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# Lecture 14 - Kinematics and Dynamics of Mechanisms ## Kinematics Deals with motion of bodies without reference to the forces that cause them. **Gruebler's Equation:** $$ DOF = 6(N-1) - \sum_{i=1}^{j} f_{i} $$ Where: - N = Number of links - J = Number of joints - $f_i$ = Number of constraints...

# Lecture 14 - Kinematics and Dynamics of Mechanisms ## Kinematics Deals with motion of bodies without reference to the forces that cause them. **Gruebler's Equation:** $$ DOF = 6(N-1) - \sum_{i=1}^{j} f_{i} $$ Where: - N = Number of links - J = Number of joints - $f_i$ = Number of constraints imposed by joint i For planar mechanisms: $$ DOF = 3(N-1) - \sum_{i=1}^{j} f_{i} $$ **Example 1:** ![Image of a four-bar linkage] N = 4 links J = 4 joints $f_i$ = 2 (1 DOF joint Revolute) $DOF = 3(4-1) - 4(2) = 1$ **Example 2:** ![Image of a slider-crank mechanism] N = 4 links J = 4 joints 3 Revolute Joints ($f_i=2$) 1 Prismatic Joint ($f_i=2$) $DOF = 3(4-1) - 3(2) - 2 = 1$ ## Dynamics Deals with relationship between forces and motion **Newton's 2nd Law:** $\sum \overrightarrow{F} = m\overrightarrow{a}$ $\sum \overrightarrow{M} = I\overrightarrow{\alpha}$ Where: - $\sum \overrightarrow{F}$ = Sum forces - $\sum \overrightarrow{M}$ = Sum moments - m = mass - I = mass moment of inertia - $\overrightarrow{a}$ = linear acceleration - $\overrightarrow{\alpha}$ = angular acceleration **Procedure:** 1. Draw Free Body Diagram (FBD) of each link 2. Write Equations of Motion 3. Solve