CAE Software of Twin-Tubes Shock Absorber Outer-Characteristic
Changcheng Zhou, Jie Meng
School of Traffic and Vehicle Engineering
Shandong University of Technology
Zibo 255049, China greatwall@sdut.edu.cn Abstract
The pathway throttle and local throttle loss of oil flow in telescopic shock absorber were analyzed, and the pathway loss coefficient and equivalent length of piston holes were studied. According to the thickness and the pre-deformation of throttle slice, and the throttle holes area, the velocity points of valve opening were researched and the analytic formulas of shock absorber velocity when valve opening were given. With the velocity points of valve opening, the model of shock absorber outer characteristic was established by piecewise linear math function. Based on this, the CAE software for shock absorber outer characteristic was developed. A practical example of simulation of shock absorber outer characteristic was given with this CAE software, and the performance test was conducted to verify the CAE software. The results show that the
CAE software is reliable, and the values simulated is close to that tested.

1. Introduction
The shock absorber of automobile is one of the most important components of suspension system, and plays a vital role in the driving process of vehicles[1,2].
The characteristic of shock absorber influences driving smoothness and ride comfortableness of vehicles[3].
The shock absorber used most commonly on automobile is telescopic shock absorber[4].
The characteristic simulation has the significance for the design of shock absorber. With the characteristic simulation, the flaws of design could be found and modified in time, so the design and develop could speed up, while also the test costs could be reduced. However, at present, both in domestic and abroad, the characteristic of shock absorber is simulated mostly with ready-made simulation

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978-1-4244-3701-6/09/$25.00 ©2009 IEEE

Yuanyi Liu
School of Mechanical Engineering
Shandong University of Technology
Zibo 255049, China liuyy@sdut.edu.cn software[5]. Utilizing this method, most of necessary parameters used in the modeling of characteristic are obtained through the experiment tests, therefore, it is hard to establish the accurate simulation model, and the simulation results is also not reliable. Though some scholars had built the model of characteristic with math function, due to the inaccurate analysis of the shock absorber damping component, and the imprecise computation of throttle slice deformation and opening size of the throttle valve with the Mechanical Design
Handbook that provides only the maximum deformation proximately formula. Thus, it is difficult to build the accuracy model, and the CAE software developed is unreliable.
In this paper, by the analysis to the pathway throttle loss coefficient and local throttle loss coefficient of piston holes, with the analytic formula of throttle slice deformation, according to the velocity points at which the valve opening, the model of shock absorber outer characteristic was established by the piecewise linear function. The CAE software of shock absorber outer characteristic was developed, the simulation example was given with the CAE software, and the performance test was conducted for to verify the CAE software. 2. Analysis of damping element
When the characteristic of shock absorber is analyzed, the throttle holes, piston hole and piston slot should be taken into account, meanwhile the local throttle loss must also be considered. The analysis on the structure, the throttle pressure and the flux of the throttle valves are as follows.

2.1. Throttle holes
Throttle slice of rebound valve is composed of superposition slices with small gaps and not, and the

rectangular cross-sections of these gaps form the throttle hole area. The types of throttle holes are determined by the ratio of throttle hole’s length and hydraulic diameter l / DHA . For single throttle hole, its area is A0i= G l A , circumference is lC 2(G  l A ) and hydraulic diameter is d HA 4 A0 i / lC . So the relation between throttle pressure and flux can be expressed as follows[6] 2 p0
.
(1)
QA0 H A0

U

Where, H is the flow coefficient of throttle orifice which is decided by the type of orifice; p0 is the throttle pressure of throttle hole, A0 is the total area of the throttle holes which can be obtained by the thickness of slice with holes h1, width of throttle hole lA and number nA, that is A0 h1l A n A .

2.2. Deformation of throttle slice
Normally, the velocity of shock absorber is quite slow, and the throttle pressure is low, thus the deformation of throttle slice could be regard as minor deflection bending deformation. According to the calculation method of the bending deformation coefficient[7], the deformation at valve mouth radius can be expressed as pf f rk Grk 3
(2)
h
Where, Grk is the deformation coefficient of throttle slice at valve mouth radius; h is the thickness of throttle slice; pf is the pressure on throttle slice.
The opening size of throttle valve is determined by the total deformation f rk and pre-deformation f rk 0 of throttle slices at valve mouth radius, that is[8]
G rk f rk  f rk 0
(3)

2.3. Throttle slot

2.4. Piston slot

pH

3
SG rk p f
.
6 Pt ln(rb / rk )

(4)

12QH Pt LH
.
3
S DhG H (1  1.5e 2 )

(5)

Where Dh is the inner diameter of cylinder, Pt is the dynamic viscosity of oil; QH is the flux of piston slot; e is eccentricity of piston, generally 0  e  1 , LH is the length of piston slot.
2.3.1. Pathway throttle Loss. Let piston orifice is smooth orifice, then the critical Reynolds number is
Rec=2300. So, the critical velocity of piston orifice is vc
(6)
vc Q R e c /d h
Where, Q is the kinematics viscosity of oil.
So, the critical velocity of shock absorber is vc Ah d hQ R e c
(7)
Vc
2
2
Sh

( Dh  d g )

Where, Sh is the annular area between piston
2
2 cylinder and piston rod, and S h S ( Dh  d g ) / 4 ; d g is the diameter of piston rod; Ah is the total area of
2
piston orifice, and Ah nhS d h / 4 .
While velocity of shock absorber VVc, the flow regime of fluid is turbulent flow, and the pathway throttle loss coefficient is as

ª 4S hV º
0.3164 «
(9)
»
¬ nhS d hQ ¼
Where, Re is Reynolds number.
Therefore, the pathway throttle-loss coefficient has relation to flowing velocity of oil. Thus, while analyzing the characteristic of shock absorber, it should be in accordance with the velocity of the shock absorber and the flow state of oil in piston hole, furthermore, with different pathway throttle-loss coefficient formulas.

Oh

When the moving velocity of shock absorber is greater than first valve opening velocity, the deformation of throttle slice at valve-mouth radius position is greater than the pre-deformation of throttle slice at this position, and the annular throttle slot formed between throttle slice upper surface and piston lower surface. So the relation between flux of throttle slot and throttle pressure is as[6]
Qf

The piston slot size has relation to the fit tolerance of piston and inner diameter of cylinder. The minimum fit clearance is G H min , and the maximum fit clearance is G H max . Thus the average fit clearance is G H =
( G H min  G H max )/2. So the throttle pressure of piston slot can be expressed as[6]


0.3164 R e 0.25

2.3.2. Local throttle loss. When oil flows through the piston hole into the cavity of the rebound valve, there are three local throttle loss, that is, respectively,

expand ] h1 , suddenly reduce ] h 2 and change direction
] h3 .
Total local throttle loss is computed with superposition principle, and then converted it into the pathway throttle loss. Therefore, the length of piston hole converted is
(10)
Le (] h1  ] h 2  ] h3 )Oh / d h
Where, ] h1 is the local throttle loss of suddenly shrinkage as oil flows from rebound chamber to piston orifices; ] h 2 is the local throttle loss of suddenly expansion when oil flows from piston orifice to rebound valve cavity; ] h 3 is the local throttle loss of direction change when oil passes through piston orifice.
It is known that, establishing the model of outer characteristic of shock absorber, the local damping loss should be converted into the equivalent length of piston hole, based on the number and the diameter of piston hole, and the velocity points.

3. Math model of outer characteristic
3.1. Velocity point of valve opening firstly
The velocity point of valve opening first is determined by the thickness of throttle slice, predeformation of slice and orifice area. An example of rebound valve is given to illustrate the procedure, the oil flow paths before the rebound valve opening firstly is shown as in the Figure 1.

-Piston hole

Q0 k1

A0H 2 p f k1 / U .

(12)

Piston holes and orifices are the series, that is
Q0 k 1 Qhk1 , so the throttle pressure of piston holes can be expressed as
128Pt Lhe Q0 k 1 phk1 .
(13)
S nh d h4
Throttle orifice and piston holes are in parallel with piston slot, therefore, the throttle pressure of piston circle slot is equal to the sum of throttle pressure of piston holes and throttle orifices, i.e. pHk1 phk1  p f k 1 .
So, the flux of piston circle throttle-slot is expressed such as
3
S DhG H (1  1.5e2 ) pHk1
.
(14)
QHk1
12Pt LH
According to the continuity theorem of oil, the flux of throttle orifices, piston holes and piston circle slot should satisfy the relation equation, i.e.
Q0 k1  QHk 1 VK 1 Sh . Thus, the velocity at which the rebound throttle-valve opening could be obtained, it is such as
3
2
A0H 2 p f k1 S DhG H (1  1.5e )( phk1  p f k1 )
(15)
VK 1 f

Sh
U
12 S h Pt LH

3.2. Velocity point of valve opening maximally
While rebound valve opening secondly, the throttle slice would touch the limit gasket and lower of shield ring as a result of deformation, so, the maximum opening size of valve is G max . It is constant and equal to the value of throttle slot, the oil paths after rebound valve opening secondly as Figure 2.

Piston slot-Piston orifice

-Piston hole
Piston slot

- Piston orifice

Figure 1 Oil paths before the rebound valve opening.
Throttle slot

Defining pre-deformation of throttle slice at valve mouth radius is f rk 0 , when the deformation is equal to the pre-deformation, the rebound valve opens first.
According to the formula of the slice deformation, the pressure on slice at this time is p fk 1 h3 f rk 0 / Grk .
(11)
When the valve opens first, the pressure on slice is equal to the throttle pressure of orifice, i.e. p f k1 p0 k1 .
The flux of orifice can be expressed as

Figure 3 Oil paths after rebound valve opening maximally.

After the throttle rebound valve being at the maximum opening size, the total deformations of throttle slice is f rk 2 G max  f rk 0 . So, the pressure on throttle slice is p fk 2

(G max  f rk 0 )h3 / Grk . For the

throttle orifice is parallel with the throttle slot of

rebound valve, so, p0 k 2 orifice is Q0 k 2

p fk 2 , the flux of throttle

A0H 2 p fk 2 / U .

Thus, the flux of

throttle slot of rebound valve is
3
SG max p fk 2
.
(16)
Qf k 2
6Pt ln(rb / rk )
Throttle orifice is parallel to throttle slot of rebound valve, and then series to piston hole, so
Qhk 2 Q fk 2  Q0 k 2 .Therefore, the throttle pressure of piston holes is
128Pt Lhe (Q fk 2  Q0 k 2 )

.
(17)
S nh d h4
So, the flux of through piston circle slot is
3
S DhG H (1  1.5e 2 )( phk 2  p0 k 2 )
.
(18)
QHk 2
12Pt LH
According to the continuity theorem of oil, when the rebound valve reaches the maximum opening size, the velocity of shock absorber is
Vk 2 f (QHk 2  Q fk 2  Q0 k 2 ) / S h .
(19)
phk 2

3.3. Piecewise characteristic linear

function

model

opening and maximum opening size. Vk1 y is the velocity at which the compression valve opens firstly,
Vk 2 y is the velocity at which the valve opens maximally. 4. CAE software and simulation example
Based on the math model of outer characteristic of shock absorber, the CAE software could be developed by the tools software, i.e. VC++.

4.1. Input windows of parameters
4.1.1. Input windows of structure parameters of shock absorber. In this input window, the structure and valves parameters of shock absorber, the oil parameters, and the slice material are inputted, such as, the inner diameter of inner tube, the diameter of piston rod, throttle area, throttle-slices thickness and their numbers, the oil density and its viscosity. They are shown, respectively, in Table 1, Table 2 and Table 3.
Table 1 Input parameters of structure

of

3.3.1. Rebound process (V>0). The outer characteristic of shock absorber in rebound process can be expressed by the following piecewise linear function[9,10] ­( p f 1  ph1 ) S h , 0  V d Vk1 f
°
Fd ®( p f 2  ph 2 ) S h , Vk1  V d Vk 2 f .
(20)
°
¯( p f 3  ph3 ) S h , V ! Vk 2 f
Where, p f 1 , p f 2 , p f 3 ; ph1 , ph 2 and ph 3 are the

throttle pressure of throttle slot of rebound valve and piston hole, when rebound valve is on three kinds of conditions that are before opening, after opening and maximum opening size.
3.2.2. Compression process (V