# Modeling and Optimization of Vehicle Suspension Employing a Nonlinear Fluid Inerter

Authors: Yujie Shen; Long Chen; Yanling Liu; Xiaoliang Zhang
Journal: Shock and Vibration
 (2016)

---

## Abstract

An ideal inerter has been applied to various vibration engineering fields because of its superior vibration isolation performance. This paper proposes a new type of fluid inerter and analyzes the nonlinearities including friction and nonlinear damping force caused by the viscosity of fluid. The nonlinear model of fluid inerter is demonstrated by the experiments analysis. Furthermore, the full-car dynamic model involving the nonlinear fluid inerter is established. It has been detected that the performance of the vehicle suspension may be influenced by the nonlinearities of inerter. So, parameters of the suspension system including the spring stiffness and the damping coefficient are optimized by means of QGA (quantum genetic algorithm), which combines the genetic algorithm and quantum computing. Results indicate that, compared with the original nonlinear suspension system, the RMS (root-mean-square) of vertical body acceleration of optimized suspension has decreased by 9.0%, the RMS of pitch angular acceleration has decreased by 19.9%, and the RMS of roll angular acceleration has decreased by 9.6%.


---

## Body

## 1. Introduction

Ride comfort is one of the most important performances of the vehicle and is greatly influenced by the vehicle suspension system set between the vehicle body and the wheels. There are mainly three kinds of suspension systems, including active suspension [1–3], semiactive suspension [4–6], and passive suspension. For active suspension and semiactive suspension, they always have a changeable suspension parameter like spring stiffness or damping coefficient to preserve the two desired aims which are vehicle handling and ride comfort. But for passive suspension system, it only has two parallel spring and damper elements. In this case, the vibration isolation performance can be improved by parameters’ optimization [7, 8] or involving new mechanical element.In [9, 10], inerter was proposed with the property that the force in the mechanical device is proportional to the relative accelerations of the two terminals. It was widely known as “J-damper” [11] in Formula One Racing since it delivered significant performance gains in handling and grip. As a new mechanical network element corresponding to the capacitor, inerter was widely used in passive network synthesis [12–14]. Also, the vibration isolation performance of conventional spring-damper isolation system can be significantly improved by involving the inerter, which has been verified in vehicle suspension [15–18], train suspension [19, 20], building suspension [21], dynamic vibration absorber [22–24], and vibration isolation [25, 26].Researches above are all concentrated on the ideal inerter, ignoring considering its nonlinearities. In [27], the nonlinearities including friction, elastic effect, and backlash of a rack-and-pinion and ball-screw inerter were taken into consideration. The above study found that the performance of vehicle suspension may be significantly degraded by the nonlinear effects. It seems important to investigate the influences of the nonlinearities of inerter on vehicle suspension system. In [28, 29], a new kind of inerter called fluid inerter was proposed and the inner force was investigated, but the effect of the nonlinear model on the suspension performance was not discussed. This paper explores a new vehicle suspension employing the nonlinear fluid inerter. The parameters of the suspension are optimized by using optimal algorithm to reduce the effect of the nonlinearities, based on the full-car model including the nonlinear inerter. The paper is arranged as follows.In Section2, a new fluid inerter is designed and the nonlinearities involving friction and damping force are taken into consideration. Experiments are finished to demonstrate the nonlinear model. Suspension system involving the nonlinear fluid inerter is built in Section 3 and the parameters of the suspension are obtained by means of QGA (quantum genetic algorithm) in Section 4. Dynamic performance of the vehicle suspension is analyzed in Section 5. Finally, some conclusions are shown in Section 6.


## 2. Design and Test of Fluid Inerter

There are many mechanical types of inerter, such as ball-screw inerter, rack-and-pinion inerter [30], hydraulic-motor inerter [31], and fluid inerter [28, 29]. Motion transferred mechanics are used in the device to enlarge the inertia force which is the most meaningful thing to inerter. The dynamic equation of inerter is(1)
F
=
b
v
˙
1
-
v
˙
2
,where F is the force in the device, v
1 and v
2 are the velocities, and b is the inertance with the unit kg. In this section, a new type of fluid inerter is designed to investigate the nonlinearities.


### 2.1. Ideal Fluid Inerter

The fluid inerter is similar to rack-and-pinion inerter or ball-screw inerter because the fluid flowing in the helical channel can be taken as a “fluid flywheel” to provide the inertia force. The prototype of the fluid inerter with the helical channel outside the cylinder is shown in Figure1.Figure 1
Prototype of fluid inerter.In Figure1, F is the force applied in the fluid inerter and x is the displacement of the piston. The hydraulic cylinder and the piston rod are the two terminals of the fluid inerter. When the piston pushes the fluid in the left of the hydraulic cylinder into the helical channel, the fluid flows through the helical channel into the right hydraulic cylinder in order to compensate the pressure loss.Firstly, the fluid in the cylinder and the helical channel is taken as nondisclosure and noncompressed. According to [29], the inertance b can be gained as(2)
b
=
m
1
+
h
/
2
π
r
4
2
S
1
S
2
2
,where m is the mass of the fluid in the helical channel, S
1 is the effective area of the hydraulic cylinder, S
2 is the section area of the helical channel, h is the pitch of the helical channel, and r
4 is the helix radius of the helical channel.S
1 can be gained by the equation(3)
S
1
=
π
r
2
2
-
r
1
2
,where r
1 is the radius of the piston rod and r
2 is the inside radius of the hydraulic cylinder.S
2 can be gained by the equation(4)
S
2
=
π
r
3
2
,where r
3 is the radius of the helical channel.m can be gained by the equation(5)
m
=
ρ
S
2
l
,where ρ is the oil density and l is the length of helical channel and can be approximated as(6)
l
=
n
h
2
+
2
π
r
4
2
+
2
l
0
,where n is the circle number of the helical channel and l
0 is the length from the inlet and outlet to the helical channel.


### 2.2. Nonlinearities Analysis of Fluid Inerter

When the forceF is applied to the fluid inerter, both the friction caused by the motion of piston rod and hydraulic cylinder f and the damping force caused by the viscosity of the fluid f
c will block the motion of the piston. The dynamic equation is(7)
F
-
f
-
f
c
=
b
x
¨
.The friction model can be described by the Column Friction Model as follows:(8)
f
=
f
0
sign
⁡
v
,where f
0 is the amplitude of the friction, v is the velocity of the piston, and sign
⁡
(
v
) is a symbolic function; it is 1 when v
>
0 and −1 when v
<
0. The direction of friction f between the piston and hydraulic cylinder is related to the velocity of the piston and has the constant value.For a hydraulic device, the damping force caused by the viscosity of the fluid is the most important impact on the dynamic performance. According to the Hagen Poiseuille Law, the flow of the fluid in the helical channelQ and the pressure drop between the two terminals Δ
p have the relation as follows:(9)
Q
=
π
r
3
4
8
μ
l
Δ
p
,in which μ is viscosity.The average flow velocity of fluid in the helical channelu is(10)
u
=
Q
S
2
=
r
3
2
8
μ
l
Δ
p
.A fluid flow entering a new channel will also experience an energy loss at the inlet and outlet of the hydraulic cylinder; the empirical formula for the resulting pressure drop is given in [32]. The pressure drop across the inlet and outlet is thus estimated to be(11)
Δ
p
1
=
0.5
ρ
u
2
2
,
Δ
p
2
=
ρ
u
2
2
.According to the volume conservation law,(12)
S
1
x
˙
=
S
2
u
.So, the damping force of the inerter can be gained as follows:(13)
f
c
=
Δ
p
+
Δ
p
1
+
Δ
p
2
S
1
=
3
ρ
S
1
3
4
S
2
2
x
˙
2
+
8
μ
l
S
1
2
r
3
2
S
2
x
˙
.The damping force is related not only to the velocity but also to the square of the velocity of the piston. The coefficient of the square of the velocity is set asC
1 and the coefficient of the velocity is set as C
2.The dynamic model of the fluid inerter can be shown in Figure2.Figure 2
Dynamic model of fluid inerter.In Figure2, z
1 and z
2 are displacements.The total force of the inerter can be gained as follows:(14)
F
=
f
0
sign
⁡
z
˙
1
-
z
˙
2
+
C
1
z
˙
1
-
z
˙
2
2
+
C
2
z
˙
1
-
z
˙
2
+
b
z
¨
1
-
z
¨
2
.


### 2.3. Experiment Identification

In this part, experiments are carried out in the INSTRON 8800 bench to test the performance of the fluid inerter. The parameters of the designed fluid inerter are shown in Table1.Table 1
Parameters of the fluid inerter.
Name
Value
Radius of piston rodr
1/m
0.012
Inner radius of the cylinderr
2/m
0.028
Radius of the helical channelr
3/m
0.005
Helix radius of channelr
4/m
0.1
Pitch of helical channelh/m
0.012
Circle number of helical channeln
14
Oil densityρ/kg⋅m−3
800
Length of transition sectionl
0/m
0.1
Viscosity of fluidμ/Pa⋅s
0.027So, the parameters of the nonlinear fluid inerter can be calculated whereb is 370 kg, C
1 is 790 N·s2·m−2, and C
2 is 4001 N·s·m−1.The structure of the bench test is shown in Figure3.Figure 3
Bench test.The displacement and the force signals can be collected by the sensors in the device. In order to gain the amplitude of friction, the displacement input is set as a triangle wave and the amplitude is 10 mm. Figure4 shows the force under 0.1 Hz condition.Figure 4
Force in 0.1 Hz.It can be noted that the force shapes like a square wave because both the damping force and the inertia force are small enough. It can be inferred that the friction force is a constant value which is 400 N. Furthermore, the fluid inerter is tested under different frequency to verify the theoretical model. The amplitude of the sinusoidal displacement input is set as 20 mm under the frequency of 0.1 Hz, 0.3 Hz, and 0.5 Hz, 10 mm under the frequency of 3 Hz, 5 Hz, and 8 Hz, and 5 mm under the frequency of 10 Hz, 12 Hz, and 15 Hz.In order to test the effectiveness of the nonlinear model, the mean value of force amplitude under the sinuous displacement input of simulation and experiment is taken into consideration. Figure5 shows the absolute difference of the force amplitude between theoretical analysis and practical experiment. It can be seen that the second data point is a little larger than the first data point and the third data point, but the absolute differences are all less than 3%, so it can be inferred that the simulation results are very close to the experiment results in low frequency. Furthermore, the absolute difference becomes a little larger with the increase of the frequency. Figure 6 shows the force of the fluid inerter between the theoretical analysis and experiment in 0.5 Hz and 12 Hz. It can be inferred that the experiment results in higher frequency coincide less well with the simulation than that in low frequency, which may be because the temperature of the fluid has increased for the vibration of the system and the fluid flow is not laminar. In conclusion, the experiment results coincide with the simulation well, which demonstrates the effectiveness of the theoretical model of fluid inerter.Figure 5
Difference between theoretical model and experiment.Figure 6
Force in time domain.
(a)
0.5 Hz
(b)
12 Hz


## 2.1. Ideal Fluid Inerter

The fluid inerter is similar to rack-and-pinion inerter or ball-screw inerter because the fluid flowing in the helical channel can be taken as a “fluid flywheel” to provide the inertia force. The prototype of the fluid inerter with the helical channel outside the cylinder is shown in Figure1.Figure 1
Prototype of fluid inerter.In Figure1, F is the force applied in the fluid inerter and x is the displacement of the piston. The hydraulic cylinder and the piston rod are the two terminals of the fluid inerter. When the piston pushes the fluid in the left of the hydraulic cylinder into the helical channel, the fluid flows through the helical channel into the right hydraulic cylinder in order to compensate the pressure loss.Firstly, the fluid in the cylinder and the helical channel is taken as nondisclosure and noncompressed. According to [29], the inertance b can be gained as(2)
b
=
m
1
+
h
/
2
π
r
4
2
S
1
S
2
2
,where m is the mass of the fluid in the helical channel, S
1 is the effective area of the hydraulic cylinder, S
2 is the section area of the helical channel, h is the pitch of the helical channel, and r
4 is the helix radius of the helical channel.S
1 can be gained by the equation(3)
S
1
=
π
r
2
2
-
r
1
2
,where r
1 is the radius of the piston rod and r
2 is the inside radius of the hydraulic cylinder.S
2 can be gained by the equation(4)
S
2
=
π
r
3
2
,where r
3 is the radius of the helical channel.m can be gained by the equation(5)
m
=
ρ
S
2
l
,where ρ is the oil density and l is the length of helical channel and can be approximated as(6)
l
=
n
h
2
+
2
π
r
4
2
+
2
l
0
,where n is the circle number of the helical channel and l
0 is the length from the inlet and outlet to the helical channel.


## 2.2. Nonlinearities Analysis of Fluid Inerter

When the forceF is applied to the fluid inerter, both the friction caused by the motion of piston rod and hydraulic cylinder f and the damping force caused by the viscosity of the fluid f
c will block the motion of the piston. The dynamic equation is(7)
F
-
f
-
f
c
=
b
x
¨
.The friction model can be described by the Column Friction Model as follows:(8)
f
=
f
0
sign
⁡
v
,where f
0 is the amplitude of the friction, v is the velocity of the piston, and sign
⁡
(
v
) is a symbolic function; it is 1 when v
>
0 and −1 when v
<
0. The direction of friction f between the piston and hydraulic cylinder is related to the velocity of the piston and has the constant value.For a hydraulic device, the damping force caused by the viscosity of the fluid is the most important impact on the dynamic performance. According to the Hagen Poiseuille Law, the flow of the fluid in the helical channelQ and the pressure drop between the two terminals Δ
p have the relation as follows:(9)
Q
=
π
r
3
4
8
μ
l
Δ
p
,in which μ is viscosity.The average flow velocity of fluid in the helical channelu is(10)
u
=
Q
S
2
=
r
3
2
8
μ
l
Δ
p
.A fluid flow entering a new channel will also experience an energy loss at the inlet and outlet of the hydraulic cylinder; the empirical formula for the resulting pressure drop is given in [32]. The pressure drop across the inlet and outlet is thus estimated to be(11)
Δ
p
1
=
0.5
ρ
u
2
2
,
Δ
p
2
=
ρ
u
2
2
.According to the volume conservation law,(12)
S
1
x
˙
=
S
2
u
.So, the damping force of the inerter can be gained as follows:(13)
f
c
=
Δ
p
+
Δ
p
1
+
Δ
p
2
S
1
=
3
ρ
S
1
3
4
S
2
2
x
˙
2
+
8
μ
l
S
1
2
r
3
2
S
2
x
˙
.The damping force is related not only to the velocity but also to the square of the velocity of the piston. The coefficient of the square of the velocity is set asC
1 and the coefficient of the velocity is set as C
2.The dynamic model of the fluid inerter can be shown in Figure2.Figure 2
Dynamic model of fluid inerter.In Figure2, z
1 and z
2 are displacements.The total force of the inerter can be gained as follows:(14)
F
=
f
0
sign
⁡
z
˙
1
-
z
˙
2
+
C
1
z
˙
1
-
z
˙
2
2
+
C
2
z
˙
1
-
z
˙
2
+
b
z
¨
1
-
z
¨
2
.


## 2.3. Experiment Identification

In this part, experiments are carried out in the INSTRON 8800 bench to test the performance of the fluid inerter. The parameters of the designed fluid inerter are shown in Table1.Table 1
Parameters of the fluid inerter.
Name
Value
Radius of piston rodr
1/m
0.012
Inner radius of the cylinderr
2/m
0.028
Radius of the helical channelr
3/m
0.005
Helix radius of channelr
4/m
0.1
Pitch of helical channelh/m
0.012
Circle number of helical channeln
14
Oil densityρ/kg⋅m−3
800
Length of transition sectionl
0/m
0.1
Viscosity of fluidμ/Pa⋅s
0.027So, the parameters of the nonlinear fluid inerter can be calculated whereb is 370 kg, C
1 is 790 N·s2·m−2, and C
2 is 4001 N·s·m−1.The structure of the bench test is shown in Figure3.Figure 3
Bench test.The displacement and the force signals can be collected by the sensors in the device. In order to gain the amplitude of friction, the displacement input is set as a triangle wave and the amplitude is 10 mm. Figure4 shows the force under 0.1 Hz condition.Figure 4
Force in 0.1 Hz.It can be noted that the force shapes like a square wave because both the damping force and the inertia force are small enough. It can be inferred that the friction force is a constant value which is 400 N. Furthermore, the fluid inerter is tested under different frequency to verify the theoretical model. The amplitude of the sinusoidal displacement input is set as 20 mm under the frequency of 0.1 Hz, 0.3 Hz, and 0.5 Hz, 10 mm under the frequency of 3 Hz, 5 Hz, and 8 Hz, and 5 mm under the frequency of 10 Hz, 12 Hz, and 15 Hz.In order to test the effectiveness of the nonlinear model, the mean value of force amplitude under the sinuous displacement input of simulation and experiment is taken into consideration. Figure5 shows the absolute difference of the force amplitude between theoretical analysis and practical experiment. It can be seen that the second data point is a little larger than the first data point and the third data point, but the absolute differences are all less than 3%, so it can be inferred that the simulation results are very close to the experiment results in low frequency. Furthermore, the absolute difference becomes a little larger with the increase of the frequency. Figure 6 shows the force of the fluid inerter between the theoretical analysis and experiment in 0.5 Hz and 12 Hz. It can be inferred that the experiment results in higher frequency coincide less well with the simulation than that in low frequency, which may be because the temperature of the fluid has increased for the vibration of the system and the fluid flow is not laminar. In conclusion, the experiment results coincide with the simulation well, which demonstrates the effectiveness of the theoretical model of fluid inerter.Figure 5
Difference between theoretical model and experiment.Figure 6
Force in time domain.
(a)
0.5 Hz
(b)
12 Hz


## 3. Suspension Involving Nonlinear Inerter

### 3.1. Nonlinear Model of Suspension Structure

In [33], a new vehicle suspension employing two springs, one damper and one inerter, was proposed based on the dynamic vibration absorber theory. But the models are all linear which are not exactly in the real condition. In this paper, vehicle suspension employing a nonlinear fluid inerter is investigated. The suspension structure involving the nonlinear fluid inerter is shown in Figure 7.Figure 7
Nonlinear model of suspension structure.In Figure7, k
1 is the stiffness of main spring, k
2 is the stiffness of deputy spring, and c is the coefficient of the damper. In this paper, the parameters are optimized by means of the method in [33], where k
1 is 23.8 kN/m, k
2 is 8.9 kN/m, and c is 887 N·s·m−1.


### 3.2. Full-Car Model

A dynamic full-car model is established in Figure8.Figure 8
Full-car model.The vehicle body vertical motion equation is(15)
m
a
z
¨
a
=
F
10
+
F
20
+
F
30
+
F
40
.The vehicle body pitching motion equation is(16)
I
y
φ
¨
=
l
r
F
30
+
F
40
-
l
f
F
10
+
F
20
.The vehicle body roll motion equation is(17)
I
x
θ
¨
=
F
20
+
F
40
-
F
10
-
F
30
d
2
,where m
a is vehicle body mass, z
a is the vertical displacement of the vehicle body, l
f is the length from the front wheel to the center of vehicle body, l
r is the length from the rear wheel to the center of vehicle body, d is the tread between left wheels and right wheels, I
x is the body roll moment of inertia, I
y is the body pitch moment of inertia, θ is the roll angle of the vehicle body, φ is the pitch angle of the vehicle body, and F
i
0 is the suspension force.The motions of four corners of vehicle body are(18)
Z
10
=
z
a
-
l
f
φ
-
d
2
θ
,
Z
20
=
z
a
-
l
f
φ
+
d
2
θ
,
Z
30
=
z
a
+
l
r
φ
-
d
2
θ
,
Z
40
=
z
a
+
l
r
φ
+
d
2
θ
,where Z
i
0 is the displacement of the four corners of the vehicle body mass.For the unsprung mass, the equations are shown as follows:(19)
m
1
Z
¨
1
=
k
t
Q
1
-
Z
1
-
F
10
,
m
2
Z
¨
2
=
k
t
Q
2
-
Z
2
-
F
20
,
m
3
Z
¨
3
=
k
t
Q
3
-
Z
3
-
F
30
,
m
4
Z
¨
4
=
k
t
Q
4
-
Z
4
-
F
40
,where m
i is unsprung mass, k
t is the tire equivalent stiffness, Q
i is the displacements of road, and Z
i is vertical displacement of the four unsprung mass.The equations of the suspension force are shown in the following:(20)
F
10
=
k
1
Z
1
-
Z
10
+
k
2
Z
1
-
Z
b
1
+
c
Z
˙
1
-
Z
˙
b
1
k
2
Z
1
-
Z
b
1
+
c
Z
˙
1
-
Z
˙
b
1
=
f
0
sign
⁡
Z
˙
b
1
-
Z
˙
10
+
b
Z
¨
b
1
-
Z
¨
10
+
C
1
Z
˙
b
1
-
Z
˙
10
2
+
C
2
Z
˙
b
1
-
Z
˙
10
F
20
=
k
1
Z
2
-
Z
20
+
k
2
Z
2
-
Z
b
2
+
c
Z
˙
2
-
Z
˙
b
2
k
2
Z
2
-
Z
b
2
+
c
Z
˙
2
-
Z
˙
b
2
=
f
0
sign
⁡
Z
˙
b
2
-
Z
˙
20
+
b
Z
¨
b
2
-
Z
¨
20
+
C
1
Z
˙
b
2
-
Z
˙
20
2
+
C
2
Z
˙
b
2
-
Z
˙
20
F
30
=
k
1
Z
3
-
Z
30
+
k
2
Z
3
-
Z
b
3
+
c
Z
˙
1
-
Z
˙
b
3
k
2
Z
3
-
Z
b
3
+
c
Z
˙
3
-
Z
˙
b
3
=
f
0
sign
⁡
Z
˙
b
3
-
Z
˙
30
+
b
Z
¨
b
3
-
Z
¨
30
+
C
1
Z
˙
b
3
-
Z
˙
30
2
+
C
2
Z
˙
b
3
-
Z
˙
30
F
40
=
k
1
Z
4
-
Z
40
+
k
2
Z
4
-
Z
b
4
+
c
Z
˙
4
-
Z
˙
b
4
k
2
Z
4
-
Z
b
4
+
c
Z
˙
4
-
Z
˙
b
4
=
f
0
sign
⁡
Z
˙
b
4
-
Z
˙
40
+
b
Z
¨
b
4
-
Z
¨
40
+
C
1
Z
˙
b
4
-
Z
˙
40
2
+
C
2
Z
˙
b
4
-
Z
˙
40
,where Z
b
i is the displacement of inerter.


## 3.1. Nonlinear Model of Suspension Structure

In [33], a new vehicle suspension employing two springs, one damper and one inerter, was proposed based on the dynamic vibration absorber theory. But the models are all linear which are not exactly in the real condition. In this paper, vehicle suspension employing a nonlinear fluid inerter is investigated. The suspension structure involving the nonlinear fluid inerter is shown in Figure 7.Figure 7
Nonlinear model of suspension structure.In Figure7, k
1 is the stiffness of main spring, k
2 is the stiffness of deputy spring, and c is the coefficient of the damper. In this paper, the parameters are optimized by means of the method in [33], where k
1 is 23.8 kN/m, k
2 is 8.9 kN/m, and c is 887 N·s·m−1.


## 3.2. Full-Car Model

A dynamic full-car model is established in Figure8.Figure 8
Full-car model.The vehicle body vertical motion equation is(15)
m
a
z
¨
a
=
F
10
+
F
20
+
F
30
+
F
40
.The vehicle body pitching motion equation is(16)
I
y
φ
¨
=
l
r
F
30
+
F
40
-
l
f
F
10
+
F
20
.The vehicle body roll motion equation is(17)
I
x
θ
¨
=
F
20
+
F
40
-
F
10
-
F
30
d
2
,where m
a is vehicle body mass, z
a is the vertical displacement of the vehicle body, l
f is the length from the front wheel to the center of vehicle body, l
r is the length from the rear wheel to the center of vehicle body, d is the tread between left wheels and right wheels, I
x is the body roll moment of inertia, I
y is the body pitch moment of inertia, θ is the roll angle of the vehicle body, φ is the pitch angle of the vehicle body, and F
i
0 is the suspension force.The motions of four corners of vehicle body are(18)
Z
10
=
z
a
-
l
f
φ
-
d
2
θ
,
Z
20
=
z
a
-
l
f
φ
+
d
2
θ
,
Z
30
=
z
a
+
l
r
φ
-
d
2
θ
,
Z
40
=
z
a
+
l
r
φ
+
d
2
θ
,where Z
i
0 is the displacement of the four corners of the vehicle body mass.For the unsprung mass, the equations are shown as follows:(19)
m
1
Z
¨
1
=
k
t
Q
1
-
Z
1
-
F
10
,
m
2
Z
¨
2
=
k
t
Q
2
-
Z
2
-
F
20
,
m
3
Z
¨
3
=
k
t
Q
3
-
Z
3
-
F
30
,
m
4
Z
¨
4
=
k
t
Q
4
-
Z
4
-
F
40
,where m
i is unsprung mass, k
t is the tire equivalent stiffness, Q
i is the displacements of road, and Z
i is vertical displacement of the four unsprung mass.The equations of the suspension force are shown in the following:(20)
F
10
=
k
1
Z
1
-
Z
10
+
k
2
Z
1
-
Z
b
1
+
c
Z
˙
1
-
Z
˙
b
1
k
2
Z
1
-
Z
b
1
+
c
Z
˙
1
-
Z
˙
b
1
=
f
0
sign
⁡
Z
˙
b
1
-
Z
˙
10
+
b
Z
¨
b
1
-
Z
¨
10
+
C
1
Z
˙
b
1
-
Z
˙
10
2
+
C
2
Z
˙
b
1
-
Z
˙
10
F
20
=
k
1
Z
2
-
Z
20
+
k
2
Z
2
-
Z
b
2
+
c
Z
˙
2
-
Z
˙
b
2
k
2
Z
2
-
Z
b
2
+
c
Z
˙
2
-
Z
˙
b
2
=
f
0
sign
⁡
Z
˙
b
2
-
Z
˙
20
+
b
Z
¨
b
2
-
Z
¨
20
+
C
1
Z
˙
b
2
-
Z
˙
20
2
+
C
2
Z
˙
b
2
-
Z
˙
20
F
30
=
k
1
Z
3
-
Z
30
+
k
2
Z
3
-
Z
b
3
+
c
Z
˙
1
-
Z
˙
b
3
k
2
Z
3
-
Z
b
3
+
c
Z
˙
3
-
Z
˙
b
3
=
f
0
sign
⁡
Z
˙
b
3
-
Z
˙
30
+
b
Z
¨
b
3
-
Z
¨
30
+
C
1
Z
˙
b
3
-
Z
˙
30
2
+
C
2
Z
˙
b
3
-
Z
˙
30
F
40
=
k
1
Z
4
-
Z
40
+
k
2
Z
4
-
Z
b
4
+
c
Z
˙
4
-
Z
˙
b
4
k
2
Z
4
-
Z
b
4
+
c
Z
˙
4
-
Z
˙
b
4
=
f
0
sign
⁡
Z
˙
b
4
-
Z
˙
40
+
b
Z
¨
b
4
-
Z
¨
40
+
C
1
Z
˙
b
4
-
Z
˙
40
2
+
C
2
Z
˙
b
4
-
Z
˙
40
,where Z
b
i is the displacement of inerter.


## 4. Optimization of the Suspension Parameters

According to [27], the nonlinearities of inerter may have a bad effect on the vehicle suspension system, so, it seems important to optimize the parameters of the suspension elements. Genetic optimal algorithm originates from the natural selection of biological evolution. It begins from a potential population and then selects the individual according to the fitness. By means of natural genetics operators combining crossover and mutation, new solution stocks will be created. The QGA (quantum genetic algorithm) is the combination of genetic algorithm and quantum computing. The quantum state vector is applied to the genetic encoding, and the evolution of chromosome is realized by quantum logic gate.The updated rules of the quantum bits are(21)
α
i
′
β
i
′
=
U
θ
i
α
i
β
i
,where α
i
′
β
i
′
T is the updated quantum bit, α
i
β
i
T is the original quantum bit, and U
(
θ
i
) is the updated rotate matrix shown in the following:(22)
U
θ
i
=
cos
⁡
θ
i
-
sin
⁡
θ
i
sin
⁡
θ
i
cos
⁡
θ
i
,where θ
i is the rotate angle and kept the default values in this paper.In order to gain the satisfied parameters of suspension, the QGA is used to optimize the parameters of suspension. The population is set as 100; the max number of the population is set as 200.k
1, k
2, and c are chosen as the optimized variables and their ranges are limited as(23)
10000
N
·
m
-
1
≤
k
1
≤
30000
N
·
m
-
1
,
10000
N
·
m
-
1
≤
k
2
≤
30000
N
·
m
-
1
,
100
N
·
s
·
m
-
1
≤
c
≤
3000
N
·
s
·
m
-
1
.The fitness is set as(24)
min
⁡
J
=
w
1
A
A
0
+
w
2
B
B
0
+
w
3
C
C
0
,where J is the fitness function of the genetic algorithm, w
1, w
2, and w
3 are the weighting coefficients and set as 0.33, A is the RMS (root-mean-square) of the vertical body acceleration, B is the RMS of the pitch angular acceleration, C is the RMS of the roll angular acceleration of the optimized suspension, A
0 is the RMS of vertical body acceleration, B
0 is the RMS of pitch angular acceleration, and C
0 is the RMS of roll angular acceleration of the original suspension.There are limitations of the suspension deflection and the dynamic tire load. For suspension deflection, there is3
σ
f
d
≤
[
f
d
], where σ
f
d is the standard deviation of suspension deflection and [f
d] is the limitation of the tire beat distance and set as 8 cm. For dynamic tire load, there is 3
σ
F
d
≤
G, where σ
F
d is the RMS of the dynamic tire load and G is the vehicle static load and set as 2500 N.The model parameters are listed in Table2.Table 2
Model parameters.
Parameter name
Value
Vehicle body massm
a/kg
770
Unsprung mass (front)m
1, m
2/kg
37
Unsprung mass (rear)m
3, m
4/kg
33
Treadd/m
1.36
Length from front axle to centroidl
f/m
0.955
Length from rear axle to centroidl
r/m
1.380
Roll moment inertiaI
x/kg⋅m2
293
Pitch moment inertiaI
y/kg⋅m2
1074
Stiffness of tirek
t/kN⋅m−1
192The optimal parameters of suspension are listed in Table3.Table 3
Optimal parameters.
Parameters
Value
Main spring stiffnessk
1/kN⋅m−1
20.1
Deputy spring stiffnessk
2/kN⋅m−1
10.2
Damper coefficientc/N⋅s⋅m−1
607


## 5. Performance Analysis

The random road model can be shown in equation(25)
Q
˙
i
t
=
-
2
π
f
cut
Q
i
t
+
2
π
G
0
V
w
t
,where Q
i
(
t
) is the displacement, G
0 is the roughness coefficient, f
cut is cut-off frequency, and V is the speed. In this study, G
0 is 5 × 10−6 m3 cycle−1 and f
cut is 0.01 Hz.In order to show the influences of the nonlinearities of the fluid inerter on the vehicle suspension system, when the speed is set as 20 m/s, Table4 shows the performance indexes among the linear suspension, the nonlinear suspension, and the optimized suspension. The linear suspension is the suspension with a linear inerter, the nonlinear suspension is the suspension with a nonlinear fluid inerter, and the optimized suspension is the suspension with a nonlinear inerter whose parameters have been optimized.Table 4
Performance indexes.
Name
Linear
Nonlinear
Optimized
Improvement
Vertical body acceleration/(m⋅s−2)
0.5230
0.5668
0.5156
9.0%
Pitch angular acceleration/(rad⋅s−2)
0.4201
0.4764
0.3815
19.9%
Roll angular acceleration/(rad⋅s−2)
0.9359
0.9478
0.8566
9.6%It is noted that the suspension performances are all degraded by the nonlinearities of the fluid inerter, the vertical body acceleration increases by 8.3%, the pitch angular acceleration increases by 13.4%, and the roll angular acceleration increases by 1.3%. In [27], the nonlinear ball-screw inerter involving friction and elastic effect was discussed. Results showed that the performance benefits were slightly degraded by ball-screw inerter nonlinearities, which is the same as the fluid inerter. So, we can conclude that the nonlinearities of the inerter will have a slight effect on the suspension performance.Furthermore, the performance indexes of the optimized suspension are all decreased compared with the nonlinear suspension. The RMS of pitch angular acceleration has dramatically decreased by 19.9%. The RMS of vertical body acceleration has decreased by 9.0% and the RMS of roll angular acceleration decreased by 9.6%. Figures9–11 show the performance indexes in the time domain.Figure 9
Vertical body acceleration.Figure 10
Pitch angular acceleration.Figure 11
Roll angular acceleration.It can be inferred that the vertical body acceleration, the pitch angular acceleration, and the roll angular acceleration are all decreased, which means the vibration isolation performance of optimized suspension is significantly superior to the nonlinear one.


## 6. Conclusion

In this paper, a new fluid inerter is designed and the nonlinearities factors including friction and damping force are analyzed. Experiments have been finished in the bench to test the fluid inerter and to verify the effectiveness of the nonlinear model of fluid inerter. Furthermore, a new vehicle suspension system employing the nonlinear fluid inerter is built and the parameters are obtained by means of quantum genetic algorithm. Results show that the dynamic performance indexes of the optimized suspension all decreased compared with the nonlinear suspension. There is a dramatic decrease which is at most 19.9% in the RMS of pitch angular acceleration; the decrease of the RMS of vertical body acceleration is 9.0% and the decrease of the RMS of roll angular acceleration is 9.6%. The vibration isolation performance is significantly improved.



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*Source: 2623017-2016-07-20.xml*