Chapter 5 Analytical Linkage Analysis

Chapter 5 Analytical Linkage Analysis 2005/4/13 機構學 (C. F. Chang) 1 Loop Closure Equations for Four-Bar Linkages ref : pp. 175-184 • We will repre...
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Chapter 5 Analytical Linkage Analysis

2005/4/13

機構學 (C. F. Chang)

1

Loop Closure Equations for Four-Bar Linkages ref : pp. 175-184 • We will represent each vector by a length ri and an angle θi • All angles are measured counterclockwise from a line that remains parallel to the fixed x axis attached to the reference frame

r p = r2 + r3 = r1 + r4 or

r2 + r3 − r4 − r1 = 0 r2 cos θ 2 + r3 cos θ 3 − r4 cos θ 4 − r1 cos θ1 = 0   r2 sin θ 2 + r3 sin θ 3 − r4 sin θ 4 − r1 sin θ1 = 0

2005/4/13

機構學 (C. F. Chang)

National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2

1

r2 cos θ 2 + r3 cos θ 3 − r4 cos θ 4 − r1 cos θ1 = 0   r2 sin θ 2 + r3 sin θ 3 − r4 sin θ 4 − r1 sin θ1 = 0

Position Analysis

r3 cos θ 3 = − r2 cos θ 2 + r4 cos θ 4 + r1 cos θ1

(a)

r3 sin θ 3 = − r2 sin θ 2 + r4 sin θ 4 + r1 sin θ1

(b)

(a)2+(b)2!

r32 = r12 + r22 + r42 + 2r1r4 (cos θ1 cos θ 4 + sin θ1 sin θ 4 ) − 2r1r2 (cos θ1 cos θ 2 + sin θ1 sin θ 2 ) + 2r2 r4 (cos θ 2 cos θ 4 + sin θ 2 sin θ 4 )

(c)

Combining the coefficient of cosθ4 and sin θ4 yields A cos θ 4 + B sin θ 4 + C = 0

(d)

where

function of θ2 (ri and θ1 are constants)

A = 2r1r4 cos θ1 − 2r2 r4 cos θ 2 B = 2r1r4 sin θ1 − 2r2 r4 sin θ 2 C = r12 + r22 + r42 − r32 − 2r1r2 (cos θ1 cos θ 2 + sin θ1 sin θ 2 ) 2005/4/13

機構學 (C. F. Chang)

3

Position Analysis (cont) A cos θ 4 + B sin θ 4 + C = 0

θ  θ  2 tan  4  1 − tan 2  4  2  2   , cos θ = sin θ 4 = 4 θ θ    1 + tan 2  4  1 + tan 2  4  2  2  

θ  θ  (C − A) tan 2  4  + 2 B tan  4  + ( A + C ) = 0  2  2

2  θ  − 2 B ± 4 B − 4(C − A)(C + A) tan  4  = 2(C − A)  2

=

− B ± B 2 − C 2 + A2 (C − A)

“± ±” corresponding to the two possible positions of the P for a given value of θ2 2005/4/13

機構學 (C. F. Chang)

National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

4

2

Position Analysis (cont)

2 2 2 θ  − B ± B − C + A tan  4  = (C − A)  2

• Special cases: " No θ4 can be found if B2-C2+A2