Question

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Problem 3 (20 points): The modified sinusoidal acceleration function for the rise function in the double dwell cam design is show in the following figure. A B/2 0B/8 1) Suppose the rise is from 7/8 B to 7, what is the value for B? ( 2 points) 2) The acceleration function in zone D Ses B) is sin ( 41 (-1)), derive the jerk function in zone D. (2 points) 3) What is the value of the acceleration when 0 = ?? (2 points) 4) For zone B and C, the acceleration function is a whole period of cosine function, what is this function? (hint: you know the period and its value at 0 = 3 (3 points) 5) Assuming the total displacement (rise) is h, list the boundary conditions (s,v,a) for @ = B. (hint: after zone D is the high dwell) (3 points) 6) Derive the velocity function in zone D. (4 points) 7) Given the angular velocity of the cam w 2 rad/s, what is the actual velocity and acceleration when 0 = 7,4 points) If Ô is constant Displacement(s) y(0) y(@) Velocity (v) (0) = yo (0) = y' Acceleration (a) (0) = y"+ y'ë (0) = y"62 Jerk (O) = y03 + 3y"ėë + y'ë y(0) = y03

Answer 1

1) Since the rise period is
40%
,
β = Δθ = π
.

2) Acceleration function,

v = sin ( 1 (4-1))

Hence, jerk function,

$y'''_D=\frac{d}{d\theta}\sin\bigg(4\pi\bigg(\frac{\theta}{\pi}-1\bigg)\bigg)=4\cos\bigg(4\pi\bigg(\frac{\theta}{\pi}-1\bigg)\bigg)$
3) Value of acceleration,

yhl@=77/8) = sin(47(-1/8)) = sin(1/2) = -1

4)

Let the function be,

Ye=y6 = a cos(b0 +

Since the period is , and the acceleration function is a whole period of a cosine function,

78/8 - 3/8 = 38/4

b x 38/4 = 1

$b=\frac{4}{3}$

Since the value of the acceleration functions in zones C and D at the point of transition must be the same,

-a = sin(-1/2) = -1

I = D

5) For $\theta=\beta$ , the boundary conditions are,

$s=h$, $v=0$ , $a=0$ .

6)

v = sin ( 1 (4-1))

Yo = 16 do = sin(40 - 47) do = cos(40 - 47) +

Since
0= f = 1
at $\theta=\beta$ ,

cos (47 – 47) C = = 1/4

Therefore,

((47 – t)s05 – 1)== qa

7)

= (8/2) = 6)

(0 = 77/8) = yp (@= 77/8) xw = 5m/s

j(@= 77/8) = y(@= 77/8) xw2 = -4m/s?