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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.
Reference Books: