Dynamic characteristics of a semi-active fractional-order inerter-based suspension with acceleration-velocity switch control

Abstract

The fractional-order differential theory is applied in the hydraulic inerter to describe its mechanical characteristic, a semi-active fractional-order (SA-FO) inerter based on the hydraulic inerter is proposed and used in the vehicle suspension to constitute the semi-active fractional-order inerter-based (SA-FOIB) suspension, which consists of two adjustable inertances. According to the mechanical characteristic of the inerter, the acceleration-velocity switch (AVS) control strategy is proposed to adjust the inertance of the SA-FO inerter. The dynamic model of the SA-FOIB suspension with AVS control is established, its dynamic response under road harmonic excitation is obtained using the averaging method, the dynamic performance under road harmonic and random excitations is analyzed and evaluated by the vehicle body acceleration, suspension dynamic deflection and wheel dynamic load, the optimized structural parameters are obtained using the genetic algorithm optimization method. The results show that for road harmonic excitation, the SA-FOIB suspension can further enhance the high-frequency dynamic performance of the passive fractional-order inerter-based (P-FOIB) suspension, which the high-frequency resonance peaks of the dynamic performance indices are smaller. For road random excitation, the SA-FOIB suspension can decrease the root-mean-square (RMS) values of the suspension dynamic deflection and wheel dynamic load compared with the P-FOIB suspension, while increases the RMS value of the vehicle body acceleration. The optimized SA-FOIB suspension further improves the dynamic performance of the suspension dynamic deflection and wheel dynamic load, which the corresponding RMS values are smaller than the unoptimized SA-FOIB suspension, while deteriorates the vehicle body acceleration.

Keywords

Semi-active fractional-order inerter semi-active fractional-order inerter-based suspension acceleration-velocity switch control dynamic characteristics

Introduction

The suspension is one of the most significant structure for a vehicle, it isolates and reduces the vibration transmitted from the road to the vehicle body, which not only supports the vehicle body, but also affects the vehicle dynamics.^1–4 Thus, the suspension research is an indispensable part in the vehicle dynamics analysis. In 2002, Smith⁵ firstly proposed a new vibration attenuation device, inerter, it is a device with two terminals, the produced equal force applied on the two terminals is proportional to the relative acceleration between them, the corresponding proportional coefficient is called inertance and is measured in kilogram.^6,7 The generation of inerter leads to the birth of the inerter-based suspension, which consists of three elements: inerter, spring and damper. Compared with the traditional two-element suspension composed of spring and damper, the inerter-based suspension has a variety of structural arrangements, which can further improve the vehicle dynamics.^8,9 According to the inertial force realization, the inerter can be divided into mechanical, hydraulic and electromagnetic ones,¹⁰ it can be also divided into passive and semi-active inerters based on whether the inertance can be changed or not.¹¹ The appearance of semi-active inerter greatly promotes the development of inerter-based suspension.

Similar to the traditional suspension, the inerter-based suspension can be divided into passive, semi-active and active ones.¹² The passive inerter-based suspension has the advantages of high reliability, easy implementation and low cost, no additional energy is required, but its improvement in vehicle dynamics is limited.^13,14 The active inerter-based suspension can adjust the force acting on the unsprung and sprung masses in real time, it has the best vibration attenuation effect in the three suspensions, but it needs a large amount of additional energy.¹⁵ The semi-active inerter-based suspension enhances the vehicle dynamics by adjusting the structural parameters, such as damping or inertance. Since there is no actuator, it greatly reduces the own energy loss and is a suspension with better comprehensive performance. Zhang et al.¹⁶ applied the semi-active damper in the inerter-based suspension and designed the skyhook, groundhook and hybrid control methods, which further improves the dynamic performance than the passive one. Chen et al.¹⁷ constructed a semi-active inerter-based suspension with both semi-active inerter and damper, and presented a force-tracking approach to adjust the inertance and damping. Hu et al.¹⁸ and Zhang et al.¹⁹ proposed an adjustable semi-active inerter and used it in the inerter-based suspension to improve the vehicle ride comfort. Wang et al.²⁰ proposed a relative-acceleration-relative-velocity control method to adjust the inertance of the semi-active inerter and enhanced the vehicle dynamics. Yang et al.²¹ presented a semi-active hydro-pneumatic inerter-based suspension based on the model predictive control method, and yield a better low-frequency dynamic performance. Zhu et al.²² designed a semi-active dual-chamber hydro-pneumatic inerter-based suspension and used the linear quadratic Gaussian controller to match its nonlinear characteristics, which improves the vehicle smoothness and handling stability. Overall, the semi-active inerter-based suspension can further obtain a better vehicle dynamic performance than the passive one.

The hydraulic inerter has the advantages of simple structure, easy adjustment of inertance and large bearing capacity, compared with the mechanical inerter. Some scholars have shown that the fluid of hydraulic inerter is similar to multi-phase media, and it is not accurate to describe the mechanical characteristic of hydraulic inerter by integer-order differential theory.²³ Westerlund²⁴ used the fractional-order differential theory to describe the multi-phase media system for hydraulic system modeling, which is more accurate than the integer-order differential theory modeling. The fractional-order calculus has a wide range of order values, which can be selected between real numbers and imaginary numbers, and can be considered as an extension of integer-order calculus. The fractional-order calculus has memory property and changeable continuity due to its own characteristic, and can better describe the dependent process of the historical development of the dynamic system. Compared with the integer-order calculus, the fractional-order calculus can describe the dynamic characteristic of the hydraulic system more accurately by using the fractional order.

In the fractional-order differential theory field, Caputo and Maindardi²⁵ established a new memory dissipation model based on the fractional-order differential theory, and proved the accuracy of the model by verifying the dissipation curves of various materials. Vinagre et al.²⁶ introduced the fractional-order differential theory into the model reference adaptive control and shown the advantage of fractional-order adaptive control. Bagley and Torviky²⁷ studied the mechanical characteristic of viscoelastic damping structure using fractional-order differential theory, which effectively improves the model accuracy. Zamani et al.²⁸ designed a fractional-order proportional-integral-derivative controller to realize semi-active control of base-isolated structure under far-field and near-field seismic excitation. You et al.²⁹ analyzed the dynamic performance and conducted the parameter design of the fractional-order traditional two-element suspension, which reduces the vehicle body vertical acceleration. Chen et al.³⁰ investigated the dynamic characteristic of one degree-of-freedom (DOF) vibration isolator with both fractional-order inerter and damper, which reduces the natural frequency and improves the vibration isolation performance of the system. In summary, it is more reasonable to use fractional-order differential theory to describe the mechanical characteristic of the hydraulic system than the integer-order differential theory, which proves the reliability of fractional-order dynamic model used in the hydraulic system. In addition, the fractional-order inerter and damper can be used to further improve the dynamic system performance.

In this paper, the fractional-order differential theory is applied in the hydraulic inerter to describe its mechanical characteristic, the semi-active fractional-order (SA-FO) inerter based on the hydraulic inerter is proposed and consists of two adjustable inertances. The order of the SA-FO inerter is fractional and ranges from 1 to 2. The SA-FO inerter is used in the vehicle suspension to constitute the semi-active fractional-order inerter-based (SA-FOIB) suspension. Due to different structural layouts of inerter, spring and damper, the SA-FOIB suspension may have various arrangements, here, the three-element parallel-connected configuration is constructed, which includes a SA-FO inerter, a passive spring and a passive damper. This structure is simple in arrangement and has high space utilization. According to the mechanical characteristic of the inerter, the acceleration-velocity switch (AVS) control strategy is proposed to adjust the inertance of the SA-FO inerter. When the AVS control strategy is used and the inertance of the SA-FO inerter changes between two adjustable inertances, the SA-FOIB suspension is actually a piece-wise nonlinear dynamic system. Thus, the averaging method is adopted to obtain the dynamic response of the SA-FOIB suspension under road harmonic excitation, which is a widely used analytical method in the dynamic solution area of the piece-wise nonlinear dynamic system. The main purpose of this paper is to show how the SA-FO inerter influences the vehicle dynamics and whether the SA-FOIB suspension can offer a better dynamic performance than the passive fractional-order inerter-based (P-FOIB) suspension. The contributions of this paper are: (a) A SA-FO inerter is proposed and used in the vehicle suspension to constitute the SA-FOIB suspension. (b) An AVS control strategy is presented to adjust the inertance of the SA-FO inerter. (c) The dynamic model of the SA-FOIB suspension with AVS control is established and its dynamic response under road harmonic excitation is obtained using the averaging method. (d) The influence of the SA-FO inerter on the vehicle suspension performance is analyzed, and compared with the P-FOIB suspension to show its benefits. (e) The structural parameters of the SA-FOIB suspension are optimized using the genetic algorithm optimization method.

The structure of this paper is: Firstly, the dynamic model of the SA-FOIB suspension with AVS control is established, its dynamic response under road harmonic excitation is obtained using the averaging method. Secondly, the dynamic performance of the SA-FOIB suspension with AVS control under road harmonic and random excitations is analyzed. Thirdly, the structural parameters optimization of the SA-FOIB suspension with AVS control is investigated. Finally, the conclusions are summarized.

Modeling of SA-FOIB suspension

Firstly, the dynamic model of the P-FOIB suspension is established and the effect of the passive fractional-order (P-FO) inerter on its dynamic performance is studied. The P-FOIB suspension is shown in Figure 1, which is a three-element parallel-connected configuration. m₁ is the sprung mass, m₂ is the unsprung mass, k_t is the tire stiffness, k₁ and c are the suspension stiffness and damping, respectively; b is the inertance of the P-FO inerter, which is passive as b is fixed, the fractional order of the P-FO inerter is μ (1<μ < 2); x₁ and x₂ are the vertical displacements of the sprung and unsprung masses, respectively; x₀ is the road excitation displacement, when considering road harmonic excitation, it is expressed as X₀cos (ωt), where X₀ is the road excitation amplitude and ω is the excitation frequency. There are different ways to define the fractional calculus, here the Caputo’s definition³¹ is adopted, which is expressed as

D^{α} [x (t)] = \frac{1}{Γ (α - n)} \int_{0}^{t} \frac{f^{n} (τ) d τ}{{(t - τ)}^{α + 1 - n}}

(1)

where D represents a differential operator, α∈R, n∈N, n-1< α<n, Γ(α-n) is a gamma function.

Figure 1.

Dynamic model of P-FOIB suspension.

The dynamic equation of the P-FOIB suspension under road harmonic excitation is

{\begin{cases} m_{1} {x_{1}}^{″} + k_{1} (x_{1} - x_{2}) + c (x_{1}^{'} - x_{2}^{'}) + b D^{μ} (x_{1} - x_{2}) = 0 \\ m_{2} {x_{2}}^{″} + k_{1} (x_{2} - x_{1}) + k_{2} (x_{2} - x_{0}) + c (x_{2}^{'} - x_{1}^{'}) - b D^{μ} (x_{1} - x_{2}) = 0 \end{cases}

(2)

The dynamic response of the P-FOIB suspension is obtained using the Laplace transform method, which is given as

{\begin{cases} X_{1} (s) = \frac{k_{2} X_{0} (k_{1} + b s^{μ} + c s)}{\begin{array}{l} k_{1} k_{2} + k_{1} m_{1} s^{2} + k_{1} m_{2} s^{2} + k_{2} m_{1} s^{2} + m_{1} m_{2} s^{4} + c k_{2} s + b k_{2} s^{μ} \\ + c m_{1} s^{3} + c m_{2} s^{3} + b m_{1} s^{μ + 2} + b m_{2} s^{μ + 2} \end{array}} \\ X_{2} (s) = \frac{k_{2} X_{0} (k_{1} + b s^{μ} + m_{1} s^{2} + c s)}{\begin{array}{l} k_{1} k_{2} + k_{1} m_{1} s^{2} + k_{1} m_{2} s^{2} + k_{2} m_{1} s^{2} + m_{1} m_{2} s^{4} + c k_{2} s + b k_{2} s^{μ} \\ + c m_{1} s^{3} + c m_{2} s^{3} + b m_{1} s^{μ + 2} + b m_{2} s^{μ + 2} \end{array}} \end{cases}

(3)

where s is the Laplace variable. The dynamic performance of the suspension is evaluated by three performance indices: vehicle body acceleration (J_VBA), suspension dynamic deflection (J_SDD) and wheel dynamic load (J_WDL), which are obtained as

J_{V B A} = \frac{k_{2} s^{2} X_{0} (k_{1} + b s^{μ} + c s)}{k_{1} k_{2} + k_{1} m_{1} s^{2} + k_{2} m_{2} s^{2} + k_{2} m_{1} s^{2} + m_{1} m_{2} s^{4} + c k_{2} s + b k_{2} s^{μ} + c m_{1} s^{3} + c m_{2} s^{3} + b m_{1} s^{μ + 2} + b m_{2} s^{μ + 2}}

(4)

J_{S D D} = \frac{- k_{2} X_{0} m_{1} s^{2}}{k_{1} k_{2} + k_{1} m_{1} s^{2} + k_{1} m_{2} s^{2} + k_{2} m_{1} s^{2} + m_{1} m_{2} s^{4} + c k_{2} s + b k_{2} s^{μ} + c m_{1} s^{3} + c m_{2} s^{3} + b m_{1} s^{μ + 2} + b m_{2} s^{μ + 2}}

(5)

J_{W D L} = \frac{- k_{2} X_{0} (k_{1} m_{1} s^{2} + k_{1} m_{2} s^{2} + m_{1} m_{2} s^{4} + c m_{1} s^{3} + c m_{2} s^{3} + b m_{2} s^{(μ + 2)} + b m_{2} s^{(μ + 2)})}{k_{1} k_{2} + k_{1} m_{1} s^{2} + k_{1} m_{2} s^{2} + k_{2} m_{1} s^{2} + m_{1} m_{2} s^{4} + c k_{2} s + b k_{2} s^{μ} + c m_{1} s^{3} + c m_{2} s^{3} + b m_{1} s^{μ + 2} + b m_{2} s^{μ + 2}}

(6)

Letting s = jω and

j^{μ} = \cos (\frac{μ π}{2}) + j \sin (\frac{μ π}{2})

, where j is the complex variable and

j^{2} = - 1

, substituting these formula into equations (4)–(6) yields

{\begin{cases} J_{V B A} (ω) = \frac{k_{2} | X_{0} | ω^{2} \sqrt{A}}{\sqrt{Δ}} \\ J_{S D D} (ω) = \frac{k_{2} | X_{0} | m_{1} ω^{2}}{\sqrt{Δ}} \\ J_{W D L} (ω) = \frac{k_{2} | X_{0} | \sqrt{B}}{\sqrt{Δ}} \end{cases}

(7)

where

A = {[k_{1} + b ω^{μ} \cos (\frac{μ π}{2})]}^{2} + {[c ω + b ω^{μ} \sin (\frac{μ π}{2})]}^{2}

(8)

B = {[(m_{1} + m_{2}) b ω^{(μ + 2)} \cos (\frac{μ π}{2}) + k_{1} m_{1} ω^{2} - m_{1} m_{1} ω^{4}]}^{2} + {[(m_{1} + m_{2}) c ω^{3} + (m_{1} + m_{2}) b ω^{(μ + 2)} \sin (\frac{μ π}{2})]}^{2}

(9)

Δ = {[k_{1} k_{2} + b k_{2} ω^{μ} \cos (\frac{μ π}{2}) - (m_{1} + m_{2}) k_{1} ω^{2} - k_{2} m_{1} ω^{2} + m_{1} m_{2} ω^{4} - (m_{1} + m_{2}) b ω^{(μ + 2)} \cos (\frac{μ π}{2})]}^{2} + {[c k_{2} ω + b k_{2} ω^{μ} \sin (\frac{μ π}{2}) - (m_{1} + m_{2}) c ω^{3} - (m_{1} + m_{2}) b ω^{(μ + 2)} \sin (\frac{μ π}{2})]}^{2}

(10)

The structural parameter values of the P-FOIB suspension is shown in Table 1. The dynamic performance indices of the P-FOIB suspension with different inertance-to-mass ratios (λ = b/m₁) and fractional orders (μ) are shown in Figures 2 and 3, respectively.

Table 1.

Structural parameter values of P-FOIB suspension.

Symbol	Definition	Value
m ₁	Sprung mass	240 kg
m ₂	Unsprung mass	36 kg
k ₁	Suspension stiffness	16 kN/m
k ₂	Tire stiffness	160 kN/m
c	Suspension damping	1000 Ns/m
b	Inertance	12 kg
X ₀	Road excitation amplitude	0.02 m

Figure 2.

Dynamic performance indices of the P-FOIB suspension with different inertance-to-mass ratios under road harmonic excitation (μ = 1.7).

Figure 3.

Dynamic performance indices of the P-FOIB suspension with different fractional orders under road harmonic excitation (λ = 0.05).

As shown in Figure 2, with the increase of the inertance-to-mass ratio, the low-frequency resonance peak of the vehicle body acceleration, the low-frequency and high-frequency resonance peaks of the suspension dynamic deflection and wheel dynamic load gradually decrease, while the high-frequency resonance peak of the vehicle body acceleration shows a trend of first decreasing and then increasing, and meanwhile, they are slightly lower than the traditional passive two-element suspension. As shown in Figure 3, with the increase of the fractional order, the high-frequency resonance peaks of the dynamic performance indices first decrease and then increase, while the corresponding low-frequency resonance peaks gradually decrease. Compared with the passive integer-order inerter-based (P-IOIB) suspension, the P-FOIB suspension has smaller high-frequency resonance peak and slightly larger low-frequency resonance peak. Thus, the SA-FO inerter is designed and the dynamic characteristic of the SA-FOIB suspension is investigated in the following part to see whether it can further improve the vehicle dynamics of the P-FOIB suspension.

The prototype of a SA-FO inerter is shown in Figure 4, which is based on a hydraulic inerter. It mainly consists of a piston rod, piston, hydraulic cylinder, block buffer, high pressure electromagnetic valve, helical slender hard-tube and high pressure hose. Due to different diameters of two helical slender hard-tudes, the SA-FO inerter can have different inertances by adjusting the high pressure electromagnetic valve in an on or off state. The SA-FOIB suspension is shown in Figure 5, which is composed of a SA-FO inerter, a passive spring and a passive damper. The inertance of the SA-FO inerter is denoted as b_semi and can be regulated, which has a regulated maximum inertance b_max and a regulated minimum inertance b_min.

Figure 4.

Prototype of a SA-FO inerter.

Figure 5.

Dynamic model of SA-FOIB suspension.

The dynamic equation of the SA-FOIB suspension under road harmonic excitation is

{\begin{cases} m_{1} x_{1}^{″} + k_{1} (x_{1} - x_{2}) + c (x_{1}^{'} - x_{2}^{'}) + b_{s e m i} D^{μ} (x_{1} - x_{2}) = 0 \\ m_{2} x_{2}^{″} + k_{1} (x_{2} - x_{1}) + k_{2} (x_{2} - x_{0}) + c (x_{2}^{'} - x_{1}^{'}) - b_{s e m i} D^{μ} (x_{1} - x_{2}) = 0 \end{cases}

(11)

Substituting

x_{1} - x_{2} = x_{r}

and using the following transformations

{ω_{1}}^{2} = \frac{k_{1}}{m_{1}}, 2 ζ_{1} = \frac{c}{m_{1}}, {ω_{2}}^{2} = \frac{k_{2}}{m_{2}}, 2 ζ_{2} = \frac{c}{m_{2}}, u = \frac{m_{1}}{m_{2}}, {ω_{r}}^{2} = \frac{k_{1} + k_{2}}{m_{2}}, λ_{s e m i} = \frac{b_{s e m i}}{m_{1}}, \frac{b_{s e m i}}{m_{2}} = u λ_{s e m i}

(12)

Equation (11) can be transformed into

{x_{1}}^{″} + {ω_{1}}^{2} x_{r} + 2 ζ_{1} {x_{r}}^{'} + λ_{s e m i} D^{μ} x_{r} = 0 {x_{1}}^{″} - {x_{r}}^{″} - {ω_{r}}^{2} x_{r} + {ω_{2}}^{2} x_{1} - 2 ζ_{2} {x_{r}}^{'} - u λ_{s e m i} D^{μ} x_{r} = {ω_{2}}^{2} x_{0}

(13)

The AVS control strategy is used to adjust the inertance of the SA-FO inerter, which is based on the relative-acceleration-relative-velocity of the sprung and unsprung masses, the AVS control strategy is shown in Figure 6 and expressed as

λ_{s e m i} = {\begin{cases} λ_{m a x} x_{r}^{''} x_{r}^{'} \geq 0 \\ λ_{m i n} x_{r}^{''} x_{r}^{'} < 0 \end{cases}

(14)

where λ_max and λ_min are the regulated maximum and minimum inertance-to-mass ratios of the SA-FO inerter, respectively (λ_max = b_max/m₁, λ_min = b_min/m₁). The inertance of the SA-FO inerter regulates to the maximum one when the signs of the relative acceleration and relative velocity for the sprung and unsprung masses are the same, and regulates to the minimum one when the signs are opposite.

Figure 6.

Schematic diagram of regulation law for AVS control strategy.

The SA-FOIB suspension is a two DOF piece-wise nonlinear dynamic system, its dynamic response under road harmonic excitation can be obtained conveniently by using the averaging method. The averaging method is also known as the constant variation method or slowly varying amplitude method, its physical essence is in each motion cycle, the motion of the nonlinear dynamic system is considered as a harmonic motion, the amplitude and phase in the second cycle only undergo slight changes compared to the first cycle, but remain unchanged for a sufficiently small period. Assuming the amplitude and phase of the dynamic response for the nonlinear dynamic system as slowly varying functions of the time, expanding the derivatives of the amplitude and phase as the Fourier series and taking their averaging values in one period, then the approximate solutions of the nonlinear dynamic system under harmonic excitation can be obtained. Thus, the approximate solutions of the sprung mass displacement x₁ and relative displacement x_r are assumed as

x_{1} = a_{1} \cos φ_{1}

(15a)

x_{r} = a_{r} \cos φ_{r}

(15b)

where φ₁ = ωt + θ₁ and φ_r = ωt + θ_r. a₁(a_r) and θ₁(θ_r) are the amplitudes and phases of the displacement x₁ and x_r, respectively. The approximate solutions of the sprung mass velocity

x_{1}^{'}

and relative velocity

x_{r}^{'}

are assumed as

{x_{1}}^{'} = - ω a_{1} \sin φ_{1}

(16a)

{x_{r}}^{'} = - ω a_{r} \sin φ_{r}

(16b)

Differentiating equations (15) and (16) with respect to t, respectively, yields

{x_{1}}^{'} = {a_{1}}^{'} \cos φ_{1} - a_{1} (ω + {θ_{1}}^{'}) \sin φ_{1}

(17a)

{x_{r}}^{'} = {a_{r}}^{'} \cos φ_{r} - a_{r} (ω + {θ_{r}}^{'}) \sin φ_{r}

(17b)

{x_{1}}^{″} = - {a_{1}}^{'} ω \sin φ_{1} - a_{1} ω (ω + {θ_{1}}^{'}) \cos φ_{1}

(18a)

{x_{r}}^{″} = - {a_{r}}^{'} ω \sin φ_{r} - a_{r} ω (ω + {θ_{r}}^{'}) \cos φ_{r}

(18b)

Combining equations (16a) and (17a), equations (16b) and (17b) and yields

{a_{1}}^{'} \cos φ_{1} - a_{1} {θ_{1}}^{'} \sin φ_{1} = 0

(19a)

{a_{r}}^{'} \cos φ_{r} - a_{r} {θ_{r}}^{'} \sin φ_{r} = 0

(19b)

Substituting equations (15a), (15b), (17a), (17b), (18a), and (18b) into equation (13) yields

{a_{1}}^{'} \sin φ_{1} + a_{1} {θ_{1}}^{'} \cos φ_{1} = \frac{E}{ω}

(20a)

E = - a_{1} ω^{2} \cos φ_{1} + {ω_{1}}^{2} a_{r} \cos φ_{r} - 2 ζ_{1} ω a_{r} \sin φ_{r} + a_{r} λ ω^{μ} \cos (φ_{r} + \frac{μ π}{2})

(20b)

{a_{r}}^{'} \sin φ_{r} + a_{r} {θ_{r}}^{'} \cos φ_{r} = \frac{F}{ω}

(21a)

F = ({ω_{1}}^{2} + {ω_{r}}^{2} - ω^{2}) a_{r} \cos φ_{r} - 2 (ζ_{1} + ζ_{2}) ω a_{r} \sin φ_{r} - {ω_{2}}^{2} a_{1} \cos φ_{1} + (1 + u) λ a_{r} ω^{μ} \cos (φ_{r} + \frac{μ π}{2}) + {ω_{2}}^{2} X_{0} \cos (ω t)

(21b)

where in equations (20b) and (21b), and the following formulas are used

D_{t}^{α} \sin ω t = ω^{α} \sin (ω t + \frac{α π}{2})

(22a)

D_{t}^{α} \cos ω t = ω^{α} \cos (ω t + \frac{α π}{2})

(22b)

Combining equation (19a) with Eq. (20a), Eq. (19b) with equation (21a), the derivatives of the amplitudes a₁ and a_r, phases θ₁ and θ_r are obtained as

{\begin{cases} {a_{1}}^{'} = \frac{1}{ω} E \sin φ_{1} \\ a_{1} {θ_{1}}^{'} = \frac{1}{ω} E \cos φ_{1} \end{cases}

(23)

{\begin{cases} {a_{r}}^{'} = \frac{1}{ω} F \sin φ_{r} \\ a_{r} {θ_{r}}^{'} = \frac{1}{ω} F \cos φ_{r} \end{cases}

(24)

Based on the solution principle of the averaging method, the two items on the right side of equations (23) and (24) are integrated in one cycle and the averaging value is obtained, then the slow-varying amplitudes and phases in one cycle are given as

{\begin{cases} {\bar{a}}_{1}^{'} = \frac{1}{2 π ω} \int_{0}^{2 π} E \sin φ_{1} d φ_{1} \\ {\bar{a}}_{1} {\bar{θ}}_{1}^{'} = \frac{1}{2 π ω} \int_{0}^{2 π} E \cos φ_{1} d φ_{1} \end{cases}

(25)

{\begin{cases} {\bar{a}}_{r}^{'} = \frac{1}{2 π ω} \int_{0}^{2 π} F \sin φ_{r} d φ_{r} \\ {\bar{a}}_{r} {\bar{θ}}_{r}^{'} = \frac{1}{2 π ω} \int_{0}^{2 π} F \cos φ_{r} d φ_{r} \end{cases}

(26)

Substituting equation (15b) into equation (14), the semi-active control law is

λ_{s e m i} = {\begin{cases} λ_{\max} {a_{r}}^{2} ω^{3} \sin φ_{r} \cos φ_{r} \geq 0 \\ λ_{\min} {a_{r}}^{2} ω^{3} \sin φ_{r} \cos φ_{r} < 0 \end{cases}

(27)

In equations (25) and (26), the integrate domain of the adjustable inertance-to-mass ratio λ is

λ_{s e m i} = {\begin{cases} λ_{\max} φ_{r} \in (0, \frac{π}{2}] \cup (π, \frac{3 π}{2}] \\ λ_{\min} φ_{r} \in (\frac{π}{2}, π] \cup (\frac{3 π}{2}, 2 π] \end{cases}

(28)

Combining equations (25), (26), and (28), setting ${\bar{a}}_{1}^{'} = 0$ , ${\bar{θ}}_{1}^{'} = 0$ , ${\bar{a}}_{r}^{'} = 0$ , ${\bar{θ}}_{r}^{'} = 0$ and ω = 2πf, the amplitude-frequency relationships of the SA-FOIB suspension can be obtained as

\frac{{ω_{1}}^{2} a_{r} \sin β}{4 π f} - ζ_{1} a_{r} \cos β + \frac{a_{r} {(2 π f)}^{μ - 1}}{2 π} ({λ \min}_{\max} \cos (\frac{μ π}{2} + β) - \frac{a_{r} {(2 π f)}^{μ - 1}}{4} ({λ \min}_{\max} \sin (\frac{μ π}{2} - β) = 0))

(29a)

- a_{1} π f + \frac{{ω_{1}}^{2} a_{r} \cos β}{4 π f} + ζ_{1} a_{r} \sin β + \frac{a_{r} {(2 π f)}^{μ - 1}}{2 π} ({λ \max}_{\min} \sin (\frac{μ π}{2} + β) + \frac{a_{r} {(2 π f)}^{μ - 1}}{4} ({λ \min}_{\max} \cos (\frac{μ π}{2} - β) = 0))

(29b)

\begin{array}{l} - (ζ_{1} + ζ_{2}) a_{r} + \frac{{ω_{2}}^{2} a_{1} \sin β}{4 π f} + \frac{(1 + u) (λ_{\max} - λ_{\min}) a_{r} {(2 π f)}^{μ - 1} \cos \frac{μ π}{2}}{2 π} \\ - \frac{(1 + u) (λ_{\max} + λ_{\min}) a_{r} {(2 π f)}^{μ - 1} \sin \frac{μ π}{2}}{4} + \frac{{ω_{2}}^{2} X_{0} \sin θ_{r}}{4 π f} = 0 \end{array}

(29c)

\begin{array}{l} \frac{({ω_{1}}^{2} + {ω_{r}}^{2} - {(2 π f)}^{2}) a_{r}}{4 π f} + \frac{- {ω_{2}}^{2} a_{1} \cos β}{4 π f} + \frac{(1 + u) (λ_{\min} - λ_{\max}) a_{r} {(2 π f)}^{μ - 1} \sin \frac{μ π}{2}}{2 π} \\ + \frac{(1 + u) (λ_{\max} + λ_{\min}) a_{r} {(2 π f)}^{μ - 1} \cos \frac{μ π}{2}}{4} + \frac{{ω_{2}}^{2} X_{0} \cos θ_{r}}{4 π f} = 0 \end{array}

(29d)

For a given excitation frequency f, the corresponding amplitude and phase can be obtained by solving the above four equations numerically. Then the three dynamic performance indices of the SA-FOIB suspension are

| J_{V B A} | = {(2 π f)}^{2} a_{1}

(30a)

| J_{S D D} | = a_{r}

(30b)

| J_{W D L} | = k_{2} \sqrt{\begin{array}{l} {(a_{1} \cos θ_{1} - a_{r} \cos θ_{r} - x_{0})}^{2} \\ + {(- a_{1} \sin θ_{1} + a_{r} \sin θ_{r})}^{2} \end{array}}

(30c)

In order to verify the accuracy of the analytical solution, the numerical solution is acquired by using the fourth order Runge-Kutta method to solve equation (13) with equation (14). The comparison results are shown in Figure 7. The analytical and numerical solutions are consistent, which indicates that the averaging method can solve the dynamic response of the SA-FOIB suspension under road harmonic excitation very well.

Figure 7.

Comparison of analytical and numerical solutions of the dynamic response for the SA-FOIB suspension (λ_max = 0.02, λ_min = 0.01).

Dynamic performance analysis of SA-FOIB suspension

Road harmonic excitation

Considering the road harmonic excitation firstly, which can show the dynamic performance of the SA-FOIB suspension in the frequency domain, the vehicle body acceleration J_VBA, suspension dynamic deflection J_SDD and wheel dynamic load J_WDL are obtained using equations (30a), (30b), and (30c), respectively. The effect of the fractional order of the SA-FO inerter on the dynamic performance of the SA-FOIB suspension is shown in Figure 8, noting that when μ = 2, the SA-FOIB suspension is equivalent to the semi-active integer-order inerter-based (SA-IOIB) suspension. As the fractional order increases, the high-frequency resonance peaks of the dynamic performance indices first decrease and then increase, while the low-frequency resonance peaks show a decreasing trend. Compared with the SA-IOIB suspension, the SA-FOIB suspension obtains smaller high-frequency resonance peaks and slightly larger low-frequency resonance peaks, the changing trend is similar to the passive one.

Figure 8.

Dynamic performance indices of the SA-FOIB suspension with different fractional orders under road harmonic excitation (λ_max = 0.2, λ_min = 0.1).

The dynamic performance indices of the SA-FOIB suspension with different λ_max and λ_min under road harmonic excitation are shown in Figures 9 and 10, respectively, the corresponding low-frequency and high-frequency resonance peaks are shown in Figures 11 and 12, respectively. As shown in Figures 9 and 11, when the minimum inertance-to-mass ratio λ_min is fixed, with the increase of the maximum inertance-to-mass ratio λ_max, the high-frequency resonance peak of the vehicle body acceleration first decreases and then increases, while the corresponding low-frequency resonance peak decreases, the low-frequency and high-frequency resonance peaks of the suspension dynamic deflection and wheel dynamic load shows a decreasing trend. Compared with the P-FOIB suspension, the low-frequency resonance peaks of the three dynamic performance indices are larger, while the high-frequency resonance peaks of the suspension dynamic deflection and wheel dynamic load are smaller, the high-frequency resonance peak of the vehicle body acceleration is first larger and then gradually smaller. Compared with the passive traditional two-element suspension (Curve 1: λ = 0 shown in Figure 9), the low-frequency and high-frequency resonance peaks of the three dynamic performance indices are smaller.

Figure 9.

Dynamic performance indices of the SA-FOIB suspension with different λ_max under road harmonic excitation (λ_min = 0, μ = 1.7, Curve 1: λ = 0, Curve 2: λ = 0.01, Curve 3: λ = 0.05, Curve 4: λ = 0.1, Curve 5: λ_max = 0.01, Curve 6: λ_max = 0.05, Curve 7: λ_max = 0.1).

Figure 10.

Dynamic performance indices of the SA-FOIB suspension with different λ_min under road harmonic excitation (λ_max = 0, μ = 1.7, Curve 1: λ = 0, Curve 2: λ = 0.1, Curve 3: λ_min = 0, Curve 4: λ_min = 0.01, Curve 5: λ_min = 0.05, Curve 6: λ_min = 0.08).

Figure 11.

Low-frequency and high-frequency resonance peaks of the dynamic performance indices for the SA-FOIB suspension with different λ_max under road harmonic excitation (μ = 1.7).

Figure 12.

Low-frequency and high-frequency resonance peaks of the dynamic performance indices for the SA-FOIB suspension with different λ_min under road harmonic excitation (μ = 1.7).

As shown in Figures 10 and 12, when the maximum inertance-to-mass ratio λ_max is fixed, with the increase of the minimum inertance-to-mass ratio λ_min, the low-frequency resonance peaks of the three dynamic performance indices all show a decreasing trend, while the corresponding high-frequency resonance peaks gradually increase. Compared with the P-FOIB suspension, the low-frequency resonance peaks of the three dynamic performance indices are larger, while the high-frequency resonance peaks are smaller. Compared with the passive traditional two-element suspension (Curve 1: λ = 0 shown in Figure 10), the low-frequency resonance peak of the vehicle body acceleration, the low-frequency and high-frequency resonance peaks of the suspension dynamic deflection and wheel dynamic load are smaller, while the high-frequency resonance peak of the vehicle body acceleration is gradually larger with the increase of the minimum inertance-to-mass ratio.

In summary, for road harmonic excitation, the SA-FOIB suspension can further improve the high-frequency dynamic performance of the P-FOIB suspension, where the high-frequency resonance peaks of the three dynamic performance indices are smaller, and can also obtain a better dynamic performance than the passive traditional two-element suspension, where both the low-frequency and high-frequency resonance peaks of the three dynamic performance indices are smaller.

Road random excitation

Then considering the road random excitation, the filtered white noise method is used to simulate the random excitation of the class C road profile, the dynamic performance of the SA-FOIB suspension under road random excitation is investigated, the root-mean-square (RMS) values of the dynamic performance indices are shown. The time domain model of the filtered white noise road roughness is

z_{r}^{'} (t) = - 2 π f_{0} z_{r} (t) + 2 π n_{0} \sqrt{G_{q} (n_{0}) u} ω (t)

(31)

where G_q (n₀) is the road roughness coefficient and is chosen as 64 × 10⁻⁶ m³, u is the vehicle velocity and is chosen as 20 m/s, n₀ is reference spatial frequency and is chosen as 0.011 m⁻¹, f₀ is the time frequency, ω(t) is the Gaussian white noise with unit intensity of 1.

The effect of the fractional order of the SA-FO inerter on the dynamic performance of the SA-FOIB suspension under road random excitation is shown in Figure 13. As the fractional order increases, the three dynamic performance indices show a trend of first decreasing and then increasing, the RMS value of the vehicle body acceleration has a minimum value when the fractional order is around 1.2, the RMS values of the suspension dynamic deflection and wheel dynamic load have the minimum values when the fractional order is around 1.9.

Figure 13.

The RMS value of the dynamic performance indices for the SA-FOIB suspension with different μ under road random excitation (λ_max = 0.1, λ_min = 0).

The dynamic performance indices of the SA-FOIB suspension with different λ_max and λ_min under road random excitation are shown in Figures 14 and 15, respectively, the corresponding detailed RMS values of the dynamic performance indices are shown in Tables 2 and 3, respectively. When the minimum inertance-to-mass ratio λ_min is fixed, with the increase of the maximum inertance-to-mass ratio λ_max, the RMS value of the vehicle body acceleration gradually increases, while the RMS values of the suspension dynamic deflection and wheel dynamic load gradually decrease. When the maximum inertance-to-mass ratio λ_max is fixed, with the increase of the minimum inertance-to-mass ratio λ_min, the RMS value of the vehicle body acceleration gradually decreases, while the RMS values of the suspension dynamic deflection and wheel dynamic load gradually increase. Compared with the P-FOIB suspension, the RMS value of the vehicle body acceleration is larger, while the RMS values of the suspension dynamic deflection and wheel dynamic load are smaller.

Figure 14.

Dynamic performance indices of the SA-FOIB suspension with different λ_max under road random excitation (μ = 1.7, λ_min = 0).

Figure 15.

Dynamic performance indices of the SA-FOIB suspension with different λ_min under road random excitation (μ = 1.7, λ_max = 0).

Table 2.

The RMS values of the dynamic performance indices for the SA-FOIB suspension with different λ_max under road random excitation (μ = 1.7, λ_min = 0).

λ _max	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
J_VBA(m/s²)	1.889	1.891	1.902	1.913	1.925	1.933	1.946	1.952	1.964	1.971	2.099
J_WDL(N)	2654	2633	2611	2593	2572	2556	2537	2515	2506	2499	2493
J_SDD(m)	0.0165	0.0163	0.0161	0.0159	0.0158	0.0154	0.0153	0.0152	0.0149	0.0148	0.0147

Table 3.

The RMS values of the dynamic performance indices for the SA-FOIB suspension with different λ_min under road random excitation (μ = 1.7, λ_max = 0).

λ _min	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
J_VBA(m/s²)	2.025	2.023	2.022	2.020	2.019	2.018	2.017	2.015	2.013	2.003	1.997
J_WDL(N)	2478	2485	2493	2501	2516	2523	2535	2548	2563	2586	2605
J_SDD(m)	0.0148	0.0149	0.0152	0.0153	0.0153	0.0154	0.0155	0.0156	0.0156	0.0157	0.0158

Structural parameters optimization of SA-FOIB suspension

In this Section, the structural parameters optimization of the SA-FOIB suspension subjects to the road random excitation is investigated, which is based on the genetic algorithm, three mainly structural parameters are optimized, the damping of the SA-FOIB suspension (c), the maximum and minimum inertance-to-mass ratios of the SA-FOIB suspension (λ_max, λ_min) and the fractional order of the SA-FO inerter (μ). The basic idea of the genetic algorithm is to simulate the natural selection mechanism of the biological evolution. Through selection, crossover, mutation and other means, it can continuously generate individuals with stronger adaptability to the environment. It is a strong robust algorithm that can solve optimization problems. The goal of structural parameters optimization of the SA-FOIB suspension is to reduce the vehicle body acceleration and wheel dynamic load, and the objective function of the dynamic performance optimization for the SA-FOIB suspension can be expressed as

f = ρ_{1} \frac{r m s (J_{V B A 1})}{r m s (J_{V B A})} + ρ_{2} \frac{r m s (J_{W D L 1})}{r m s (J_{W D L})}

(32)

where ρ₁ and ρ₂ are the weight coefficients of the vehicle body acceleration and wheel dynamic load, respectively, if one of the two dynamic performance indices is more important, the corresponding weight coefficient is larger, here the vehicle body acceleration and wheel dynamic load are the same important, thus the two weight coefficients are equal and chosen as 0.5. The rms (J_VBA1) and rms (J_WDL1) are the RMS values of the vehicle body acceleration and wheel dynamic load for the optimized SA-FOIB suspension, respectively; the rms (J_VBA) and rms (J_WDL) are the RMS values of the vehicle body acceleration and wheel dynamic load for the reference P-FOIB suspension, respectively. The current limit travel range of the suspension for most passenger vehicles is 0.07-0.09 m, here it is taken as 0.08 m, and the suspension dynamic deflection should be within 1/3 of the limit travel. As a constraint, the RMS value of the suspension dynamic deflection should be less than 0.027 m during optimization. The other structural parameters of the SA-FOIB suspension used in the optimization simulation are shown in Table 1, the initial population number and genetic algorithm iteration are chosen as 50. The parameter optimization ranges are chosen as: c∈[800 Ns/m, 5000 Ns/m], λ_max∈[0, 0.1], λ_min∈[0, 0.1], μ∈[1, 2], the vehicle velocity is chosen as 20 m/s and the class C road profile is chosen.

The calculation process of the structural parameters optimization for the SA-FOIB suspension is as follows:

(a) Genetic algorithm initialization. Population size, chromosome length, crossover and mutation probability and stopping rules are set through the MATLAB Genetic Algorithm Toolbox.

(b) Sorting the mutated population by fitness continuously, and selecting individuals with high fitness as new populations. When the results meet the termination conditions, the optimal structural parameters of the SA-FOIB suspension (c, λ_max, λ_min, μ) are calculated.

The convergent process of the genetic algorithm is shown in Figure 16, when the genetic algorithm iteration reaches at 18 iteration and the average change of fitness values is very small, the optimal structural parameters of the SA-FOIB suspension are obtained as: c = 1408.8 Ns/m, λ_max = 0.0407, λ_min = 0.0123, μ = 1.79. The dynamic performance of the optimized SA-FOIB suspension is compared with three other suspensions: the traditional two-element suspension (λ = 0), the P-FOIB suspension (λ = 0.1, μ = 1.7) and the unoptimized SA-FOIB suspension (λ_max = 0.1, λ_min = 0, μ = 1.7). The dynamic performance indices for four types of suspensions are compared in Figure 17, the corresponding RMS values are shown in Table 4. For the traditional two-element suspension, the RMS values of the three dynamic performance indices are 1.9244 m/s², 0.0158 m and 2666.7 N, respectively. The corresponding RMS values for the P-FOIB suspension is 1.9840 m/s², 0.0154 m and 2569.2 N, respectively, which improves the suspension dynamic deflection and wheel dynamic load by 2.53 % and 3.66 %, respectively, while worsens the vehicle body acceleration by 3.09 % compared with the traditional two-element suspension. For the unoptimized SA-FOIB suspension, the RMS values of the three dynamic performance indices are 1.9517 m/s², 0.0159 m and 2660.3 N, respectively, compared with the P-FOIB suspension, the RMS value of the vehicle body acceleration decreases by 1.63 %, while the RMS values of suspension dynamic deflection and wheel dynamic load increase by 3.25 % and 3.55 %, respectively. When using the genetic algorithm optimization, for the optimized SA-FOIB suspension, the RMS values of the three dynamic performance indices are 2.1140 m/s², 0.0126 m and 2135.3 N, respectively, which improves the suspension dynamic deflection and wheel dynamic load by 20.75 % and 19.73 %, respectively, while worsens the vehicle body acceleration by 8.32 % compared with the unoptimized SA-FOIB suspension, it obtains the smallest suspension dynamic deflection and wheel dynamic load among the four suspensions.

Figure 16.

The convergent process of the genetic algorithm.

Figure 17.

Comparison of the time domain response of the dynamic performance indices for different types of suspensions.

Table 4.

The RMS values of the dynamic performance indices for different types of suspensions.

Suspension	c (Ns/m)	λ	λ _max	λ _min	μ	J_VBA(m/s²)	J_SDD(m)	J_WDL(N)
Traditional two-element	1000	-	-	-	2	1.9244	0.0158	2666.7
P-FOIB	1000	0.1	-	-	1.7	1.9244	0.0154	2569.2
SA-FOIB	1000	-	0.1	0	1.7	1.9244	0.0159	2660.3
Optimized SA-FOIB	1408.8	-	0.0407	0.0123	1.79	2.1440	0.0126	2135.3

In order to show the effectiveness of the optimized SA-FOIB suspension more clearly, the comparison of the power spectral density of the dynamic performance indices for different types of suspensions is shown in Figure 18, which is expressed in the logarithm coordinate. It can be further seen that compared with the other three suspensions, the optimized SA-FOIB suspension decreases the vehicle body acceleration, suspension dynamic deflection and wheel dynamic load in the high-frequency band, while increases them in the low-frequency band which is below 1 Hz.

Figure 18.

Comparison of the power spectral density of the dynamic performance indices for different types of suspensions.

Conclusion

In this paper, the SA-FO inerter is proposed and used in the vehicle suspension to constitute the SA-FOIB suspension, which consists of two adjustable inertances. Based on the mechanical characteristic of the inerter, the AVS control strategy is proposed to adjust the inertance of the SA-FO inerter. The dynamic model of the SA-FOIB suspension with AVS control is established, its dynamic response under road harmonic excitation is obtained using the averaging method, the dynamic performance under road harmonic and random excitations is analyzed, the optimized structural parameters are obtained. The conclusions are summarized as follows:

(1) The P-FOIB suspension has smaller high-frequency resonance peaks and slightly larger low-frequency resonance peaks for the dynamic performance indices than the P-IOIB suspension. The SA-FOIB suspension is a two DOF piece-wise nonlinear dynamic system, its dynamic response under road harmonic excitation is obtained using the averaging method, the analytical solution is consistent with the numerical solution, which verifies the effectiveness of the averaging method for solving fractional-order piece-wise nonlinear dynamic system.

(2) When the SA-FOIB suspension is under road harmonic excitation, it can further enhance the high-frequency dynamic performance of the P-FOIB suspension, which the high-frequency resonance peaks of the three dynamic performance indices are smaller. For road random excitation, its RMS value of the vehicle body acceleration is larger, while the RMS values of the suspension dynamic deflection and wheel dynamic load are smaller, compared with the P-FOIB suspension.

(3) The optimized structural parameters of the SA-FOIB suspension is obtained using the genetic algorithm optimization method, the optimized SA-FOIB suspension has smaller suspension dynamic deflection and wheel dynamic load than the traditional two-element, P-FOIB and unoptimized SA-FOIB suspensions, which are smallest among the four suspensions, while the vehicle body acceleration is larger.

Footnotes

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 work was supported by the National Natural Science Foundation of China (Grant No. 12172153, 51805216), Natural Science Foundation of Jiangsu Province (Grant No. BK20231360), National Key Laboratory of Automotive Chassis Integration and Bionics (Jilin University) (Grant No. 20210212), Major Project of Basic Science (Natural Science) of the Jiangsu Higher Education Institutions (22KJA410001), Changzhou Science and Technology Program (CZ20240019) and the project funded by the Youth Talent Cultivation Program of Jiangsu University.

ORCID iD

Yong Wang

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