1. ME321 Kinematics and Dynamics of Machines
Steve Lambert
Mechanical Engineering,
U of Waterloo
9/23/2018

2. Kinematics and Dynamics
Position Analysis
Velocity Analysis
Acceleration Analysis
Force Analysis
We will concentrate on four-bar linkages
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3. Velocity Analysis Can use vector methods or instantaneous centres
Vector equations can be expressed in general form, or specialized for planar problems
Graphical Solutions
Vector Component Solutions
Complex Number Solutions
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4. Vector Equations
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5. Vector Equations for Velocity
Differentiate Position Vector with respect to Time
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6. Vector Velocity Equation
Where:
= Total absolute velocity of point
= Absolute velocity of local origin
= Relative velocity in local system
= Angular velocity of Local System
= Position of point in local system
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7. Planar Velocity Equations
Assume:
Motion is restricted to the XY plane
Local frame is aligned with and fixed to link
Therefore:
 becomes the angular velocity of the link
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8. Planar Velocity Equations
Becomes:
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9. Application to Four-Bar Linkages
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10. Graphical Solution
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11. Velocity Image A’B’ is the velocity image of link AB
And then the velocity of point C, VC, can be obtained directly from the figure as the vector O’C’
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12. Vector Component Solution
But:
and
Giving:
or:
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13. Instant Centres
An instant centre is a point at which there is no relative velocity between two links in a mechanism, at a particular instant in time
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14. Kennedy’s Theorem
Kennedy’s theorem states: the three instant centres of three bodies moving relative to one another must lie along a straight line.
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15. Kennedy’s Theorem
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16. Instant Centre Velocity Analysis
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