Example 2.11: The 2-inch crank on the wiper mechanism in Figure 2.85 rotates counterclockwise (CCW) at...

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Question

Example 2.11: The 2-inch crank on the wiper mechanism in Figure 2.85 rotates counterclockwise (CCW) at 15 rpm and decelerates

Answer 1

Answer #1

Grashof's Criterion:

smallest Link: s = 2'' - crank.

Longest Link: l = 14'' - fixed link.

p = 13''; q = 3.5''

p+q = 13 + 3.5 = 16.5

s+l = 2+14 = 16

So, s+l < p+q - Grashof's critereion.

So, This is a crank-rocker mechanism.

Mobility:

No. of links = 4

No. of one degree of freedom joints = 4

Mobility = 3*(n-1) - 2j1

M = 3*(4-1) - 2*(4)

M = 1. So, this is a one-degree of freedom system.

The actuator can be placed at the crank since it can rotate completely,

Position Analysis:

$\tiny AB = 2cos\theta_2 i +2sin\theta_2 j \\ BC = 13cos\theta_3 i +13 sin\theta_3 j \\AD = 14 i \\ DC = 3.5cos\theta_4 i +3.5sin\theta_4 j$

$\tiny Vector\;Loop = AB+BC = AD+DC$

Separate i and j components.

$\tiny 2cos\theta_2+ 13cos\theta_3 = 14 + 3.5cos\theta_4 \\ \\ 2sin\theta_2+ 13sin\theta_3 = 3.5sin\theta_4$

$\tiny \theta_2 = 180-50 = 130^0$

$\tiny 2cos(130^0)+ 13cos\theta_3 = 14 + 3.5cos\theta_4 \\ \\ 2sin(130^0)+ 13sin\theta_3 = 3.5sin\theta_4$

$\tiny -13cos\theta_3+ 3.5cos\theta_4 = -12.05 \\ \\ 13sin\theta_3 -3.5sin\theta_4 = -1.532$

solving, we get:

$\tiny \theta_3 = -16.21^0;\theta_4 = -143.19^0$

Position of X:

Length of DX = 16+8 = 24''

Angle made by DX with x-axis:

$\tiny (-143.19+90) + 70+90 = 106.81^0$

$\tiny AX = AD+D X \\ \Rightarrow AX = 14 i +24*(cos(106.81^0) i +sin (106.81^0) j) \\ \Rightarrow AX = 7.059 i +22.974$

X(7.059,22.974)

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Answer 2