semiactive suspension systems strongly depended on the control method and control rule of the algorithm programs [78]. To enhance the control performance, advanced control methods using the Adaboost algorithm and machine learning were applied [910]. In the above studies, the dynamic model was established to calculate the vibration equations of the automobile. Then, these vibration equations were built and simulated to compute the automobile’s acceleration responses in the time region. The root mean square values of these acceleration responses were then computed to assess the automobile’s ride comfort based on ISO 26311:1997 [11].
However, ISO 26311 showed that the ride comfort and health of the driver were also strongly affected by the vehicle’s vibration excitations in the frequency region [11], especially at the excitations in the low frequency from 0.5 to 10 Hz of the road surface when the vehicle is moving. From the random excitations of the road surface built based on ISO 8068 [12], the interaction models of the vehicle and random road surface were established and studied the vibration of the vehicle or cab in the low frequency region [1314]. Besides, the effect of the design parameters of the isolation systems on the vehicle’s vibrations in the low frequencies was also evaluated [1516]. The results indicated that the density of resonant frequencies and resonant amplitudes of automobilerobot’s accelerationfrequency response appeared very much in the low frequency region, especially at excitations from 0.5 to 4.0 Hz. This not only affected the driver’s health but also strongly affected the durability of the automobilerobot’s structures and road surfaces. Thus, the resonant frequencies and resonant amplitudes in the automobilerobot’s accelerationfrequency response in this excitation range needed to be minimized. These resonant frequencies and resonant amplitudes were directly impacted by the design parameters and operation parameters of the automobilerobots such as the stiffness, mass, speed, and road surface, etc. Therefore, the effect of the design parameters and operation parameters of the automobilerobots on the driver’s health and the durability in automobilerobot’s structures under different frequency excitations need to be researched and analyzed. However, this issue has not been considered in the existing research.
In this study, a dynamic model of the automobilerobot is established to calculate its vibration equations in the time region. Based on the theory of the Laplace transfer function [17], the automobilerobot’s vibration equations in the time region are transformed and converted to the automobilerobot’s vibration equations in the frequency region. Then, the effect of the automobilerobot’s design parameters and operation parameters on the characteristic of the automobilerobot’s accelerationfrequency is simulated and analyzed to evaluate the automobilerobot’s ride comfort as well as the durability of the automobilerobot’s structures in the frequency region. Enhancing the working performance of the automobilerobot is the goal of this study.
2. Automobilerobot’s mathematical model
2.1. Calculating the vibration equations of the automobilerobot in the time region
In order to compute an automobilerobot’s vibration equations, based on its actual structure, a 2D automobilerobot dynamics model is established and shown in Fig. 1, where four degrees of freedom of the automobilerobot including the automobilerobot body’s vertical vibration, automobilerobot’s pitch vibration, front axle’s vibration, and rear axle’s vibration are defined by $z$, $\phi $, ${z}_{1}$, and ${z}_{2}$, respectively. The mass of the automobilerobot’s body, frontaxle, and rearaxle are also defined by $m$, ${m}_{1}$, and ${m}_{2}$, respectively. The stiffness and damping parameters of the front and rear axles are also defined by {${c}_{1}$ and ${k}_{1}$} and {${c}_{2}$ and ${k}_{2}$}. The stiffness and damping parameters of front and rear tires are also defined by {${c}_{t1}$ and ${k}_{t1}$} and {${c}_{t2}$ and ${k}_{t2}$}. ${l}_{\mathrm{1,2}}$ and ${q}_{\mathrm{1,2}}$ are the distances and vibration excitations of the automobilerobot and tires.
From the automobilerobot’s dynamics model shown in Fig. 1, its vibration equations are then written by:
Fig. 1The dynamic model of the automobilerobot
In the research of the automobilerobot’s vibration, the automobilerobot’s vibration in the time region is mainly applied for assessing the automobilerobot’s comfort. However, based on ISO 26311:1997 [11], the automobilerobot’s vibration responses in the frequency region also greatly affected the ride comfort and structure in the automobilerobot’s systems. Therefore, in this study, the vibration characteristic of the automobilerobot in the frequency region will be researched and evaluated under different operation conditions of the automobilerobot.
2.2. Calculating the vibration equations of the automobilerobot in the frequency range
To establish the automobilerobot’s vibration equations in the frequency region as well as evaluate the vibration characteristic of the car in the frequency region, based on the automobilerobot’s vibration equation in the time region in Eq. (1), the Laplace transfer function [17] is then used to convert Eq. (1) in the time region ($t$) to the image function ($s$) in the frequency region with the excitation frequency of $\omega $. Herein, $\omega =2\pi f$ and $s=d/dt$.
The theory of the Laplace transfer function is described by: If a vibration function of $n\left(t\right)$ operates and depends on the variable time of $t>0$ in its operation range defined by {$a$ and $b$}, based on the method of the Laplace transfer function, the image function of $n\left(t\right)$ defined by $N\left(s\right)$ is expressed as follows:
Or:
Similarly, based on the theory of the Laplace transfer function, the derivative equations of the image function of $n\left(t\right)$, $\dot{n}\left(t\right)$, and $\ddot{n}\left(t\right)$ are also written by [17]:
From the dynamic model of the car in Fig. 1, at the initial condition of the automobilerobot moving when $t=0$, the vibration responses of the automobilerobot’s and front/rear wheel axles are equal to zero ($z\left(t\right)=0$, $\phi \left(t\right)=0$, ${z}_{1}\left(t\right)=0$, and ${z}_{2}\left(t\right)=0$). Therefore, the derivative equations of their image function at the initial condition when $t=0$ are also equal to zero ($N\left(0\right)=0$).
Based on the Laplace transfer function in Eqs. (3) and (4), the derivative equations of the automobilerobot body’s vertical vibration$z\left(t\right)$, automobilerobot body’s pitch vibration $\phi \left(t\right)$, front axle’s vibration ${z}_{1}\left(t\right)$, and rear axle’s vibration ${z}_{2}\left(t\right)$ calculated in Eq. (1) at the time region are described by the image functions ($s$) of $Z\left(s\right)$, $\mathrm{\Psi}\left(s\right)$, ${Z}_{1}\left(s\right)$, and ${Z}_{2}\left(s\right)$ in the frequency region as follows:
Thus, the automobilerobot’s vibration equation of Eq. (1) in the time region is rewritten by the automobilerobot’s vibration equation at the frequency range via the theory of Laplace functions as follows:
or:
By dividing Eq. (6) by ${Q}_{1}\left(s\right)$, the matrix of Eq. (6) has been rewritten by:
where $s=i\omega $,${s}^{2}={\omega}^{2}\text{,}$${a}_{11}={m\omega}^{2}+\left({k}_{1}+{k}_{2}\right)+i({c}_{1}+{c}_{2})\omega \text{,}$${a}_{12}={a}_{21}=\left({k}_{1}{l}_{1}+{k}_{2}{l}_{2}\right)+i({c}_{1}{l}_{1}+{c}_{2}{l}_{2})\omega $, ${a}_{31}={a}_{13}={k}_{1}i{c}_{1}\omega $, ${a}_{41}={a}_{14}={k}_{2}i{c}_{2}\omega $, ${a}_{22}={I\omega}^{2}+\left({k}_{1}{l}_{1}{l}_{1}+{k}_{2}{l}_{2}{l}_{2}\right)+i({c}_{1}{l}_{1}{l}_{1}+{c}_{2}{l}_{2}{l}_{2})\omega $, ${a}_{32}={a}_{23}={{k}_{1}l}_{1}i{c}_{1}{l}_{1}\omega $, ${a}_{42}={a}_{24}={{k}_{2}l}_{2}i{c}_{2}{l}_{2}\omega \text{,}$${a}_{33}={{m}_{1}\omega}^{2}+\left({k}_{1}+{k}_{t1}\right)+i({c}_{1}+{c}_{t1})\omega \text{,}$${a}_{34}={{m}_{2}\omega}^{2}+\left({k}_{2}+{k}_{t2}\right)+i({c}_{2}+{c}_{t2})\omega $, ${b}_{3}={k}_{t1}+i{c}_{t1}\omega $, and ${b}_{4}={k}_{t2}+i{c}_{t2}\omega $, respectively.
Let ${T}_{z}=Z\left(s\right)/{Q}_{1}\left(s\right)$, ${T}_{\phi}=\Psi \left(s\right)/{Q}_{1}\left(s\right)$, ${T}_{z1}={Z}_{1}\left(s\right)/{Q}_{1}\left(s\right)$, and ${T}_{z2}={Z}_{2}\left(s\right)/{Q}_{2}\left(s\right)$, thus, ${T}_{z}$, ${T}_{\phi}$, ${T}_{z1}$, and ${T}_{z2}$ are defined as the vibration’s transfer functions from the road to the automobilerobot body and front/rear axles, respectively.
Based on the calculated results in Refs [17], the result of the acceleration amplitude obtained via ${T}_{n}=$ {${T}_{z}$, ${T}_{\phi}$, ${T}_{z1}$, and ${T}_{z2}$} in Eq. (7) under road’s excitations ${Q}_{1}\left(s\right)$ are written as follows:
2.3. Road’s excitations on car’s wheels
When the automobilerobot is traveling on the road, the vibration excitation of the road described by the harmonic function with its wavelength from 5 m to 10 m and its height from 0.01 m to 0.012 m greatly affects the automobilerobot’s ride comfort and structure [12, 1819]. This harmonic function mainly causes resonant vibrations in the automobilerobot’s suspension system. Thus, this excitation is used to evaluate the vibration characteristic of the automobilerobot at the frequency range. The road surface’s vibration equation using the harmonic surface at time region has been described as:
With the frequency and wavelength of the road defined by $L$ and $l$, Eq. (9) is then rewritten in the traveling direction of $X$ as follows:
With an unchanged speed of the automobilerobot ($v$), thus, $X=vt$. Both Eqs. (9) and (10) are then rewritten by:
The basic length of the automobilerobot is defined by (${l}_{1}+{l}_{2}$), as shown in Fig. 1, thus, the vibration excitation at the rear tire (${q}_{2}$) calculated based on the vibration excitation at the front tire is expressed by:
From the ratio of ${q}_{2}/q$ calculated based on Eqs. (11) and (12), the Laplace transformation ${T}_{q}$ of ${q}_{2}/{q}_{1}$ is then described by:
Eq. (13) is then used as the vibration excitation of the automobilerobot to evaluate the characteristic of the automobilerobot’s vibrations in the frequency region.
3. Simulation and analysis result
Based on the automobilerobot’s excitations using the road’s harmonic function with ${q}_{0}=$10 mm and the road’s wavelength $l=$8 m as well as the dynamic parameters of the automobilerobot listed in Table 1, the vibration characteristic of the automobilerobot in the frequency region under the different operation conditions is then simulated and analyzed.
Table 1Automobilerobot’s dynamic parameters
Parameters  Values  Parameters  Values  Parameters  Values 
$m$ (kg)  1384  ${k}_{1}$ (N/m)  90880  ${c}_{1}$ (Ns/m)  7733 
${m}_{1}$_{}(kg)  66  ${k}_{2}$ (N/m)  93884  ${c}_{2}$ (Ns/m)  9804 
${m}_{2}$ (kg)  87  ${k}_{t1}$_{}(N/m)  193211  ${c}_{t1}$ (Ns/m)  2000 
$I$ (kg.m^{2})  11632  ${k}_{t2}$_{}(N/m)  226422  ${c}_{t2}$ (Ns/m)  2000 
${l}_{1}$ (m)  1.35  ${l}_{2}$ (m)  1.604  ${q}_{0}$ (mm)  10 
3.1. Automobile’s vibration characteristic under different stiffness of the suspension system
To evaluate the effect of stiffness parameters in the automobilerobot’s systems on the characteristic of the accelerationfrequency in the automobilerobot, three different stiffness parameters of the automobilerobot’s suspension system including $K=\left[80\%,100\%,120\%\right]\times \{{k}_{\mathrm{1,2}},{k}_{t\mathrm{1,2}}\}$ are simulated when the automobilerobot is traveling on the road surface with the harmonic function of ${q}_{0}=$ 10 mm and wavelength $l=$ 8 m at $v=$ 20 m/s. Results in the accelerationfrequency of the automobilerobot’s body in the vertical and pitching vibrations have been shown in Figs. 2(a) and 2(b).
Fig. 2The response of the automobilerobot body’s accelerationfrequency under different stiffness values
a) The vertical accelerationfrequency
b) The pitching accelerationfrequency
The simulation results show that both the responses of the accelerationfrequency of the automobilerobot’s body in the vertical and pitching directions are significantly affected by the different stiffness coefficients of the automobilerobot’s suspensions and wheels. Resonant frequencies in the vertical and pitching direction of the automobilerobot in the low frequency region appeared at 1.1 Hz, 1.3 Hz, and 1.5 Hz when the stiffness parameters were reduced by 80 %$K$, used by 100 %$K$, and increased by 120 %$K$, respectively. Additionally, the accelerationfrequency amplitude in the vertical and pitching direction of the automobilerobot at low frequencies is also depended on stiffness coefficients in the automobilerobot’s suspension systems and wheels. The automobilerobot’s accelerationfrequency amplitudes are increased with the increase of the stiffness parameters and vice versa. These results mean that the $K$ of the automobilerobot’s suspensions and wheels not only influences the amplitude but also influences the resonantfrequency of the automobilerobot’s acceleration frequency in both the vertical and pitching direction. In order to ameliorate the automobilerobot’s comfort as well as ensure the durability in automobilerobot’s structures, the designed parameters in the stiffness of automobilerobot’s suspensions and tires need to be chosen to minimum the amplitude of automobilerobot’s acceleration frequency at resonant frequencies.
3.2. Automobile’s vibration characteristic under different mass
The analysis results in Section 3.1 show that the automobilerobot’s accelerationfrequency amplitudes and resonant frequencies are affected by the stiffness parameters of the automobilerobot. Besides, based on the formula used to determine the resonant frequency of the system, the resonant frequency is calculated by ${f}^{2}=K/M$. Thus, the automobilerobot’s mass ($M$) is also influenced the automobilerobot’s accelerationfrequency characteristic. To clearly this issue, the automobilerobot’s different mass including $M=\left[80\%,100\%,120\%\right]\times \{m{,m}_{1},{m}_{2}\}$ are also simulated under the same excitation of the road surface in Section 3.1. The results of the accelerationfrequency of the automobilerobot’s body in the vertical and pitching vibrations are plotted in Figs. 3(a) and 3(b).
The simulation results indicate that both the responses of the accelerationfrequency of the automobilerobot’s body in the vertical and pitching directions are also significantly affected by the different mass in automobilerobot’s body and front/rearaxles. The resonant frequencies in the vertical and pitching direction of the automobilerobot in the low frequency region are appeared at 1.5 Hz, 1.7 Hz, and 1.9 Hz when the automobilerobot’s mass is increased by 120 %$M$, used by 100 %$M$, and reduced by 80 %$M$, respectively.
Fig. 3The response of the automobilerobot body’s accelerationfrequency under different load conditions
a) The vertical accelerationfrequency
b) The pitching accelerationfrequency
Besides, the amplitude of the accelerationfrequency in the vertical and pitching direction of the automobilerobot in the low frequency region is also dependent on the automobilerobot’s mass. The automobilerobot’s accelerationfrequency amplitudes are increased when the automobilerobot’s mass is reduced and vice versa. This also means that the automobilerobot’s mass not only influences amplitudes but also influence resonantfrequencies of automobilerobot’s acceleration frequency in both the vertical and pitching direction. In order to ameliorate automobilerobot’s comfort and ensure durability in automobilerobot’s structures, in the design process of the automobilerobot, both the mass $M$ and stiffness $K$ of the automobilerobot’s systems should be calculated and chosen to minimize the amplitude of the accelerationfrequency at the resonant frequencies.
3.3. Automobile’s vibration characteristic under road’s different wavelengths
In the automobilerobot’s condition traveling on the pavement, the road wavelength can affect the automobilerobot’s ride comfort. To clear this issue, three different wavelengths of the road including $l=$6 m, $l=$8 m, and $l=$10 m at the same excitations of the road in Section 3.1 are simulated, respectively. The results of the accelerationfrequency of the automobilerobot’s body in the vertical and pitching vibrations are plotted in Figs. 4(a) and 4(b).
Fig. 4The response of the automobilerobot body’s accelerationfrequency under road’s different wavelengths
a) The vertical accelerationfrequency
b) The pitching accelerationfrequency
The simulation results in both Figs. 4(a) and 4(b) show that the resonant frequencies of the automobilerobot’s body in the vertical and pitching vibrations unchange and appear at 1.4 Hz, 2.1 Hz, and 8.5 Hz under the different values of the road wavelength. This means that the road wavelength not influences characteristics of the automobilerobot’s acceleration frequency. However, the amplitude of the accelerationfrequency in the vertical and pitching direction of the automobilerobot in the low frequency region is changed and affected by the road’s different wavelengths. Their amplitude is increased when the road’s wavelength is reduced and vice versa. Thus, to reduce the amplitude of the accelerationfrequency in the vertical and pitching direction of the automobilerobot, the road’s wavelength needs to be increased. This means that the pavement’s roughness needs to be decreased or the pavement’s surface quality needs to be enhanced. This issue is also proven and recommended in existing studies [20].
4. Conclusions
With automobilerobot (autonomous driving car), investigating frequency response to evaluate ride comfort and structural safety is a necessary issue. This study uses the complexdomain method for evaluating automobilerobot’s vibrations in the frequency region. The study can be summarized as follows:
The design parameters of the stiffness, mass, and road wavelength remarkably affect to characteristics of the automobilerobot’s acceleration frequency.
To reduce resonant amplitudes of the automobilerobot’s acceleration frequency in both vertical and pitching directions, stiffness parameters in the automobilerobot’s suspensions and tires should be reduced while the mass of the automobilerobot’s body should be increased. However, the reduction of the stiffness of the automobilerobot can lead to reduce the stability and safety of movement of the automobilerobot. To solve this issue, the automobilerobot’s suspension systems are researched and replaced by using air suspension systems or active suspension systems.
The resonant amplitude of the accelerationfrequency in the vertical and pitching direction of the automobilerobot is significantly affected by the road wavelength, thus, to reduce this resonant amplitude, the pavement’s roughness needs to be decreased or the pavement’s surface quality needs to be enhanced.
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About this article
The authors have not disclosed any funding.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
The calculation of vibration equations and simulation are performed by Jia Yujie. The analysis of the paper results is performed by Nguyen Vanliem.
The authors declare that they have no conflict of interest.