# Introduction to Mechanisms

Yi Zhang

with

Susan Finger

Stephannie Behrens

Table of Contents

1 Physical Principles

This chapter introduces the basic physical principles behind mechanisms as well as basic concepts and principles required for this course.

1.1 Force and Torque

1.1.1 Force

Force: an agent or influence that, if applied to a free body results chiefly in an acceleration of the body and sometimes in elastic deformation and other effects. Every day we deal with forces of one kind or another. A pressure is a force. The earth exerts a force of attraction for all bodies or objects on its surface. To study the forces acting on objects, we must know how the forces are applied, the direction of the forces and their value. Graphically, forces are often represented by a vector whose end represents the point of action.

A mechanism is what is responsible for any action or reaction. Machines are based on the idea of transmitting forces through a series of predetermined motions. These related concepts are the basis of dynamic movement.

1.1.2 Torque

Torque: Something that produces or tends to produce rotation and whose effectiveness is measured by the product of the force and the perpendicular distance from the line of action of the force to the axis of rotation.

Consider the lever shown in Figure 1-1. The lever is a bar that is free to turn about the fixed point, A, called the fulcrum; a weight acts on the one side of the lever, and a balancing force acts on the other side of the lever.

Figure 1-1 A lever with balanced forces

To analyze levers, we need to find the torques of the forces acting on the lever. To get the torque of force W about point A, multiply W by l1, its distance from A. Similarly F x l2 is the torque of F about fulcrum A.

1.2 Motion

Motion: a change of position or orientation.

1.2.1 Motion Along a Straight Path

We begin our study of motion with the simplest case, motion in a straight line.

Position and displacement along a line

The first step in the study of motion is to describe the position of a moving object. Consider a car on an east-west stretch of straight highway. We can describe the displacement of the car by saying "the car is 5 kilometers west of the center town". In this description, we specified two factors, the original point of measure and the direction of the displacement.

Velocity

We can define the velocity of an object moving steadily as its displacement per unit time:

(1-1)

where t = t2 - t1 is the time interval during which the displacement occurred. When velocity varies, we can let the time interval become infinitesimally small, thus

(1-2)

Acceleration

Acceleration is the variation of the velocity in a unit time period. If the velocity changes in a constant rate, then we can describe the acceleration by

(1-3)

More generally, acceleration is

(1-4)

1.2.2 Linear Motion in Space

The picture becomes more complicated when the motion is not merely along a straight line, but rather extends into a plane. Here we can describe the motion with a vector which includes the magnitude and the direction of movement.

Position vector and displacement vector

The directed segment which describes the position of an object relative to an origin is the position vector, as d1 and d2 in Figure 1-2

Figure 1-2 Position vector and displacement vector

If we wish to describe a motion from position d1 to position d2, for example, we can use vector d1, the vector starts at the point described by d1 and goes to the point described by d2, which is called the displacement vector.

(1-5)

Velocity vector

For a displacement d occurring in a time interval t, the average velocity during the interval is

(1-6)

Clearly Vave has the direction of d.

In the limit as delta t approaches zero, the instantaneous velocity is

(1-7)

The direction of V is the...

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