6.position analysis
TRANSCRIPT
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Kinematic Analysis: Scope
•Need to know the dynamic forces to be able to compute stresses in the components
• Dynamic forces are proportional to acceleration (Newton second law)
• Goal shifts to finding acceleration of all the moving parts in the assembly
•In order to calculate the accelerations:
• need to find the positions of all the links , for all increments in input motion
• differentiate the position eqs. to find velocities, diff. again to get accelerations
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Kinematic Analysis: Coordinate systems
Helps to measure some parameters within a link,
Independent of its rotation
Position, defined
through a vector
Moves with its origin on the link, // to GCS
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Displacement
•Displacement of a point is the change in its position
•Can be defined as the straight line distance between the initial and final position
•of a point which has moved in the reference frame.
•Displacement is not necessarily the same as the path length, from its initial to
•final position
The position of B with respect to A =
Absolute position of B minus that of A;
(absolute implying the origin of the GCS.)
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Displacement: Particle versus a Rigid Body
Translation, Rotation, and Complex motion
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Displacement: Particle versus a Rigid Body
Translation, Rotation, and Complex motion
(order of translation and rotation is not important)
Total displacement = Translation component + Rotation component
The new absolute position of B w.r.t. the origin at A is:
Chasles theorem
Any displacement of a rigid body is equivalent to the sum of a translation of one point on
that body and a rotation of the body about an axis through that point
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Graphical Position Analysis
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Graphical Position Analysis
Crossed: If two links adjacent to the shortest link cross one another
(-): Accuracy; repeatability, generality
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Algebraic Position Analysis of Linkages
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Displacement: Vector Loop Representation of Linkage
•Links are drawn as position vectors which form a vector loop.
•This loop closes on itself making the sum of the vectors around the loop, zero.
We’ll represent vectors by complex-number-notations
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Displacement: Vector Loop Representation of Linkage
Representation of vectors by complex-number-notations
Complex numbers refers to a set of number of the form a+ib,
where a: real part & b: imaginary part
Real
Imaginary
b
a
a+ibThey may be represented by an ordered pair (a,b), getting away
with the use of I, with the implicit understanding that a & b,
symbolize real & imaginary parts, respectively.
Real
Imaginary
(0,0)
(1,0)
(0,1)
(0,0) plays the role
of zero in CNS
(1,0) plays the role
of unity in CNS
(0,1) acts like the
square-root of -1
In CNS:
(0,1)(0,1)= (-1,0)
Complex Numbers
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Displacement: Vector Loop Representation of Linkage
Representation of vectors by complex-number-notations
benefits
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Displacement: Vector Loop Representation of Linkage
Representation of vectors by complex-number-notations
A
BMultiply A with (0,1), get B
(1,0) (0,1)=B(0,1)
(a+ib)(c+id)-(ac-bd, ad+bc)
Real
Imaginary
(0,0)
(1,0)
(0,1)
OC
Multiply B with (0,1), get B
(0,1) (0,1)=C(-1,0)
(-1,0)
D
(0,-1)
Multiply C with (0,1), get D
(-1,0) (0,1)=D(0,-1)
Multiply D with (0,1), get A
(0,-1) (0,1)=D(1,-0)
(0,1) = 0+j(1) = j = an operator which rotates a given vector counter-clockwise by 90º
Vector rotations in the complex plane
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Displacement: Vector Loop Representation of Linkage
Representation of vectors by complex-number-notations
Vector rotations in the complex plane
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Displacement: Vector Loop Representation of 4bar Linkage
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?
Displacement: Vector Loop Representation of 4bar Linkage
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Displacement: Vector Loop Representation of 4bar Linkage
?
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Displacement: Vector Loop Representation of 4bar Linkage
?
Now what?
Double Angle formulas
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Displacement: Vector Loop Representation of 4bar Linkage
?
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Displacement: Vector Loop Representation of 4bar Linkage
?
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Displacement: Vector Loop Representation of 4bar Linkage
(link lengths; non-grashoff where input angle goes beyond toggle positions)
Why?
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Displacement: Vector Loop Representation of 4bar Linkage
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Displacement: VLR of offset 4bar Slider Crank LinkageP2
(Offset: the slider axis does not pass through the crank pivot)
Option1
R2,R3, Rs
Rs: Varying magnitude
& direction
Option2
R1,R2, R3,R4
R1: Varying magnitude
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Displacement: VLR of offset 4bar Slider Crank LinkageP2
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Displacement: VLR of an Inverted Slider Crank LinkageP3
All slider linkages will have at least one link whose effective length between joints will vary
as the link moves.{d,θ3}?
{b,θ3, θ4}?
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Displacement: VLR of an Inverted Slider Crank LinkageP3
{b,θ3, θ4}?
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Displacement: VLR of an Inverted Slider Crank LinkageP3
{b,θ3, θ4}?
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Displacement: VLR of an Inverted Slider Crank LinkageP3
{b,θ3, θ4}?
Separating
linear terms
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Displacement: VLR of an Inverted Slider Crank LinkageP3
{b,θ3, θ4}?
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Displacement: VLR of an Inverted Slider Crank LinkageP3
{b,θ3, θ4}?
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Displacement: VLR of an Inverted Slider Crank LinkageP3
{b,θ3, θ4}?
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Displacement: VLR of an Inverted Slider Crank LinkageP3
{b,θ3, θ4}?
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Link 4 (c)
Link 1 (d)
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Link 4 (c)
Link 1 (d)
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