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Abstract

The analysis of random vibration of a vehicle with hysteretic nonlinear suspension under road roughness excitation is a fundamental part of evaluation of a vehicle’s dynamic features and design of its active suspension system. The effective analysis method of random vibration of a vehicle with hysteretic suspension springs is presented based on the pseudoexcitation method and the equivalent linearisation technique. A stable and efficient iteration scheme is constructed to obtain the equivalent linearised system of the original nonlinear vehicle system. The power spectral density of the vehicle responses (vertical body acceleration, suspension working space and dynamic tyre load) at different speeds and with different nonlinear levels of hysteretic suspension springs are analysed, respectively, by the proposed method. It is concluded that hysteretic nonlinear suspensions influence the vehicle dynamic characteristic significantly; the frequency-weighted root mean square values at the front and rear suspensions and the vehicle’s centre of gravity are reduced greatly with increasing the nonlinear levels of hysteretic suspension springs, resulting in better ride comfort of the vehicle.

Keywords

Hysteretic nonlinear suspension random vibration pseudoexcitation method equivalent linearisation

Introduction

In recent years, vehicle vibration induced by a rough road profile, including its measurement and elimination, has received significant interest.^1–3 Random vibration is generated when vehicle traverses a rough road, which not only affects driving safety and ride comfort but also has a great influence on integrity of components and service life of a vehicle. Therefore, it is of great significance for characteristic simulation and structural optimal design to study random vibration of a vehicle under irregular excitation from a road surface.

Generally in the vibration analysis of vehicle systems, suspension elements are modelled as linear springs and dampers for simplicity of analysis. Nevertheless, the force–displacement characteristic of springs exhibit nonlinear behaviour especially of hysteretic type in reality, and modelling of deteriorating hysteretic behaviour is increasingly important. Many models have been developed to describe hysteretic restoring force, such as bilinear hysteresis model, Ramberg–Osgood model and Bouc–Wen hysteresis model.^4–6 The Bouc–Wen auxiliary differential equation model is widely used for its good applicability, straightforward description of various hysteretic characteristics by adjusting hysteresis parameters properly and simple solution of differential equations of motion.⁷ And nonlinear suspension elements such as hysteretic stiffness elements were modelled by the Bouc–Wen model in many literatures.^8–12

In cases where the stiffness coefficient varies with responses, the analysis becomes complicated because the resulting equations of motion involve nonlinearities. The study of random vibration of nonlinear systems has always been a subject of great interest for researchers, for example, Fokker–Planck–Kolmogorov (FPK) equation method,¹³ stochastic averaging methods,¹⁴ equivalent linear method,¹⁵ equivalent nonlinear system method¹⁶ and Monte Carlo method,^17,18 were developed in recent decades.

The equivalent linear method substitutes the original nonlinear system with a linear system according to a criterion and can solve the original nonlinear system approximately, which is applied more extensively in engineering. Nevertheless, when the equivalent linearisation technique is used to analyse the random vibration of a vehicle with nonlinear hysteretic suspension, the Lyapunov differential equations always need to be solved after linearisation, and the high-order covariance matrices of displacement, velocity and hysteretic displacement of each degree of freedom (DOF) must be computed, resulting in a more complicated computing process and a higher computational expenses consumed in the linearisation iteration procedures.

As a highly efficient and accurate algorithm, the pseudoexcitation method (PEM) has been developed to analyse the random vibration of linear time–invariant system in recent two decades, which transforms a stationary random vibration analysis into a deterministic harmonic analysis and has the wider applicability in engineering fields.^19–22 An approach combining the PEM with the equivalent linearisation technique is developed to realise the nonlinear random vibration analysis of a vehicle with hysteretic suspension springs in this article. It is shown that the covariance responses of a small number of DOFs need to be computed as the solution of Lyapunov equations is replaced by the PEM, and an appropriate algorithm can be established to achieve equivalent linearisation iterative solution for random vibration of a nonlinear vehicle system and meanwhile the solving process is greatly simplified.

The main purpose of this article is to investigate the random vibration response of a half-car model with hysteretic nonlinear suspensions. The Bouc–Wen model of hysteretic suspension spring and its equivalent linearisation model is presented; the dynamic model of a half-car with hysteretic suspension springs is established, and its corresponding state space equation is derived; and the general random vibration analysis approach of a vehicle with hysteretic suspension springs is established based on the PEM and the equivalent linearisation technique; a Sedan is selected for numerical simulation and the power spectral density (PSD) of the vehicle responses (vertical body acceleration, suspension working space and dynamic tyre load) at different speeds and with different nonlinear levels of hysteretic suspension springs are discussed, respectively, and the vehicle ride comfort is evaluated in a further discussion. Numerical results show that random vibration analysis of the vehicle with hysteretic nonlinear elements can be carried out effectively using the PEM and the equivalent linearisation technique, and hysteretic nonlinear suspension elements influence the vehicle’s dynamic characteristics significantly with the frequency-weighted root mean square (RMS) values of the vehicle body reduced greatly, resulting in better ride comfort of the vehicle.

The Bouc–Wen model of hysteretic nonlinear suspension spring and its equivalent linearisation

The restoring force of a hysteretic nonlinear element can be represented as

q (x, \overset{\cdot}{x}, t) = g (x, \overset{\cdot}{x}) + z (t)

(1)

where $q (x, \overset{\cdot}{x}, t)$ is restoring force, $g (x, \overset{\cdot}{x})$ is the non-hysteretic component, which is a function of displacement and velocity; $z (t)$ is the hysteretic component satisfying the following nonlinear differential equation⁶

\overset{\cdot}{z} = \frac{1}{η} [A \overset{\cdot}{x} - ν (γ | \overset{\cdot}{x} | z {| z |}^{n - 1} - β \overset{\cdot}{x} {| z |}^{n})]

(2)

where $γ$ and $β$ are parameters controlling the shape of the hysteresis loop; A determines the original stiffness of hysteretic displacement; $ν$ and $η$ determine hysteresis strength and stiffness; and $n$ sets the transitional smoothness from elastic area to plastic area. Selecting these parameters properly, a hysteretic system of different energy consumption level can be expressed by equation (2).

The hysteretic nonlinear suspension spring in this article is considered to have a restoring force of the Bouc–Wen type; when $n = η = ν = 1$ and a hysteretic spring subjected to stationary random excitation, the nonlinear differential equation (2) can be rewritten as

\begin{matrix} α kx + (1 - α) kz = p (t) \\ \overset{\cdot}{z} = A \overset{\cdot}{x} - γ | \overset{\cdot}{x} | z - β \overset{\cdot}{x} | z | \end{matrix}

(3)

where $k$ and $α$ are hysteretic suspension spring parameters, and $p (t)$ is excitation assumed as a zero-mean stationary random process.

An equivalent linear system can be used to represent the original nonlinear system approximately according to a criterion (the mean square value of error, that is, the difference between nonlinear and linear equations, is minimum) when the hysteretic system is under stationary random load, and the governing differential equation of which is

\overset{\cdot}{z} + c_{e} \overset{\cdot}{x} + k_{e} z = 0

(4)

where the equivalent damping coefficient and stiffness are¹⁵

\begin{matrix} c_{e} = \sqrt{\frac{2}{π}} [γ \frac{E [\overset{\cdot}{x} z]}{σ_{\overset{\cdot}{x}}} + β σ_{z}] - A \\ k_{e} = \sqrt{\frac{2}{π}} [γ σ_{\overset{\cdot}{x}} + β \frac{E [\overset{\cdot}{x} z]}{σ_{z}}] \end{matrix}

(5)

and where $σ_{z}$ is standard deviation of hysteresis displacement, $σ_{\overset{\cdot}{x}}$ is standard deviation of relative velocity of the node at the two ends of the hysteresis element; $E [\overset{\cdot}{x} z]$ is the covariance between hysteresis displacement and relative velocity.

Therefore, the dynamic equations of a vehicle can be established according to linear multi-body dynamic theory as the hysteretic suspension spring modelled by the Bouc–Wen model are linearised from equations (4) and (5).

The dynamic model and state space equation of vehicle

A 4-DOF half-car model is shown in Figure 1. As the constitutive relation of the front and rear suspension springs is given by equation (3), the equations of motion of the half-car model can be derived from Lagrange equation as

\begin{matrix} M {\overset{\cdot\cdot}{x}}_{c} = c_{1} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) + α_{1} k_{1} (x_{1} - x_{2}) \\ - (1 - α_{1}) k_{1} z_{f} + c_{2} ({\overset{\cdot}{x}}_{3} - {\overset{\cdot}{x}}_{4}) \\ + α_{2} k_{2} (x_{3} - x_{4}) - (1 - α_{2}) k_{2} z_{r} \end{matrix}

(6)

\begin{matrix} I_{c} {\overset{\cdot\cdot}{θ}}^{=} a [c_{1} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) + α_{1} k_{1} (x_{1} - x_{2}) \\ - (1 - α_{1}) k_{1} z_{f}] - b [c_{2} ({\overset{\cdot}{x}}_{3} - {\overset{\cdot}{x}}_{4}) \\ + α_{2} k_{2} (x_{3} - x_{4}) - (1 - α_{2}) k_{2} z_{r}] \end{matrix}

(7)

\begin{matrix} m_{1} {\overset{\cdot\cdot}{x}}_{1} = c_{1} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) + α_{1} k_{1} (x_{2} - x_{1}) \\ - (1 - α_{1}) k_{1} z_{f} + k_{t 1} (h_{f} - x_{1}) \end{matrix}

(8)

\begin{matrix} m_{2} {\overset{\cdot\cdot}{x}}_{3} = c_{2} ({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{x}}_{3}) + α_{2} k_{2} (x_{4} - x_{3}) \\ - (1 - α_{2}) k_{2} z_{r} + k_{t 2} (h_{r} - x_{3}) \end{matrix}

(9)

\begin{matrix} {\overset{\cdot\cdot}{x}}_{2} = [c_{1} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) + α_{1} k_{1} (x_{1} - x_{2}) \\ - (1 - α_{1}) k_{1} z_{f}] δ + [c_{2} ({\overset{\cdot}{x}}_{3} - {\overset{\cdot}{x}}_{4}) \\ + α_{2} k_{2} (x_{3} - x_{4}) - (1 - α_{2}) k_{2} z_{r} + u_{2}] η \end{matrix}

(10)

\begin{matrix} {\overset{\cdot\cdot}{x}}_{4} = [c_{1} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) + α_{1} k_{1} (x_{1} - x_{2}) \\ - (1 - α_{1}) k_{1} z_{f}] η + [c_{2} ({\overset{\cdot}{x}}_{3} - {\overset{\cdot}{x}}_{4}) \\ + α_{2} k_{2} (x_{3} - x_{4}) - (1 - α_{2}) k_{2} z_{r}] ζ \end{matrix}

(11)

\begin{matrix} δ = 1 / M + a^{2} / I, η = 1 / M - ab / I \\ ζ = 1 / M + b^{2} / I \end{matrix}

(12)

where $m_{1}$ and $m_{2}$ are the front and rear unsprung masses, respectively; $M$ is the body mass; $I_{c}$ is the mass moment of inertia of the vehicle about its centre of gravity; $c_{1}$ and $c_{2}$ are front and rear suspension damping coefficients; $k_{1}$ and $k_{2}$ are the front and rear hysteretic suspension stiffnesses; $k_{t 1}$ and $k_{t 2}$ are the front and rear tyre stiffnesses; a and b are the horizontal distances from the front and rear axles to the centre of gravity; $x_{1} and x_{3}$ are absolute displacements of the front and rear unsprung masses, respectively; $x_{c}$ is the absolute displacement of the centre of gravity; $x_{2}$ and $x_{4}$ are absolute displacements of the front and rear end of the vehicle body; $z_{r}$ and $z_{f}$ are hysteretic displacements of the front and rear suspension springs; $θ$ is angular rotation of the vehicle about its centre of gravity; $h_{f}$ and $h_{r}$ are units step road inputs at front and rear wheels; $α_{1} and α_{2}$ are the nonlinear level of hysteretic front and rear suspension springs.

Figure 1.

Half-car model with hysteretic suspension springs.

The stiffnesses of the front and rear hysteretic suspension springs in the equations of motion are modelled using the Bouc–Wen model according to section ‘The Bouc–Wen model of hysteretic nonlinear suspension spring and its equivalent linearisation’, and the hysteretic displacements in equations (6)–(11) satisfy the following differential equations of equivalent linear system

\begin{matrix} {\overset{\cdot}{z}}_{f} + c_{f} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) + k_{f} z_{f} = 0 \\ {\overset{\cdot}{z}}_{r} + c_{r} ({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{x}}_{3}) + k_{r} z_{r} = 0 \end{matrix}

(13)

where $c_{f}$ and $c_{r}$ are equivalent damping coefficients corresponding to the front and rear suspension springs; and $k_{f}$ and $k_{r}$ are equivalent stiffness coefficients of the front and rear suspensions, respectively, and

\begin{matrix} c_{f} = \sqrt{\frac{2}{π}} [γ_{f} \frac{E [({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) z_{f}]}{σ_{({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1})}} + β_{f} σ_{z_{f}}] - A_{f} \\ k_{f} = \sqrt{\frac{2}{π}} [γ_{f} σ_{({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1})} + β_{f} \frac{E [({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) z_{f}]}{σ_{z_{f}}}] \\ c_{r} = \sqrt{\frac{2}{π}} [γ_{r} \frac{E [({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{x}}_{3}) z_{r}]}{σ_{({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{x}}_{3})}} + β_{r} σ_{z_{r}}] - A_{r} \\ k_{r} = \sqrt{\frac{2}{π}} [γ_{r} σ_{({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{x}}_{3})} + β_{r} \frac{E [({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{x}}_{3}) z_{r}]}{σ_{z_{r}}}] \end{matrix}

(14)

Introducing the state vector

x = {[\begin{matrix} z_{1} & z_{2} & z_{3} & z_{4} & z_{5} & z_{6} & z_{7} & z_{8} & z_{f} & z_{r} \end{matrix}]}^{T}

where

\begin{matrix} z_{1} = x_{1} - h_{f}; z_{2} = x_{2} - x_{1}; z_{3} = x_{5}; z_{4} = x_{6} \\ z_{5} = x_{3} - h_{r}; z_{6} = x_{4} - x_{3}; z_{7} = x_{7}; z_{8} = x_{8} \\ {\overset{\cdot}{x}}_{1} = x_{5}; {\overset{\cdot}{x}}_{2} = x_{6}; {\overset{\cdot}{x}}_{3} = x_{7}; {\overset{\cdot}{x}}_{4} = x_{8} \end{matrix}

Combining equations (6)–(11) and (13), the equations of motion for the vehicle can be rewritten in state space form as

\overset{\cdot}{x} = Ax + Df

(15)

With

\begin{matrix} A = [\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}] \\ A_{11} = [\begin{matrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 1 & 0 & 0 & 0 & 0 \\ \frac{- k_{t 1}}{m_{1}} & \frac{α_{1} k_{1}}{m_{1}} & \frac{- c_{1}}{m_{1}} & \frac{c_{1}}{m_{1}} & 0 & 0 & 0 & 0 \\ 0 & - δ α_{1} k_{1} & δ c_{1} & - δ c_{1} & 0 & - η α_{2} k_{2} & η c_{2} & - η c_{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 1 \\ 0 & 0 & 0 & 0 & \frac{- k_{t 2}}{m_{2}} & \frac{α_{2} k_{2}}{m_{2}} & \frac{- c_{2}}{m_{2}} & \frac{c_{2}}{m_{2}} \\ 0 & - η α_{1} k_{1} & η c_{1} & - η c_{1} & 0 & - ζ α_{2} k_{2} & ζ c_{2} & - ζ c_{2} \end{matrix}] \\ A_{12} = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ \frac{(1 - α_{1}) k_{1}}{m_{1}} & 0 \\ δ (1 - α_{1}) k_{1} & η (1 - α_{2}) k_{2} \\ 0 & 0 \\ 0 & 0 \\ 0 & \frac{(1 - α_{2}) k_{2}}{m_{2}} \\ η (1 - α_{1}) k_{1} & ζ (1 - α_{2}) k_{2} \end{matrix}] \\ A_{21} = [\begin{matrix} 0 & 0 & c_{f} & - c_{f} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{r} & - c_{r} \end{matrix}] \\ A_{22} = [\begin{matrix} - k_{f} & 0 \\ 0 & - k_{r} \end{matrix}] \\ D = {[\begin{matrix} - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{T} \\ f = \overset{\cdot}{w} = {\begin{matrix} {\overset{\cdot}{h}}_{f} \\ {\overset{\cdot}{h}}_{r} \end{matrix}} \end{matrix}

(16)

The PEM-based random vibration analysis of a vehicle with hysteretic suspension springs

Road roughness

The displacement PSD of the road profile is given by the study of Thompson and Davis²³

S_{r} (Ω) = G_{r} Ω^{- 2}

(17)

where $G_{r}$ is road irregularity coefficient and $Ω$ is spatial frequency.

Supposing vehicle travels at constant speed $v,$ equation (17) can be expressed in time–frequency domain as

S_{r} (ω) = \frac{1}{2 π v} S_{r} (\frac{ω}{2 π v}) = \frac{R_{r}}{ω^{2}}, R_{r} = 2 π G_{r} v

(18)

where $ω$ denotes circular frequency.

The amount of road excitation imposed at the vehicle tyres depends on both the road roughness coefficient and vehicle velocity. It is assumed that the rear wheel follows the same path as the front wheel but with a constant time delay $t_{s},$ so that

{\overset{\cdot}{h}}_{r} (t) = {\overset{\cdot}{h}}_{f} (t - t_{s}), t_{s} = \frac{(a + b)}{v}

(19)

The PEM of random vibration analysis for vehicle

Two problems need to be considered in computational procedure, one is computing efficiency as the conventional linear process requires iterative iteration to compute the equivalent coefficients (parameters in equation (14)), and the other is the coherent effect between the front and rear wheels as it is assumed that when the vehicle travels at constant speed, the rear wheel follows the same path as the front wheel but with a delay phase.

As the PEM is a highly efficient algorithm for analysing stochastic vibration of linear system, which transforms a stationary random vibration analysis into a deterministic harmonic analysis, it can easily solve these two problems by constructing the pseudoexcitation as equation (20) and analysing the corresponding pseudoresponse and can realise spectrum analysis by vector multiplication calculation of the pseudoresponse. Moreover, the PEM can deal with the coherent stochastic loads of the system by constructing proper pseudoexcitation. Combined with the PEM and the equivalent technique, the detailed procedure of the stochastic vibration analysis of the nonlinear vehicle system are as follows.

Introducing the pseudoexcitation¹⁹

\tilde{f} (t) = φ \sqrt{S_{r} (ω)} e^{i ω t}

(20)

where $i = \sqrt{- 1}$ . There is a constant phase between the road excitation on the front and rear tyres considering the travelling wave effect, and the phase vector can be expressed as

φ = {\begin{matrix} 1 & e^{- i ω t_{s}} \end{matrix}}^{T}

(21)

Substituting equation (20) into equation (15)

{\overset{\cdot}{\tilde{x}}}^{=} A \overset{\cdot}{\tilde{x}} + D \tilde{f} (t)

(22)

The stable solution of the vehicle system under pseudoharmonic excitation can be obtained as

\tilde{x} = {(i ω I - A)}^{- 1} D φ \sqrt{S_{r} (ω)} e^{i ω t}

(23)

For any vehicle responses $z_{i} and z_{j}$ , the auto spectrum and cross spectrum based on the PEM are given by

\begin{matrix} G_{z_{i} z_{i}} (ω) = {\tilde{z}}_{i}^{*} {\tilde{z}}_{i}^{T} \\ G_{z_{j} z_{j}} (ω) = {\tilde{z}}_{j}^{*} {\tilde{z}}_{j}^{T} \\ G_{z_{i} z_{j}} (ω) = {\tilde{z}}_{i}^{*} {\tilde{z}}_{j}^{T} \end{matrix}

(24)

Accordingly, the variance and covariance of the vehicle responses can be expressed as

\begin{matrix} σ_{z_{i}}^{2} = 2 \int_{0}^{\infty} G_{z_{i} z_{i}} (ω) d ω \\ σ_{z_{j}}^{2} = 2 \int_{0}^{\infty} G_{z_{j} z_{j}} (ω) d ω \\ E [z_{i} z_{j}] = 2 \int_{0}^{\infty} G_{z_{i} z_{j}} (ω) d ω \end{matrix}

(25)

The equivalent linearisation iterative procedures

The basic idea of the equivalent linearisation technique is the mean square value of error, that is, the difference between the nonlinear and linear equations is minimum; therefore, the equivalent linear system approximates the original nonlinear system step by step in terms of a iteration scheme.²⁴ Based on the foregoing discussion, the implementation process of the random vibration analysis of a vehicle with hysteretic suspension springs using the PEM and the equivalent linearisation technique are as follows:

Suppose the initial equivalent coefficient in equation (13) $X_{k} = [c_{f}^{k}, k_{f}^{k}, c_{r}^{k}, k_{r}^{k}]$ and the coefficient matrix $A_{k}$ in equation (15) can be obtained from equation (16), then the initial linear system can be established.

Construct the pseudoexcitation from equation (20); equations (22)–(24) are employed to determine the stationary random vibration power spectrum of the equivalent linear system, and variance and covariance of the vehicle responses could be calculated from equation (25).

Calculate the equivalent coefficient $X_{k + 1} = [c_{f}^{k + 1}, k_{f}^{k + 1}, c_{r}^{k + 1}, k_{r}^{k + 1}] in terms of$ equation (14), and then calculate the coefficient matrix $A_{k + 1}$ in equation (15) again.

The norm $| | X_{k + 1} | |_{p} and | | X_{k} | |_{p}$ of coefficients vector $X_{k + 1}$ and $X_{k}$ are computed, respectively, and the convergence error is defined as

err = \frac{({‖ {X_{k + 1}}_{p} - X_{k} ‖}_{p})}{{‖ X_{k} ‖}_{p}}

(26)

If $err > ε$ (a very small positive number), replace $X_{k} with X_{k + 1}$ , and iterate the steps (1)–(4) or else stop iteration and output the computing results.

In addition, the following iterative scheme is applied to deal with the problem as the iteration results cannot obtain the stable solution in some condition¹⁶

\begin{matrix} X_{k + 1} = τ ψ (X_{k}) + (1 - τ) X_{k} \\ = X_{k} + τ (ψ (X_{k}) - X_{k}), 0 < τ < 1 \end{matrix}

(27)

where $ψ (\cdot)$ is iteration implicit function.

Vehicle ride comfort analysis

The ISO2631-1: 1997(E) standard is adopted to evaluate the ride comfort of a vehicle, and the frequency-weighted RMS of body acceleration response can be expressed as

a_{w} = {[\int_{0.5}^{80} w_{k}^{2} (f) G_{a} (f) df]}^{1 / 2}

(28)

where f denotes the frequency in Hz; $G_{a} (f)$ is the single-sided PSD of the vertical body acceleration response with $G_{a} (f) = 2 π G_{a} (ω)$ ; $w_{k} (f)$ is the frequency-weighted function satisfying

w_{k} (f) = {\begin{matrix} \begin{matrix} 0.5 (0.5 < f < 2) \\ f / 4 (2 < f \leq 4) \end{matrix} \\ \begin{matrix} 1 (4 < f \leq 12.5) \\ 12.5 / f (12.5 < f < 80) \end{matrix} \end{matrix}

(29)

Numerical results

The main parameters of the half-car model are from Ford Granada Sedan;²⁵ $M = 690 kg$ , $I = 1222 kg m^{2}$ , $m_{1} = 40.5 kg$ , $m_{2} = 45.4 kg$ , $c_{1} = 1500 N s / m$ , $c_{2} = 1500 N s / m, k_{1} = 17000 N / m$ , $k_{2} = 22000 N / m$ , $k_{tf} = 192, 000 N / m$ , $k_{tr} = 192, 000 N / m$ , $a = 1.25 m$ , $b = 1.51 m$ . As mentioned earlier in equation (2), A determines the original stiffness of hysteretic displacement, $γ$ and $β$ are parameters controlling the shape of the hysteresis loop. The essential experiment study aimed at the hysteretic suspension system must be carried out to obtain accurate parameters of the Bouc–Wen model. In this article, the parameters of Bouc–Wen model are referred to the study of Narayanan and Raju⁸ and are assumed to be $A_{f} = A_{r} = 1.5$ , $γ_{f} = γ_{r} = 0.5$ , $β_{f} = β_{r} = 0.5$ .

The PSD curves of the vehicle responses (vertical body acceleration, suspension working space and dynamic tyre load) at different speeds with different nonlinear levels of hysteretic suspension springs are discussed, respectively; the RMS values of the vehicle responses, the frequency-weighted RMS values of the front and rear suspensions, and the vehicle’s centre of gravity are also calculated separately.

Vehicle model with linear suspension springs at different speeds

First, the random vibration responses of the half-car model with linear suspensions at 15, 20 and 30 m/s are computed. $α_{1}$ and $α_{2}$ in equations (6)–(11) equal to 1.0. The road roughness coefficient is selected as $5 \times 10^{- 6} m^{3} / cycle$ . Figure 2 illustrates the PSD curves of vertical body acceleration, suspension working space and dynamic tyre load at different speeds. Table 1 is the main PSD peak frequencies of the vehicle responses with the PEM and the conventional random vibration analysis method in the study of Yu and Lin;²⁵ it can be seen that the resonant frequencies of the vehicle body and the bounce frequencies of the front and rear wheels with the two methods are remarkable close, which also verified the correctness of the PEM. Meanwhile, as the PEM transforms a stationary random vibration analysis of time-invariant linear system into a deterministic harmonic analysis, no matter how complex the road excitation is, the PSD of the vehicle responses could be obtained provided with the PSD of the surface roughness, resulting in high-effective random vibration analysis of vehicle.

Figure 2.

PSD of vehicle responses: (a and b) vertical body acceleration (BA) at front and rear suspension, (c and d) front and rear suspension working spaces (SWS) and (e and f) front and rear dynamic tyre loads (DTL) (the black, red and blue line corresponding to v = 15, 20 and 30 m/s, respectively).

Table 1.

PSD peak frequencies of the vehicle responses (Hz).

$α_{1} = α_{2} = 1.0$	Front body	Rear body	Front wheel	Rear wheel
PEM	1.04	1.30	10.45	10.70
Yu and Lin²⁵	1.01	1.27	10.43	10.88

PSD: power spectral density; PEM: pseudoexcitation method.

Table 2 gives the RMS values of the vehicle responses. It shows that the RMS values of vertical body acceleration, suspension working space and dynamic tyre load all increase with vehicle speeds. Similarly, the frequency-weighted RMS values of vertical acceleration at the front and rear suspension and the vehicle’s centre of gravity all increase as the vehicle speed increases.

Table 2.

RMS values of linear vehicle’s responses.

Speed (m/s)	Body acceleration (m/s²)		Suspension working space (mm)		Dynamic tyre load (kN)
Speed (m/s)	Front	Rear	Front	Rear	Front	Rear
15	1.411	1.821	14.01	13.50	0.925	0.969
20	1.634	2.103	16.29	15.56	1.080	1.130
30	1.988	2.555	20.09	18.94	1.336	1.394

RMS: root mean square.

Vehicle model with different nonlinear levels of hysteretic suspension springs

The vehicle speed is now selected as $20 m / s$ , and the road roughness coefficient is $5 \times 10^{- 6} m^{3} / cycle$ . The nonlinear level of the front and rear hysteretic suspension springs are the same and are set as 0.9, 0.7, 0.5. The random vibration responses of the vehicle with hysteretic suspension springs are investigated. Figure 3 depicts the PSD curves of vertical body acceleration, suspension working space and dynamic tyre load for vehicle with different nonlinear values of hysteretic suspension springs. It shows that the hysteretic nonlinear suspension influences the vehicle dynamic responses greatly, especially in the frequency range close to the bounce frequencies of the vehicle body; as the nonlinear level increases, the front and rear vertical body accelerations are decreased significantly compared with the same kinds of responses of the vehicle with linear suspension springs; it can be seen from Figure 3(d) that the suspension working spaces of the rear suspension are decreased to some degree, while Figure 3(c) and (e) indicates that the working spaces and dynamic tyre loads of the front suspension are decreased first, and increased, and then decreased sharply with the increase in nonlinear level of hysteretic suspension springs.

Figure 3.

PSD of vehicle responses: (a and b) vertical body acceleration (BA) at front and rear suspension, (c and d) front and rear suspension working spaces (SWS) and (e and f) front and rear dynamic tyre loads (DTL) (the purple, blue, red and black line corresponding to α₁ = α₂ = 1.0, 0.9, 0.7, 0.5, respectively).

It also shows from Table 3 that the bounce frequencies of the vehicle body attenuated greatly with increasing the nonlinear level of hysteretic suspension springs, and the bounce frequencies of the front and rear wheels change slightly.

Table 3.

PSD peak frequencies of the vehicle responses (Hz).

$α_{1} = α_{2}$	Front body	Rear body	Front wheel	Rear wheel
1.0	1.04	1.30	10.45	10.70
0.9	0.90	1.12	10.48	10.72
0.7	0.48	0.61	10.55	10.79
0.5	0.50	0.64	10.63	10.88

PSD: power spectral density.

Table 4 gives the RMS values of the vehicle responses with different nonlinear levels of hysteretic suspension springs. As can be seen, the RMS values of the vehicle responses are more or less reduced as the nonlinear levels of hysteretic suspension springs increases except that the RMS value of the front suspension working space is increased slightly when $α_{1} = α_{2} = 0.7 .$ It is observed from Table 5 that the frequency-weighted RMS values at the front and rear suspensions, and the vehicle’s centre of gravity are reduced apparently with the increase in nonlinear levels of hysteretic suspension springs, resulting in better vehicle ride comfort.

Table 4.

RMS values of nonlinear vehicle’s responses.

$α_{1} = α_{2}$	Body acceleration (m/s²)		Suspension working space (mm)		Dynamic tyre load (kN)
$α_{1} = α_{2}$	Front	Rear	Front	Rear	Front	Rear
0.9	1.537	1.927	16.18	15.29	1.077	1.124
0.7	1.408	1.696	16.30	15.07	1.072	1.116
0.5	1.368	1.630	11.11	10.12	1.038	1.070

RMS: root mean square.

Table 5.

Frequency-weighted RMS values of body acceleration (m/s²).

Position	Linear	Nonlinear $(α_{1} = α_{2} = α)$
Position	Linear	α = 0.9	Reduction (%)	α = 0.7	Reduction (%)	α = 0.5	Reduction (%)
At front suspension	1.411	1.366	3.19	1.304	7.58	1.284	9.00
At rear suspension	1.779	1.692	4.89	1.579	11.24	1.545	13.15
Centre of gravity	1.040	0.988	5.00	0.919	11.63	0.899	13.56

RMS: root mean square.

Conclusion

The PEM is an effective method to analyse the random vibration of a system, which transforms a stationary random vibration analysis of time-invariant linear system into a deterministic harmonic analysis. The random vibration analysis on a half-car model with hysteretic suspension springs under road irregularity is made using the PEM and the equivalent linearisation technique:

The hysteretic characteristic of the nonlinear suspension spring is represented by the Bouc–Wen model, and a general random vibration analysis of a vehicle with hysteretic nonlinear suspension springs is carried out.

The random vibration of a half-car model is analysed, the PSD of the vehicle responses (vertical body acceleration, suspension working space and dynamic tyre load) at different speeds and with different nonlinear levels of hysteretic suspension springs are discussed and the vehicle ride comfort is also evaluated. The numerical results show that the PSD of a road surface can be used to evaluate the PSD of vehicle responses accurately and efficiently.

Hysteretic nonlinear suspension springs influence the vehicle dynamic responses significantly; and the frequency-weighted RMS values of the front and rear body, and the vehicle’s centre of gravity are reduced remarkably with increasing the nonlinear levels of hysteretic suspension springs, resulting in better ride comfort of the vehicle.

Footnotes

Handling Editor: Elsa de Sa Caetano

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by National Natural Science Foundation of China (no. 51405399).

ORCID iD

Chunrong Hua

Zhi Wen Lu

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Random vibration of vehicle with hysteretic nonlinear suspension under road roughness excitation

Abstract

Keywords

Introduction

The Bouc–Wen model of hysteretic nonlinear suspension spring and its equivalent linearisation

The dynamic model and state space equation of vehicle

The PEM-based random vibration analysis of a vehicle with hysteretic suspension springs

Road roughness

The PEM of random vibration analysis for vehicle

The equivalent linearisation iterative procedures

Vehicle ride comfort analysis

Numerical results

Vehicle model with linear suspension springs at different speeds

Vehicle model with different nonlinear levels of hysteretic suspension springs

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References