MECH 335 - Lecture Pack 2

Position Analysis

• Useful indices for Position Analysis:

a) Types of Mechanism:

• Grashof vs. Non-Grashof:

• Grashof: at least 1 pair of adjacent links is capable of full rotation

• Test for Grashof mech: “Grashof’s Law”

• “The sum of the shortest and longest links’ lengths is less than the sum of the lengths of the remaining two links”

MECH 335 Lecture Notes © R.Podhorodeski, 2009

Position Analysis


a) Types of Mechanism:

• 4-bar mechanisms can be classified based on Grashof criterion

• For those that meet the criterion, the identity of the shortest link defines the mechanism type


Position Analysis

• Example: Types of 4-bar

– Grashof 4-bar with rshort = r2


Crank-Rocker

Position Analysis


– Grashof 4-bar with rshort = r1


Drag-Link

Position Analysis


– Grashof 4-bar with rshort = r3 (or 4)


Rocker-Rocker

Note that the extreme angular positions of the crank and follower do not necessarily coincide

Position Analysis


– Any non-Grashof 4-bar is a Triple-Rocker


Again, extreme angular positions of the crank and follower do not necessarily coincide

Also, there are points where the motion of the mechanism diverges, i.e. one or more outputs are possible for a given crank input

Which way will link 3 rotate when the input is rotated CW from this position?

How about link 4?

Position Analysis


b) Inversions of a mechanism

• Inversions change which of the mechanism’s links is fixed

• An n-link mechanism has n inversions


Position Analysis

• Example: Inversions of the slider-crank

– Standard slider-crank


Position Analysis


– Freeing the slide, and fixing link 2


Position Analysis


– Or, fixing link 3 (note: 4 is still free to rotate)


Position Analysis


– Or, fixing 4 (1 slides, but does not rotate, w.r.t. 4)


Position Analysis


c) Transmission (γ) and Deviation (δ) angles

• Transmission Angle (γ): The acute angle between the directions of the velocity-difference vector of the driving point , and the absolute velocity vector of the driven point

• Deviation Angle (δ): The angle between the absolute direction of travel of the driven point and the direction of the force exerted by the driving link


INPUT

VB3

A1,2

B2,3

3

1

2

44

INPUT

A1,2

B2,3

3

1

2

44

γVB

2/A

2

INPUT

A1,2

B2,3

3

1

2

44

Position Analysis

• Example:

– Transmission Angle (γ):


The acute angle between the directions of:

and

INPUT

VB3

A1,2

B2,3

3

1

2

44

γVB

2/A

2

the velocity-difference vector of the driving point

the absolute velocity vector of the driven point

INPUT

A1,2

B2,3

3

1

2

44

FB2→B3

INPUT

VB3

A1,2

B2,3

3

1

2

44

INPUT

A1,2

B2,3

3

1

2

44

Position Analysis

• Example:

– Deviation Angle (δ):


The angle between:

and

INPUT

VB3

A1,2

B2,3

3

1

2

44

FB2→B3

δThe absolute direction of travel of the driven point

the direction of the force exerted by the driving link

Displacement Analysis

• Any single-loop, planar mechanism will have two unknowns (dependent variables)

• Need a method to solve for these variables in terms of known mechanism parameters

• 2 main methods presented:

– Graphical

– Analytical



• Example: 4-bar mech.– Vectors Ri used to

represent links

– All link lengths known

– Input (independent) variable is crank angle θ2

– What are the dependent variables?


A

B

A0 B0R1

R2

R3

R4

θ2


• Example: 4-bar mech.– Vectors Ri used to

represent links

– All link lengths known

– Input (independent) variable is crank angle θ2

– What are the dependent variables?• Need a method to solve for

these variables


A

B

A0 B0R1

R2

R3

R4

θ2

θ3

θ4



A

A0 B0R1

R2

θ2

• Solution lies at the intersection of the possiblepositions of links 3 & 4

• Graphical (approximate) method



A

A0 B0

θ2


• Note presence of 2 possible solutions!




A

B

A0 B0

θ2

θ3

θ4


• Note presence of 2 possible solutions!

• Unknown angles are measured from drawing




• Analytical methods

– Graphical methods inexact, inconvenient for repeated analysis

– 2 Analytical methods presented:

• Geometric analysis

• Loop-closure

4

• Example 1: Geometric analysis



A

B

A0 B0

R1

R2

R3

R4

θ2

arg(D)Δ

θ3

φ

γ

Dθ4

• Loop Closure – Based Displacement Analysis

– Represent each link by a vector,

– Sum vectors to close the loop

– Split the real and imaginary components of this equation to yield two equations by applying:



4

• Example 2: Loop Closure Analysis



A

B

A0 B0

R1

R2

R3

R4

θ2

θ3

θ4

4

• Example 2: Loop Closure Analysis



A

B

A0 B0

R1

R2

R3

R4

θ2

θ3

θ4

EQUATION FROM REAL PARTS

EQUATION FROM IMAGINARY PARTS

4

• Example 3: Loop Closure, offset slider-crank



4




2

3

1

0

4




2

3

1

0

• Solution is correct, but “weak”, since multiple angles have the same sine




2

3

1

0

• A stronger solution can be found by writing:

4

• Iterative Loop-Closure based displacement solution

– If the dependent variables are inaccurate

– An iterative algorithm for closure can then be based on:



and

• Example: Newton-Raphson iterative closure



2

3

1

0

Loop closure error:

Input:Unknowns: ,

i.e.




If is correct, Estimate of can be improved by iteration, i.e.

Input:Unknowns: ,

i.e.

2

3

1

0




For this example:

Then, from (3):Input:Unknowns: ,

i.e.

2

3

1

0

Input:Unknowns: ,

i.e.




Inverting (4), substituting into (5), and then into (2) gives:

Where is given in (1).Equation (6) would be used to iteratively improve the estimates of , until:

2

3

1

0

• Numerical Example:

» Suppose we start by guessing:

» Analysis tolerance:



2

3

1

0

• Numerical Example:

• Now we iterate from that starting point



2

3

1

0

MECH 335 - Lecture Pack 2

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