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

Kinematics and Dynamics Position Analysis Velocity Analysis Acceleration Analysis Force Analysis We will concentrate on four-bar linkages 9/23/2018

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 9/23/2018

Vector Equations 9/23/2018

Vector Equations for Velocity Differentiate Position Vector with respect to Time 9/23/2018

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 9/23/2018

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 9/23/2018

Planar Velocity Equations Becomes: 9/23/2018

Application to Four-Bar Linkages 9/23/2018

Graphical Solution 9/23/2018

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’ 9/23/2018

Vector Component Solution But: and Giving: or: 9/23/2018

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 9/23/2018

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. 9/23/2018

Kennedy’s Theorem 9/23/2018

Instant Centre Velocity Analysis 9/23/2018