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Position, Velocity and Acceleration Analysis

Complex Algebra Method

Dr. A-Alabduljabbar

3-4-2006

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Introduction

Displacement: Consider the displacement of a point from

position P (represented by rp) to P' (represented by rp'), in the x-y coordinate system. The change in the position of the point, r, is found by:

r = rp’- rp (1)

Equation (1) is a vector equation which can be resolved into the two coordinates, and solved for the unknown displacement (magnitude and direction) provided that both magnitudes and directions of rp and rp' are defined.

rp

rp'

O

Drp

P

P'

y

x

3

Consider the motion of a point from position P to P' during a time period of t as shown in the figure. The change is position is give by r.

The velocity is defined as the change in position with respect to time, and we can define the average velocity

When the time change becomes very small, the limit is taken to get the real (instant) velocity:

Velocity

rp

rp'

O

Drp

P

P'

y

x

dt

d

tt

rrv

lim

0

tave

r

v

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Acceleration

Similarly, the acceleration is defined as the change in velocity with respect to time, and we can define the average acceleration

When the time period becomes very small, the limit is taken to get the instant acceleration :

vp

vp'

O

DvpP

P'

y

x

dt

d

tt

vva

lim

0

tave

v

a

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Methods for Solution Solution for the unknown quantities in a certain mechanism can be

obtained by different methods: Analytical Methods:

Trigonometry Analysis Considering the geometry: magnitudes and Angles of each vector

Vector Algebra Standard vector analysis: addition and subtraction of vectors

Complex Algebra Representation of vectors as complex numbers consisting of

magnitudes and angles.

Graphical methods: Using graphical tools.

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Complex Algebra Method The vector is represented as a

complex number which includes the magnitude and direction of the vector.

r = r ejq

= r cos q +j r sinq Where r: magnitude of r;

q: Direction of r (from +ve x-axis);

j: (-1)1/2;

and e: Base of the natural logarithm. To get velocity, differentiate the

position vector. To get acceleration, differentiate the

velocity vector.

r

q

r cosq

jr sinq

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Velocity

jjj ererredt

d

dt

d

)(r

v

Where:

w: is the angular velocity

)();( dt

dr

dt

dr

Acceleration

jjjj

jj

ejrererjer

ejrerdt

d

dt

d

22

)(

a

va

Where:

a: is the angular acceleration

2

2

2

2

;)(dt

d

dt

rdr

dt

dr

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An off-set crank slider mechanism

Find the displacement of the slider (link 4) in terms of the dimensions of the other links and their positions.

2

3

4

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Solution using Complex Algebra MethodAlgebraic methods have similar procedure for solution, as follows:

Procedure for Solution:1. Form one closed vector loop for the mechanism.2. Write vector equation representing the loop: Loop-Closure

Equation.3. Represent each vector with corresponding magnitudes and

angles on the graph.4. Determine known and unknown quantities, and also constant and

variable quantities.5. Solve the equation. 6. For velocity, differentiate the LCE once 7. For acceleration, differentiate the LCE twice

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An off-set crank slider mechanism

r1

q2

2

3

4

r2

r3

r4

Given links 2 and 3 dimensions, the input angle q2, angular speed w2 , and acceleration a2 ; determine the slider position, velocity and acceleration.

Solution

Draw the vector closure loop Write the LCE Represent each vector Determine known and unknown variables Solve LCE for displacement of slider For velocity and acceleration, differentiate

LCE as needed.

e

w2a2