Abstract

This paper aims to investigate the property and behavior of the hysteretic nonlinear energy sink (HNES) coupled to a half vehicle system which is a nine-degree-of-freedom, nonlinear, and semiactive suspension system in order to improve the ride comfort and increase the stability in shock mitigation by using the computer simulation method. The HNES model is a semiactive suspension device, which comprises the famous Bouc–Wen (B-W) model employed to describe the force produced by both the purely hysteretic spring and linear elastic spring of potentially negative stiffness connected in parallel, for the half vehicle system. Nine nonlinear motion equations of the half vehicle system are derived in terms of the seven displacements and the two dimensionless hysteretic variables, which are integrated numerically by employing the direct time integration method for studying both the variables of vertical displacements, velocities, accelerations, chassis pitch angle, and the ride comfort and driver safety, respectively, based on the bump and random road inputs of the pseudoexcitation method as excitation signal. Simulation results show that, compared with the HNES model and the magnetorheological (MR) model coupled to the half vehicle system, the ride comfort and stability have been evidently improved. A successful validation process has been performed, which indicated that both the ride comfort and driver safety properties of the HNES model coupled to half vehicle significantly improved.

1. Introduction

To achieve both ride comfort and driver safety is the main objective of the vehicle industry, and there are two ways to achieve ride comfort of high quality; to this end, the first way is to minimize the axis and angular acceleration of the gravity center of the vehicle body, while the second way is to maintain the tire contacts with the ground while the tire strikes a bump. Minimizing the vertical displacement of the vehicle can facilitate the ride stability.

Passive, active, and semiactive suspension vehicle systems are three available classifications of suspensions which depend on the ability of the suspension system to absorb, add, or extract energy. More recently, one of the oldest passive vibration control devices is the tuned mass damper (TMD) which was proposed by Watts in 1883 [1] and patented by Frahm [2], consisting of a mass, a spring, and a viscous damper which is attached to a primary vibrating system in order to suppress undesirable vibrations. The property of the TMD can be significantly reduced for the environmental influences, and other external parameters may alter its properties and disturb its tuning [3]. To help reduce the vibration, a large oscillating mass is generally required. Both construction and placement are rather difficult. In light of the above drawbacks, to overcome the disadvantages of the TMD, new nonlinear strategies in vibration absorption have been introduced, and among them, researchers investigated the targeted energy transfer (TET) through different designs of NES, so they have gained more attention and are more prevalent [4]. According to [5], definitely, the NES is a nonlinear attachment coupled to a linear system which dissipates energy irreversibly through nonlinear stiffness elements. To help dissipate energy effectively, compared with the TMD, the NES does not need to be tuned to a special frequency. The performance of nonsmooth NES in the presence of gravity has also been investigated where the pure cubic nonlinearity is broken in vertical configurations as the gravity induces asymmetry to the system [6, 7].

The concept of hysteretic nonlinear energy sink (HNES) was proposed by Tsiatas and studied by Tsiatas and Charalampakis in 2018 [8]. The HNES is the modification of the NES [9, 10], i.e., apart from a small mass and a nonlinear elastic spring of the duffing oscillator, the HNES also comprises a purely hysteretic spring and a linear elastic spring of potentially negative stiffness, connected in parallel. The BoucWen (B-W) model is used to describe the force produced by the purely hysteretic and linear elastic springs and has been investigated in NES dynamics previously [11, 12].

Eltantawie has studied the mechanical behavior of the magnetorheological damper (MRD) as a semiactive device of the vehicle suspension system for decentralized neurofuzzy control method previously. In this paper, the HNES model is employed as a semiactive suspension device and coupled to the linear half vehicle system in order to improve the ride comfort and increase the stability.

This study proposes a comprehensive design by using the HNES model as the semiactive suspension device, and the motion equations of the vehicle system which consist of the HNES model have been established, followed by employing the new direct time integration method, which is presented for solution of motion equations describing the dynamic response of multi-degree-of-freedom, structural linear and nonlinear systems in 2014 [13], for the solution of the system of first-order ordinary differential equations which represent semidiscrete diffusion equations in 2016 [14], and for solution of the motion equations of the vehicle model by means of the computer simulation method.

The rest of this paper is organized as follows: Section 2 describes and analyzes both half vehicle system and the HNES coupled to a primary model. The differential equations of the HNES coupled to a primary model and the semiactive suspension system are analyzed by employing the new direct time integral method, which be formulated in Section 3. The responses of the displacements, velocity, acceleration, and chassis pitch angle in the time domain are simulated by the parameters of half vehicle system in Section 4. The ride comfort and drive safety of half vehicle system are solved numerically, which are described in the Section 5. The main conclusions and findings are drawn in Section 6.

2. Half Vehicle and the HNES Model

2.1. Half Vehicle Coupling with the HNES Model

In this paper, a semiactive suspension system of the half vehicle model, which consists of sprung mass supported on front and rear suspensions as shown in Figure 1, is proposed. Both the front and rear suspensions which are considered as unsprung masses with the front and rear tire mass are connected to their tire axles, respectively. Damper C_f, elastic device K_f, and HNES model (details of the HNES model is described in Section 2.2) are connected through chassis mass M₂ and front tire directly in parallel, same as the front suspension, damper C_r, elastic device K_r, and HNES model connected through chassis mass M₂ and rear tire in parallel. The model is a nine-degree-of-freedom system and represented by the displacements of vertical driver U₅, vertical vehicle body U₄, body pitch U₃, front and rear tire deflections U_f1 and U_r1 and U_f2 and U_r2 of the mass M₁, and the displacements U_Zf and U_Zr which indicates the BoucWen model (Figure 1).

Figure 1 

Half vehicle system coupling with the HNES model.

The mathematical model is derived by applying Newton’s second law; the motion equations can be formulated as follows:

Equations (1)–(11) can be rewritten in the matrix form as follows:where are the BoucWen damper forces of the front and rear of the half vehicle system, respectively, denote matrices of mass, damper, and spring stiffness, are vectors of acceleration, velocity, and displacements, and denote excitation vector coefficient matrix and the road inputs vector of the half vehicle system, respectively. For the definition of matrices and vectors , refer to Appendix A, and for expressing intuitively in the figures of this paper, the subscript of symbols are represented by , respectively.

2.2. Dynamic Responses of Single-Degree-of-Freedom Primary System Coupling with HNES

The schematic representation of single-degree-of-freedom linear primary system coupling with HNES, which is considered as the HNES model, is shown in Figure 2. The HNES model is constituted apart from a small mass m₂ and a nonlinear elastic spring of the duffing oscillator; it also comprises a purely hysteretic spring and a linear elastic spring of potentially negative stiffness, which constitute the BoucWen model, connected in parallel. The BoucWen model is used to describe the force produced by the purely hysteretic and linear elastic springs. In light of the above, the single-degree-of-freedom (SDoF) primary system coupling with HNES (the HNES model) is constituted from the BoucWen model, a linear damp C₂, and a cubic nonlinear spring K_NL in parallel. Damper C₁ and elastic linear spring K_EL are connected to mass m₁ in parallel. The BoucWen model, damper C₂, and cubic nonlinear spring K_NL are connected between the masses m₂ and m₁ of the constituted attachments of the HNES in parallel.

Figure 2 

The SDoF primary system coupling with HNES (the HNES model).

Assume that the rear tire travels over the same path as the front tire except for a time delay which can be mentioned as equation (25) in this study. The excitation function of road inputs , which is a stationary random process, acts on the center of the tire and road surface contact point. The BoucWen model, which was introduced by Bouc [15] and extended by Wen [16], is employed to describe the force produced by both the purely hysteretic and linear elastic springs.

The SDoF primary system coupling with HNES (the HNES model) is shown in Figure 2; the hysteresis force is expressed by referring to [17, 18]:where k > 0, D > 0, t is the time history of the input variable, and z(t) is a dimensionless hysteretic variable which is governed by the following differential equation:where n > 0, denotes the signum function, x is the displacement, k is the initial stiffness, is the ratio of postyield to preyield stiffness, D indicates the yield displacement, the dimensionless exponential parameter n governs the abruptness of transition between preyield and postyield response, and the values of are set in Table 1 [8].

The HNES model is derived by applying Newton’s second law, and the equations of motion can be formulated as follows:

In this investigation, only the direct impulsive force of the primary system is considered, and thus, the initial conditions of the problem are as follows:where are the mass, displacement, stiffness of spring, and damping coefficient of the linear primary system, respectively, and are the mass, displacement, damping coefficient, and cubic nonlinear stiffness of the HNES’s attachments.

3. State Space Form of the Motion Equations and the Excitation of Road Inputs

3.1. State Space Form

Equations (15)–(18) can be reorganized into state space form as follows:

Together with the pertinent initial conditions, the following can be expressed:where denotes the initial value.

Equations (19)–(21) can be rewritten as follows:where , denotes the mass matrix which consists of and function is already expressed in equation (20).

It should be noteworthy that equation (22) is a typical nonlinear initial value problem, which are solved numerically using the new direct time integration method introduced by Katsikadelis [13], and that for multidegree of freedom systems is described as follows:where denotes the responses of the displacement vector, of which elements are the value in different time histories for . T is the simulation period, and h is the sampling interval. can be solved by coupling equations (22) and (23) with k = 1, 2, …, T/h.

Figure 3 depicts the responses of displacements X₁ and X₃ of the masses m₁ and m₂ of primary system coupling with HNES against the impulse magnitude for  = 0.35 m/s, h = 0.01, and T = 500 seconds (Figure 3(a)) and  = 2 m/s, h = 0.01, and T = 500 seconds (Figure 3(b)). It can be observed that displacements X₁ and X₃ against time are similar in shape during 500 seconds, and the amplitude changes more slowly over the entire time domain, and the maximum values of amplitudes are X₁ = 2.1 m and X₂ = 2.3 m (Figure 3(b)) and X₁ = 0.12 m and X₃ = 0.01 m (Figure 3(a)).

(a)

(b)

(a)
(b)

Figure 3 

The response of displacements of the primary system coupling with the HNES model for  = 0.35 m/s, h = 0.01, and T = 500 seconds (a) and  = 2 m/s, h = 0.01, and T = 500 seconds (b).

3.2. Excitation of Road Inputs

3.2.1. Bump Road Inputs

The bump road excitation method is one of the simple inputs for a vehicle, which has the equations as follows [19]:where i (= 1, 2) represents the front and rear tire, respectively, denotes the amplitude of bump road excitation, simulation period T = 4 seconds, the bump road excitation for the rear tire is assumed to be that of the front but with a time delay (seconds) of , and sampling interval h = 0.01 seconds in this study. Time delay is formulated as follows:where symbol denotes the vehicle velocity and L₁ and L₂ are the front and rear wheelbase; the excitation signals for bump road inputs are shown in Figure 4(a).

(a)

(b)

(a)
(b)

Figure 4 

Two types of excitation signal for (a) the bump road inputs, where simulated period T = 4 seconds, sampling interval h = 0.01 second, and vehicle velocity  = 20 m/s, and (b) the random road inputs, where simulated period T = 20 seconds, sampling interval h = 0.01 seconds, and vehicle velocity  = 20 m/s.

The road excitation of rear tire inputs is formulated as

3.2.2. Random Excitation of Road Surface Roughness

The road power spectral density (PSD) about the spatial frequency is G_q(n), n₁ <n < n₂(n = f/v), where is the average vehicle velocity and n is the spatial frequency. The road surface roughness variance of by using the frequency spectrum properties of average stationary random process can be expressed as follows [20]:

Dividing the area (n₁, n₂) into m smaller intervals, the frequency of the power spectral density value is G_q (n_mid-i) instead of G_q (n) at each small interval of the central part, where . Throughout, the area between the value of the formula after discretization approximation can be written as follows:

Corresponding to each small range, we need to find the frequency for and the standard deviation of sine wave function, and such sine wave function can be expressed asThen, the corresponding sine wave function in each area is added up to obtain the road random displacement input on the time domain:where is uniformly distributed random numbers between , of which probability density and distribution function can be referred to in Appendix B. And, t denotes the time history which can be normalized by (s is the total distance traveled by the vehicle in this study). According to reference [21], the spatial frequency n of road is distributed between [0.011, 2.83] m^-1. In this study, considering both the relevant regulations and practical life experience, the range of vehicle velocity m/s (36–108 km/h).

Relationships between time frequency and spatial frequency of the vehicle can be formulated as equation f = vn (Hz). Therefore, the range of road spatial frequency in this paper can be expressed as follows:

Here, denote the spatial frequency range at lowest velocity and at the highest velocity and symbol denotes a mathematical operation for the intersection of two sets.

The random excitation signal of road surface roughness is shown in Figure 4(b).

4. Simulation Response in the Time Domain

Equations (1)–(11) can be altered into equation (12) which is a nonlinear equation and can be written using the direct time integration method numerically as follows:where , h is the sampling interval time (second), and size (I) = [9 × 9] is the unit matrix.

can be solved by combining equations (32) and (33) with n = 1, 2, …, T/h. can be solved using equation (33) which is a linear equation and then substituting into equation (32) which is a nonlinear equation and can be solved to yield . The function “fsovle” of Matlab 2017a has been employed to obtain the numerical results of both equations.

5. Performance Index and Numerical Simulation

5.1. Response of Half Vehicle System Coupling with the HNES Model

By querying the power spectral density coefficient (PSD) of standard road surface excitation [22] and referring a 10-degree-of-freedom (DOF) passenger vehicle model determine six state variables related to the longitudinal, lateral, and heave velocities and roll, pitch, and yaw rates [23]. Equations (32)–(34) can be solved using the software programming Matlab 2017a numerically. The parameters of half vehicle system coupling with the HNES model are given in Table 2.

Figure 5 depicts the response of vertical displacements of driver U₅ and chassis U₄ and pitch angle of chassis U₃ against the road inputs for the bump excitation, of which simulation period T = 10 seconds, sampling interval h = 0.01, and different velocities of the vehicle  = 1.1256 m/s (Figure 5(a)) and  = 20 m/s (Figure 5(b)), respectively. From these figures, it can be observed that for the bump road excitation with the amplitude A = 0.11 m, the chassis (sprung mass) reached a maximum of 0.05 m (Figure 5(a)) and 0.09 m (Figure 5(b)). Also when the inputs dropped to zero, the chassis (sprung mass) vertical displacement went down for only −0.001 m (Figure 5(a)) and −0.01 m (Figure 5(b)) which is almost zero. The overall amplitude achieved by the chassis is equal to that against the front and rear tire inputs. The response of the angular displacement of chassis had a maximum peak value of 0.021 rad (Figure 5(a)) and 0.04 rad (Figure 5(b)). However, when the rear tire of the vehicle received the input of road excitation, the maximum displacement response of the driver is –0.024 m while the vehicle velocity is 1.1256 m/s, and –0.022 m while the vehicle velocity is 20 m/s . Both values are smaller than the response values of the model in reference [24]. The output data obtained from Figure 5(a) for the maximum displacement of the vehicle body against time is 0.036 m, and the maximum angular displacement of the vehicle pitch angle is 0.02 rad which is smaller than of the model mentioned in reference [24].

(a)

(b)

(a)
(b)

Figure 5 

Responses of displacement of the half vehicle system while the excitation is bump road inputs. The vector represents the displacements of the driver U₅, the chassis U₄, and the pitch angle U₃, i.e., , respectively. Simulation period T = 10 seconds, sampling interval h = 0.01 seconds, vehicle velocity  = 1.1256 m/s (the time delay ) and (b) same as (a) except for the vehicle velocity  = 20 m/s (the time delay ).

The solution data obtained from the programs of the motion equations of the half vehicle system for velocity of the drive V₅, chassis (sprung mass) V₄, and angular velocity of the chassis V₃ are plotted against time in Figure 6. It can be grasped that for bump excitation road inputs, the peak value is 0.11 m, simulation period T = 10 seconds, sampling interval h = 0.01 seconds, and different velocities of the vehicle  = 1.1256 m/s (Figure 6(a)) and  = 20 m/s (Figure 6(b)), respectively. When the front tire of the vehicle passed road excitation q¹ (t), the maximum velocity responses of the vehicle body are 0.21 m/s (a) and 0.272 m/s (b), respectively. When the rear tire excitation q² (t) received by the vehicle gradually approaches 0 m , the velocity of the vehicle body only decreases by –0.08 m/s (a) and –0.18 m/s (b). These two values gradually tend to be stable after 10 seconds. The response of the driver velocity V₅ stabilizes faster than others.

(a)

(b)

(a)
(b)

Figure 6 

Responses of velocities of half vehicle system coupling with the HNES model while excitation is bump road inputs and the velocity of vehicle is 1.1256 m/s (a) and 20 m/s (b). The vector presents the velocities of the driver, the chassis, and the pitch angle.

The responses of the displacements , velocities , and the accelerations against the road inputs were recorded for both the front and rear tire random excitation road inputs in Figures 7–9. It is obvious that the amplitude of displacement, velocity, and acceleration against time for the vehicle velocity in Figure 7(a), Figure 8(a), and Figure 9(a) is smaller than that in Figure 7(b), Figure 8(b), and Figure 9(b), respectively. It can be seen that the displacement of vehicle body had a series random value, of which the maximum is 0.51 m, against the time for road inputs of random excitation, while the velocity of the vehicle is 20 m/s. It can be observed that the response of the displacement, velocity, and the acceleration of half vehicle system coupling with the HNES model against the road inputs for random excitation is very consistent with the vehicle theory, while the velocity is in the range [1.1256, 20] m/s. Therefore, from the dynamic response analysis, it can be known that it is reasonable and reliable to apply the HNES model to the vehicle system.

(a)

(b)

(a)
(b)

Figure 7 

Responses of displacements of half vehicle system coupling with the HNES model while the excitation is random road inputs and the velocity of vehicle is 1.1256 m/s (a) and 20 m/s (b).

(a)

(b)

(a)
(b)

Figure 8 

Responses of velocities of half vehicle system coupling with the HNES model while the excitation is random road inputs and the velocity of vehicle is 1.1256 m/s (a) and 20 m/s (b).

(a)

(b)

(a)
(b)

Figure 9 

Responses of accelerations of half vehicle system coupling with the HNES model while the excitation is random road inputs and the velocity of vehicle is 1.1256 m/s (a) and 20 m/s (b).

5.2. Performance of the Half Vehicle System Coupling with the HNES Model

Another simulation program has created with the aid of the semiactive suspension of half vehicle system coupling with the HNES model to obtain the drive safety and the ride comfort property. The drive safety and the quality of the contact between the tire and the road are described by the RMS (root mean square) value of the forces, which are dynamically exchanged by the tire and the road that is defined in [21]:

Ride comfort can be formulated as

Drive safety can be formulated as

The steady-state load on the tire can be formulated aswhere i represents the front and rear suspensions, respectively, and j = 1, 2.

Figure 10 shows the ratio of dynamic loads to static loads for the front (rear) tire, and the square of the ratio of acceleration to gravity acceleration against the road inputs is recorded for bump road excitation in Figure 10(a) and road inputs for random excitation in Figure 10(b) while the vehicle velocity  = 1.1256 m/s (the delay time is 2.5 seconds). It is obvious that the ratio of dynamic loads to static loads of the front tire had a maximum peak value of 0.48 against the road inputs for bump excitation in the time range of [0.5, 0.75] seconds, of which rear tire had a maximum peak value of 0.058 after the time delay of 2.5 seconds in Figure 10(a). The square of ratio of acceleration to gravity acceleration against the road inputs for bump excitation had a maximum peak value of 0.041 in the time range of [0.7, 0.75] seconds. The maximum peak value of 0.58 was reached after the delay time of 2.5 seconds in Figure 10(a). The ratio of dynamic loads to static loads of the front (rear) tire, and the square of the ratio of acceleration to gravity acceleration against the road inputs for random excitation is very small which tends to zero as shown in Figure 10(b).

(a)

(b)

(a)
(b)

Figure 10 

Responses of the ratio of dynamic loads and static loads of the front and rear tire against road inputs and the square of ratio of acceleration and gravity acceleration against the road inputs (a) for bump excitation and (b) for random inputs. The vehicle velocity  = 1.1256 m/s (the time delay  = 2.5 second), the simulation period T = 20 seconds, and sampling interval (h) = 0.01 seconds.

It can be observed from Tables 3 and 4 that the responses of displacements, velocities, and accelerations of the half vehicle system coupling with the HNES model are smaller. The ride comfort and driver safety properties are lower than those in the semiactive suspension of the MR model with the same parameters. However, ride comfort parameters of the half vehicle system coupling with the HNES model have only been achieved for bump road inputs.

6. Conclusions

In this paper, the behavior and performance of a semiactive suspension consisting of the HNES for the half vehicle system has been proposed and investigated in shock mitigation. A successful validation process has been performed to solve the half vehicle system coupling with HNES using the same parameters to compare the developed model to the MR model. The response of the half vehicle system coupling with HNES is solved numerically using the direct time integration method (analog equation method). The model is simulated against two kinds of road inputs which are bump road inputs and random road inputs. The great performance of the model is evident as the velocity of the vehicle changes from 1.1256 m/s to 20 m/s. The main conclusions that can be drawn from this investigation are as follows:(i)The HNES model as a semiactive suspension coupled to the linear half vehicle system was studied(ii)The performance of semiactive suspension in half vehicle system coupling with the HNES model has a significant potential of reducing the road input disturbance(iii)The responses of displacements, velocities, and accelerations of the HENS model as a semiactive suspension in the half vehicle system are much better than those of the MR model as a semiactive suspension system for the same parameter in the time domain ()(iv)The ride comfort and the drive safety of the HNES as a semiactive suspension in the half vehicle system both against the bump road inputs and random road inputs are much higher than those of the MR model given in the study [23]

Appendix

A. Matrices M, K, and C and Vectors F, Q, and U in equation (12)

Matrices M, K, and C and vectors F, Q, and U are