Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

Introduction

Fundamentals of Analytical Method

For analytical kinematic (and dynamic) analysis, the following vector fundamentals are necessary:

1. Convention for describing angle of a vector

2. Projecting a vector onto a Cartesian reference frame

3. Types of vectors

4. Time derivatives of vectors

Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

There are infinite ways to describe angle of a vector. We adopt the convention to describe the angle of a vector with respect to the positive x-axis in a counter-clockwise (CCW) direction.

For vector R (with a magnitude R) the angle is measured as:

Angle of A Vector

R

x

y

1. From the base (tail) of the vector draw a line in the positive x-axis

2. In a counter-clockwise direction, starting from this line, draw an arc until it reaches the vector

This is the angle .

Regardless of the orientation of a vector, this process provides the angle in such a way that simplifies any further analysis.

►

Angle of A Vector

►

Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

Examples:

1. Vector R1

2. Vector R2

3. Vector R3

4. Vector R4

Angle of A Vector

R1

x

y

1 = 36.4o

►

Angle of A Vector (cont.)

►

R2

2= 160.7o

►

R3

3= 215.6o

R4

4= 310.2o

►

Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

For vector R with a magnitude R and an angle , if the angle is described according to the “convention”, it is a simple matter to determine its x-y components.

Vector Projection

►

R

x

y

R cos

R sinThe x component is:

R(x) = R cos (a)

The y component is:

R(y) = R sin (b)

Equations (a) and (b) are always valid, regardless of the angle being in the first quadrant (0-90 degrees) or the second quadrant (90-180 degrees) or so on.

►

Projecting A vector onto x-y Axes

Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

1. Constant magnitude and constant angle

2. Constant magnitude and variable angle

3. Variable magnitude and constant angle

4. Variable magnitude and variable angle

Types of Vectors

►

►

Types of Vectors

R4

R1

R1

R1

R4

R3

R2

R3

R2

R2

R4

►

►

The magnitude and the angle of a vector may be constant and/or variable. Therefore we may consider four types of vectors:

Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

Constant magnitude and constant angle

Position components:

R(x) = R cos

R(y) = R sin

First derivative (velocity components):

R(x) = d R(x) / dt = 0

R(y) = d R(y) / dt = 0

Second time derivative (acceleration components):

R(x) = d2 R(x) / dt2 = 0

R(y) = d2 R(y) / dt2 = 0

Time Derivatives

Time Derivatives of Vectors

R

.

.

..

..

Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

Constant magnitude and variable angle

Position components:

R(x) = R cos

R(y) = R sin

First derivative (velocity components):

R(x) = d R(x) / dt = R (sin)

R(y) = d R(y) / dt = R (cos)

V t

Where = d/ dt is the angular velocity of the link.

Second time derivative (acceleration components):

R(x) = d2 R(x) / dt2 = R (sin)R (cos)

R(y) = d2 R(y) / dt2 = R (cos)R (sin)

At An

Where = d/ dt is the angular acceleration of the link.

Time Derivatives

Time Derivatives of Vectors (cont.)

R

R.

.

..

..

Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

Variable magnitude and constant angle

Position components:

R(x) = R cos

R(y) = R sin

First derivative (velocity components):

R(x) = d R(x) / dt = R cos

R(y) = d R(y) / dt = R sin

V s

Where R = dR / dt = V s is the rate of change in the magnitude of the link.

Second time derivative (acceleration components):

R(x) = d2 R(x) / dt2 = R cos

R(y) = d2 R(y) / dt2 = R sin

A s

Where R = d2R / dt2 = A s is the second time derivative of the magnitude of the link.

Time Derivatives

Time Derivatives of Vectors (cont.)

R

.

.

..

..

.

..

.

.

..

..

Fundamentals of Analytical Analysis 1

P. Nikravesh, AME, U of A

Variable magnitude and variable angle

Position components:

R(x) = R cos

R(y) = R sin

First derivative (velocity components):

R(x) = d R(x) / dt = R cos R (sin)

R(y) = d R(y) / dt = R sinR (cos)

V s V t

Second time derivative (acceleration components):

R(x) = d2 R(x) / dt2 = R cos R (sin)R (cos) 2R (sin)

R(y) = d2 R(y) / dt2 = R sinR (cos)R (sin)2R (cos)

As At An Ac

Time Derivatives

Time Derivatives of Vectors (cont.)

R

R

.

.

..

..

.

.

..

..

.

.