THEORY OF MACHINES804319-3

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Lecture 4

Position Analysis

POSITION ANALYSIS

Introduction Coordinate Systems Position and Displacement Translation, Rotation, and Complex

Motion Graphical Position Analysis of Linkages Algebraic Position Analysis of Linkages Vector Loop Representation of Linkages Complex Numbers as Vectors

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INTRODUCTION

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A principal goal of kinematic analysis is to determine the accelerations of all the moving parts in the assembly

Design

Stresses

Static Forces DynamicForces

Acceleration

Graphical Approach

Algebraic Approach

Velocity

Graphical Approach

Algebraic Approach

Position

Graphical Approach

Algebraic Approach

COORDINATE SYSTEMS

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Global or absolute coordinate system Local coordinate systems

POSITION AND DISPLACEMENT

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PositionThe Position of a point in the plane can be defined by the use of a Position vector.The attributes of the position vector can be expressed in either Polar or Cartesian coordinates.Each form is directly convertible into the other by: The Pythagorean theorem: And the trigonometry:

POSITION AND DISPLACEMENT

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Displacement

Displacement is defined as “the straight line distance between the initial and final position of a point which has moved in the reference Frame.”

Position difference equation:

RBA = RB - RA

This expression is read: The position of B with respect to A is equal to the (absolute) position of B minus the (absolute) position of A, where absolute means with respect to the origin of the global reference frame.

POSITION AND DISPLACEMENT

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CASE 1: One body in two successive positions =>position difference

CASE 2: Two bodies simultaneously in separate positions => relative position

TRANSLATION, ROTATION, AND COMPLEX MOTION

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Translation“All points on the body have the same displacement.” Curvilinear Translation Rectilinear Translation

RA′A = RB′B.

TRANSLATION, ROTATION, AND COMPLEX MOTION

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Rotation “Different points in the body undergo different

displacements and thus there is a displacement difference between any two points chosen”.

The link now changes its angular orientation in the reference frame, and all points have different displacements.

RB′B = RB′A - RBA

TRANSLATION, ROTATION, AND COMPLEX MOTION

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Complex Motion The total complex displacement of point B is defined

by the following expression:Total displacement = translation component + rotation component

The new absolute position of point B referred to the origin at A is:

RB′′B = RB′B + RB′′B′

RB′′A = RA′A + RB′′A′

TRANSLATION, ROTATION, AND COMPLEX MOTION

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Euler's theorem:“The general displacement of a rigid body with one point fixed is a rotation about some axes.”

Chasles' theorem:“Any displacement of a rigid body is equivalent to the sum of a translation of any one point on that body and a rotation of the body about an axis through that point.

GRAPHICAL POSITION ANALYSIS OF LINKAGES

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GRAPHICAL POSITION ANALYSIS OF LINKAGES

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ALGEBRAIC POSITION ANALYSIS OF LINKAGES

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The coordinates of point A are:

The coordinates of point B are found using the equations of circles about A and 04.

ALGEBRAIC POSITION ANALYSIS OF LINKAGES

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VECTOR LOOP REPRESENTATION OF LINKAGES

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An alternate approach to linkage position analysis

creates a vector loop (or loops) around the linkage. This loop closes on itself making the sum of the

vectors around the loop zero.

COMPLEX NUMBERS AS VECTORS

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There are many ways to represent vectors. They may be defined in Polar coordinates, by their magnitude and angle, or in Cartesian coordinates as x and y components.

We can represent vectors by unit vectors or by complex number notation.

COMPLEX NUMBERS AS VECTORS

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Euler identity:

COMPLEX NUMBERS AS VECTORS

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COMPLEX NUMBERS AS VECTORS

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