Post on 23-Oct-2020
Vol.47,No.6,November 2016. 1367
研究論文
Modeling of an Attenuation Characteristics of the Damper Used for Micro Vibration of a Car Body
Hiroki Nakamura Toshiaki Kamo Hideki Ohsawa Shota Fukushima Hiroshi Sakanoue Toru Yamazaki
The present damper developed for a micro vibration suppression of a car body, that uses an oil hydraulic pressure control valve for generating
damping force, is able to produce a velocity-independence nearly constant damping force unlike an ordinary viscous damper and hence to
produce a large damping force in a very small velocity range. However, a particular account of the generation mechanism of a damping force has
not been given so far.
This paper presents firstly the derivation of equation for attenuation characteristics; nonlinearity of the present damper with a built-in oil
hydraulic pressure control valve as an attenuation mechanism. Next, linear attenuation coefficient equivalent to the non-linear damping force is
driven using a describing function method and then the validity of modeling of damping mechanism and its linearization technique is verified
on the basis of comparison of simulations with experiments.
: Damping for micro-vibration (B3)
(1) 1
Fig.1 Performance damper installed on a car
*2016 6 8 2016 5 25
1) 4) 6) (221-8686 3-27-1)
2) 3) 5) ( ) (438-8501 2500)
2
2
Fig.2 Components of test performance damper
Conduit High pressure nitrogen gas
Free piston
Oil chamber
Valve plate Piston rod
Piston head Spring for cancellation
of gas pressure
20134020164634⬅TOPに戻る⬅TOPに戻る
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1368 自動車技術会論文集
車体制振ダンパーの減衰特性のモデル化
3
0 1 2 3 4 50
900
Velocity [mm/s]
Dam
pin
g F
orc
e [
N]
Fig.3 Measurements of the damping force vs piston speed
characteristics of test dampers
f(t) m
x
k F (1)
2
F
2
2( )
d xm F kx f t
dt(1)
4
pc
p
Flow rate
Pre
ssu
re d
iffe
ren
ce
Fig.4 General characteristics of pressure control valve
4 3
F
p Q
p dx/dt F
Fig.5 Valve plate structure for pressure control valve with a
built-in test damper
5 2.1
5
(1)
P0
pc p
High
Middle
Low
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Vol.47,No.6,November 2016. 1369
車体制振ダンパーの減衰特性のモデル化
(2) P0
p (dx/dt 0 )
(dx/dt 0 )
(3) F p
1
Table 1 Symbol and term of parameters for pressure control valve
Symbol Term
Number of valve (flow path)
Valve opening
Discharge angle from the valve
Density of the fluid
Flow rate
Discharge coefficient of the valve
Diameter of the conduit
6
x
Ah Ar
k -x0 dx/dt 0
dx/dt 0 2
Fig.6 Modeling of pressure control valve for generating constant
damping force
0)()( 000 PApPAxxk hr (2)
0 0 0 0r hkx A P A P (3)
(1)
F (2) (3)
0 : rdx
F A pdt
(4)
0 : rdx
F A pdt
(5)
(4) (5)
sgn rdx
F A pdt
(6)
p Q
6 dx/dt 0
7
Q
2d v
pQ nC Dx (7)
Cd
0.65 0.7
kv xv, xv.0
a v
Cu (9)
2
.0cos ( )v v vap vQ k x x (8)
.02 sin cos ( )u d v v v vap C C px k x x (9)
Fig.7 Model of a plate–type pressure
control valve
7 90°
(9)
xv
±dx/dt
Ah p Ar
Piston head
Control orifice
Conduit
Valve plate
Control orifice
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1370 自動車技術会論文集
車体制振ダンパーの減衰特性のモデル化
.0( )v v vap k x x (10)
, p xv
.0( )v v v v vc
k x x k xp p
a a(11)
pc xv=0
p
pc (11)
.0v vc
k xp
a (12)
(11) xv p
.0v vv
v
ap k xx
k (13)
(7) Q p
.02 2
2( )
v vd v d
v
d c
v
ap k xp pQ nC Dx nC D
k
a pnC D p p
k
(14)
Q p
a, kv,
4
Q p 14
Q dx/dt
dt
dxAQ r (15)
(6) F p
F dx/dt
F
F
0sgndx dx
F Fdt dt
(16)
F0 pc
(=Arpc)
0
0
Velocity [mm/s]
Dam
pin
g f
orc
e [N
]
Fig.8 Correlation between velocity and damping force
(16) 8 0
F0
F0
(17) X0
tXx sin0 (17)
(16)
F ceq
dt
dxcF eq (18)
16 dx/dt>0 dx/dt0
1 (16)
(18)
dxdt
dxcdx
dt
dxF eq0 (19)
ceq F0 X0
0
0
4eq
Fc
X (20)
f
F0
-F0
Grad.:
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Vol.47,No.6,November 2016. 1371
車体制振ダンパーの減衰特性のモデル化
( )
k x x k x
tsin
dx
2
2sineq
d x dxm c kx f t
dt dt (21)
9
3
3 39
(a) 0.5mm 10, 20, 30, 40, 50Hz 5
(b) 10Hz 0.5, 0.7, 0.9, 1.0mm 4
(c) 20Hz 0.5, 0.7, 0.9, 1.0mm 4
Fig.9 Sinusoidal excitation experimental equipment for test damper
0.5mm 40Hz 10
a b
11
10 11
dx/dt
2.2
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09-0.5
0
0.5
Time [s]
Dis
pla
cem
ent
[mm
](a) displacement
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0
Time [s]
Velo
cit
y [
mm
/s]
(b) velocity
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0
Time [s]
Dam
pin
g f
orc
e [
N]
(c) damping force
Fig.10 Time history waveforms of displacement and damping force
at 40Hz sinusoidal excitation of 0.5mm amplitude
8
P0
1 3.2
High
Middle
Low
High
Middle
Low
High
Middle
Low
1000
-1000
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1372 自動車技術会論文集
車体制振ダンパーの減衰特性のモデル化
-800 -400 0 400 800
0
Velocity [mm/s]
Dam
pin
g f
orc
e [N
]
Fig.11 Correlation between velocity and damping force
ceq (20)
ceq
2 11
RMS RMS
3
0.5mm 10Hz 50Hz
Table2 Damping coefficients obtained from experimental data
[Ns/m]
Type Input frequency
10Hz 20Hz 30Hz 40Hz 50Hz
High 26921 15746 11385 9652 9005
Middle 19980 11421 8601 7435 7054
Low 10066 6068 4904 4410 4284
3 3 11
F0 11
3 F0 X0
(20)
4
10Hz
10Hz
Table3 Damping coefficient calculated by Eq. (20) [Ns/m]
Input frequency
10Hz 20Hz 30Hz 40Hz 50Hz
High 31093 15895 10829 8296 6776
Middle 18878 9759 6719 5199 4288
Low 8731 4678 3327 2651 2246
Table4 Relative error between experiment and proposed model
Type Input frequency
10Hz 20Hz 30Hz 40Hz 50Hz
High 17.4 4.1 0.5 8.8 19.2
Middle 3.7 11.4 17.7 25.2 34.1
Low 9.5 16.7 24.5 31.4 38.8
30Hz
25%
(20)
(1)
(2)
(3) (2)
High
Middle
Low
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