ME321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo 9/17/2018

Kinematics and Dynamics Position Analysis Velocity Analysis Acceleration Analysis Force Analysis We will concentrate on four-bar linkages 9/17/2018

Four-Bar Linkages What type of motion is possible? s q l p 9/17/2018

Grashof’s Criteria Used to determine whether or not at least one of the links can rotate 360o the sum of the shortest and longest links of a planar four-bar mechanism cannot be greater than the sum of the remaining two links if there is to be continuous relative rotation between the two links. s + l < p + q s q l p 9/17/2018

Grashof’s Criteria s + l > p + q s + l < p + q Non-Grashof Mechanism s + l < p + q Grashof Mechanism 9/17/2018

Grashof Mechanisms (s+l < p+q) Crank-Rocker Rocker-Crank Shortest link pinned to ground and rotates 360o 9/17/2018

Grashof Mechanisms (s+l < p+q) Drag-Link - Both input and output links rotate 360o Double-Rocker - Coupler rotates 360o 9/17/2018

Change-Point Mechanism S+l = p+q s q l p 9/17/2018

Non-Grashof Mechanisms Four possible triple- rockers Coupler does not rotate 360o p s q l 9/17/2018

Transmission Angle One objective of position analysis is to determine the transmission angle,  Desire transmission angle to be in the range: 45o <  < 135o  output link input coupler 9/17/2018

Position Analysis Given the length of all links, and the input angle,in, what is the position of all other links? Use vector position analysis or analytical geometry output link input coupler in 9/17/2018

Vector Position Analysis ‘Close the loop’ of vectors to get a vector equation with two unknowns Three possible solution techniques: Graphical Solution Vector Components Complex Arithmetic 9/17/2018

Graphical Solution Draw ground and input links to scale, and at correct angle Draw arcs (circles) corresponding to length of coupler and output links Intersection points represent possible solutions 9/17/2018

Vector Component Solution  R x, j y, i O2 O4 2 3 4 ‘Close the loop’ to get a vector equation: 9/17/2018

Vector Component Solution (con’t) Rewrite in terms of i and j component equations: These represent two simultaneous transcendental equations in two unknowns:  3 and 4 Must use non-linear (iterative) solver 9/17/2018

Complex Arithmetic Represent (planar) vectors as complex numbers  x iy R Write loop equations in terms of real and imaginary components and solve as before 9/17/2018

Analytical Geometry Examine each mechanism as a special case, and apply analytical geometry rules For four-bar mechanisms, draw a diagonal to form two triangles Apply cosine law as required to determine length of diagonal, and remaining angles A B O2 O4 2 3 4    9/17/2018

Limiting Positions for Linkages What is the range of output motion for a crack-rocker mechanism? 9/17/2018