6.position analysis

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

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

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.)

Displacement: Particle versus a Rigid Body

Translation, Rotation, and Complex motion

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

Graphical Position Analysis

Graphical Position Analysis

Crossed: If two links adjacent to the shortest link cross one another

(-): Accuracy; repeatability, generality

Algebraic Position Analysis of Linkages

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

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

Displacement: Vector Loop Representation of Linkage

Representation of vectors by complex-number-notations

benefits

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

Displacement: Vector Loop Representation of Linkage

Representation of vectors by complex-number-notations

Vector rotations in the complex plane

Displacement: Vector Loop Representation of 4bar Linkage

?

Displacement: Vector Loop Representation of 4bar Linkage

Displacement: Vector Loop Representation of 4bar Linkage

?

Displacement: Vector Loop Representation of 4bar Linkage

?

Now what?

Double Angle formulas

Displacement: Vector Loop Representation of 4bar Linkage

?

Displacement: Vector Loop Representation of 4bar Linkage

?

Displacement: Vector Loop Representation of 4bar Linkage

(link lengths; non-grashoff where input angle goes beyond toggle positions)

Why?

Displacement: Vector Loop Representation of 4bar Linkage

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

Displacement: VLR of offset 4bar Slider Crank LinkageP2

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}?

Displacement: VLR of an Inverted Slider Crank LinkageP3

{b,θ3, θ4}?

Displacement: VLR of an Inverted Slider Crank LinkageP3

{b,θ3, θ4}?

Displacement: VLR of an Inverted Slider Crank LinkageP3

{b,θ3, θ4}?

Separating

linear terms

Displacement: VLR of an Inverted Slider Crank LinkageP3

{b,θ3, θ4}?

Displacement: VLR of an Inverted Slider Crank LinkageP3

{b,θ3, θ4}?

Displacement: VLR of an Inverted Slider Crank LinkageP3

{b,θ3, θ4}?

Displacement: VLR of an Inverted Slider Crank LinkageP3

{b,θ3, θ4}?

Link 4 (c)

Link 1 (d)

Link 4 (c)

Link 1 (d)