Engineering mechanics by A.Vinoth Jebaraj

Prepared by Dr.A.Vinoth Jebaraj

Force Action of one body on the other (push or pull)

Point of Application

Direction

Magnitude

What is the need of knowing MECHANICS?

Mechanics Deals with forces

Mechanics

Mechanics of Rigid Bodies

Mechanics of Deformable Bodies

Mechanics of Fluids

Statics Dynamics

kinematics

kinetics

Studying External effect offorces on a body such asvelocity, acceleration,displacement etc.

Studying Internal effect offorces on a body such asstresses (internal resistance),change in shape etc.

Rigid body mechanics

Deformable body mechanics

Statics

Deals with forces and its effectswhen the body is at rest

Dynamics

Deals with forces and its effects whenthe body is in moving condition

Truss Bridge IC Engine

Rigid body mechanics

Actual structures and machines are never rigid under the action ofexternal loads or forces.But the deformations induced are usually very small which does notaffect the condition of equilibrium.

Negligible deformation (no deformation) under the action of forces.Assuming 100% strength in the materials. Large number of particlesoccupying fixed positions with each other.

Particle Mechanics

Treating the rigid body as a particle which is negligible insize when compared to the study involved. (very smallamount of matter which is assumed as a point in a space).

Example: studying the orbital motion of earth

Types of forces Concurrent coplanar forces

Collinear forces

Non Concurrent coplanar (Parallel)

Concurrent non-coplanar

Components of a Force

Plane ForcePlane Force

Space ForceSpace Force

Couple Two equal and opposite forces are acting at some distance forming a couple

How Rotational Effect will change with

distance?

Free body diagram

Isolated body from the structure of machinery which shows all the forces andreaction forces acting on it.

Examples for free body diagram

Parallelogram law:

Two forces acting on a particle can be replaced by the singlecomponent of a force (RESULTANT) by drawing diagonal of theparallelogram which has the sides equal to the given forces.

Parallelogram law cannot be proved mathematically . It is anexperimental finding.

The two vectors can also be added by head to tail by using triangle law.

Triangle law states that if three concurrent coplanar forces are acting at apoint be represented in magnitude and direction by the sides of a triangle,then they are in static equilibrium.

Lami’s Theorem states that if three concurrent coplanarforces are acting at a point, then each force is directlyproportional to the sine of the angle between the other twoforces.

Lami’s theorem considering onlythe equilibrium of three forcesacting on a point not the stressacting through a ropes or strings

The principle of transmissibility isapplicable only for rigid bodies notfor deformable bodies

F1

F2

F5F4

F3

A B

E

D

C

Polygon Law of Forces

“If many number of forces acting at a point can be represented as a sides of

a polygon, then they are in equilibrium”

Equivalent Couples

Supports and Reactions

FrictionFriction is a force [Tangential force]that resists the movement of slidingaction of one surface over the other.

Few examples where friction force

involved

Theory of Dry Friction

Uneven distribution of friction forceand normal reaction in the surface.

Microscopic irregularities producesreactive forces at each point of contact.

The distance ‘x’ is to avoid “tippingeffect” caused by the force ‘P’ so thatmoment equilibrium has been arrivedabout point ‘O’.

Limiting static frictional force: when this value is reached then the body will be inunstable equilibrium since any further increase in P will cause the body to move.

At this instance, frictional force is directly proportional to normal reaction on thefrictional surface.

Where μs coefficient of static friction

When a body is at rest, the angle that the resultant force makes with normal reaction isknown as angle of static friction.

Where μk coefficient of kinetic friction

When a body is in motion, the angle that the resultant force makes with normal reactionis known as angle of kinetic friction.

Laws of Dry Friction

Laws of Dry Friction

Laws of Dry Friction

Analysis of Trusses

Trusses Stationary, fully constrained structures inwhich members are acted upon by two equal andopposite forces directed along the member.

Frames Stationary, fully constrained structures inwhich atleast one member acted upon by three or moreforces which are not directed along the member.

Machines Containing moving parts, always containat least one multiforce member.

Applications of Trusses

Electric Tower

BridgeRoof support

Cranes

A framework composed of members joined at their ends to form arigid structure is called a truss.

Rigid Structure

Rigid Non-collapsible and deformation ofthe members due to induced internal strains isnegligible.

Axially Loaded Members

Types of TrussesPlane Trusses

Bridge Trusses Roof Trusses

Space Trusses

Internal and External Redundancy

External Redundancy More additional supports

Internal Redundancy

If m + 3 = 2j, then the truss is statically determinate structure

If m + 3 > 2j, then the truss is redundant structure (staticallyindeterminate structure)[more members than independent equations]

If m + 3 < 2j, then the truss is unstable structure (will collapse underexternal load)[deficiency of internal members]

For statically determinate trusses, ‘2j’ equations for a truss with ‘j ‘ joints is equal tom+3 (‘m’ two force members and having the maximum of three unknown supportreactions)

Method of Joints

Special Conditions

Zero Force Members

These members are not useless.

They do not carry any loads under the loading conditions shown,but the same members would probably carry loads if the loadingconditions were changed.

These members are needed to support the weight of the truss andto maintain the truss in the desired shape.

Method of sections

When a particle moves along a curve other than a straight line, then the particle is incurvilinear motion.

Curvilinear Motion.

Velocity of a particle is a vector tangent to the path of the particle

Acceleration is not tangent to the path of the particle

The curve described by the tip of v is called thehodograph of the motion

Tangential and Normal Components

Tangential component of the acceleration is equal to the rate of change of the speed ofthe particle.

Normal component is equal to the square of the speed divided by the radius ofcurvature of the path at P.

Radial and Transverse components

The position of the particle P is defined by polar coordinates r and θ. It is thenconvenient to resolve the velocity and acceleration of the particle into componentsparallel and perpendicular to the line OP.

Unit vector er defines the radial direction, i.e., the direction in which P would move if rwere increased and θ were kept constant.

The unit vector eθ defines the transverse direction, i.e., the direction in which P wouldmove if θ were increased and r were kept constant.

Where -er denotes a unit vector of sense opposite to that of er

Using the chain rule of differentiation,

Using dots to indicate differentiation with respect to t

To obtain the velocity v of the particle P, express the position vector r of P as theproduct of the scalar r and the unit vector er and differentiate with respect to t:

Differentiating again with respect to t to obtain the acceleration,

The scalar components of the velocity and the acceleration in the radial and transversedirections are, therefore,

In the case of a particle moving along a circle of center O, have r = constant and

Kinetics of Particles

Work Energy Method Work of a force & Kinetic energy of particle.

In this method, there is no determination of acceleration.

This method relates force, mass, velocity and displacement.

Work of a Constant Force in Rectilinear Motion

Work of the Force of Gravity

Work of the Force Exerted by a Spring

Kinetic Energy of a particle

Consider a particle of mass m acted upon by a force F and moving along a path which iseither rectilinear or curved.

When a particle moves from A1 to A2 under the action of a force F, the work of the force Fis equal to the change in kinetic energy of the particle. This is known as the principle ofwork and energy.

Dynamic Equilibrium Equation

ΣF - ma = 0 The vector -ma, of magnitude ‘ma’ and of direction opposite to that of the acceleration,is called an inertia vector.

The particle may thus be considered to be in equilibrium under the given forces and theinertia vector or inertia force.

When tangential and normal components are used, it is more convenient to representthe inertia vector by its two components -mat and –man.

Principle of Impulse and Momentum

Consider a particle of mass m acted upon by a force F. Newton’s second law can beexpressed in the form

where ‘mv’ is the linear momentum of the particle.

The integral in Equation is a vector known as the linear impulse, or simply theimpulse, of the force F during the interval of time considered.

Vectorial addition of initialmomentum mv1 and the impulseof the force F gives the finalmomentum mv2.

Definition: A force acting on a particle during a very short time interval that is large

enough to produce a definite change in momentum is called an impulsive force and the

resulting motion is called an impulsive motion.

When two particles which are moving freely collide with one another, then the totalmomentum of the particles is conserved.

KINEMATICS OF RIGID BODIES Investigate the relations existing between the

time, the positions, the velocities, and the accelerations of the various particles

forming a rigid body.

Various types of rigid-body motion Various types of rigid-body motion

Translation A motion is said to be a translation if any straight line inside the

body keeps the same direction during the motion.

Rectilinear translation (Paths are straight lines)

Curvilinear translation(Paths are curved lines)

Rotation about a Fixed Axis Particles forming the rigid body move in parallelplanes along circles centered on the same fixed axis called the axis of rotation.

The particles located on the axis have zero velocity and zero acceleration

Rotation and the curvilinear translation are not the same.

General Plane Motion Motions in which all the particles of the body move in

parallel planes.

Any plane motion which is neither a rotation nor a translation is referred to as a

general plane motion.

Examples of general plane motion :

Motion about a Fixed Point The three-dimensional motion of a rigid body

attached at a fixed point O, e.g., the motion of a top on a rough floor is known as

motion about a fixed point.

General Motion Any motion of a rigid body which does not fall in any of the

categories above is referred to as a general motion.

Example:

Translation (either rectilinear or curvilinear translation)

Since A and B, belong to the same rigid body, the derivative of rB/A is zero

When a rigid body is in translation, all the points of the body have the same velocity andthe same acceleration at any given instant.

In the case of curvilinear translation, the velocity and acceleration change in directionas well as in magnitude at every instant.

Rotation about a fixed axis

Consider a rigid body which rotates about a fixed axis AA’

‘P’ be a point of the body and ‘r’ its position vectorwith respect to a fixed frame of reference.

The angle θ depends on the position of Pwithin the body, but the rate of change Ѳ isitself independent of P.

The velocity v of P is a vectorperpendicular to the plane containing AA’and r.

The vector

It is angular velocity of the body and is equalin magnitude to the rate of change of Ѳ withrespect to time.

The acceleration ‘a’ of the particle ‘P’

α is the angular acceleration of abody rotating about a fixed axis is avector directed along the axis ofrotation, and is equal in magnitudeto the rate of change of ‘ω’ withrespect to time

Two particular cases of rotation

Uniform Rotation This case is characterized by the fact that the angular acceleration is zero. The angular velocity is thus constant.

Uniformly Accelerated Rotation n this case, the angular acceleration is constant

General plane motion The sum of a translation and a rotation

Absolute and relative velocity in plane motion

Any plane motion of a slab can be replaced by a translation defined by the motion of anarbitrary reference point A and a simultaneous rotation about A.

The absolute velocity vB of a particle B of the slab is

The velocity vA corresponds to the translation of the slab with A, while the relative

velocity vB/A is associated with the rotation of the slab about A and is measured with

respect to axes centered at A and of fixed orientation

Consider the rod AB. Assuming that the velocity vA of end A is known, we propose to

find the velocity vB of end B and the angular velocity ω of the rod, in terms of the

velocity vA, the length l, and the angle θ.

The angular velocity ω of the rod in its rotation about B is the same as in its rotationabout A.

The angular velocity ω of a rigid body in plane motion is independent of the referencepoint.

Absolute and relative acceleration in plane motion

For any body undergoing planar motion, there always exists a point in the plane of

motion at which the velocity is instantaneously zero. This point is called the

instantaneous center of rotation, or C. It may or may not lie on the body!

Instantaneous Centre

As far as the velocities are concerned, the slab seems to rotate about the instantaneouscenter C.

If vA and vB were parallel and having same magnitude the instantaneous center C would beat an infinite distance and ω would be zero; All points of the slab would have the samevelocity.

If vA = 0, point A is itself is the instantaneouscenter of rotation, and if ω = 0, all the particleshave the same velocity vA.

Concept of instantaneous center of rotation

At the instant considered, the velocities of all the particles of the rod are thus the same asif the rod rotated about C.

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