1

Kinematic Analysis of Kinematic Analysis of Planar LinkagesPlanar Linkages

Chapter 4Chapter 4

22

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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33

Position AnalysisPosition Analysis

44

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?

3

55

Vector Loop: 4 BarVector Loop: 4 Bar

66

4

77

88

Vector Loop: Offset SliderVector Loop: Offset Slider--CrankCrank

5

99

Vector Loop: 6 BarVector Loop: 6 Bar

1010

Vector Loop: Examples…Vector Loop: Examples…

6

1111

Point PathPoint Path

How do we determine the kinematics of point P?

1212

Point PathPoint Path

How do we determine the kinematics of point P?

7

1313

Solving position equations?Solving position equations?

Nonlinear equation solver, NewtonNonlinear equation solver, Newton--Raphson iterationRaphson iteration

1414

Multiple Solutions…Multiple Solutions…

Initial guesses important for position solution

8

1515

Multiple Solutions…Multiple Solutions…

Other position possible?

1616

Rolling Contact Eqn…Rolling Contact Eqn…

9

1717

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

1818

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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1919

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)

2020

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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2121

Rolling Contact Equation 3Rolling Contact Equation 3

ρB

rB

rA rC

ρBrB

∆rA

∆θB

∆θC

ρB(∆θB – ∆θC) = ∆rA

(∆θB – ∆θC)

2222

Rolling Contact ExampleRolling Contact Example

Kinematic Analysis

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2323

Rolling Contact ExampleRolling Contact Example

Kinematic Analysis

2424

Vector Loop: Joints…Vector Loop: Joints…