FOUNDATION of MECHANICS 1 FOUNDATION of MECHANICS 1 Presentation03: Kinematics analysis of mechanisms Outline • Four-bar linkage: introduction; velocity and acceleration analyses (graphical approach). approach). • Crank-slider mechanism: position, velocity, and acceleration analyses (graphical and analytical approaches) (graphical and analytical approaches). • General analytical approach: the matrix formulation. • Elements for the analytical study of Relative Motions.
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FOUNDATION of MECHANICS 1FOUNDATION of MECHANICS 1
Presentation03: Kinematics analysis of mechanisms
Outline
• Four-bar linkage: introduction; velocity and acceleration analyses (graphical approach).approach).
• Crank-slider mechanism: position, velocity, and acceleration analyses (graphical and analytical approaches)(graphical and analytical approaches).
• General analytical approach: the matrix formulation.
• Elements for the analytical study of Relative Motions.
Acceleration: 2 2'( )( )d s k qs k q q q k q k qdt q
( ) i ( )
2
cos( ) sin( )cos( ) cos( )
sin( ) tan( ) cos( ) cos( ) tan( )sin( )Bs r r r r
sin( ) tan( ) cos( ) cos( ) tan( )sin( )B r r r r
RELATIVE MOTION
Position
0 1 1 0 1 1 1 1 0 0 0 0( - ) ( - ) ( - ) x y x yP O P O O O i j i j
1 Velocity
P
y1
20( - )
Pd P O
dt v
y0
jO1
x1j1 i1 1 1
0 0 0 0 1 1 1 1 1 1x y x y x yd ddt dt
i ji j i j
j0
i0 x0O01 1( )O r T rP O v v v v
RELATIVE MOTION
0 1 1 0 1 1 1 1 0 0 0 0( - ) ( - ) ( - ) x y x yP O P O O O i j i j
0 0 0 0 1 1 1 1 1x y x y ( )P P O
v i j i j
0 1 1 0 1 1 1 1 0 0 0 0( ) ( ) ( ) y yj j
Acceleration
1 1 1
0 0 0 0 1 1 1 1 1 1 1( )x y x y x y ( )P
d d d P OP Odt dt dt
i ja i j i j
2( ) ( ) 2
dt dt dt
P O P O
a a v1 1( ) ( ) 2O r rP O P O
a a v
a a aT r C a a a
KINEMATICS: SUMMARY
T i /P bl M h dTopic/Problem
• Kinematics of a particle
Methods
Cartesian planar vectors; p
• Kinematics of a rigid body
pComplex Numbers
Cartesian planar vectors;Kinematics of a rigid body(Rivals theorem, Instant Centre ofRotation, Kennedy-Aronhold theorem, Rotational/Translational/Rolling motions)
Cartesian planar vectors; Complex Numbers
g )
• Kinematic analysis of mechanisms(Position, Velocity, and Acceleration analyses,
Graphical approach;Analytical approaches:(Position, Velocity, and Acceleration analyses,