Sugar Milling Research Institute Theory of Machines Dr. Richard C. Loubser Sugar Milling Research Institute Contact: 031 2731352 rloubser@smri.org www.rloubser.za.net

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Lecture 1 Definitions Degrees of Freedom Conventions Angles Symbols

Definitions Theory of machines: Kinematics and dynamics of multi-body systems Kinematics: Description of position, velocity and acceleration Dynamics: Forward problem: response of system under influence of given forces Inverse problem: forces required to give specific response

Definitions Rigid body Kinematic pair System of particles where the relative position between particles does not change Link, element, member Large motions Kinematic pair Two links joined together by a joint Revolute joint Slider

Definitions Mechanism: series of links joined together to produce a specific motion Kinematic chains Open chains Closed chains Machine: assemblage of mechanisms

Definitions Planar mechanism: Mechanism which moves in plane or parallel planes Spatial mechanism: Mechanism moving in space

Coordinate systems Cartesian Generalised [x,y,z]’ [q1, q2, q3, q4, q5,… qN]’

Symbols  constraint equation  angle (time dependant) L length of link (fixed) S variable distance    fixed angles

Angles Angles are measure counter clockwise from +ve X axis B B q L q AB=[Lcosq,Lsinq]T A A

Degrees of Freedom Y q X [X,Y,]’ f=3

Degrees of Freedom Pin joint p Y q X [X,Y,]’ f=3-2

Degrees of Freedom Slider s Y q X [X,Y,]’ f=3-1=2

Degrees of Freedom Guide g Y q X [X,Y,]’ f=3-2

Four bar mechanism 3 2 4 f=3(n-1)-2p-s-2g 1 n=4 p=4 s=0 g=0

Four bar mechanism A 3 B 2 4 C e O d OA=OC+CB+BA non-linear

Four bar mechanism A 3 B 2 4 C e O d

Four bar mechanism A 3 B 2 4 C e O d

Grashoff’s Rule A 3 B 2 4 C e O d L+S≤P+Q L=Longest link S=Shortest Link P,Q = other links S is fully rotatable