Report

ME321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo 7/17/2015 Kinematics and Dynamics Position Analysis Velocity Analysis Acceleration Analysis Force Analysis F ma We will concentrate on four-bar linkages 7/17/2015 Four-Bar Linkages What type of motion is possible? q l s p 7/17/2015 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 q l s p 7/17/2015 Grashof’s Criteria l q l s s p q p 7/17/2015 s+l<p+q s+l>p+q Grashof Mechanism Non-Grashof Mechanism Grashof Mechanisms (s+l < p+q) l l q q s s p Crank-Rocker p Rocker-Crank Shortest link pinned to ground and rotates 360o 7/17/2015 Grashof Mechanisms (s+l < p+q) s q l p q p s l 7/17/2015 Drag-Link Double-Rocker - Both input and output links rotate 360o - Coupler rotates 360o Change-Point Mechanism S+l = p+q l s q p 7/17/2015 Non-Grashof Mechanisms •Four possible triplerockers •Coupler does not rotate 360o s q p l 7/17/2015 Transmission Angle • One objective of position analysis is to determine the transmission angle, • Desire transmission angle to be in the range: 45o < < 135o 7/17/2015 coupler input link output link 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 coupler output link input link in 7/17/2015 Vector Position Analysis • ‘Close the loop’ of vectors to get a vector equation with two unknowns • Three possible solution techniques: RB / A RA s O2 • Graphical Solution • Vector Components • Complex Arithmetic 7/17/2015 RB RO2 / O4 O4 RA RB / A RO4 / O2 RB 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 7/17/2015 Vector Component Solution y, i R 3 x, j 4 2 O4 O2 ‘Close the loop’ to get a vector equation: R2 cos 2iˆ sin 2 ˆj R3 cos3iˆ sin3 ˆj R1iˆ R4 cos 4iˆ sin 4 ˆj 7/17/2015 Vector Component Solution (con’t) Rewrite in terms of i and j component equations: R2 cos 2 R3 cos 3 R1 R4 cos 4 R2 sin 2 R3 sin 3 R4 sin 4 • These represent two simultaneous transcendental equations in two unknowns: 3 and 4 •Must use non-linear (iterative) solver 7/17/2015 Complex Arithmetic • Represent (planar) vectors as complex numbers R Rei Rcos i sin iy R x • Write loop equations in terms of real and imaginary components and solve as before 7/17/2015 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 7/17/2015 B 3 A 2 O2 4 O4 l 2 a 2 b 2 2ab cos Limiting Positions for Linkages • What is the range of output motion for a crack-rocker mechanism? 7/17/2015