A study on fully passive isolation systems with low natural frequency

Content Acknowledge i ii Content iii List of figure vi List of table x Chapter I Introduction 1 1.1 Mechanism of an isolation system 2 1.2 Background of this thesis 5 1.2.1 Special isol

공학박사 학위논문

울산대학교 대학원 기계자동차 공학부 Luu Thanh Tung

낮은 고유진동수를 가지는 전 페시브

진동절연 시스템에 대한연구

A Study on fully passive isolation systems with

low natural frequency

지도교수 박 성 태

이 논문을 공학박사 학위 논문으로 제출함

2006년 12 월

울산대학교 대학원 기계자동차 공학부 Luu Thanh Tung

낮은 고유진동수를 가지는 전 페시브

진동절연 시스템에 대한연구

A Study on fully passive isolation systems with

low natural frequency

Acknowledge

First, I must thank my supervisor, Professor Sung Tae Park for his unlimited supports and encouragement throughout this work I also thank Professors, Do Joong Kim, Chang Myung Lee, Byung Ryong Lee and Dok Young Lee for their service on my thesis advisory committee

I could send my special thanks to my wife, Thi Diem Hien Nguyen, for her untiring labor and supporting everything in my life during the time I have stayed far from home Her helps must have been energy which helps me to pass the difficult research

I am thankful my Professor Hong Ngan Nguyen for her encouragement in HoChiMinh City University of Technology I would thank my Vietnamese and Korean friends for helping me

to finish the experimental work and research

I also thank my younger brother, Thanh Tra Luu, who gives me his affection and encouragement during my research in Korea I would like to send my respectful thanks to my parents and mother-in-law for all their work

Summer 2006

Luu Thanh-Tung

국문 요약

진동에 민감한 장치의 기초로부터 전달되는 진동에너지를 감소시키기 위해서 진동절연 시스템이 사용된다 특히 이러한 진동절연 시스템은 낮은 가격과 함께

높은 신뢰성을 구비조건으로 한다 여러 진동절연 시스템들 중 본 논문에서 연구한 수동식 진동절연 시스템이 좋은 후보에 재량된다

본 연구는 수동식 시스템의 고유진동수를 낮춤으로서 고유진동수보다 높은 주파수 영역에서 진동절연 특성을 개선하기 위해 필요한 해석적 방법을 제안하고 그 응용을 다룬다 시스템에 저장된 에너지가 시스템의 고유진동수에 영향을 미치는 중요한 요소인 것에 기초하여 시스템의 고유진동수를 낮추기 위한 새로운 방법으로

“포텐셜법(Potential Method)” 을 제안하였다 이 방법은 낮은 고유진동수의 새로운 진동절연 시스템을 얻기 위해 일반 진동절연 시스템에 불안정 시스템을 보조시스템으로 추가시킨다

이 방법은 먼저 1.7Hz의 고유진동수를 가지는 초기 버스 운전석 의자의 진동절연 시스템을 1Hz의 고유진동수를 가지는 새로운 진동절연 시스템으로 개선하기 위해 적용되었다 두 번째로 영의 고유진동수를 가지는 진동절연 시스템을 찾기 위해 부강성 시스템에 적용되었으며, 그 결과는 0.8Hz의 고유진동수를 가지는 산업용

방진 테이블의 종 방향 절연에 사용될 수 있다 마지막으로 포텐셜법은 영의 고유진동수를 가지는 횡 진동절연 시스템에 적용되어 종 방향 시스템과 함께 3차원 진동절연 시스템에 적용되었다

Content

Acknowledge i

ii

Content iii

List of figure vi

List of table x Chapter I Introduction 1

1.1 Mechanism of an isolation system 2

1.2 Background of this thesis 5

1.2.1 Special isolation systems with low natural frequency 5

1.2.1.1 Lacoste spring 5

1.2.1.2 Air spring 7

1.2.1.3 Non-linear spring 8

1.2.1.4 Special isolation system 10

1.2.2 Disadvantages of the current special isolation systems 11

1.3 Objective and outline of this thesis 11

Chapter II Potential method 14

2.1 Introduction 15

2.2 Linearized equation of motion 15

2.2.1 Linearized equation of motion of a simple pendulum system 16

2.2.2 Linearized equation of motion of a mass-spring system 17 2.3 General linearized equation of motion of single degree of freedom systems 18

2.4 Method to reduce the natural frequency of single degree of freedom Systems 21

2.5 Conclusion 25 Chapter III New design of a suspension system for a driver’s seat 26

3.1 Introduction 27

3.2 Current driver’s seat 29

3.3.2 Calculation of period at the ideal equilibrium point 34

3.3.5 Elastic characteristic of the new isolation system 39

Chapter IV Negative stiffness systems and optimal isolation systems 42

5.2 Application of the negative stiffness system for real product 64

7.2.1 Analyses of the horizontal isolation system 87

7.2.2 Equation of motion of the horizontal isolation system and

problems about the weight of the isolation system 89

7.2.4 Experiments of horizontal isolation system 92

7.3.1 Choice of the vertical isolation system 93

List of Figures

Fig 1.3 Transmissibility from base to mass with different excitation frequency

a/ The stiffness is kept fixed and the damping coefficient is changed

b/ the damper coefficient is kept fixed, the stiffness value is changed

Fig 1.7 Force - displacement characteristic of elastic buckling in Euler springs 9

Fig.2.2 A vibration system consists of a mass and a spring 17

a/ Difference D(x) between the two potential energies of the two

systems I and II

b/ Convex functionD ( x)

Fig 2.4 A new method to increase the convexity of D ( x) Line 1: the potential

function of the original isolation system Line 2: the potential energy of the

expected isolation system Line 3: The potential energy of the added system 23 Fig 2.5 Types of the added system is the upper position The original isolation

system and added system is connected together to become the expected isolation

a/ inverse pendulum

b/ System having negative stiffness

c/ a mass making from steel in magnetic field

Fig 3.1 A seat suspension modeled as a single degree of freedom system with

Fig 3.2 Transmissibility as a function of exciting frequency for different values

Fig 3.3 Seat structure 30

a/ Present seat

b/ Mathematical model of the seat

Fig 3.4 Description the application of the potential method 32

a/ when the mass increase and the pawl hit the stopper

b/ the pawl and a ratchet wheel fix together at new position

Fig 3.8 The experimental results of time versus amplitude with different δo 36 (a) δo = 80mm

(b) δo = 60mm

(c) δo = 40mm

(d) δo = 20mm

Fig 3.10 Frequency response function for new and original suspension system

Fig 4.2 Diagram of a type of negative stiffness system 44

Fig 4.3 Decrease of the natural frequency, in turn, system (1), (2) and (3) 46

Fig 4.4 Typical isolation systems use the potential method or negative stiffness 47

a/ the added system use normal spring

b/ the added system use leaf spring

c/ the added system combine normal spring and a bar

Fig 4.5 Structure with the added system using normal spring 48

a/ Structure with added system using leaf spring

b/ The pin-end leaf spring

a/ Real model for low natural frequency experiment

b/ Real model structure

Fig 4.9 Transmissibility between base and mass 59

a/ The supplement system stiffness K2equals 800N/m

b/ K2equals 450N/m

c/K2equals 160N/m

d/ K2equals 155N/m

Fig.5.2 Anti-vibration table with the gross load capacity of 50Kg 64

a/ Practical structure of the anti-vibration table

b/ Detailed structure of the isolator in the anti-vibration table

a/ Real structure of the anti-vibration table

b/ Detailed structure of all system

c/ Joint between the main system and the negative stiffness system

Fig 5.4 Resulting transfer function shows a natural frequency that is approximately

Fig 6.1 Structures of the isolation system using the active negative stiffness spring 70 Fig 6.2 A system consists of two springs connected in series 71

a/ System with a normal spring

b/ System with a leaf spring

c/ System with a bar and a normal spring

Fig 6.4 A isolation system includes the main springK0connected to the system

which contains the normal springK1 and the SWNS connected in series 74

Fig 6.5 Experimental model; a/ Real model; b/ Structure of the model 76 Fig 6.6 The transmissibility between the base and the mass at different amplitudes

and different initial deformationδ The solid line is at 0 δ0 =1.6L, the thick solid

line is atδ0 =2L, and the dashed line is the isolation system without the SWHNS 81

a/ K = 90N/m

b/ K= 100N/m

c/ K=105N/m

Fig 6.7 The characteristic of the natural frequency of the non-linear isolation

Fig 7.1 Three types of the horizontal isolation systems 85

a/ Counter-Spring

b/ Geometrical

c/ Counter-Weight

Fig 7.5 Geometrical schematic The suspension point chosen to give a straight-line

Fig 7.6 Comparison between theoretical and numerical formula 89 Fig 7.7 The Scott-Rusell linkage 1- mass, 2- rigid beam, 3- small wire 90 Fig 7.8 A tension spring is added to reduce the stored energy 91

Fig 7.10 Natural frequency vs the gap between the suspension point and

straight line The dash line is theoretical one, the solid line is experimental one 93

List of tables

Table 3.3 Comparison on natural frequency between theoretical and experimental

Table 3.4 Comparison on the natural frequency between the theoretical

Chapter 1:

Introduction

1.4 Mechanism of an isolation system

Isolation systems are used to absorb vibration energy which is transferred from base to

mass A simple structure of an isolation system is shown in Fig 1.1 in which the spring is an

element to support the mass The spring element could be an actual compression, tension or

torsion spring or an equivalent system that stores potential energy Because of its structure

the spring might cause a lot of obstacles in an attempt to decrease natural frequency

Fig 1.1 A simple isolation system

The equation of motion for the structure in Fig 1.1 is written as follows:

m x&&+ &x+kx =F ( y) (1.1)

where k is the spring stiffness, m the mass, c the damping coefficient and F(y) the

excitation force

Fig 1.2 shows the transmissibility from the base to the mass The vibration amplitude of

the mass will reach maximum at the resonance frequency Moreover, as the excitation

frequency increases, the amplitude decreases in the high frequency range

Fig 1.2 Transmissibility from the base to the mass

The resonant frequency can be written as follows [23]:

In Eq 1.3, it can be seen that the natural frequency of the system with constant mass will decrease as stiffness decreases

Parameters that affect the transmissibility of the isolation system are the damping and the stiffness as shown in Fig 1.3 Fig 1.3a shows the transmissibility when the stiffness is kept unchanged and the damping coefficient is changed and fig 1.3 b illustrate the transmissibility when the damper coefficient is kept unchanged and the stiffness value is changed

Fig 1.3.a shows that the transmissibility is low around the resonant frequency and becomes bad at higher frequencies when the stiffness is fixed and the damping coefficient is

increased This means the increase or the decrease of the damping coefficient does not bring the expected anti-vibration by the change of natural frequency

a/

b/

Fig 1.3 Transmissibility from base to mass with different excitation frequency values a/ The stiffness is kept unchanged and the damping coefficient is changed b/ the damper coefficient is kept unchanged and the

stiffness value is changed

Meanwhile, Fig 1.3b shows that when the damping coefficient is fixed and the stiffness of the spring is decreased, the natural frequency reduces and the isolation system obtains anti-vibration effect So, the ideal solution is that the natural frequency of the spring should be obtained as low as possible The lower the resonance frequency is, the narrower the base of the resulting peak becomes From Eq 1.3, when the stiffness decreases, the natural frequency decreases Nevertheless, the spring element will not permit the system to obtain low natural frequency by its structure characteristic The natural frequency and the static deflection Δstatic

of the spring have an inverse quadratic relation as follows:

2

n static

1.5 Background of this research

1.2.1 Special isolation system with low natural frequency

As discussed before even thought the spring-mass system can not reach low natural frequency, the natural frequency needs to be decreased to obtain high isolation effect To overcome this problem, some special systems for the vibration isolation have been introduced:

1.2.1.1 Lacoste spring

Many conventional seismometers have utilized the ``zero-length" spring invented by La Coste in the 1930's which is illustrated in Fig 1.4 Although the spring was originally designed for use in gravimeters (La Coste Romberg types being very popular), it has been popularly used in vertical seismometers of the conventional non force-balance type

Randall [36] explained the working principle of the system as follows: Assuming that the system has no damping and the boom and the spring are weightless, the equation of motion is given by:

θ φ θ 2θ&&

0)sin( )(

mgb − − − = (1.5)

where g is the magnitude of the gravitational field, k the spring constant, L0 the length of the unstretched spring and, φ and θ the angles as indicated

Fig1.4 La Coste spring

To analyze this system, it is convenient to employ the law of sine:

(φ θ) (π θ) cosθ

2sinsin

L L

1 (1.7)

where L e is the length of the spring at equilibrium when y = 0

For small motions, [ L e] { [ ( )L L e] }

e

φ

&& (1.8)

Since the parenthetic expression multiplied by y involves the angular frequency of the

simple harmonic motion,ω2, the period of the motion is inversely proportional to sin φ In

principle, the period can be extended by giving D→ 0 from the relation sin(φe) = D/Le However, as D Æ 0 in Fig 1.4, the system tends to have an equilibrium state which at best is approximately neutral, and which at worst is unstable Therefore, it practically is very difficult to work with such a system having a period T > 30s because of material limitations associated with elasticity

From Eq (1.8), it is seen that T can be lengthened by letting L0 → 0 Working with zero-length spring and φe ≈ π/4 has been a popular combination that is widely used in the World Wide Standard Network

In this isolation system, the mass and the stiffness of the “zero length” spring is firstly determined When the mass is changed, the isolation system must be designed again This is

an obstacle of the isolation system using the Lacost spring So this isolation system is not good for a system with alternating loads

2

= (1.9)

where P is pressure of compressed air, A the area of the piston and V the initial volume In

Eq 1.9, it is noted that increasing the air volume results in a lower natural frequency that is

necessary condition for the efficiency of high isolation:

The air spring has a lot of advantages as follows:

a Efficient reduction of vibration

Like a compressible cushioning medium, the air strings provide a very low degree of stiffness, i.e a very "soft" cushion The system's inherent frequency reaches below 3 Hz when combined with auxiliary volumes even well under 1 Hz There is even a higher degree of isolation for low inherent frequencies

b Constant design height

The operating height of the air springs can be adjusted via the air pressure inputted, regardless of the load capacity There is no static deflection as with other spring components A user-friendly, automatically functioning control system guarantees a uniform operating height, even in the case of alternating loads

c Load dependent isolation properties

The spring stiffness is proportional to the load capacity The vibration isolation is virtually independent of the carrying force, i.e constant inherent frequencies even with fluctuating loads The broad load range of air springs provides possibilities for standardizing the spring types in assembly families

d Avoiding of resonance effects

Air spring isolation systems can be designed with two inherent frequencies so that resonance can be avoided when the excitation frequency undergoes constant changing due to switching

e Lateral stability

The air springs have very good lateral stability

f Low overall height

The air springs have a very low installation height

g Isolation of structure-borne noise

The air springs provide a very high degree of structure-borne noise isolation

h Maintenance-free and long service life

The air springs are made of materials of high dynamic strength Materials are resistant to environmental influences and aging factors

However, the air spring has some disadvantages The isolation system using air spring needs some componentsfor the operation of the air spring such as filters, a compressor, hose, etc

The meaning of “non-linear” itself shows the characteristics of this system The stiffness

of the system is not constant in its operating range and makes the system obtain a very low natural frequency value at a determined operating range Taking the advantage, it is possible

to design the system with very low natural frequency But because non-linear springs are multiform, it could be possible to illustrate not all of non-linear spring types but a special kind of the non-linear spring system, Euler spring

The Euler spring is well known that a column of an elastic bar will support a load with no deflection until it suddenly starts to buckle at some critical value of load (dependent on its modulus) This is a sort of wall-shaped nonlinearity that is required in the force-displacement characteristic to provide zero static energy Fig 1.6 shows two cases of buckling columns – one in Fig 1.6a having pinned ends and the other in Fig 1.6b having rigidly fixed or clamped ends

Fig 1.6 Types of Euler spring

Fig 1.7 Force - displacement characteristic of elastic buckling in Euler springs

Fig 1.7 shows the force-displacement characteristic of the Euler spring [9] The stiffness and the natural frequency of the Euler spring are written in Eq (1.9) and Eq (1.10) respectively:

l

mg k

1.2.1.4 Isolation system with special structures

There are several special isolation systems that are based on many kinds of elastic components [17], [35] Refer to Fig 1.8, due to the spring that is combined with an inverted pendulum, the natural frequency is decreased when the inverted pendulum is added to the main spring [35]

The natural frequency of the combined system is written as follows:

mgl K

n

ω (1.12)

where m is the mass of the inverted pendulum and M the mass of the horizontal pendulum

After the inverted pendulum is added, the natural frequency of the new isolation system is decreased Isolation systems similar to that in Fig 1.8 are called the combined systems By choosing exactly parameters of the inverted pendulum, the natural frequency of the combined isolation system can be obtained as low as possible

1.2.2 Disadvantages of the current special isolation systems

All of special isolation systems presented above have advantages and disadvantages when they are applied in practical system The coil spring is the most popular because of low cost, easy manufacture and simple structure However, in order to increase the effectiveness of the anti-vibration, the natural frequency must be reduced [6…11] The coil spring can not obtain the low natural frequency because of obstacles in its structure The spring having the lower natural frequency than 1 Hz is really cumbersome As analyses in [3], the length of spring of the system with the natural frequency of 1 Hz must be at least 0.5m, which is unfeasible According to the reference [68], the air spring can achieve very low natural frequency When using big air tanks, the natural frequency depending on the initial volume can reach 0.1

Hz Because of the advantage, the air spring is used in most modern luxury cars However, there is a shortcoming in this kind of the system The isolation system becomes expensive and complicated because it needs to accompany with some other devices such as tank, compressor etc

The non-linear spring is also employed in a widespread manner However, it only obtains low natural frequency at the design point When it operates out of the point, the result may be unexpected Therefore, the non-linear spring is suitable for unchanged masses

The other isolation systems, although they can obtain low natural frequency, are complicated and big Therefore, the new isolation system should provide benefits such as effectiveness of the anti-vibration, a simple structure and cheapness and high reliability So, the main purpose of this thesis is to find out such passive isolation systems

1.3 Objective and outline of this research

The general objective of this research is to find out general methodology to decrease the natural frequency of the passive isolation systems Some specific purposes are:

- To find out the general method to decrease the natural frequency of a single degree of freedom system, which is called potential method

- To make an application of the potential method on the negative stiffness with low natural frequency

- To propose a new method reducing the size of the negative stiffness system for commercial purpose

- To propose a high negative stiffness system when the original isolation systems have high positive stiffness

- To make an application of the potential method on the horizontal isolation system that requires the zero-natural frequency

-

In this research, the main point of the study is to find out isolation systems with the lowest natural frequency My work begins with analyses of the nature of the natural frequency that helps researchers easily create isolation systems with very low natural frequency All of the isolation systems are based on the structures of single degree of freedom systems The contents of the thesis are organized as follows:

In chapter 2, the relation between the natural frequency and system parameters will be discovered In order to do this, a general form of the equation of motion with single degree of freedom systems is determined Based on the general form of the equation of motion, the main factor influencing the natural frequency is found And a new method called “potential method” is introduced

Chapter 3 solves a problem related to the isolation of a driver’s seat Because the isolation system of the current seat has a high natural frequency, the vibration attenuation from the chassis to the driver is weak To solve this problem, a new model is proposed and analyzed theoretically and experimentally And the result of the new system is compared with the one

of the old system Also, this chapter will introduce a design to make the new isolation system not pass the threshold that comes from the change of the equilibrium point

Chapter 4 presents the analysis and the design of the combined system with the added system which is called negative stiffness system When the combined system is constructed, the stiffness can reach very low However, most of the combined systems are non-linear ones, which prevents the isolation system from obtaining the zero stiffness This chapter will show how to achieve a linear combined system The experimental results prove the possibility of achieving the zero stiffness of the isolation system, which makes the natural frequency of this

Chapter 5 shows an application on a commercial anti-vibration table By the change of the direction of the negative stiffness system analyzed in chapter 4, the new isolation system will be achieved with a smaller size and high competition

In chapter 6, an isolation system with zero stiffness can be produced by combining a negative stiffness system and the original system as analyzed in chapter 4 In the case of the original isolation system having a positive high negative stiffness, the added negative system will become bulky In this chapter, the combination of a normal spring and negative stiffness elements can produce a high negative stiffness This chapter will use the mathematical analyses used in chapter 4 to construct a high negative stiffness linear system Characteristics

of the new isolation system using the highly negative stiffness linear system will be shown through experimental results

Chapter 7 introduces the isolation system in three-dimensional space in which the horizontal isolation system is mainly studied Producing the horizontal isolation system is also based on the potential method shown in chapter 2 in which the optimal isolation system has its potential energy unchanged with any movement of the system When a pendulum is used in horizontal isolation, the system will be difficult to obtain a very low natural frequency So, a new horizontal isolation system using a straight line structure is introduced

In this structure, the load works on a flat plane in order to make the potential energy unchanged However, because of the weight of load bearing elements, the natural frequency

of the isolation is difficult to reach very low value A new method using zero length spring is applied for the load bearing elements to make the natural frequency of the horizontal isolation system can reach zero exactly By using such horizontal isolation system and by using the vertical isolation system with negative stiffness linear system, a new isolation system in three dimensions is introduced Experiment proves the effectiveness of vibration attenuation in both horizontal and vertical directions

Chapter 8 makes a summary on this research Advantages and disadvantages of the isolation system are all listed This chapter also presents application fields in which the potential method and the proposed isolation systems can be applied

Chapter 2:

Potential method

2.5 Introduction

Many isolators are modeled as single degree of freedom systems One of the most important factors of an isolator is the natural frequency [23], [24] The lower the natural frequency is, the more quickly the vibration will be attenuated at higher frequencies However, reducing the natural frequency is not a simple work because the factors influencing the natural frequency are different case by case Let’s consider four examples to illustrate the difference among single degree of freedom systems:

First, in a simple pendulum, the pendulum length and the acceleration of gravity are two important factors influencing the natural frequency [23] By increasing the length of the pendulum or by moving the system to a place where the acceleration of gravity is small, the natural frequency will decrease

Second, a spring-mass system has two factors influencing the natural frequency The natural frequency of the mass-spring system will decrease when the mass increases or/and the stiffness decreases

Third, in the horizontal isolator for gravitational wave detection [9], the lower the stored energy is, the lower the natural frequency is The straight line structure introduced makes the stored energy approximately zero In this case, the stored energy is an important factor

Finally, a quasi-zero stiffness isolator is made by combining a common spring with negative stiffness elements Total stiffness of the system can reach zero to make natural frequency zero approximately [17]

Here, some questions can be raised: why are factors which influence the natural frequency different even though the pendulum, the spring-mass system, the gravitational wave detection system, and quasi-zero system are all single degree of freedom systems? What is the common factor influencing the natural frequency? If the factor can be found, a general method to reduce the natural frequency will be suggested

The following sections are devoted to find out the general form of the equations of motion of isolation systems In section 2.2 and 2.3, the equations of motion and the natural frequencies of two different single degree of freedom systems are derived and presents that the stored potential energy can be main factor influencing the natural frequency Section 2.4

is devoted to find the general method which reduces the natural frequency of the single degree of freedom system and the conclusion is given in section 2.5

2.6 Equation of motion and natural frequency

2.2.1 Linearized equation of motion of a simple pendulum system

Fig 2.1 A simple pendulum

With a simple pendulum given in Fig 2.1, the linearized equation of motion and the natural frequency are:

Also, it is easy to recognize that the potential function is written in the new coordinate:

l

mgx mgl

U

2)cos1(

x

U x

m&& (2.5)

where, m is the bob mass

2.2.2 Linearized equation of motion of a mass-spring system

A vibrational isolation system with a mass and a spring is shown in Fig 2.2

Fig 2.2 A vibration system with a mass and a spring

The potential function of the system is given:

where K is stiffness of the spring and x the displacement from the equilibrium point

The equation of motion is written:

=0

∂

∂+

x

U x

M&& (2.7)

where M is the mass

Note that the forms of the equation of motion in (2.5) and (2.7) are similar although the structures of the two systems are different Thus, it is necessary to find a general form of the equation of motion to discover the main factor influencing the natural frequency

2.7 General linearized equation of motion of single degree of freedom systems

Because the focus of the research only concentrates on reducing the natural frequency, the damper of the isolation system is ignored in the equation of motion From [29], the potential function only depends on the position of the mass in the potential field Moreover, the potential function is an even function since most of the systems of single degree of freedom

vibrate symmetrically about the equilibrium point [23] Defining the potential function U(x) where x is a generalized displacementcoordinate, the kinetic energy function is written [7]:

2 2

2

12

1mv c I cω

T = + (2.8)

where, m is the mass, I the moment of inertia about mass center, c v the velocity of the c

center of gravity and ω the angular velocity

In general, the single degree of freedom systems consist of a lot of elements From Eq (2.8), the kinetic function is written in the following forms:

2 2

)()()

()( ⎥⎦⎤ + ⎢⎣⎡ ⎥⎦⎤

i i

i i i

i i

i i i

m

T ( ) 2(&) ( ) 2(&) (2.10)

where, the functions g i ( x& and ) k i ( x& consist of the first or higher degree terms in ) x&, m is i

the mass of the element i and I is the moment of inertia of the element i i

Applying the Lagrange’s law into the holonomic conservative system with single degree

of freedom, the equation of motion can be derived in two forms of either Eq (2.11) or (2.12):

i i

i i i

i i

i i i

i i

i i i

i i i

i i

i i i

i i

i i i

i i i i

i i i

i i

i i i

i i

i i i i i

i i i

m

i

i i i i i i i

i i i i

i i i i

i i i i

i i

i i

i i

x

x f

x

x g

x

x h

x

x k

where a and b are constants The linearized equation is performed by considering small

motions about equilibrium and by ignoring any non-linear terms of x& and x

In the Eq (2.11), the second and third terms are ignored because they are small nonlinear terms of x& In similar, the second and third terms in Eq (2.12) are ignored In Eq (2.11) and (2.12), the coefficient of the acceleration x&& must be linearized to become a constant by ignoring non-linear terms or by considering non-linear terms as constants for small motions

As a result, the linearized equation of motion is:

=0

∂

∂+

x

U x

A&& (2.14)

where A that consists of inertia terms is a constant

After integration and some algebra manipulation from Eq (2.14), the period is:

where xmax and xmin are two extreme points respectively It is noted that the natural

frequency is the inverse of the period From Eq (2.15), the natural frequency depends on A consisting of inertia terms When A increases, the natural frequency will reduce However,

this is not good for most of isolation systems because it makes the isolation system heavy Besides, in some systems such as the pendulum, inertia has no influence on the natural frequency

Note that when the isolation system is stable, the potential function is a convex one [26] Moreover, in linearized systems with single degree of freedom, the potential function is an

even one and it achieves the lowest value at the equilibrium point Thus, U(x) is less than or

equal to U(xmax) Let’s give two systems of single degree of freedom I and II that have the same value of A, and different potential functions U I(x) and U II(x) Also define a function )

( x

D satisfied with:

D(x)=U I(x)−U II(x) (2.16)

Applying the Eq (2.15) for systems I and II and calculating the difference between two

periods of two systems, the result is given:

max max

max max

U U

x U x U x

U x U x

U

x D x

D A T

(2.17)

where T I is the period of the system I, and T II the period of the system II

In Eq (2.17), the denominator is positive and the difference of the two periods is positive when the numerator is positive In other words, in two systems with same inertia and working range, if D(x) is a convex function, the natural frequency of the system I will be higher than that of the system II Note that D(x) represents the difference of two stored potential energies

of two system I and II When this difference increases, the convexity of D(x) increases As a

result, the difference of two periods will increase Therefore it is concluded that the stored potential energy be a main factor which influences the natural frequency That is, the lower the stored potential energy is, the lower the natural frequency is, which is also mentioned in [9]

2.4 General method to reduce the natural frequency of single degree of freedom systems

In the previous section, when the convexity of D(x) increases, the difference between

two natural frequencies of the above two systems I and II will increase The graphical

analysis of Eq (2.17) is given in Fig 2.3

In Eq (2.4), It is can be seen that when the length of the pendulum is increased, the stored

potential energy at any point x in the working range will decrease In other words, the

convexity of D(x) increases, it makes the natural frequency of the pendulum reduced In Eq (2.6), it is clear that when stiffness of the spring decreases, the reduction of the stored

potential energy which makes the convexity of D(x) increase will be occurred That means

the natural frequency will decrease

Fig 2.4 A new method to increase the convexity of D(x) Line 1: the potential function of the original isolation system Line 2: the potential energy of the expected isolation system Line 3: The potential energy of

the added system

In Fig 2.4, line 1 shows the potential function of the original isolation system Line 2 shows the potential function of the expected isolation system whose natural frequency is as low as expected As above analyses have shown, the natural frequency of the system given by line 2 is lower than that of the system given by line 1 Line 3 illustrates the potential function

of the system that is added to the original isolation system When the added system is connected to the original isolation system, the sum of two potential functions, one of the original isolation system and the other of the added system, equals that of the expected system It means the sum value of line 1 and 3 equals the value of line 2 It is noted that the potential function of the expected system must be a convex function to satisfy with the stability condition [8]

At this point the questions can be arisen what is the added system illustrated in line 3, And what is the characteristic of the added system When the potential function of the added system is a concave function, the minimum value at the equilibrium point can be achieved Also note that in order to satisfy the above requirements the added system is an unstable one

In addition, the potential function of the added system must be the difference of two potential functions, one of the original isolation system and the other of the expected isolation system

However, in practice, the added system is sometimes chosen first and the expected isolation system will be determined later

a/

b/

c/

Fig 2.5 Types of the added system is the upper position The original isolation system and added system is

connected together to become the expected isolation system in lower figure

a/ inverse pendulum b/ system having negative stiffness c/ a mass making from steel in magnetic field

Some examples to reduce the natural frequency are introduced in Fig 2.5 In these examples, the added systems in the upper position of Fig 2.5 are combined with the original isolation systems to construct the expected systems in the lower position of Fig 2.5 Through experiments and mathematical analyses, it has been proved that the natural frequency remarkably reduces

In [9], the system is analyzed that the natural frequency of the combined system in Fig 2.5a can achieve 0.3Hz The system in the upper position of Fig 2.5b is a negative stiffness element

Following mathematical analyses, when the negative stiffness element is combined with the original isolation system, the natural frequency of the combined system can reach 0 Hz

In [10], the system in the upper position of Fig 2.5c is used to generate an infinite negative stiffness that is combined with a system having high positive stiffness to make systems having zero-stiffness As a result, the natural frequency reaches zero

In the upper position of Fig 2.5, the potential functions of these systems satisfy the potential function given as line 3 of Fig 2.4 When these systems are connected to the original isolation system, they become the systems in the lower position of Fig 2.5 These systems are the expected isolation systems which are given as line 2 of Fig 2.4 With this combination, the natural frequency can reduces as shown above and the natural frequency of the expected system is determined from the Eq (2.15)

2.6 Conclusion

This chapter shows that the stored potential energy is the most common factor which influences the natural frequency of the single degree of freedom systems From this factor, the new method to reduce the natural frequency is introduced It also shows that most of the unstable systems can be used as the added systems, which suggests new ideas to decrease the natural frequency of the single degree of freedom systems

Chapter 3:

New design of a suspension system for driver’s seat

A common suspension system of a driver seat consists of mass, spring and damper as illustrated in Fig 3.1

Fig 3.1 A seat suspension modeled as a single degree of freedom system with stiffness K, mass M and

damping C

With a small damping coefficient, this suspension system shown in Fig 3.2 will be almost resonated at the natural frequency The lower the natural frequency is, the narrower the bandwidth becomes and the vibrations will be attenuated faster at higher frequencies Thus the ideal solution to attenuate the vibration from base is to take the natural frequency as low

as possible [3], [11] However, reducing the natural frequency is not a simple work as discussed in chapter 1

Human body is least tolerant of the vertical vibration in the excitation frequency range between 4 and 8 Hz [1] In reality, the natural frequency of most suspension system of the driver seat is between 2 and 4 Hz [3-7] With such frequency range, the transmissibility between chassis and seat in the range 4 to 8 Hz is still large (more than one) as in Fig 3.2