Question

Question 3. A crank shaft mechanism is shown in Figure 3. Link B is rotating with a constant angular velocity wgåg in the fixed reference frame A as shown in the figure. For this mechanism: (a) Using the vector kinematic equations and methods taught in this [20 marks) module, obtain the expressions for velocity and acceleration of point in the fixed frame A, in terms of wb.lg, lc, and e.

x=3

Answer 1

Let na obliquity ratio = less o- angle made by crank with IDC or line of shoke (08) Let go = angle made by connecting rod (PQ) with line of shooke (06) velocity of the slider: (8) let "v" be the displacement of slider from IDC after t seconds. le del Vr - R 1) sh on OB + x = 89': 09'- 09 - CHP'I+ Po) - (BR+ RO) x = ( la +18) - (PO COSø + OP COSO) x = lctlB - lc coso - lp coso x - lp [(i-cose) +n (i-cosø)] From triangles PBR and PRO, PR=le sind - le sine or sin = sino We know that; cosp=/(1- (1-sin?) (1- Expand the above expression by binomial theorem we we get t..... ( Neglect 2 h Therefore coso- no I sinzo sin²o h2 soy? ha 1 cosy - i- 1-Ź xsino higher terms) h2 (3 - (-cosø)= sino 2n² x=l6(1-coso) + xx sinzo -cose)+ sin 26 (4 2n2 2n Substinete equation 3 in 1 we get 0[1-cose) + [p-coso) Differentiate the above equation w.nit O; dx-lo sino + 1x2sino. [sino + (sino.cos– es (sino + (2 sino cose = sin 20 ) sin20 2n 2n do

Voo ido dt do 2h Therefore velocity of slides & w.rit o is given as; Vg da dxxdo) - dx xwe at) Substitute equation in above equation; Ves = wala sino + since ... here n-f%ce) Substituting ; 18 Also; nale 57 IB Vo = 25.13/sino + sin 20 (sine (velocity of point & sad 16cm - 21 (1500) =157.1 polls 0.16m WB 2 60 3.56 z = 16 7.12 ms

d Vs x WB -X dva de • Acceleration of the slider (8) i ag= dva dva do dt © do dt do Differentiation velocity of slider (V8) equation 6 w.ato; wel caso + cos 20.6 [ coset zu Substituting the value of dus in equation 7 above, do w² cose + cos20 ( cose Substituting the given values; w=1574 vedis , lg-0.16m [ceso + 2 ) WB LB /cesot cosso h Ag = n n= 3.56 as= (3948.86) cose + cos20 deceleration of point) 3.56 (m/s)