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(c) 0 (d) 1 (e) 1 43 The area between the curves y = sin x and y = cos x over the interval [ /2, 9 /4] is (a) 5 2 1 (b) 4 2 1 (c) 2 5 (d) 5 2 + 1 (e) 2 + 1 44 If 3 f (x) dx = 2 and (a) 4 (b) 5 (c) 5 (d) 4 (e) 2 45 If (a) (b) (c) (d) (e)

(1227)

This is a second-order linear differential equation with variable coef cients Figure 1223 shows the acceleration response of a cycloidal cam with T = 015 when damping is not taken into account Note that when F = 0 (corresponding to an in nitely rigid shaft), the curve resembles that of nominal acceleration With F increasing, the amplitudes of the residual vibration increase markedly The reason for this phenomenon is that at big values of F, ie, with a relatively exible shaft, a large amount of energy is stored in windup and bending during the rst half of the cam lift During the second half, with the decreasing slope of the cam, the shaft relaxes and acts as a catapult At the increased angular velocity of the cam, the deceleration of the follower may appreciably exceed the nominal value, so that heavy vibrations will persist after the end of the cam rise Figure 1224 shows a plot of the residual amplitude of acceleration versus T, with F as a varying parameter for an undamped cycloidal cam Damping would change the effect very little

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50 40 48 46 47

1245 Multi-Degree-of-Freedom System A complex dynamic model with eleven degrees of freedom was studied by Kim and Newcombe (1981) For more on multi-degree-of-freedom systems, see Pisano and Freudenstein (1983) Kim and Newcombe (1981) also presented a study to take into account the random nature of exibility and fabrication errors Figure 1225 shows the schematic model consisting of driving torsional and driven translational subsystems Figure 1226 shows the three-degree-of-freedom torsional subsystem model Figure 1227 displays the eight-degree-of-freedom translational subsystem model The latter is divided into two subsystems according to the vertical motion (y axis) and the horizontal motion (x axis) of the parts Using Newton s second law, we can write the following sets of equations of motion For translational movement of the follower (see Fig 1227) along the y axis direction

The lxe*drx can itself be integrated by parts as was done in the previous problem Use the result of problem 10-16 to write

46 The area between the parabolas y = x 2 + 1 and y = x 2 + 4x + 7 equals (a) 64/3 (b) 61/3 (c) 64/5 (d) 59/2 (e) 63/7 47 The area between the parabola y = x 2 8 2x and the x-axis is (a) 52 (b) 40 (c) 12 (d) 36 (e) 24

FIGURE 1223 Acceleration response for cycloidal cam with T = 015, undamped, and different degrees of driveshaft exibility

M I y5 + c f 1 ( y5 - y4 ) + k f ( y5 - y4 ) = 0 1 M f y4 - c f 1 ( y5 - y4 ) + c f y4 + kvc ( y4 - y3 ) - k f ( y5 - y4 ) = 0 2 1 M fc y3 + c f y3 + krb ( y3 - y2 ) - kvc ( y4 - y3 ) + Fp + krs y3 = 0 2 Mr y2 - krb ( y3 - y2 ) - Fn cos a = 0 Mc y1 + cvs y1 + kvs y1 + Fn cos a = 0 along the x-axis M fc x3 - krb ( x2 - x3 ) + khc x3 = 0 Mr x2 + krb ( x2 - x3 ) + Fn sin a = 0 Mc x1 + chs x1 + khs x1 - Fn sin a = 0 For torsional movement of the drive shaft (see Fig 1226) (1229)

48 If (a) (b) (c) (d) (e)

(1228)

50 40 30 20

Jx2eXcix = x 2 e x - 2 ( x e X - e x ) = x 2 e x - 2 x e X + 2 e X = e X ( x 2 - 2 x + 2 )

49 The limit limx 0 (a) (b) (c) (d) (e) 1/2 1/2 1/3 1/3 1

10 8 6 4

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