神奈川大学工学部機械工学科...the present damper developed for a micro vibration...

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

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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




(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




.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.:




( )

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




-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


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


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


神奈川大学 工学部機械工学科...the present damper developed for a micro vibration...

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神奈川大学工学部機械工学科...the present damper developed for a micro vibration...