Semi-Active Damping Control of Vehicles Based on Negative Stiffness

Suspensions

Zhijie Li and Shaoping Shen

Department of Automation, Xiamen University, Xiamen, Fujian, China

Keywords: Suspension System, Damping Control Algorithm, Matlab/Simulink.

Abstract: In order to design a suspension system with better ride comfort, the concept of negative stiffness is

introduced into the suspension system by analyzing the functional characteristics of elastic elements, and the

damping control algorithm is designed according to the functional characteristics of the spring suspension,

simulated and analyzed using Matlab/Simulink tools, and the results are compared with the passive

suspension, negative stiffness passive suspension and skyhook damping semi-active control suspension. The

results show that the new proposed algorithm combined with the negative stiffness suspension can

effectively improve the vehicle ride comfort.

1 INTRODUCTION

As a product of the civilization of our time, the

number of automobiles has been increasing day by

day since its creation in the late 19th century.

According to the website of the Chinese Ministry of

Public Security, the number of cars and motor

vehicles in China has also been gradually increasing

in recent years. In modern society, the car is not only

a means of travel, but also a way of daily life in

pursuit of a higher quality of life and better sensory

enjoyment. Therefore, improving the comfort and

safety of cars has become one of the important goals

of car design in modern society (Zhang, 2021).

As one of the most important parts of a car, the

suspension connects the body to the axle, and it is

not only the medium for transmitting all the forces

and moments between the wheels and the body, but

also plays an important role in cushioning and

suppressing the shock and vibration caused by the

unevenness of the road. Therefore, a well-designed

suspension can effectively improve the ride comfort

of the car (Olugbade,2021). Scholars at home and

abroad have focused more on the fault tolerance

capability of the strategy in the study of semi-active

suspension control strategy, while less attention has

been paid to the improvement of the overall

combined performance of the suspension control

strategy and the suspension. In this paper, based on

the previous research, we propose a negative

stiffness semi-active suspension damping algorithm

in combination with the vehicle suspension

performance.

2 DESIGN OF NEGATIVE

STIFFNESS SUSPENSION

The analysis of the spring characteristics of the

vehicle suspension (Zhang, 2017) shows that the

deflection of the vehicle suspension spring has two

parts: one is the static deflection, which is mainly

caused by the vehicle itself and the vehicle load, and

the other is the dynamic deflection, which is mainly

caused by the vibration. The main function of the

static deflection is to support the vehicle itself and its

load, and the main function of the dynamic

deflection is to transmit the vibration of the unsprung

part to the whole vehicle body. The expectation is

that the vibration of the unsprung part will be

reduced in the process of transferring it to the

vehicle body, which requires that the spring stiffness

of the unsprung part be as small as possible, because

the smaller the spring stiffness, the more the

vibration energy will be reduced. Such a spring

structure can be designed with the stiffness

characteristics shown in Figure 1. This ensures that

the spring has a small (or even negative) stiffness

(Shahadat, 2010) in the vibration region, while still

having a large stiffness coefficient to carry the

overall vehicle load. The spring stiffness

characteristics can be obtained by connecting a

spring with negative stiffness characteristics in

parallel with the normal spring operating at the

balance point, and the positive and negative stiffness

characteristics of the spring are shown in Figure 2.

126

Li, Z. and Shen, S.

Semi-Active Damping Control of Vehicles Based on Negative Stiffness Suspensions.

DOI: 10.5220/0012150800003562

In Proceedings of the 1st International Conference on Data Processing, Control and Simulation (ICDPCS 2023), pages 126-132

ISBN: 978-989-758-675-0

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

Figure 1: Negative stiffness characteristics of the

suspension concept.

Figure 2: Spring positive and negative stiffness characteristics.

When the incremental deformation of the elastic

element after the force is in the same direction as the

incremental load, the stiffness is positive and the

elastic element is said to have positive stiffness

characteristics, as shown in Figure 2(a); on the

contrary, when the incremental deformation of the

elastic element after the force is in the opposite

direction of the incremental load, the elastic element

is said to have negative stiffness characteristics, as

shown in Figure 2(b). A spring element with a small

range of negative stiffness characteristics can be

connected in parallel with a spring element with

positive stiffness characteristics at the vibration

balance point, allowing the suspension system to

have a small range of negative stiffness

characteristics at the vibration balance point. A steel

plate spring in parallel with a membrane air spring

gives us the desired negative spring stiffness

characteristics, with a local negative stiffness

characteristic at the design point and a load-

deflection curve roughly as shown in Figure 3. In

order to facilitate the analysis of the suspension

vertical dynamics, we established the two-degree-of-

freedom quarter-vehicle suspension model in Figure

According to Newton's second law we can

obtain the following set of kinetic equations:

()() 0

()()() 0

ss s u s u switchs

uu su su tur switchs

mx g x x c x x c x

mx gx x c x x k x x c x

+−+ −+ =





−−− −+ −− =



   

(1)

Where 𝒎

𝒔

and 𝑚



are the spring loaded mass

and unsprung mass, respectively; 𝑥



and 𝑥



are the

spring loaded displacement and unsprung

displacement, respectively;

is the tire vertical

stiffness;

is the damper damping coefficient;

switch

is the variable switching control damping;

is the displacement excitation of the road surface

to the wheels;

()gx

is the restoring force of the

suspension system with negative stiffness

characteristics, and the expression can be fitted

according to the experimental simulation results

(preconceived).

3 DESIGN OF DAMPING

CONTROL ALGORITHM

3.1 Skyhook Control

The skyhook damper control algorithm was first

proposed by Karnopp in the United States (Karnopp,

1974), using an imaginary dampener connected with

the skyhook damper to suppress the vibration of the

body. The principle is to design and install an ideal

skyhook damper between the body and the sky, with

the sky remaining absolutely stationary. The ideal

skyhook damper can suppress the vertical motion of

the body, thus making the vehicle more stable and

improving the comfort of the ride and smoothness of

the vehicle during the driving process.

Semi-Active Damping Control of Vehicles Based on Negative Stiffness Suspensions

127

Figure 3: Spring design load-deflection curve. Figure 4: Vehicle suspension model.

Figure 5 shows the quarter vehicle suspension

dynamics model with ideal skyhook damper

damping control. where

and

are the spring

loaded mass and unsprung mass, respectively; 𝑥



and

are the spring loaded displacement and unsprung

displacement, respectively; 𝑘



and 𝑘



are suspension

stiffness and tire vertical stiffness, respectively;

is the damper damping coefficient; 𝑐



is the

skyhook damper damping coefficient; and 𝑥



is the

displacement excitation of the road surface to the

wheel. The kinetic equation can be expressed as

()() 0

()()()0

ss s s u s s u skys

uu s s u s s u t u r

mx k x x c x x c x

mx k x x c x x k x x

+−+−+=







−−−−+−=





   

  

(2)

In practical applications where the vehicle

cannot exert this ideal force, a controllable actuator

is generally used in the system to simulate the

skyhook damper control force. The actuation rules

for the skyhook damper damping coefficient

are developed by measuring the relative velocity of

motion of the spring loaded and unsprung masses,

and the rules are as follows:

max

min

, ( ) 0

ss u

sky

ss u

cxxx

−≥





−<



 

(3)

Where, 𝑐



and

min

represent the maximum and

minimum damping coefficients that skyhook

dampers can produce, respectively.

3.2 Improved Skyhook Control

According to the mechanical properties of the

damper, we can know that the damping force is

always in the opposite direction of its relative motion

speed and proportional to its size. The main role of

the dampers in the suspension is to absorb energy to

reduce the relative speed displacement changes

between the body and the wheels, thus playing a role

in energy consumption and attenuation of vibration.

(Chen, 2010) In the process of evaluating the ride

comfort and smoothness of the vehicle, we pay more

attention to the magnitude of the vertical

displacement, vertical velocity and vertical

acceleration of the spring-loaded part of the vehicle.

Skyhook damping control is designed to install an

ideal skyhook damper between the body and the sky,

which is essentially achieved by suppressing the

vertical velocity of the vehicle body. It is based on

the relationship between the car body vertical speed

and the suspension speed relative to the car body

speed to act. Since it takes some time for the

suspension speed to change with respect to the

vehicle speed, this can make the damping action

behavior lag in time, which will lead to the method

not better improve the ride comfort.

In this paper, the main purpose is to improve

the ride comfort, and the improved canopy damping

control algorithm is proposed for the negative

stiffness suspension model mentioned in the

previous paper. The damping adjustment is based on

different speed relationship, so as to achieve the

effect of suppressing the vehicle vibration and

improving the vehicle ride comfort. Figure 6 shows

the relationship between the body vertical velocity

and the suspension vertical velocity.

ICDPCS 2023 - The International Conference on Data Processing, Control and Simulation

128

Figure 5: Vehicle suspension model.

Figure 6: The relationship between the action of parameters



and



We can discuss in two cases: the first case is

when the spring-loaded velocity 𝑥



and the unsprung

velocity 𝑥



are in the same direction (quadrant I and

III in Fig. 6), at this time the car body and the

suspension show the tendency to move in the same

direction, which we can understand as encountering

the convex block road or concave block road, since

there is already a negative stiffness characteristic

restoring force to prompt the suspension to act after

the disturbance, considering that the restoring force

will increase the dynamic travel of the suspension,

the speed change difference between the suspension

and the body is reduced by increasing the coefficient

of the damper, so that the vibration transmitted by

the suspension to the body can be reduced, which

can directly reduce the speed change of the body and

the travel of the body. The second case is when the

spring-loaded velocity 𝑥



is opposite to the unsprung

velocity 𝑥



(quadrant II and IV in Figure 6), when

the car body and the suspension show a tendency to

move in the opposite direction, which can be

interpreted as the back-range condition when

encountering a bumpy road or a concave road, and

the difference in velocity between the suspension

and the car body is reduced by reducing the damper

coefficient. The speed change difference between the

suspension and the car body is reduced, so that the

vibration of the car body can be reduced when the

suspension returns to its original state, which can

also directly reduce the speed change of the car

body. Considering the simplicity of the algorithm,

we attribute the case of the presence of 0 to the first

case. According to the above discussion, the

expression of the damping coefficient in the

improved skyhook control algorithm is:

max

min

, 0

sky

cxx

≥









(4)

Bringing equation (4) into equation (1), we can

obtain a dynamics expression for the application of

the improved damping control strategy on a negative

stiffness suspension, and we call it new negative

stiffness improved damping control strategy.

4 NEGATIVE STIFFNESS

SEMI-ACTIVE SUSPENSION

PERFORMANCE ANALYSIS

In order to analyze the effectiveness of the algorithm

improved in this paper on the role of the designed

negative stiffness suspension, this paper uses the

more widely used passive suspension, skyhook semi-

active control suspension as a comparison, while for

better quantitative analysis, we add the negative

stiffness passive suspension as a comparison. The

body vibration displacement, body vibration velocity

and body vibration acceleration are used as

indicators for time domain analysis.

4.1 Random Pavement Excitation

Response

Random pavement is the closest model to the real

pavement, and the response analysis under random

pavement excitation is an important method to

comprehensively examine the overall performance

of the suspension. (Hongbin, 2011) We use the

random pavement excitation curve fitted by the finite

bandwidth white noise as the input of the simulated

pavement, as shown in Figure 7, and the time

domain curves of each index for different control

methods of different suspensions under the random

pavement excitation are shown in Figures 8-10.

Semi-Active Damping Control of Vehicles Based on Negative Stiffness Suspensions

129

Figure 7: C-class pavement model. Figure 8: Vehicle body displacement response.

Figure 9: Vehicle body speed response. Figure 10: Vehicle body acceleration response

Table 1: Calculation results of suspension indexes.

Algorithm

Amplitude(m) Velocity(m s

-1

) Acceleration(m s

-2

)

RMS Improve RMS Improve RMS Improve

Passive 0.0154

——

0.0851

——

0.7157

——

Negative 0.0158 -2.60% 0.0303 64.39% 0.1534 78.57%

Skyhook 0.0137 11.04% 0.0538 36.78% 0.4194 41.40%

New 0.011 28.57% 0.0157 81.55% 0.0942 86.84%

As can be seen from the figure, the improved

damping control strategy combined with the negative

stiffness suspension can effectively attenuate the

body vibration amplitude, while the speed and

acceleration of the body vibration are also

effectively suppressed. In order to be able to

quantitatively analyze, we calculate the root mean

square (RMS) values of body vibration amplitude

response, body speed response and body acceleration

response indexes and the calculation results of

optimization degree are shown in Table 1.

According to the data analysis in the table, we

can know that different suspension control methods

have different effects on body vibration amplitude,

body speed and body acceleration suppression. In

terms of body vibration amplitude suppression,

compared with the passive suspension, the negative

stiffness suspension has a slight deterioration of

2.60%, the skyhook semi-active control suspension

has an 11.04% improvement, and the new improved

damping control negative stiffness suspension has a

28.57% improvement. In terms of body speed

suppression, compared to the passive suspension,

0246810

time/s

-0.02

-0.01

0.01

0.02

0.03

0.04

0.05

Amplitude/m

C-class pavement model

road

Xs(m)

dXs(m/s)

ddXs(m/s2)

ICDPCS 2023 - The International Conference on Data Processing, Control and Simulation

130

there is a 64.39% improvement for the negative

stiffness suspension, a 36.78% improvement for the

skyhook semi-active control suspension, and an

81.55% improvement for the new improved damping

control negative stiffness suspension. In terms of

body acceleration suppression, compared to the

passive suspension, the negative stiffness suspension

has a 78.57% improvement, the skyhook semi-active

control suspension has a 41.40% improvement, and

the new improved damping control negative stiffness

suspension has an 86.84% improvement. It can be

seen that our proposed new improved damping

controlled negative stiffness suspension system has

some or greater improvement compared to other

suspension systems under random road excitation.

4.2 Impact Response of Bumpy

Pavement

Bump pavement is usually used for suspension

impact tests, which mainly simulate road conditions

such as speed bumps or potholes on the road. (Khot,

2017) We use the following equation to simulate the

clod pavement excitation.

000

[1 cos( ( ) )] ,

()

0 ,

tt t tt T

other

ωπ



+−−≤≤+









(5)

Where 𝐴 is the peak height of the bump

pavement, 𝑡



is the start time of the bump pavement

(s), and

is the sine wave period (s), which is the

duration of the bump pavement. The excitation

curves are shown in Fig. 11, and the time domain

curves of each index for different control methods of

different suspensions under bump pavement

excitation are shown in Figs. 12-14.

As can be seen from the figure, under the action

of bump road excitation, improved damping control

strategy combined with negative stiffness suspension

has good performance relative to passive suspension,

skyhook semi-active suspension and negative

stiffness passive suspension in terms of attenuating

the vibration amplitude of the body and the vibration

speed of the body. However, at this time in terms of.

Figure 11: Bump pavement model. Figure 12: Vehicle body displacement response.

ure 13: Vehicle bod

eed res

onse. Fi

ure 14: Vehicle bod

acceleration res

onse.

0246810

time/s

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Bump pavement model

road

0246810

time

(

)

-0.04

-0.02

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Vehicle body displacement response

road

passive

negative

skyhook

new

dXs(m/s)

ddXs(m/s2)

Semi-Active Damping Control of Vehicles Based on Negative Stiffness Suspensions

131

body acceleration there will be high-frequency jitter

vibration phenomenon, due to the high frequency,

the vibration can consume most of the energy after it

is transmitted to the human body through the seat,

the follow-up will be dedicated to the study. This

shows that the combination of improved damping

control strategy and negative stiffness suspension

system can provide better ride comfort and driving

smoothness under the road conditions through the

bumpy road

5 CONCLUSIONS

In this paper, with the main purpose of improving

vehicle ride comfort, the concept of negative

stiffness is introduced into the suspension system by

analyzing the functional characteristics of elastic

elements, and the damping control algorithm is

designed according to the functional characteristics

of the spring suspension, and the following

conclusions are obtained after simulation and

analysis:

The new improved damping control negative

stiffness suspension has superior vibration

suppression performance compared to other

suspension systems on normal random road surfaces,

effectively reducing the root mean square value of

vehicle amplitude and significantly improving the

comfort of the vehicle ride.

When the new improved damping control

negative stiffness suspension passes through the

bump road surface such as the acceleration belt or

the pit, the vehicle body vibration amplitude is also

obviously smaller than other suspensions, and its

callback time is longer, which can better reduce the

vibration energy of vehicle body vibration and

improve the ride comfort.

ACKNOWLEDGMENTS

This research is supported by the National Natural

Science Foundation (NNSF) of China under Grants

61333008, 61603320, 61733017, 61673327 and

Xiamen Key Lab. Of Big Data Intelligent Analysis

and Decision.

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