ME 321 Kinematics and Dynamics of Machines Steve

ME 321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo 2/22/2021

Kinematics and Dynamics l l Position Analysis Velocity Analysis Acceleration Analysis Force Analysis We will concentrate on four-bar linkages 2/22/2021

Velocity Analysis l l 2/22/2021 Can use vector methods or instantaneous centres Vector equations can be expressed in general form, or specialized for planar problems Ô Graphical Solutions Ô Vector Component Solutions Ô Complex Number Solutions

Vector Equations 2/22/2021

Vector Equations for Velocity Differentiate Position Vector with respect to Time 2/22/2021

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 2/22/2021

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 2/22/2021

Planar Velocity Equations Becomes: 2/22/2021

Application to Four-Bar Linkages 2/22/2021

Graphical Solution 2/22/2021

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’ 2/22/2021

Vector Component Solution But: and Giving: or: 2/22/2021

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 2/22/2021

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. 2/22/2021

Kennedy’s Theorem 2/22/2021

Instant Centre Velocity Analysis 2/22/2021