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Kinematic Analysis of Kinematic Analysis of Planar LinkagesPlanar Linkages
Chapter 4Chapter 4
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Vector Loop MethodVector Loop Method
General method to perform kinematic analysis of General method to perform kinematic analysis of linkages, applicable from the simple to the complexlinkages, applicable from the simple to the complexDetermine the state of motion (position, velocity, Determine the state of motion (position, velocity, acceleration) of all other links or points of interestacceleration) of all other links or points of interestVLMVLM�� Attach vectors to the mechanism links to form a closed loopAttach vectors to the mechanism links to form a closed loop�� Choose vectors to ‘measure’ variables of interestChoose vectors to ‘measure’ variables of interest�� Write the vector loop equation (VLE)Write the vector loop equation (VLE)�� Break the vector equation into scalar componentsBreak the vector equation into scalar components�� Differentiate position equations to determine velocitiesDifferentiate position equations to determine velocities�� Differentiate again to determine acceleration unknownsDifferentiate again to determine acceleration unknowns
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Position AnalysisPosition Analysis
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Vector Loop MethodVector Loop Method
Do I have enough loops and good Do I have enough loops and good vectors?vectors?�� # of unknowns = # independent scalar eqns?# of unknowns = # independent scalar eqns?�� Do the variables include all those we wish to Do the variables include all those we wish to
determine?determine?
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Vector Loop: 4 BarVector Loop: 4 Bar
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Vector Loop: Offset SliderVector Loop: Offset Slider--CrankCrank
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Vector Loop: 6 BarVector Loop: 6 Bar
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Vector Loop: Examples…Vector Loop: Examples…
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Point PathPoint Path
How do we determine the kinematics of point P?
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Point PathPoint Path
How do we determine the kinematics of point P?
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Solving position equations?Solving position equations?
Nonlinear equation solver, NewtonNonlinear equation solver, Newton--Raphson iterationRaphson iteration
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Multiple Solutions…Multiple Solutions…
Initial guesses important for position solution
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Multiple Solutions…Multiple Solutions…
Other position possible?
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Rolling Contact Eqn…Rolling Contact Eqn…
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Rolling Contact Rolling Contact Equation Equation (Simple Case)(Simple Case)
ρA
ρB
rA
rC
rB
ρA
ρB
rA
rC
rB
∆θA
∆θB
ρA ∆θA = -ρB ∆θB
∆θx = θx – θxi
(∆θB)
(∆θA)
θB = -λ ∆θA + φ
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Rolling Contact Rolling Contact Equation Equation (Another Simple Case)(Another Simple Case)
ρA
ρB
rA
rC
rB
ρA
ρB
rA
rC
rB
∆θA
∆θC
∆θB
∆θA = ∆θC = ∆θB
∆θx = θx – θxi
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Rolling Contact Rolling Contact Equation Equation (General Case)(General Case)
ρA
ρB
rA
rC
rB
ρA
ρB
rA
rC
rB
∆θA
∆θC
∆θB
ρA(∆θA – ∆θC) = -ρB(∆θB – ∆θC)
∆θx = θx – θxi
(∆θB – ∆θC)
(∆θA – ∆θC)
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Rolling Contact Equation 2Rolling Contact Equation 2
ρB
rB
ρA rC rA
ρB
rB
ρArC
rA∆θC
∆θA
∆θB
ρB(∆θB – ∆θC) = ρA(∆θA – ∆θC)
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Rolling Contact Equation 3Rolling Contact Equation 3
ρB
rB
rA rC
ρBrB
∆rA
∆θB
∆θC
ρB(∆θB – ∆θC) = ∆rA
(∆θB – ∆θC)
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Rolling Contact ExampleRolling Contact Example
Kinematic Analysis
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Rolling Contact ExampleRolling Contact Example
Kinematic Analysis
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Vector Loop: Joints…Vector Loop: Joints…
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