Problem

The gear reducer is subjected to the couple moments shown. Determine the resultant couple...

The gear reducer is subjected to the couple moments shown. Determine the resultant couple moment and specify its magnitude and coordinate direction angles.

Step-by-Step Solution

Solution 1

Step #1 of 3

Draw the free body diagram of the given system and represent the couple moments on it.

M₂=60

M₁=50

30^o

Step #2 of 3

Write the moment $M_{1}$.

$$ M_{1}=50 \mathrm{j} \mathrm{N} \cdot \mathrm{m} $$

Write the moment $M_{2}$.

$$ M_{2}=60 \cos 30^{\circ} \mathbf{i}+60 \sin 30^{\circ} \mathbf{k} $$

Calculate the resultant moment $M_{R}$ as follows.

$$ \begin{aligned} M_{R} &=M_{1}+M_{2} \\ &=50 \mathbf{i}+60 \cos 30^{\circ} \mathbf{i}+60 \sin 30^{\circ} \mathbf{k} \\ &=51.96 \mathbf{i}+50 \mathbf{j}+30 \mathbf{k} \end{aligned} $$

Calculate the magnitude of the resultant moment as follows.

$$ \begin{aligned} M &=\left|M_{R}\right| \\ &=\sqrt{(51.96)^{2}+(50)^{2}+(30)^{2}} \\ &=78.10 \mathrm{~N} \cdot \mathrm{m} \end{aligned} $$

Thus, the resultant couple moment is $78.10 \mathrm{~N} \cdot \mathrm{m}$

Step #3 of 3

Calculate the unit vector of the resultant moment is

$\begin{aligned} u &=\frac{M_{R}}{M} \\ &=\frac{51.96 \mathbf{i}+50 \mathbf{j}+30 \mathbf{k}}{78.10} \\ &=0.665 \mathbf{i}+0.64 \mathbf{j}+0.384 \mathbf{k} \end{aligned}$

Calculate the general unit vector of force is as follow

$u=\operatorname{Cos}\left(\alpha_{1}\right)+\operatorname{Cos}\left(\beta_{1}\right)+\operatorname{Cos}(\gamma)$

The direction angles of the resultant moments can be determined as follow

Compare unit vector of resultant moment and general unit vector.

$\cos \alpha_{1}=0.665$

$\alpha_{1}=\cos ^{-1}(0.1665)$

$\quad=48.31^{*}$

And

$\cos \beta_{1}=0.64$

$\begin{aligned} \beta_{1} &=\cos ^{-1}(0.64) \\ &=50.2^{*} \end{aligned}$

Also,

$\cos \gamma_{1}=0.384$

$\begin{aligned} \gamma_{1} &=\cos ^{-1}(0.384) \\ &=67.4 \mathrm{l}^{*} \end{aligned}$

Thus, the direction angles of the resultant force are $\alpha_{1}=48.31^{\circ}, \beta_{1}=50.2^{\circ}$, and

$\gamma_{1}=67.41^{\circ}$

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