Introduction to mechanism and kinematics

Objectives

Up on completion of this chapter, the student will be able to

1. Explain the need for kinematic analysis of mechanism.2. Define the basic components that comprise a mechanism.3. Draw the kinematic diagram from a view of a complex mechanism.4. Compute the number of degrees of freedom of a mechanism.5. Identify a four bar mechanism and classify it according to its possible motion.6. Identify a slider crank mechanism.

Mechanism and machinesIntroductionThe subject theory of machine deals with the study of motion and forces in machinery devices that provide a wide variety of functions. The subject matter treated inhere includes synthesis and analysis of machinery.The study of a mechanism involves its analysis as well as synthesisAnalysis is the study of motions and forces concerning different parts of an existing mechanism, where as synthesis involves the design of its different partsIn a reciprocating engine, the displacement of the piston depends up on the length of the connecting rod and the crank. It is independent of the bearing strength of the parts or whether they are able to withstand the forces or not. Thus, for the study of motion, it is immaterial if a machine part is made of mild steel, cast iron or wood. It is not necessary to know the actual shape and area of cross section of the part. The study of mechanisms, therefore, can be divided in to the following disciplines.KinematicsIt deals with the relative motions of different parts of a mechanism without taking in to consideration forces producing the motions.It also deals with the way things move. Formally defined, it is the study of the geometry of motion.Kinematic analysis involves

- Determination of position- Determination of displacement, rotation, speed, velocity and acceleration.

DynamicsIt involves the calculations of forces impressed up on different parts of a mechanism. The forces can be either static or dynamic.Another major task in designing machinery is determining the effect of the forces acting in the machineTerminologies (basic definition)Mechanism: - is the mechanical portion of a machine that has the function of transferring motion and

from a power source to an output.If a number of bodies are assembled in such a way that the motion of one causes constrained and predictable motion to the others, it is known as a mechanism. A mechanism transmits and modifies a motion.Machines: - are devices used to alter transmit and direct forces to accomplish a specific objective.

A machine is a mechanism or combination of mechanisms which, Apart from imparting definite motions to the parts, also transmits and modifies the available mechanical energy in to some kind of desired work.

A machine may also be defined to be a combination of resistant bodies whose relative motions are completely constrained, by means of which the natural energies at our disposal may be transformed in to any special from of works.

In the first place, then, a machine is a combination of bodies- a single body cannot constitute a machine. In each of what are often called the ‘simple machine’ for example the lever, wheel and axle etc… there are at least two bodies, in some more than two. The mare bar which we call a lever does not in itself constitute a machine either simple or otherwise.

n. 1.

a. A device consisting of fixed and moving parts that modifies mechanical energy and transmits it in a more useful form.

b. A simple device, such as a lever, a pulley, or an inclined plane, that alters the magnitude or direction, or both, of an applied force; a simple machine.

2. A system or device for doing work, as an automobile or a jackhammer, together with its power source and auxiliary equipment.

3. A system or device, such as a computer, that performs or assists in the performance of a human task: The machine is down.

4. An intricate natural system or organism, such as the human body.

5. A person who acts in a rigid, mechanical, or unconscious manner.

6. An organized group of people whose members are or appear to be under the control of one or more leaders: a political machine.

7.

a. A device used to produce a stage effect, especially a mechanical means of lowering an actor onto the stage.

b. A literary device used to produce an effect, especially the introduction of a supernatural being to resolve a plot.

8. An answering machine: Leave a message on my machine if I'm not home.

adj.

Of, relating to, or felt to resemble a machine: machine repairs; machine politics.

v., -chined, -chin·ing, -chines.

v.tr.

To cut, shape, or finish by machine.

v.intr.

To be cut, shaped, or finished by machine: This metal machines easily.

Planar mechanism: - a mechanism its motion is limited to two dimensional spaces.Plane motion when a body moves so that any one selection parallel to the first moves also in its own plane, and the motion of the body is said to be ‘co-plane’ ‘coplanar’ or we may call it simply ‘plane’.

Mechanism terminology

Linkage: - is a mechanism where all parts are connected together to form a closed chainLinks: - are individual parts of a mechanism. They are considered rigid body (bodies) and are connected with other links to transmit motion and forces.Link also defined as resistant body or a group of resistant bodies with rigid connections preventing their relative movements.A link is also known as kinematic link or elementLink can be classified in to binary, ternary, quaternary etc, depending up on their ends on which revolute or turning pairs can be placedSimple link: - is a rigid body that contains only two joins, which connect it to other links.Complex link: - a rigid body that contains more than two joints.Frame: - it serves as the frame of reference for the motion of all other parts. The frame is typically a part that exhibits no motion.Joint: - is a movable connection between links and allows relative motion between the links.The two primary joint also called full joint, are the revolute and sliding joint

The revolute also called a pin or a hinge joint. The sliding joint is also called a piston or a prism joint.

Higher order joint: - permits for both rotation and sliding between the two links that it connects [Also called half joint]. Cam and gear connections are higher order joint.Crank: - is a simple link that is able to complete a full rotation about a fixed center.Rocker: - is a simple link that oscillates through an angle, reversing its direction at certain intervals.A rocker arm: - is a complex link, containing three joints, that is pivoted near its center.A bell crank: - is similar to a rocker arm, but is bent in the center.A point of interest: - is a point on a link where the motion is special interest. Once kinematic analysis is performed, the displacement, velocity, and accelerations of that point are determined.Rigid and resistant bodiesA body is said to be rigid if under the action of forces, it does not suffer any distortion or the distance between any two points it remain constant

Resistant bodies are those which are some semi rigid bodies which are normally flexible, but under certain loading conditions acts as rigid bodies for the limited purpose.Actuator: - is the component that drives the mechanism.Common actuator includes

Motor(electric and hydraulic) Engine, cylinder(hydraulic and pneumatic) Solenoid.

Kinematic pairKinematic pair or simply a pair is a joint of two links having relative motion between themTypes of kinematic pairsKinematic pairs can be classified according to

Nature of contact Nature of mechanical constraint Nature of relative motion

i. Kinematic pair according to nature of contacta. Lower pair: - a pair of links having surface or area contact between the members. The contact

surfaces of the two links are similar.E.g. nut turning on a screw

b. Higher pair:- when a pair has a point or a line contact between the links. The contact surfaces of the two links are dissimilarE.g. wheel rolling on a surface, ball and roller bearing

II. Kinematic pairs according to nature of mechanical constrainta. Closed pair

When the elements of a pair are held together mechanically all the lower pairs and some of the higher pairs are closed pairs.

b. Unclosed pairWhen two links of a pair are in contact either due to force of gravity or some spring action, they constitute an unclosed pairs.

iii. Kinematic pairs according to nature of relative motiona. Sliding pair

If two links have a sliding motion relative to each other, they form a sliding pairb. Turning pair

When one link has a turning or revolving motion relative to the other, they constitute a turning or revolving pair

c. Rolling pairWhen the links of a pair having a rolling motion relative to each other, they form a rolling pair.E.g. ball bearing

d. Screw pair(helical pair)If two mating links have a turning as well as sliding motion between them, they form a screw joint.

e. Spherical pairWhen one link in the form of a sphere turns inside a fixed link, it is a spherical pair.

E.g. the ball and socket joint

Kinematic diagram: - it shows the schematic diagram of a mechanism. Kinematic diagram should be drawn to scale proportional to the actual mechanism.

Simple linkComplex link

Sliding jointPin joint

Simple link with two points of interest and their schematic representation

Kinematic inversion: utilizing alternate link to serve as the fixed link is termed as kinematic inversion The relative motion of the links doesn’t change with the selection of a frame, the choice of a frame link is often not important.Inversion also defined as, if in a mechanism the link which was fixed is allowed to move and another link becomes fixed, the mechanism is said to be inverted. The inversion of a mechanism does not change the motion of its links relative to each other, but does change their absolute motion. The number of possible kinematic inversions is equal to the number of links in a mechanism.Transmission of motionMotion is transmitted from one member to another in three ways:

a. By direct contact between two members;b. Through an intermediate link or a connecting rod;c. By flexible connector such as belt or chain.

In transmission of motion one element of the mechanism must be a driver and another element must be a driven element or a follower.Kinematic chainIt is an assembly of links in which the relative motions of the links is possible and the motion of each relative to the other is definite.A redundant chain doesn’t allow any motion of a link relative to the otherLinkage, mechanism and structureA linkage is obtained if one of the links of a kinematic chain is fixed to the ground.Mobility The number of degrees of freedom of a mechanism is also called the mobilityAn important property in mechanism analysis is the number of degree of freedom of the mechanism. The degree of freedom: - is the number independent inputs required to precisely position all links of the mechanism with respect to the ground.It can also be defined as the number of actuators needed to operate the mechanism.An unconstrained rigid body moving in space can describe the following independent motion

1. Translational motion along any other mutually perpendicular axes x, y, z2. Rotational motion about three axes

Thus a rigid body possesses six degree of freedom.Degree of freedom of a pair is defined as the number of independent relative motions, both translational and rotational, a pair can haveDegree of freedom=6-number of restraintDegree of freedom of a mechanism in space can be determined as follow

n= number of links in a mechanismF= degree of freedomP1= number of pairs having one degree of freedomP2= number of pairs having two degree of freedom

In a mechanism, one link is fixed. ThereforeNumber of movable links=n-1Number of degree of freedom of (n-1) movable links=6(n-1)

Each pair having one degree of freedom imposes 5 restraints on the mechanism reducing its degree of freedom by 5P1

Each pair having two degree of freedom imposes 4 restraints on the mechanism reducing its degree of freedom by 4P2

Similarly other pairs having 3, 4, and 5 degrees of freedom reduce the degree of freedom of the mechanism. Thus

F=6(n-1)-5P1-4 P2-3 P3-2P4-1P5

For planar mechanism the degree of freedomF=3(n-1)-2P1-P2

This is known as Grubler equation Most of the linkages are expected to have one degree of freedom so that one input to any of the links, a constrained motion of other is obtained

1=3(n-1)-2P1P1=1.5*n-2

If n and P1 are whole number, the relation can be satisfied only if n is evenFor turning pair the following relations are valid

F=n-(2L+1)P1=n+ (L-1)

Where L- number of loops in a linkage

L F P11 n-3 n2 n-5 n+13 n-7 n+24 n-9 n+35 n-11 n+4

For example, if a linkage, there are 4 loops and 11 links, its degree of freedom will be 2 and number of joints 14If a linkage has 3 loops, it will require 8 links to have one degree of freedom, 9 links to have 2 degree of freedomGrubler’s equationDegree of freedom for planar mechanism joined with common joints can be calculated through Grubler’s equation

F=degree of freedom=3(n-1)-2*Jp-2Jh

Where n=total number of links in the mechanism Jp= total number of primary joints (pins or sliding joints) Jh= total number of higher joints (cam or gear joint)

The four bar linkage

Four bar linkage is the simplest and most common linkage It is a combination of four links, one being designated as frame, and connected by four pin joint.

The mobility of a four bar mechanism consists of n=4, Jp=4 Jh=0F=3(4-1)-2*4 F=1Because the four bar mechanism has one degree of freedom, it is constrained or fully operated with one driverThe pivoted link that is connected to the driver or power source called the input link. The other pivoted link that is attached to the frame is designated the output link or follower.The coupler or connecting arm “couple” the motion of the input link to the output link.Grashof’s criterionThe following nomenclature is used to describe the length of the four links

S=length of the shortest linkL=length of the longest linkP=length of one of the intermediate linkq=length of the other intermediate link

Grashof’s theorem states that a four bar mechanism has at least one revolving link if

Conversely, the three non-fixed links merely rock if

Four bar mechanism fall in to one of the five categories listed as follow

case criterion shortest link category1 s+l<p+q frame double crank2 s+l<p+q slide crank rocker3 s+l<p+q coupler double rocker4 s+l=p+q any change point5 s+l<p+q any triple rocker

1. Double crank or crank-crank

It has the shortest link of the four bar mechanism configured as the fixed link or the frame.If one of the pivoted links is rotated continuously, the other pivoted link will also rotate continuously.If double crank mechanism is also called a drag link mechanism

2. Crank rockerIt has the shortest link of the four bar mechanism configured adjacent to the frame. If this shortest link is continuously rotated the output link will oscillate between limits. Thus the shortest link is called the crank; the output link is called the rocker.

3. Double rocker The double rocker or rocker-rocker, it has the link opposite the shortest link of the four bar mechanism configured as frame. In this configuration neither link connected to the frame will be able to complete a full revolution. Thus, both input and out put links are constrained to oscillate between limits and called rocker. However, the coupler is able to complete a full revolution.

4. Change point mechanismThe change point mechanism can be positioned such that all the links become collinear. The most familiar type of change point mechanism is a parallelogram linkage. The frame and the coupler are the same length, and the two pivoting links are the same length. In that collinear configuration, the nption becomes indeterminate. The motion may remain in a parallelogram arrangement. Or cross in to anti parallelogram, or butterfly, arrangement, for this reason, the change point is called a singularity configuration.

5. Triple rockerIn a triple rocker linkage, none of its links are able to complete a full revolution. All three moving links rock

Deltoid linkageIn a deltoid linkage the equal links are adjacent to each other. When any of the shorter link is fixed, a double crank mechanism is obtained in which one revolution of the longer link causes two revolution of the other shorter link.

Slider crank mechanism Another mechanism that is commonly encountered is a slider crank. This mechanism also consists of a combination of four links, with one being designated as the frame. This mechanism, however, is connected by three pin joints and one sliding joints.

The mobility of a slider crank mechanism is represented by the following relationn=4, Jp= (3pins+1sliding) Jh=0F=3(4-1)2*4 F=1

Because the slider crank mechanism has one degree of freedom, it is constrained or fully operated with one driverIn general, the pivoted link connected to the frame is called the crank. This link is not always capable of completing a full revolution. The link that translates is called the slider. The coupler or connecting rod “couples” the motion of the crank to the slider.Simple machineA simple machine is a device that enables us to do work more easily. Simple machines help us accomplish a task faster or more conveniently

i. Machine transform energy ii. Some machines transfer energy from one place to anotheriii. Multiply forceiv. Multiply speedv. Change direction of force

Type of simple machine lever Wedge Inclines plane Pulley Wheel and axle Screw

Laws of leverFE+LE=FL+LL

Theory of simple machinei. Mechanical advantage

MA=Load/Effortii. Velocity ratio

VR=velocity of the effort/velocity of the loadVR=distance of effort/distance of the load

iii. EfficiencyEfficiency=[used work output/useful work input]*100%

If VR=MA then the ideal machineEquivalent mechanism

This can be done according to some set rules so that the new mechanisms also have the same degree of freedom and are kinematically similar

When D lies at infinity so link C moves perpendicular to link4.Spring in place of turning pairs

To find the degree of freedom of such a mechanism, the spring has to be replaced by the binary linkCam pairs In place of turning pairA cam pair has two degree of freedom. For linkage with one degree of freedom, application of grubler’s equation yields F=3(n-1)-2P1-P21=3n-3-2P1-1P1=3n-5/2This shows that to have one cam pair in a mechanism with one degree of freedom, the number of links and turning pairs should be as ……………………….

Slider crank mechanism When one of the pairs of a four bar chain is replaced by a sliding pair, it becomes a single slider crank chain or simply a slider crank chain. It is also possible to replace two sliding pairs of a four bar chain to get a double slider crank chainIn a slider may be passing through the fixed pivoted O or may be displaced. The distance e between the fixed pivot O and the straight line path of the slider is called the off-set and the chain so formed an off-set slider crank chain.Different mechanisms obtained by fixing different links of a kinematic chain are known as its INVERSION

First inversionThis inversion is obtained when link 1 is fixed and link 2 and 4 are made the crank and slider respectively Application:- reciprocating engine, reciprocating compressor

Second inversionFixing of link 2 of a slider crank chain results in the second inversion

Application: - whit worth quick return mechanism, rotary engine

Third inversionBy fixing link three the slider crank mechanism, third inversion is obtained

Application:- oscillating cylinder, crank and slotted lever mechanism

Fourth inversionIf link 4 of the slider crank mechanism is fixed, the fourth inversion is obtained

Application: - hand pump

Four bar linkageTransmission angleThe angle µ between the output link and the coupler is known as transmission angleIf link AB id the input link the force applied to the output link DC is transmitted through the coupler is when µ is 900. If links BC and DC becomes coincident, the transmission angle is zero and the mechanism would lock or jamFrom cosine law we have