Trusses

• Consist entirely of straight two-force members, connected at joints
• Example on board: decompose truss into two-force members and joints, and show how forces meet at joints

Trusses: Method of Joints:-

• Typically used to find forces in all or several of the members
• Each joint is a particle
• Particle equilibrium in 2D:

• For each joint, we have 2 equations, therefore, we can solve for 2 unknowns
• Must start process at a joint with only 2 unknown forces
• Find a joint with only two unknown forces

-First, may need to draw an FBD of the entire truss and find support reactions

• Draw FBD of selected joint

-Draw each force along the direction of the member
-Draw in tension (away from joint)
-Find angle of force from truss geometry
-Resolve angled forces into x, y components

• Apply equilibrium, to solve for 2 unknown forces

A + sign: the force is tension (T)

A – sign: the force is compression (C)

• Find the next joint that has only 2 unknown forces and repeat the process

–Typically this is adjacent to the prior joint

• Repeat with additional joints until all member forces are known.
• Remember to specify (T) or (C) for each force!

Analysis of Structures

•Trusses

–Designed to support loads

–Consist entirely of two-force members

•Frames

–Designed to support loads

–Include one or more multi-force members

•Machines

–Designed to transmit and/or modify forces

–Include one or more multi-force members

Two-Force Members:-

• Pinned at both ends (both joints)
• No applied forces between joints
• No applied moments
• Line of action of forces is directed along a line drawn between the two joints 

Vectors

The vectors can be solved by

1. Law of sine and law of cosines (two forces)
2. Graphically
3. Equilibrium

• Table
• Sum of values

Types of Forces(Loads)

1.Point loads - concentrated forces exerted at point or location

2.Distributed loads - a force applied along a length or over an area.  The distribution can be uniform or non-uniform.

Resultant Forces-

If two forces P and Q acting on a particle A may be replaced by a single force R, which has the same effect on the particle.

Resultant Forces-

• This force is called the resultant of the forces P and Q and may be obtained by constructing a parallelogram, using P and Q as two sides of the parallelogram.  The diagonal that pass through A represents the resultant.
• This is known as the parallelogram law for the addition of two forces.  This law is based on experimental evidence,; it can not be proved or derived mathematically. 
• For multiple forces action on a point, the forces can be broken into the components of x and y.
  

Principle of Transmissibility

The principle of transmissibility states that the condition of equilibrium or of motion of a rigid body will remain unchanged if a force F action at a given point of the rigid body is replace by a force F’ of the same magnitude and the same direction, but acting at a different point, provided that the two forces have the same line of action.

Line of action

Scalar Quantity,Vector Quantities

Scalar Quantity has magnitude only (not direction) and can be indicated by a point on a scale. Examples are temperature, mass, time and dollars.

Vector Quantities have magnitude and direction.  Examples are wind velocity, distance between to points on a map and forces.

Collinear : If several forces lie along the same line-of –action, they are said to be collinear.

Coplanar When all forces acting on a body are in the same plane, the forces are coplanar.

Type of Vectors

Free Vector - is vector which may be freely moved creating couples in space.

Sliding Vector - forces action on a rigid body  are represented by vectors which may move or slid along their line of action.

Bound  Vector or Fixed Vector - can not be moved without modifying the conditions of the problem.

Addition of Couples

• Consider two intersecting planes P1 andP2 with each containing a couple
  

•  Resultants of the vectors also form a couple
  

•  By Varigon’s theorem
  

• Sum of two couples is also a couple that is equalto the vector sum of the two couples

Moment of a Couple

• Two forces F and -F having the same magnitude, parallel lines of action, and opposite sense are said to form a couple.
•  Moment of the couple,
  

•  The moment vector of the couple is independent of the choice of the origin of the coordinate axes, i.e., it is a free vector that can be applied at any point with the same effect.
• Two couples will have equal moments if
  

•  the two couples lie in parallel planes, and
• the two couples have the same sense or the tendency to cause rotation in the same direction.

Concurrent Force Systems

A concurrent force system contains forces whose lines-of action meet at some one point.

Forces may be tensile (pulling).

Forces may be compressive (pushing)

Force exerted on a body has two effects:

•    The external effect, which is tendency to change the motion of the body or to develop resisting forces in the body
•    The internal effect, which is the tendency to deform the body.
• If the force system acting on a body produces no external effect, the forces are said to be in balance and the body experience no change in motion is said to be in equilibrium. The process of reducing a force system to a simpler equivalent stem is called a reduction. The process of expanding a force or a force system into a less simple equivalent system is called a resolution.
• A force is a vector quantity that, when applied to some rigid body, has a tendency to produce translation (movement in a straight line) or translation and rotation of body.  When problems are given, a force may also be referred to as a load or weight.
  Characteristics of force are the: magnitude,direction(orientation) and point of application.

The Laws of Dry Friction. Coefficients of Friction

• Block of weight W placed on horizontal surface. Forces acting on block are its weight and reaction of surface N.
•  Small horizontal force P applied to block. For block to remain stationary, in equilibrium, a horizontal component F of the surface reaction is required. F is a static-friction force.
•  As P increases, the static-friction force F increases as well until it reaches a maximum value Fm.
  

•  Further increase in P causes the block to begin to move as F drops to a smaller kinetic-friction
  

Friction

• In preceding lectures, it was assumed that surfaces in contact were either frictionless (surfaces could move freely with respect to each other) or rough (tangential forces prevent relative motion between surfaces).
• Actually, no perfectly frictionless surface exists. For two surfaces in contact, tangential forces, called friction forces, will develop if one attempts to move one relative to the other.
•  However, the friction forces are limited in magnitude and will not prevent motion if sufficiently large forces are applied.
• The distinction between frictionless and rough is, therefore, a matter of degree.
• There are two types of friction: dry or Coulomb friction and fluid friction. Fluid friction applies to lubricated mechanisms. The present discussion is limited to dry friction between nonlubricated surfaces.

Addition of Forces by Summing Components 2

• Wish to find the resultant of 3 or more concurrent forces
  
• Resolve each force into rectangular components
  
• The scalar components of the resultant are equal to the sum of the corresponding scalar components of the given forces

• To find the resultant magnitude and direction
  

MECHANICS

Introduction

The objective for the current chapter is to investigate the effects of forces on particles:
- replacing multiple forces acting on a particle with a single equivalent or resultant force
 - relations between forces acting on a particle that is in a state of equilibrium

The focus on particles does not imply a restriction to miniscule bodies. Rather, the study is restricted to analyses in which the size and shape of the bodies is not significant so that all forces may be assumed to be applied at a single point.

Resultant of Two Forces

• force: action of one body on another; characterized by its point of application, magnitude, line of action, and sense.
• Experimental evidence shows that the combined effect of two forces may be represented by a single resultant force.
• The resultant is equivalent to the diagonal of a parallelogram which contains the two forces in adjacent legs.
• Force is a vector quantity.

Vectors

• Vector: parameters possessing magnitude and direction which add according to the parallelogram law. Examples: displacements, velocities, accelerations.
• Scalar: parameters possessing magnitude but not direction. Examples: mass, volume, temperature
• Vector classifications:

1. Fixed or bound vectors have well defined points of application that cannot be changed without affecting an analysis.
2. Free vectors may be freely moved in space without changing their effect on an analysis.
3. Sliding vectors may be applied anywhere along their line of action without affecting an analysis.

• Equal vectors have the same magnitude and direction.
• Negative vector of a given vector has the same magnitude and the opposite direction.

Rectangular Components of a Force: Unit Vectors

• May resolve a force vector into perpendicular components so that the resulting parallelogram is a rectangle.Fx & Fy are referred to as rectangular vector components                                                

• Vector components may be expressed as products of the unit vectors with the scalar magnitudes of the

vector components

Thevenin’s Theorem

Thevenins theorem states theat “Any linear, bilateral network having a no of voltage sources and resistances can replaced by a simple equivalent circuit consisting of single voltage source in series with a resistance, where the value of the voltage source is equal to the open circuit voltage across the two terminals of the network and the resistance is equal to the equivalent resistance measured between the terminals with all the energy sources replaced by their internal resistances”.
In many pratical applications it is always not necessary to analyze the complete circuit. It requires that the voltage or current or power in only one resistance of a circuit be found. The use of this theorem provides a simple, equivalent circuit, which can be substituted for the original network.

Superposition Principle

In any linear bilateral network, the current flowing in any element/branch when two or more sources are present is equal to the algebraic sum of the currents flowing in individual elements/branches when individual sources are acting alone, while the other sources are non-operative, that is while considering the effect of individual sources other ideal voltage sources and ideal current sources in the network are replaced by ‘short circuit’ and ‘open circuit’ across their terminals respectively.

Norton’s Theorem

It states that “Any two terminal linear network with current sources, voltage sources and resistances can be replaced by an equivalent circuit consisting of a current source in parallel with a resistance”.
The value of the current source is the short circuit current between the two terminals of the network and the resistance of the equivalent resistance measured between the terminals of the network with all the voltage sources replaced by their internal resistances and current sources by open circuit.

Maximum Power Transfer Theorem

This theorem is used to find the value of load resistance for which there would be maximum amount of power transfer from source to load.
Maximum power transfer theorem states that “In any linear bilateral network the maximum power will be delivered by the load, when the load resistance is equal to the source resistance.
i.e RL = RS.

IL = Vth /(RL+Rth) [RS = Rth]
Pmax = Vth
2/4Rth

D.C. Motor Characteristics

The performance of a d.c. motor can be judged from its characteristic curves known as motor characteristics, following are the three important characteristics of a d.c. motor:
(i) Torque and Armature current characteristic (Ta/Ia) It is the curve between armature torque Ta and armature current Ia of a d.c. motor. It is also known as electrical characteristic of the motor.
(ii) Speed and armature current characteristic (N/ia) It is the curve between speed N and armature current Ia of a d.c. motor. It is very important characteristic as it is often the deciding factor in the selection of the motor for a particular application.
(iii) Speed and torque characteristic (N/Ta) It is the curve between speed N and armature torque Ta of a d.c. motor. It is also known as mechanical characteristic.

Speed of a D.C. Motor

Therefore, in a d.c. motor, speed is directly proportional to back e.m.f. Eb and
inversely proportional to flux per pole .

Shaft Torque (Tsh)

The torque which is available at the motor shaft for doing useful work is known as shaft torque. It is represented by Tsh.

The total or gross torque Ta developed in the armature of a motor is not available at the shaft because a part of it is lost in overcoming the iron and frictional losses in the motor. Therefore, shaft torque Tsh is somewhat less than the armature torque Ta. The difference Ta - Tsh is called lost torque.
Clearly,

As stated above, it is the shaft torque Tsh that produces the useful output. If the speed of the motor is N r.p.m., then,

Armature Torque of D.C. Motor

Torque is the turning moment of a force about an axis and is measured by the product of force (F) and radius (r) at right angle to which the force acts.
T = F x r
In a d.c. motor, each conductor is acted upon by a circumferential force F at a distance r, the radius of the armature. Therefore, each conductor exerts a torque, tending to rotate the armature. The sum of the torques due to all armature conductors is known as gross or armature torque (Ta).
Let in a d.c. motor
r = average radius of armature in m

l = effective length of each conductor in m
Z = total number of armature conductors
A = number of parallel paths
i = current in each conductor = Ia/A
B = average flux density inWb/m2
f = flux per pole inWb
P = number of poles
Force on each conductor, F = B i l newtons
Torque due to one conductor = F x r (newton- metre)
Total armature torque, Ta = Z F r (newton-metre)
= Z B i l r
Now i = Ia/A, B = /a where a is the x-sectional area of flux path per pole at
radius r. Clearly,
   a = 2  r l /P.

Since Z, P and A are fixed for a given machine,

 

Hence torque in a d.c. motor is directly proportional to flux per pole and armature
current.
(i) For a shunt motor, flux f is practically constant.

(ii) For a series motor, flux f is directly proportional to armature current Ia
provided magnetic saturation does not take place.

Types of D.C. Motors

•  Shunt Motor
•  Series Motor
•  Compound Motor

Shunt DC Motor

Series DC Motor

Compound DC Motor

Back or Counter E.M.F.

When the armature of a d.c. motor rotates under the influence of the driving torque, the armature conductors move through the magnetic field and hence e.m.f. is induced in them as in a generator.
The induced e.m.f. acts in opposite direction to the applied voltage V(Lenz’s law)and in known as back or counter e.m.f. Eb.
The back e.m.f. Eb = P ZN/60A) is always less than the applied voltage V, although this difference is small when the motor is running under normal conditions.
The electric work done in overcoming and causing the current to flow against Eb is converted into mechanical energy developed in the armature.
It follows, therefore, that energy conversion in a d.c. motor is only possible due to the production of back e.m.f. Eb.

Net voltage across armature circuit = V - Eb

If Ra is the armature circuit resistance, then,

Since V and Ra are usually fixed, the value of Eb will determine the current drawn by the motor. If the speed of the motor is high, then back e.m.f.
Eb = P ZN/60A is large and hence the motor will draw less armature current and vice versa.

Significance of Back E.M.F.

The presence of back e.m.f. makes the d.c. motor a self-regulating machine i.e., it makes the motor to draw as much armature current as is just sufficient to develop the torque required by the load.
Armature current,
(i) When the motor is running on no load, small torque is required to overcome the friction and windage losses. Therefore, the armature current Ia is small and the back e.m.f. is nearly equal to the applied voltage.
(ii) If the motor is suddenly loaded, the first effect is to cause the armature to slow down. Therefore, the speed at which the armature conductors move through the field is reduced and hence the back e.m.f. Eb falls. The decreased back e.m.f. allows a larger current to flow through the armature and larger current means increased driving torque. Thus, the driving torque increases as the motor slows down. The motor will stop slowing down when the armature current is just sufficient to produce the increased torque required by the load.
(iii) If the load on the motor is decreased, the driving torque is momentarily in excess of the requirement so that armature is accelerated. As the armature speed increases, the back e.m.f. Eb also increases and causes the armature current Ia to decrease. The motor will stop accelerating when the armature current is just sufficient to produce the reduced torque required by the load. It follows, therefore, that back e.m.f. in a d.c. motor regulates the flow of armature current

i.e., it automatically changes the armature current to meet the load requirement.

Construction of D.C. Generator

    The d.c. generators and d.c. motors have the same general construction. In fact, when the machine is being assembled, the workmen usually do not know whether it is a d.c. generator or motor. Any d.c. generator can be run as a d.c.motor and vice-versa. All d.c. machines have five principal components viz.,

(i) Field System
(ii) Armature Core
(iii)Armature Winding
(iv) Commutator
(v) Brushes

(i) Field system

The function of the field system is to produce uniform magnetic field within which the armature rotates. It consists of a number of salient poles (of course, even number) bolted to the inside of circular frame (generally called yoke). The yoke is usually made of solid cast steel whereas the pole pieces are composed of stacked laminations. Field coils are mounted on the poles and carry the d.c. exciting current. The field coils are connected in such a way that adjacent poles have opposite polarity.

The m.m.f. developed by the field coils produces a magnetic flux that passes through the pole pieces, the air gap, the armature and the frame. Practical d.c. machines have air gaps ranging from 0.5 mm to 1.5 mm. Since armature and field systems are composed of materials that have high permeability, most of the m.m.f. of field coils is required to set up flux in the air gap. By reducing the length of air gap, we can reduce the size of field coils (i.e. number of turns).

(ii) Armature core

The armature core is keyed to the machine shaft and rotates between the field poles. It consists of slotted soft-iron laminations (about 0.4 to 0.6 mm thick) that are stacked to form a cylindrical core. The laminations are individually coated with a thin insulating film so that they do not come in electrical contact with each other. The purpose of laminating the core is to reduce the eddy current loss. The laminations are slotted to accommodate and provide mechanical security to the armature winding and to give shorter air gap for the flux to cross between the pole face and the armature “teeth”.

(iii) Armature winding

The slots of the armature core hold insulated conductors that are connected in a suitable manner. This is known as armature winding. This is the winding in which “working” e.m.f. is induced. The armature conductors are connected in series-parallel; the conductors being connected in series so as to increase the voltage and in parallel paths so as to increase the current. The armature winding of a d.c. machine is a closed-circuit winding; the conductors being connected in a symmetrical manner forming a closed loop or series of closed loops.

Armature

• The armature windings start and finish at a point on the armature called the commutator.

(iv) Commutator

A commutator is a mechanical rectifier which converts the alternating voltage generated in the armature winding into direct voltage across the brushes. The commutator is made of copper segments insulated from each other by mica sheets and mounted on the shaft of the machine. The armature conductors are soldered to the commutator segments in a suitable manner to give rise to the armature winding. Depending upon the manner in which the armature conductors are connected to the commutator segments, there are two types of armature winding in a d.c. machine viz., (a) lap winding (b) wave winding. Great care is taken in building the commutator because any eccentricity will cause the brushes to bounce, producing unacceptable sparking. The sparks may bum the brushes and overheat and carbonise the commutator.

(v) Brushes

The purpose of brushes is to ensure electrical connections between the rotating commutator and stationary external load circuit. The brushes are made of carbon and rest on the commutator. The brush pressure is adjusted by means of adjustable springs. If the brush pressure is very large, the friction produces heating of the commutator and the brushes. On the other hand, if it is too weak, the imperfect contact with the commutator may produce sparking. Multipole machines have as many brushes as they have poles. For example, a 4- pole machine has 4 brushes. As we go round the commutator, the successive brushes have positive and negative polarities. Brushes having the same polarity are connected together so that we have two terminals viz., the +ve terminal and the -ve terminal.