OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
Fig. 4. 4-bar Linkages with clearance
From the close loop vector relation
L
c
c
iθ1
iθ2
iθ3
iθ4
iγ
2 iγ
3
1 e + L2e
+ L3e
+ L4e
+ r2e
+ r3
e =
Fig .5. 4-bar Linkage with clearance: vector loop
By separating Eq. (7), into its real part and imaginary part and using trigonometric relations,
joint clearance can be expressed as a function of θ and γ
2 2 respectively:
0
⎛
2 ⎞
c −1⎜
− B ± B − 4AC
θ =
⎟
3 2 tan
(8)
⎜ 2A
⎟
⎝
⎠
c
⎡ L
⎤
1
− L2
cosθ2
− r2
cosγ
2
− L3
cosθ3
− r3
cosγ
3
⎢
⎥
⎣
L4
⎦
c −1
θ
4
= cos
(9)
(7)
c c
3 and θ 4
θ with
Where the superscript c denotes the value with clearance. A , B,
C terms are given, with joint clearance
respectively
A = −( L1
+ L3
)( 2L2
cosθ2
+ 2r2
cos γ 2 ) + 2r3
cos γ 3
+ 2L2r2
cos( θ2
− γ 2 ) + 2L2r3
cos( θ2
− γ 3) + 2r2r3
cos( γ 2 − γ 3)
2 2 2 2 2 2
+ 2L1L
3 + L1
+ L2
+ L3
− L4
+ r2
+ r
(10)
3
B = 4L
( L sin θ + r sin γ + r γ )
(11)
2 2 2 2 3 3
C = ( L3
− L1
) 2L2
cosθ
2 + 2r2
cos γ 2 + 2r3
cos γ 3 +
2L2r2
cos θ 2 − γ 2 + 2L2r3
cos θ 2 − γ 3 + 2r2
r3
cos γ 2 − γ 3
2
1
+ L
+ L
3 sin
( ) ( ) ( )
2
2
+ L
2
3
− L
2
4
2
2
+ r
2
3
+ r
− 2L
L
As shown in Fig.4, the position of the coupler point ( x y)
clearance, respectively:
c
x
c
( θ ) + r cosγ
+ A′
P ( θ + β )
P = L2 cos
2 2 2
cos
3
c
c
( θ ) + r sinγ
+ A′
P ( θ + β )
Py
= L2 sin 2 2 2 sin 3
Where
c
x
clearance.
P ,
c
P denote the
y
X and Y
1 3
(12)
P , relative to the crank pivot A
1
is given with joint
(13)
(14)
coordinate values for the path of the coupler point in considering the joint
In the kinematic analysis of the 4-bar mechanism with double joint clearance, it is necessary to determine the
position of mass centre for moving links and then their corresponding velocities and accelerations. So in the
case of joint clearance these positions are derived from the vector representation of the mechanism in Fig.5.
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