Research Article
Hybrid modeling and verification of
disk-stacked shock absorber valve
Advances in Mechanical Engineering
2018, Vol. 10(2) 1–12
Ó The Author(s) 2018
DOI: 10.1177/1687814018756398
journals.sagepub.com/home/ade
Jingli Xu, Jinli Chu and Hongwei Ma
Abstract
The mathematical model of valve deformation of hydraulic shock absorber is established using the hybrid method of
large deflection and beam deflection. The model can be used for design purposes and helps in developing damping valve
based on single disk and shim stack. The model of the hydraulic shock absorber with valve system consists of shim stack
and a spring preloaded disk is developed using AMESim. The correctness of the mathematical model of the valve deformation based on the hybrid method and the simulation model using AMESim is verified by the shock absorber bench
test, which proves the method proposed in this article is reasonable and reliable. It is significant for reference in the
design and development of hydraulic shock absorber.
Keywords
Valve deformation, shock absorber, AMESim simulation, test verification
Date received: 8 September 2017; accepted: 7 December 2017
Handling Editor: Nima Mahmoodi
Introduction
The development of modern vehicle chassis is very fast,
and the performance of vehicle chassis is mainly influenced by an appropriate design of suspension systems.1
The shock absorber is an important part in a car’s suspension, and its damping force characteristic plays a
very important role in the safety and the comfort of the
vehicle.2 The throttle valve is an important part in the
shock absorber, and its bending deformation directly
influences the damping force characteristic of the shock
absorber, which affects the performance of the suspension, that is, the ride comfort and the handling stability
of the vehicle.3 Each valve is characterized by different
orifices, located in series or in parallel, which can have
constant or variable areas. In particular, the typical
variation (which can be observed between low- and
middle-high values of piston speed) of the slope of the
force versus speed diagram of the shock absorber is
due to deformable valves, which are completely closed
until a pre-defined pressure drop is reached between
their ports.
A single valve can be viewed as a circular thin disk.4
At present, for the deformation of these types of components, there are two main computational theories:
small deflection theory and large deflection theory.
Small deflection theory only applies to the small deformation of the valve, which means that the maximum
deformation of the valve can not exceed 1/5 of its thickness. When the maximum deformation of the valve is
between 1/5 and 5 of its thickness, this range belongs to
large deformation, and the valve has geometric nonlinearity. Therefore, it is required to calculate via large
deflection theory. So far, there have been numerous
studies on the valve deflection of hydraulic dampers.
School of Mechanical and Electrical Engineering, Wuhan University of
Technology, Wuhan, People’s Republic of China
Corresponding author:
Jinli Chu, School of Mechanical and Electrical Engineering, Wuhan
University of Technology, 205 Luoyu Road, Hongshan District, Wuhan
430070, Hubei, People’s Republic of China.
Email: 2471653554@qq.com
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2
At early stages, Duym et al.5 and Ventsel and
Krauthammer6 studied theoretical models of the disk
stack prediction and interpretation of the fatigue wear
process. After that, existing approaches to valve system
modeling and measurement data interpretation are
being continually improved7 and the finite element
modeling approach is being implemented to support
development of more advanced models. YJ Chen8 and
CC Zhou et al.9 developed a precise mathematical
model of the valve deflection using the small deflection
theory, but the valve deformation of shock absorber
belongs to the large deformation in the actual working
condition. With the reduction of the thickness of the
valve and the increase of the working load, the error of
valve deformation based on deflection theory is also
increasing. YD Li and J Li10 use the Chien perturbation
method to solve the valve deformation, but the error is
larger. C C Zhou and L Gu11 utilized finite element
method (FEM) to solve valve deformation problem.
Though the solution was more accurate, finite element
method can not be used in modeling and analysis of the
whole shock absorber. LP He et al.12 have overcome
these shortcomings using perturbation method to combine FEM to solve the valve deformation; however,
only the maximum deformation of the valve can be
obtained. P Czop et al.13 developed the simplified and
advanced models for damper valve system. The valve
deflection error obtained from the simplified linear
model is large, P Czop et al. used iterative method to
solve the problem of simplified the nonlinear model,
however, the calculation process is complicated and it
is easy to cause errors in process. Meanwhile, P Czop
et al. Used finite element method to solve advanced
model, and the solution is accurate. A Farjoud et al.14
studied the method of calculating the deflection of shim
stack using energy method and verified the correctness
of the model by the shock absorber bench test.
This article will discuss a mathematical model of
shock absorber valve based on the hybrid method of
beam deflection and large deflection; the model can be
applied to analyze single disk deflection and shim stack
deflection. The form of this model is simple and easy to
solve. And the accuracy of the model is verified by FEM
simulation. The hydraulic shock absorber with valve
system consists of shim stack and a spring preloaded
disk is modeled and simulated using AMESim,18 and
the correctness of the mathematical theory and simulation is verified by bench test of the shock absorber.
Modeling of valve deformation
Modeling of single valve deformation based on beam
deflection theory
Figure 1(a) shows a single valve mechanical model
under uniform load q, the model can be regarded as a
Advances in Mechanical Engineering
Figure 1. Model of the single valve: (a) Single valve model
under uniform load, (b) The thin disc of several sectors.
Figure 2. The diagram of relationship between deflection and
load of the valve.
thin disk with fixed inner ring and unconstrained outer
ring and its thickness is d. It can be divided into several
sectors with minimal angle (du), as shown in Figure
1(b), which can be equivalent to several variable-width
cantilever beams with the same deflection. Therefore,
the valve deformation can be calculated by the beam
deflection theory.
The diagram of relationship between deformation
and load of the valve after the transformation is shown
in Figure 2.
As shown in Figure 2, a coordinate system is established where the center of the ring is the origin of the
coordinate. The bending moment equation of the section of radius r is
Mr = qdu
r 3 r1 2 r
r1 3
+
6
3
2
ð1Þ
where q is pressure differential across the damping valve
for a given flow rate (Pa), du is the angle of the sector
area (rad), and r1 is the outer radius of the valve (m).
According to the formula of the inertia moment of
the rectangular section in material mechanics,16 the section inertia moment at the radius r is
Ir =
rdu d3
12
Xu et al.
3
Using the integral method in material mechanics to
calculate the beam deflection,16 the deformable valve
deflection fr at any radius r is
3
3
2
2
3
r1 3 ln rr2
r
r
r
r
r
r
2
2 5= Ed3
fr = 12q4 1 + 1 +
18
2
18
2
3
2
valve (m); b1 , b2 is the coefficient related to a, and
u, A1 , A2 , B1 , B2 , C1 , C2 , D1 , D2 are undetermined coefficients, see Li and Li.10
Making vm = frmax = 12qCr1 =(Ed3 ), then q=d3 =
vm E=12Cr1 ; by substituting it into equation (2), the
equation for the valve deflection based on the hybrid
method can be written as
Making
Fr = vm Cr =Cr1
Cr =
r1 3 ln
r3 r12 r
r 2 r2
+ 1 +
18
2
2
3
r
r2
r23
18
as the deflection coefficient of the deformable valve at
any radius, the deflection formula of the deformable
valve obtained based on the beam deflection theory can
be written as
3
fr = 12qCr = Ed
where Cr1 is the deformation coefficient of the valve at
the outer radius.
Making g = Cr1 =Cr as the correction factor of deformation at radius r, then Fr = vm =g; by substituting it
into equation (3) and changing the form, the analytic
formula of the hybrid method is obtained
q=
ð2Þ
where E is Young’s modulus (Pa) and d is thickness of
the valve (m).
4Eb1
4Eb
d3 gFr + 4 2 dðgFr Þ3
2
m Þ
r1
3r12 ð1
Because beam deflection theory is derived from the
derivation of elastic mechanics theory, it is only suitable
for elastic deformation under small load. In the actual
working condition, the valve deformation of the shock
absorber belongs to small deformation only near the
inner edge, and the deformation of valve port belongs
to large deformation and it is closely related to damping
force characteristics. Therefore, a mathematic model of
the valve deformation based on the hybrid method of
beam deflection and large deflection is proposed.
According to the mechanical diagram of the deformable valve shown in Figure 1, based on the large deflection theory, the deformation equation of the
deformable valve at the outer edge is obtained by the
second-order perturbation solution of the Chien perturbation method11
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v i3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vm
m
+ b2
3ð1 u2 Þ
3ð1 u2 Þ
d
d
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3r14 ð1 u2 Þ 3ð1 u2 Þ
Q=
q,
4Ed4
2B1
C1
b1 = A1 +
p 4ða D1 Þ2 + C12
4Eb1
4Eb
,K = 4 2
2
m Þ
r1
3r12 ð1
the simplified hybrid method is
q = J d3 gFr + KdðgFr Þ3
a=
The actual shock absorber valve system is generally
disks in cylindrical or pyramidal stacks. In this article,
the model of cylindrical shim stack deflection is mainly
studied. Considering the shim stack assembly shown in
Figure 3. The shim stack consists of n disks stacked one
ð3Þ
2B2
C2
,
p 4ða D2 Þ2 + C22
r2
, 0:3 a 0:8
r1
where vm is the deformation of the valve at the outer
radius (m); u is Poisson’s ratio; d is the thickness of the
ð6Þ
Modeling of shim stack deformation based on hybrid
method
Q = b1
b2 = A2 +
ð5Þ
Let
J=
Modeling of single valve deformation based on hybrid
method
ð4Þ
Figure 3. The force diagram of cylindrical shim stack with
preloaded spring.
4
Advances in Mechanical Engineering
upon the other. A single valve geometry model shown
in Figure 1. The first valve plate is subjected to the fluid
pressure acting thereon, resulting in bending deformation, the force is transmitted to contact with another
valve, repeating the transmission of force, making the
entire shim stack bending deformation. Inter-shim friction is not investigated within this model, although it is
suspected to have a contribution to the overall valve
hysteresis.
It is known from the literature9 that each shim
deflection of the shim stack is equal, and then the
deflections and loads of the shim stack shown in
Figure 3 have the following relation
Fr1 = Fr2 = = Fri = Frn = Fr
8
q q1, 2 = Jd31 gFr + Kd1 ðgFr Þ3
>
>
>
>
>
q q2, 3 = Jd32 gFr + Kd2 ðgFr Þ3
>
< 1, 2
>
qi2, i1 qi1, i = Jd3i gFr + Kdi ðgFr Þ3
>
>
>
>
>
:
qn1, n qpre = Jd3n gFr + Kdn ðgFr Þ3
iX
=n
d3i gFr + K
i=1
iX
=n
di g 3 Fr 3
ð7Þ
Equation (7) is a special form of cubic equation with
one variable Fr , and the solution of Fr is solved by
Caldon’s formula
A=K
iX
=n
ð8Þ
B=J
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
3
u
3 q q
q qpre 2
Kg 3 Fr 3
t
pre
dE =
+
2J gFr
2J gFr
3J
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
s
u
u
2
3 3
3
3 q q
q qpre
Kg Fr
t
pre
+
+
+
2J gFr
2J gFr
3J
ð10Þ
The theoretical calculation and finite element simulation are carried out based on the single damping valve,
the model of the valve shown in Figure 1 is established
by ANSYS, and its boundary constraint and load are
shown in Figure 1(a), and the parameters of simulation
and theoretical calculation are listed in Table 1.
Because the deformation of the damping valve is
large deformation in the actual working condition, the
solution control is set to large deformation when simulating. The results are shown in Figure 4.
Figure 4(a) shows the deformation distribution on
the valve; the deformation results are illustrated via
color contour representing the varying deflection
regions. The maximum deflection is 2.9895e–04m,
according to equation (6) and the related parameters in
Table 1; the calculated maximum deflection of the valve
is 2.971e–04m, and the relative deviation is 0.6188%
and the error is very small. Figure 4(b) shows the
deflection diagram of the valve at any radius, where the
origin of the horizontal axis represents the inner radius
of the valve. From Figure 4(b), it can be seen that the
deflection at the valve port is 1.7653e–04m. According
to equation (6), substituting related parameters in
Table 1, the calculated deflection at the valve port is
di g 3
i=1
iX
=n
where dE is equivalent thickness of shim stack.
The equivalent thickness of the shim stack dE is
obtained by solving formula (9)
Numerical simulation verification
i=1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
u
q q 2
3 q q
B 3
t
pre
pre
+
Fr =
2A
2A
3A
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
u
q q 2
3 q q
B 3
t
pre
pre
+
+
+
2A
2KA
3A
ð9Þ
With equation (10), the shim stack can be equivalent
to a single valve to calculate its deformation.
Where q is pressure acting on the shim stack for a given
flow rate, qi1, i (i = 2, . . . , n) is the interaction force
between the shims, di (i = 1, 2, . . . , n) is the thickness of
single different shim, and qpre is spring preload from
spring preloaded disk.
The above n equations are summed to get a model of
the shim stack deflection, which is
q qpre = J
q qpre = J d3E gFr + KdE g 3 Fr 3
Table 1. The main parameters of the valve plate.
d3i g
Parameter (units)
Value
Inside radius of the valve plate r2 (m)
Radius of the valve mouth rk (m)
Outside radius of the valve r1 (m)
Valve thickness d (m)
Modulus (Pa)
Uniform load (Pa)
0.0055
0.0085
0.01
0.0003
206E + 09
3E + 06
i=1
After the value of Fr is obtained, according to the
principle that the deflection of the valve remains the
same when the thickness of the shim stack is equal to
the thickness of a single valve, the shim stack equivalent thickness formula is
Xu et al.
5
Figure 4. Simulation results of the valve deformation based on FEM: (a) deformation distribution on the valve and (b) deflection
diagram of the valve.
1.6904e–04m, and the relative deviation is 4.24%. Fully
meet the practical engineering requirements.
At present, the small deflection theory is widely used
to calculate the deformation of damping valve,17
because only the elastic deformation is considered, and
the calculated deformation is larger than the actual
deformation of the damping valve. In order to compare
the deflection calculated by different methods more
intuitively, the deflection curve of the valve based on
FEM, the small deflection method, and the hybrid
method is plotted, as shown in Figure 5.
From Figure 5, the valve deformation based on the
hybrid method has good agreement with the simulation
result of FEM, and the error gradually decreases with
the increase of deformation. The deformation calculated by small deflection method only has good agreement with the simulation result in a relatively small
deformation range, but the error also increases with the
increase of deformation. Because only the elastic deformation is considered in the small deflection method, the
calculated valve deformation is larger than the actual
deformation of the damping valve, which shows that
the hybrid model can be used to calculate the valve
deformation at any radius, and the small deflection
Figure 5. Valve deflection diagram of three methods.
—: deflection based on hybrid method; – –: deflection based on small
deflection method; ——
: deflection based on FEM.
6
Advances in Mechanical Engineering
Figure 6. Structure of the shock absorber.
1: Comp. valve; 2: accumulator; 3: lower working chamber; 4: piston rod;
5: spring preloaded disk; 6: upper working chamber; 7: bottom valve; 8:
Comp. check valve; 9: Reb. valve; 10: piston; 11: Reb. check valve.
method is only applicable to calculate small deformation of the damping valve.
Figure 7. Physical model diagram of the twin-tube hydraulic
damper.
Simulation using AMESim and test
verification of the shock absorber
Structure and principle of the shock absorber
The shock absorber studied in this article is a twin-tube
hydraulic shock absorber, and its structure is shown in
Figure 6.
The shock absorber has two main working strokes:
recovery stroke and compression stroke. The red thick
arrows in Figure 6 indicate the direction of the piston
rod in the recovery stroke. When the wheel is stretched
relative to the body, the piston rod moves outward relative to the working cylinder, and oil flows in the shock
absorber, as indicated by the red thin arrows in Figure 6.
The damping force caused by friction in the oil and flow
through different damping valve hinders the relative
movement of the wheel and body. Therefore, the damper
plays the role of reducing damping and buffering vibration to improve ride comfort of vehicle.
The motion principle of the shock absorber in the
compression stroke is similar to that of the recovery
stroke, and only the directions of the piston rod and oil
flows are opposite to that of the recovery stroke.
Fdf = (p1 p2 )Srd + p2 Sg + Ffriction sgn(_x)
where Sg is the area of the piston rod (m2); Srd is the
annular area between piston cylinder and piston rod
(m2); Ffriction is friction (N); x_ is piston rod speed (m/s).
By changing the form of the formula, the above
damping force equation can also be written as follows
Fdf = p12 Srd + p32 Sg p3 Sg + Ffriction sgnðx_ Þ
where p12 = p1 p2 is the pressure drop between chamber 1 and chamber 2; p32 = p3 p2 is the pressure drop
between chamber 3 and chamber 2.
The friction mechanism is very complex, according
to the damper motion state, there is a mutual conversion between dynamic friction and static friction. The
speed of the shock absorber is much higher than the static friction speed range. Therefore, the damper friction
force can be simplified to a constant; we found the friction force is 40 N when the speed is 0.001 m/s, the value
of the friction force can be substituted in equation (11).
Equation (12) permits to compute gas pressure by
adopting an adiabatic approximation
p3 = pdyn = The mathematical model of the shock absorber
The physical model shown in Figure 7 is established
according to the structure diagram of the twin-tube
hydraulic shock absorber, as shown in Figure 6.
Taking the recovery stroke as an example, the parametric model of the damper is established. The Fdf indicates the total damping force in the recovery stroke. p1 ,
p2 , p3 indicate the pressure of chamber 1, chamber 2,
and the accumulator of the shock absorber, respectively. The equation of the damping force during recovery stroke can be expressed as
ð11Þ
n
pstatic Vstatic
n
Vstatic + Sg x
ð12Þ
where x is piston displacement (m); pstatic and Vstatic ðm3 Þ
are gas pressure and volume of the accumulator in the
initial, respectively (Pa); pdyn is the actual pressure of
the accumulator (Pa); and n is a polytrophic index
related to the speed of change of gas state, here n = 1:4.
1.
Damping characteristic before the valve opens.
Figure 8 shows hydraulic circuit diagram before the
damping valve opens. The damping valve is completely
Xu et al.
7
Qhf =
pDh d3hf ð1 + 1:5e2 Þphf pDh Vh dhf
12mt Lhf
2
ð14Þ
where Dh is hydraulic diameter of the piston (m), e is
the eccentricity of the piston (m), Lhf is the length of the
piston gap (m), mt is dynamic viscosity of oil fluid, and
Vh is piston rod speed (m/s.)
The constant orifice is a small rectangular orifice in
a symmetric form, and its equivalent diameter d is computed according to the equal velocity flow method
d = ab=2ða + bÞ
Figure 8. Hydraulic circuit diagram before the damping valve
opens.
closed until the pressure drop between Port 1 and Port
2 reaches the pre-defined pressure. The piston rod speed
is less than the initial opening valve speed; the valve
does not have deformation, oil flows from upper chamber to lower chamber through piston orifices, constant
orifices and piston gap, resulting in throttling pressure.
In addition, because the recovery chamber has a piston
rod, the fluid flowing into the compression chamber
from the recovery chamber is not enough to fill the volume increased by the compression chamber, a certain
negative pressure is generated in the compression chamber, the pressure difference between chamber 3 and
chamber 2 deforms the compensating valve, and the oil
flows into chamber 2 from chamber 3. The stiffness of
the compensating valve is small and can be considered
as a check valve, which can be deformed by the small
oil pressure. Only in this way can oil flow to chamber 2
in time, in order to avoid distortion of the compression
stroke.
According to the pressure drop and flow relationship across two orifices in series or in parallel,15 there is
the following relationships between flow and pressure
of piston
Q1 = Qhf + Qhk , Qhk = Qck , phf = p12 = phk + pck ð13Þ
where Q1 = Vh Srd is total flow rate from chamber 1
flow to chamber 2, Qhf is flow rate through the piston
gap, phf is pressure difference at both ends of the piston
gap, Qhk is flow rate through the piston orifices, phk is
pressure difference at both ends of the piston orifices,
Qck is flow rate through the constant orifices, and pck
is pressure difference at both ends of the constant orifices. p12 is pressure drop between chamber 1 and chamber 2.
According to fluid mechanics theory, the pressure–
flow relationship can be written in the following form
ð15Þ
where a, b are width and length of rectangular orifice,
respectively. After the equivalent constant-orifice can
be regarded as a thick-walled small hole. According to
the theory of fluid mechanics, we can obtain the relationship between flow and pressure
rQ2ck
ð16Þ
2 A 2
2Cde
ck
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where Cde = Cd = 1 + jCd2 is equivalent flow coefficient, Cd is flow coefficient and j is local resistance coefficient; Ack is the total area of the constant orifices.
The piston orifices distribute on the piston evenly,
the diameter of each orifice dhk and the numbers of orifice nhk are standardized, when the length lhk is four
times bigger than diameter dhk, the piston orifice is slender orifice. The relationship between flow rate and pressure can be written as
pck =
phk =
Qhk
KAhk
ð17Þ
2
=37:5mt lhke ; lhke is the stacking length of
where K = dhk
the piston orifices (m).
The following relations can be obtained by equations
(13), (14), (16), and (17)
A1 B2 2 p12 2 + ð2A1 B1 B2 + A2 B2 1Þp12
+ A1 B1 2 + A2 B1 = 0
ð18Þ
where
A1 =
r
2 A2
2Cde
ck
B1 = Vh Srd +
, A2 =
1
,
KAhk
pDh d3hf ð1 + 1:5e2 Þ
pDh Vh dhf
, B2 =
2
12mt Lhf
We can utilize equation (18) to calculate the equation between the velocity Vh of piston rod and the pressure drop p12 ; similarly, according to the relationship
between the flow rate and the pressure drop of bottom
valve, we can also obtain the relationship between the
pressure drop p32 and the piston rod speed Vh ,
8
Advances in Mechanical Engineering
The relationship between the throttle pressure drop
pjf and the valve deformation Fr is
8
<
pjf ppre = J
:
iP
=n
d3i gFr + K
i=1
Fr = gvm
iP
=n
di g 3 Fr 3
i=1
ð22Þ
where d is the thickness of the Reb. valve (m).
Combining equations (21) and (22), the following
equation can be obtained
qffiffiffiffi
8
pjf + A2 B1
pjf
>
< B1 + B2 1A2 B2 P1 = A3 v3m pjf
iP
=n
iP
=n
>
: pjf = J
d3i gFr + K
di g 3 Fr 3 + ppre
i=1
ð23Þ
i=1
3
3
A3 = pCrk
=½6Cr1
mt
Figure 9. Hydraulic circuit diagram of the damping valve opens
first.
combining the gas equation shown in (12), we get relationship between damping force and velocity.
2.
Damping characteristic after the valve opens
first.
Before and after the Reb. valve opens first, hydraulic
circuit diagram of the bottom valve remains unchanged,
so there is only the piston hydraulic circuit diagram
after the valve opens first, the diagram is shown in
Figure 9.
According to the pressure drop and flow relationship across two elements in series or in parallel15, the
equations of flow rate and pressure of hydraulic circuit diagram shown in Figure 9 are
Q1 = Qhf + Qhk , Qhk = Qjf + Qck ,
p12 = phf = phk + pck , pck = pjf
ð19Þ
When the pressure drop between Port 1 and Port 2
reaches the pre-defined pressure, the valve deforms at
the valve port, creating a circular gap, the relationship
of flow rate Qjf through the gap and pressure can be
written as
Qjf =
pd3f pjf
6mt ln r1f =rkf
ð20Þ
where df is the deformation of valve port (m), rkf is the
radius of valve port (m), and r1f is the outer radius of
the valve (m).
The following equation can be obtained by equations
(14), (16), (17), (19), and (20)
pd3f pjf
p + A2 B1
= B1 + B2 jf
1 A2 B2
6mt ln r1f =rkf
rffiffiffiffiffiffi
pjf
P1
ð21Þ
where
ln (r1f =rkf ).
Combining equations (14) and (16)–(18), the following relationship can be obtained
p12 = pjf + AB1 =ð1 A2 B2 Þ
ð24Þ
Combining equations (23) and (24), the relationship
between the velocity and the pressure drop p12 between
chamber 1 and chamber 2 can be obtained, p32 and p3
are known, then the relationship between the damping
force and the velocity is obtained.
3.
Damping characteristic of the valve opens to the
maximum.
The damping characteristic of the valve fully opens is
similar to that after the valve opens first, and the
hydraulic circuit is shown in Figure 9. The Reb. valve
has achieved maximum deformation Frmax and the leakage area of the Reb. valve has reached the maximum
value. The damping characteristic in the compression
stroke is similar to that in the recovery stroke, so it is
not introduced in detail.
Modeling and simulating of the hydraulic shock
absorber based on AMESim
The deformation of the damping valve directly affects
the damping force characteristic in form of the force–
displacement (F-S) and the force–velocity (F-V) diagram. Therefore, the curves of the F-S and the F-V of
the shock absorber can be used to verify the correctness
of the mathematical model of the valve deformation
based on the hybrid method. According to structure
and principle of the shock absorber, a model of the
hydraulic shock absorber is developed using AMESim,
as shown in Figure 10. The spring of varying rate in
AMESim is used to simulate the damper valve system.
The stiffness of the shim stack and a spring preloaded
disk are calculated using the hybrid method, the overall
stiffness is taken as the stiffness of the damper valve
Xu et al.
9
Figure 10. Simulation model of the hydraulic shock absorber based on AMESim.
A: displacement excitation; B: piston gap and chamber 1 and chamber 2; C: Reb. valve and its constant orifices + Reb. check valve + piston
orifices; D: Comp. valve and its constant orifices + Comp. check valve + bottom valve orifices; E: accumulator; F: oil/gas properties.
system and is set as the spring stiffness in AMESim,
and spring preload is set in AMESim to simulate the
spring preload of the actual damper valve system. The
other parameters of the simulation model are set
according to the actual parameters of the shock absorber and the above mathematical models, and the main
parameters of the shock absorber are listed in Table 2.
Simulation AMESim and test verification
According to QC/T545-1999(shock absorber bench test
standard), we input related data into the simulation
model and the experiment, then load sinusoidal excitation. Table 3 shows the specific parameters.
Some measurements are carried out on a Mechanical
Testing&Simulation (MTS) test rig. Through testing
the shock absorbers by giving different excitation conditions, Figure 11 shows the damper and the test rig of
the damper used in this study, we obtained the multiworking indicator diagram of the shock absorber, the
diagram was shown in Figure 12(a). By setting batch
parameters in AMESim, we obtained the multi-frequency indicator diagram of the shock absorber, which
was shown in Figure 12 (b).
From Figure 12(a) and Figure 12(b), it can be seen
that the indicator diagram of the model is in fairly good
Table 2. Main parameters of the shock absorber.
Parameter (units)
Value
Piston diameter (mm)
Piston rod diameter (mm)
Piston gap (mm)
Maximum lift of the Reb. valve (mm)
Maximum lift of the Comp. valve (mm)
Oil density (kg/m3)
Absolute viscosity of oil (10–3 Pa s)
The volume modulus of oil (MPa)
Inner diameter of compensation cavity (mm)
Inflatable pressure (MPa)
Reb. valve spring preload (N)
Comp. valve spring preload (N)
Flow coefficient of the constant orifice
Inner diameter of the working cylinder (mm)
Outer diameter of the working cylinder (mm)
30
15
0.03
0.3
0.3
860
6.7
1700
43
0.4
150
60
0.72
30
32
agreement with the measurements. There are only small
differences existing locally, this may be due to the simplification of the shock absorber during the simulation
modeling. Such as the frictional force of the piston master cylinder is not taken into account.
The velocity input characteristics determine whether
the test and simulation are performed statically or
10
Advances in Mechanical Engineering
Table 3. Parameters of the loaded excitation.
Stroke (mm)
Velocity (m/s)
Frequency (Hz)
50
50
50
50
50
50
50
0.02
0.052
0.131
0.262
0.393
0.524
1.0
0.130
0.330
0.840
1.670
2.490
3.330
6.340
Figure 11. The test rig of the shock absorber.
dynamically. During static simulations and tests, different constant velocities are entered. Table 3 shows
the specific input conditions. Figure 13 shows the static
simulations and tests results.
As shown in Figure 13, static damping curves of the
simulation and measurement are composed of three
parts: low-speed zone, the shape of the low-speed zone
is mainly determined by the size of the constant orifice;
the knee zone (the region on the right of the knee
point), the shape of the knee zone is determined by the
opening characteristics of the valve; high-speed zone,
starts where the curve blends into a straight line, the
maximum size of the opening valve and the preload of
the spring are main factors determining the slope of the
zone.
As can be seen from Figure 13, the trend of F-V
curves in simulation and test are basically the same for
the three zones. The simulation and test results are
highly consistent for the knee point of the curve (the
valve-opening point of the damper). The slope of the
curve is slightly different only in the high-speed zone;
the simplification of the valve model and the simulation
model are the main factors resulting in the difference.
Because the static curve cannot well simulate the
dynamic process of the actual work of the damper, the
dynamic simulation and measurement of the damper
are carried out. In AMESim, the input excitation is set
to a standard sine signal. The excitation frequency is
6.34 Hz; the amplitude is 6 0.025 m. The test input is
same as the simulation input. The dynamic velocity
characteristic curve obtained by the simulation and the
test are shown in Figure 14.
Similar to the static damping curve shown in Figure
13, the dynamic curve can also be divided into three
zones, for the three different zones, the dynamic damping curve trends in simulation and test are basically
consistent. Unlike the static damping force curve, the
dynamic damping force curve is circular; the cycle is
divided into four different phases: rebound acceleration
(Reb. acc), rebound deceleration (Reb. dec), compression acceleration (Comp. acc), and compression deceleration (Comp. dec). According to Figure 14, the
acceleration force values are lower compared the deceleration ones around zero velocity, where the movement
changes direction, this separation is called ‘‘Hysteresis.’’
Figure 12. Indicator diagrams of the shock absorber: (a) simulation curve based on AMESim and (b) test curve.
Xu et al.
11
the shock absorber with valve system consists of shim
stack and a spring preloaded disk is developed using
AMESim, and the measurement is carried out by damper test rig. The following conclusions are obtained:
1.
Figure 13. Static velocity characteristic curve.
2.
3.
The mathematical model for the valve deformation based on the hybrid method of beam deflection and large deflection is simple and easy to
calculate, and the accuracy is high. The model
can be used for design purposes and helps in
developing damping valve based on single disk
and shim stack, which is beneficial to the whole
modeling and analysis of shock absorber in
practical engineering.
The simulation results show high agreement
with experimental results, the proposed
AMESim model of shock absorber with valve
system consists of shim stack and a spring preloaded disc has high accuracy, which has a certain reference for the development of shock
absorber.
The simulation results and the experimental
results have some small errors in the local area,
mainly because the shock absorber is simplified
in modeling. In practical work, the shock absorber is very complex and needs to be studied and
improved continuously to improve design level
of shock absorber.
Declaration of conflicting interests
Figure 14. Dynamic velocity characteristic curve.
According to Figure 14, both the simulation model and
the test have hysteresis characteristics in the middleand low-speed regions, and the only difference is in the
relatively low hysteresis level estimated by the simulation model, compared to the actual characteristics of
the shock absorber. This is because the simulation
model ignores some major factors that cause hysteresis,
such as leakage and elastic deformation of master cylinder and friction of inner-shim.
The above analysis shows that the shock absorber
simulation model based on AMESim is reasonable and
accurate. The model simulates the actual shock absorber well, It also shows that it is feasible to calculate the
deformation of the shock absorber valve by the hybrid
method.
Conclusion
In this article, based on the hybrid method of beam
deflection and large deflection, the mathematical model
of the valve deformation is established, the model of
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
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