Step 1

We have given the following data:

The angular velocity of rod AB is ${\omega _{AB}} = 2\;{\rm{rad}}/{\rm{s}}$.

The angular acceleration of rod AB is ${\alpha _{AB}} = 8\;{\rm{rad}}/{{\rm{s}}^2}$.

We are asked to calculate the angular velocity and angular acceleration of rod CD at the instant shown.

 
Step 2

Draw a labeled diagram of the given system.

Here, ${\omega _{CD}}$ is the angular velocity of rod CD and ${\alpha _{CD}}$ is the angular acceleration of rod CD.

 
Step 3

The fixed and rotating X-Y and x-y coordinate systems are set to coincide with origin at A as shown in the diagram. Here, the x-y coordinate system is attached to link AC.

The velocity of fixed point A in the vector form is given by:

\[{\overrightarrow v _A} = 0\;{\rm{m}}/{\rm{s}}\]

The acceleration of fixed point A in the vector form is given by:

\[{\overrightarrow a _A} = 0\;{\rm{m}}/{{\rm{s}}^2}\]

The angular velocity of rod AB in vector form is given by:

\[\Omega = {\omega _{AB}} = 2{\bf{k}}\;{\rm{rad}}/{\rm{s}}\]

The angular acceleration of rod AB in vector form is given by:

\[\dot \Omega = {\alpha _{AB}} = 8{\bf{k}}\;{\rm{rad}}/{{\rm{s}}^2}\]

The distance of slider C relative to point A is given by:

\[{\overrightarrow r _{C/A}} = \left\{ {1.5{\bf{i}}} \right\}\;{\rm{m}}\]

The velocity ${\left( {{v_{C/A}}} \right)_{xyz}}$ of slider C with respect to moving reference is given by:

\[{\left( {{{\overrightarrow v }_{C/A}}} \right)_{xyz}} = {\left( {{v_{C/A}}} \right)_{xyz}}{\bf{i}}\]

The acceleration ${\left( {{a_{C/A}}} \right)_{xyz}}$ of slider C with respect to moving reference is given by:

\[{\left( {{{\overrightarrow a }_{C/A}}} \right)_{xyz}} = {\left( {{a_{C/A}}} \right)_{xyz}}{\bf{i}}\]
 
Step 4

The distance of point C from point D can be represented as:

\[\begin{array}{c} {\overrightarrow r _{C/D}} = \left\{ { - 1{\bf{i}}} \right\}\;{\rm{m}}\\ = \left\{ { - {\bf{i}}} \right\}\;{\rm{m}} \end{array}\]
 
Step 5

Considering the motion of collar C in the fixed system, the velocity of the collar C can be expressed as:

\[{\overrightarrow v _C} = {\omega _{CD}} \times {\overrightarrow r _{C/D}}\]

Substitute the value of ${\overrightarrow r _{C/D}}$ in the above equation:

\[\begin{array}{c} {\overrightarrow v _C} = - {\omega _{CD}}{\bf{k}} \times \left( { - {\bf{i}}} \right)\\ = {\omega _{CD}}{\bf{j}} \end{array}\]
 
Step 6

Considering the motion of collar C in the fixed system, the acceleration of the collar C can be expressed as:

\[{\overrightarrow a _C} = {\alpha _{CD}} \times {\overrightarrow r _{C/D}} - \omega _{CD}^2{\overrightarrow r _{C/D}}\]

Substitute the value of ${\overrightarrow r _{C/D}}$ in the above equation:

\[\begin{array}{c} {\overrightarrow a _C} = - {\alpha _{CD}}{\bf{k}} \times \left( { - {\bf{i}}} \right) - \omega _{CD}^2\left( { - {\bf{i}}} \right)\\ {\overrightarrow a _C} = \omega _{CD}^2{\bf{i}} + {\alpha _{CD}}{\bf{j}} \end{array}\]
 
Step 7

Applying the relative velocity equation, the velocity of collar C is given by:

\[{\overrightarrow v _C} = {\overrightarrow v _A} + \Omega \times {\overrightarrow r _{C/A}} + {\left( {{{\overrightarrow v }_{C/A}}} \right)_{xyz}}\]

Substitute the values of parameters in above equation:

\[\begin{array}{c} \left( {{\omega _{CD}}{\bf{j}}} \right) = \left( 0 \right) + \left( {2{\bf{k}}} \right) \times \left( {1.5{\bf{i}}} \right) + {\left( {{v_{C/A}}} \right)_{xyz}}{\bf{i}}\\ {\omega _{CD}}{\bf{j}} = {\left( {{v_{C/A}}} \right)_{xyz}}{\bf{i}} + 3{\bf{j}} \end{array}\]......(1)
 
Step 8

On equating the ${\bf{j}}$ components of equation (1), we get:

\[{\omega _{CD}} = 3\;{\rm{rad}}/{\rm{s}}\]

On equating the ${\bf{i}}$ components of equation (1), we get:

\[{\left( {{v_{C/A}}} \right)_{xyz}} = 0\;{\rm{m}}/{\rm{s}}\]
 
Step 9

Applying the relative acceleration equation, the acceleration of collar C is given by:

\[{\overrightarrow a _C} = {\overrightarrow a _A} + \dot \Omega \times {\overrightarrow r _{C/A}} + \Omega \times \left( {\Omega \times {{\overrightarrow r }_{C/A}}} \right) + 2\Omega \times {\left( {{{\overrightarrow v }_{C/A}}} \right)_{xyz}} + {\left( {{{\overrightarrow a }_{C/A}}} \right)_{xyz}}\]

Substitute the values of parameters in the above equation:

\[\begin{array}{c} \left( {\omega _{CD}^2{\bf{i}} + {\alpha _{CD}}{\bf{j}}} \right) = \left\{ \begin{array}{l} \left( 0 \right) + \left( {8{\bf{k}}} \right) \times \left( {1.5{\bf{i}}} \right) + \left( {2{\bf{k}}} \right) \times \left( {\left( {2{\bf{k}}} \right) \times \left( {1.5{\bf{i}}} \right)} \right)\\ + 2\left( {2{\bf{k}}} \right) \times \left( 0 \right) + {\left( {{a_{C/A}}} \right)_{xyz}}{\bf{i}} \end{array} \right\}\\ \omega _{CD}^2{\bf{i}} + {\alpha _{CD}}{\bf{j}} = \left\{ {\left( {{{\left( {{a_{C/A}}} \right)}_{xyz}} - 6} \right){\bf{i}} + 12{\bf{j}}} \right\} \end{array}\]

Substitute the value of ${\omega _{CD}}$ in the above equation:

\[\begin{array}{c} {\left( 3 \right)^2}{\bf{i}} + {\alpha _{CD}}{\bf{j}} = \left\{ {\left( {{{\left( {{a_{C/A}}} \right)}_{xyz}} - 6} \right){\bf{i}} + 12{\bf{j}}} \right\}\\ 9{\bf{i}} + {\alpha _{CD}}{\bf{j}} = \left( {{{\left( {{a_{C/A}}} \right)}_{xyz}} - 6} \right){\bf{i}} + 12{\bf{j}} \end{array}\]......(2)
 
Step 10

On equating the ${\bf{j}}$ components of equation (2), we get:

\[{\alpha _{CD}} = 12\;{\rm{rad}}/{{\rm{s}}^2}\]

On equating the ${\bf{i}}$ components of equation (2), we get:

\[\begin{array}{c} 9\;{\rm{m}}/{{\rm{s}}^2} = {\left( {{a_{C/A}}} \right)_{xyz}} - 6\;{\rm{m}}/{{\rm{s}}^2}\\ {\left( {{a_{C/A}}} \right)_{xyz}} = 15\;{\rm{m}}/{{\rm{s}}^2} \end{array}\]