Lecture 1 Introduction OK

Chapter 1. Introduction to Dynamics 1

Chapter 1Introduction to Dynamics

Chapter Objectives

To review two parts of mechanics: statics and dynamics

To introduce main concepts of dynamics

To give axioms of dynamics

To examine the standard procedures for solving dynamical problems

1.1 Introduction

The objective of this text book is to show the reader the second part of Mechanics that is how

to predict the motion and forces causing that motion of physical systems modeled as

collections of particles and rigid bodies. As stated in the first text book, Mechanics is a branch

of the physical sciences that is concerned with the state of rest or motion of bodies subjected

to the action of forces. The mechanics of rigid bodies is divided into two areas: statics and

dynamics. Statics which has been already discussed in the first course is concerned with the

equilibrium of a body that is either at rest or moves with constant velocity. The foregoing

treatment in this second course is concerned with dynamics which deals with the accelerated

motion of a body. Hence, it is assumed that the student has previously studied the statics of

particles and rigid bodies and is familiar with necessary topics commonly treated in statics,

such as vector algebra, concept of equivalent set of force vectors, equilibrium of force

systems, center of mass, moments of inertia, products of inertia etc.

Here the subject of dynamics will be presented in two parts: kinematics, which treats only the

geometric aspects of the motion, and kinetics, which is the analysis of the forces causing the

motion. To develop these principles, the dynamics of a particle will be discussed first,

followed by topics in rigid-body dynamics in two and then three dimensions.


Historically. dynamics is relatively recent subject compared with statics. The principles of

dynamics developed when it was possible to make an accurate measurement of time. Galileo

Galilei (1564-1642) was one of the first major contributors to this field. His work consisted of

experiments using pendulums and falling bodies. The most significant contributions in

dynamics, however, were made by Isaac Newton (1642-1727), who is noted for his

formulation of the three fundamental laws of motion and the law of universal gravitational

attraction. Newton’s famous work was published in the first edition (1687) of his

Philosophiae naturalis principia mathematica, which is generally recognized as one of the

greatest of all recorded contributions to knowledge. Shortly after these laws were postulated,

important techniques for their application were developed by Euler, D'Alembert, Lagrange,

and others.

There are many problems in engineering whose solutions require application of the . principles

of dynamics. Typically the structural design of any vehicle, such as an automobile or airplane,

requires consideration of the motion to which it is subjected. This is also true for many

mechanical devices. such as motors, pumps, movable tools, industrial manipulators, and

machinery. Furthermore, predictions of the motions of artificial satellites, projectiles, and

spacecraft are based on the theory of dynamics. With further advances in technology, there

will be an even greater need for knowing how to apply the principles of this subject and

students whose interests lead them into one or more of activities in branches of advanced

technology will find a constant need for applying the fundamentals of dynamics.

1.2 Basic Concepts of Dynamics

The following basic quantities were given in the first course Statics. They are summarized

here along with additional comments of special relevance to the study of dynamics.

Space. Space is the geometric region occupied by bodies whose positions are described by

linear and angular measurements relative to a coordinate system. Our daily experiences give

us an intuitive notion of space and the locations, or positions, of points in space. In a three

dimensional space we need three independent coordinates. For two-dimensional problems our

space requires only two independent coordinates. The basic frame of reference for the laws of

Newtonian mechanics is primary inertial system which is an imaginary set of rectangular

axes assumed to have no translation or rotation in space. A reference frame in which the

Newton’s first law of motion is valid is called inertia system.

Time. Time is a measure of the succession of events and is considered an absolute quantity in

Newtonian mechanics.


Mass. Mass is a property of matter by which we can compare the action of one body with that

of another. We can define it as a quantitative measure of the inertia of matter which is its

resistance to a change in velocity. Mass is also the property which gives rise to gravitational

attraction.

Force. Force is the vector action of one body on another. This quantity has been thoroughly

treated in the first course Statics Keep in mind that we can distinguish forces according to

various criteria: internal and external forces, active and reaction forces etc.

Particle. A particle can be defined as a body which has a mass but its size can be neglected.

Also, when the dimensions of a body are irrelevant to the description of its position or the

action of applied forces the body may be considered as a particle. When a body is idealized as

a particle, the principles of mechanics reduce to a rather simplified form since the geometry of

the body will not be involved in the analysis of the problem

Rigid Body. A rigid body can be considered as a combination of a large number of particles

in which all the particles remain at a fixed distance from one another both before and after

applying a load. As a result, the material properties of any body that is assumed to be rigid

will not have to be considered when analyzing the forces acting on the body. In most cases the

actual deformations occurring in structures, machines, mechanisms, and the like are relatively

small, and the rigid-body assumption is suitable for analysis. On the other hand, if the

problem is one of examining the internal stresses in the body due to changing dynamic loads,

then, the deformable characteristics of the body would have to be examined and the body

could no longer be considered a rigid body.

Mechanical system. Mechanical system consists of particles and rigid bodies. We can

distinguish free and constrained mechanical systems. The interaction between the

components of the mechanical systems is expressed only by forces. In constrained

mechanical system the motion of particles and rigid bodies are restricted by constraints

in positions and velocities.

1.3 Axioms of Dynamics

Five following basic laws or axioms of dynamics create the basis of dynamics. The first three

laws were stated by Newton as laws of motion. In modern terminology we can express them

as follows:

First dynamic axiom

When the sum of the forces acting on a particle is zero, its velocity is constant. In particular if

the particle is initially stationary, it will remain stationary.


Second dynamic axiom

When the sum of the forces acting on a particle is not zero, the sum of the forces is equal to

the rate of change of the linear momentum of the particle. If the mass is constant, the sum of

the forces is equal to the product of the mass of the particle and its acceleration

Third dynamic axiom

The forces of action and reaction between interacting bodies are equal in magnitude, opposite

in direction and collinear.

These laws have been verified by counterless physical measurements. The first two laws hold

for measurement in an absolute inertia frame of reference which has no acceleration. Hence,

the first law gives us the tool for recognizing the inertia frame.

Newton’s second law forms the basis for most of the analysis in dynamics. For a particle of

mass m subjected to a resultant force F the law may be stated as:

(1.1)

where a is the resulting acceleration measured in an inertia frame of reference. If we define

the linear momentum of the particle as

(1.2)where v is the velocity of the particle, the law may be stated as

(1.3)

The third law constitutes the principle of action and reaction with which we should be

thoroughly familiar from our work in statics.

Fourth dynamic axiom

The acceleration of a particle acted upon by several forces is the sum of accelerations which

each force gives the particle individually.

This axiom is commonly known as the principle of superposition.

Fifth dynamic axiom

The constrained body can be considered as a free body when it is released from constraints

and subjected by corresponding reactive forces instead.

This law allows us to use all above basic laws, stated for free bodies and particles, even in

cases when the particles and bodies are constrained.

1.4 Units

Unit systems have been discussed in the first course. We will give a brief summary due to

their importance in solving engineering problems. The SI system of units has become nearly


standard throughout the world. In the USA, US Customary units are also used.

In SI system units, length is measured in meters (m) and mass in kilograms (kg). Time is

measure in seconds (s). Meters, kilograms and seconds are called the base units of the SI

system. Force is measured in newtons (N). Since these units are related by Newtons’s second

law one newton is the force required to give an object of one kilogram mass an acceleration of

one meter per second squared:

Hence, the newton is called a derived unit.

In US Customary units, length is measured in feet (ft) and force is measured in pounds (lb).

Time is measured in seconds (s). These are the base units of the US Customary system. In this

system, mass is a derived unit. The unit of mass is the slug, which is the mass of material

accelerated at one foot per second squared by a force of one pound, i.e.

In both SI and US Customary units, angles are normally expressed in radians (rad). It is

defined by the ratio of the part of the circumference of the circle subtended by the angle to the

radius r of the circle. Since there are 360 degrees (360o) in a complete circle and the complete

circumference of the circle is 2πr, 360o equal 2π rad.

In this text book we will use commonly the SI system of units.

1.5 Basic Problems of Dynamics

Dynamics is considered to be more involved than statics since both the forces applied to a

body and its motion must be taken into account. Also, many applications require using

calculus, rather than just algebra and trigonometry. In any case, the most effective way of

learning the principles of dynamics is to solve problems.

We can distinguish two basic problems of dynamics

1. Forward problem: in such problem the motion of the body is known and the task is to

determine the forces causing that motion

2. Inverse problem: in this case the forces acting on the body and the initial condition of

the body motion are known and the task is to specify the motion of the body.

To be successful at problem solving, it is necessary to present the work in a logical and

orderly manner as suggested by the following sequence of steps:

1. Read the problem carefully and try to correlate the actual physical situation with the

theory studied.


2. Draw any necessary diagrams and tabulate the problem data.

3. Establish a coordinate system and apply the relevant principles, generally in

mathematical form.

4. Solve the necessary equations algebraically as far as practical; then, use a consistent

set of units and complete the solution numerically. Report the answer with no more

significant figures than the accuracy of the given data.

5. Study the answer using technical judgment and common sense to determine whether

or not it seems reasonable.

6. Once the solution has been completed, review the problem. Try to think of other ways

of obtaining the same solution.

In applying this general procedure, do the work as neatly as possible. Being neat generally

stimulates clear and orderly thinking and vice versa.

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Lecture 1 Introduction OK