1.1 Basic definition

Resistant Body: A body is said to be resistant if it is capable of transmitting the required force with negligible deformation. Bodies which do not suffer appreciable distortion or change in physical form due to forces acting on them are resistant bodies. Apart from rigid bodies, there are some semi-rigid bodies which are normally flexible, but under certain loading conditions act as rigid bodies for the limited purposes. e.g. belt, fluids in hydraulic press, springs.

Kinematic Link or Element: A kinematic link is a resistant body or an assembly of resistant bodies which go to make a part or parts of a machine connecting other parts which have motion relative to it. Each link or element may consist of several parts which are manufactured as a separate unit.

   Fig-1.1 Kinematic Link

1.2 Types of Link

• Binary Link
• Ternary Link
• Quarternary

 

Fig 1.2 - Types of Link

 

1.3 Classification of link:

         (i) Rigid links: which does not undergo any deformation while transmitting motion e. g. crank, connecting rod

Fig-1.3.a Rigid Link

(ii) Flexible links: Which while transmitting motion is partly deformed e.g. belt, ropes, chain, springs etc.

Fig-1.3.b Flexible Link

(iii) Fluid links: Which deformed by having fluid in a closed vessel and the motion is transmitted through the fluid by pressure

Fig-1.3.c Fluid Link

1.4 Difference between machine and structure

Machine	Structure
The parts of machine move relative to one another	The members of a structure do not move relative to each other
Machine converts the available enery in to some useful work	In a structure no energgy is converted inot useful work
Links of a machine may transmit both power and motion	Members off a structure can transmit forcces only
Machine can have more than one mechanism	It does not have any mechanism

1.5 Mechanism and machines

Mechanism: When one element or link of a kinematic chain is fixed, the result is a mechanism. If a different link of the same chain is made the fixed link, the result is a different mechanism. The mechanisms are of the following types:

• Simple Mechanism: A mechanism which has four links.
• Compound Mechanism: A mechanism which has more than four links.
• Complex Mechanism: It is formed by the inclusion of ternary or higher order link to a simple mechanism.
• Planar Mechanism: When all the links of the mechanism lie in the same plane.
• Spatial Mechanism: When the links of the mechanism lie in different planes

Machine: When a mechanism is required to transmit power or to do some particular kind of work, the various elements or links have to be designed so as to carry with safety the forces to which they are subjected. The arrangement then becomes a machine.

 1.5.1 Difference between Machine and Mechanism

Mechanism	Machine
Mechanism is a combination of various links which are capable of having relative motion with respect to one another.	Machine is the combination of number of such mechanism used to carry out a particular task
It is related to the motion only	It is related to the energy

1.6 Kinematic pair

Two links or elements of a machine when in contact with each other in such a way that their relative motion is completely constrained forms a kinematic pair. It may be classified according to the following:

• Type of relative motion
• Type of contact
• Type of mechanical constraint

According to relative motion

 

Fig-1.4.a Kinematic pair as per relative motion

According to type of contact

1. Lower pair: The two elements have surface contact and when relative motion takes place, the surface of one slide over the surface of the other element. Eg. Nut turning on a screw 
2. Higher pair: They have line or point contact and the pair must be forced closed in order to provide completely constrained motion. Eg. Cam and follower, ball and roller bearing. 

Fig-1.4.b Kinematic pair as per type of contact

 

According to nature of Mechanical constrained

1. Closed Pair: When the elements of a pair are held together mechanically, it is known as closed pair. All lower pair and some of higher pairs are closed pairs
2. Unclosed Pair: When two links of a pair are in contact either due to force of gravity or some spring action, they constitute an unclosed pair.

Fig-1.4.c Kinematic pair as per type of closure

1.7 Types of Constrained Motion:

• Completely constrained motion: When the motion between pair takes place in a definite direction irrespective of the direction of force applied, the motion is said to be a completely constrained motion. e.g. square bar in a square hole or shaft with collars at each end of a circular hole.

Fig 1.5.a Completely constrained motion

• Successfully Constrained Motion: When the constrained motion between a pair is not completed by itself but by some other means, it is said to be successfully constrained motion.

Fig 1.5.b Successfully constrained motion

• Incompletely Constrained Motion: When the motion between pair can take place in more than one direction, it is said to be incompletely constrained motion. e.g. Circular shaft in a circular hole.

Fig 1.5.c Incompletely constrained motion

1.8 Kinematic chain

A kinematic chain is a combination of kinematic pairs in which each element or links form a part of two pairs and in which the relative motion is completely constrained.

The general equation is

$l= 2p-4 \quad \text{or } \quad l=\frac{2}{3}(j+2)$

When a number of links are connected in space such that the relative motion of any point on a link with respect to any point on the other link follows a law, the chain is called a kinematic chain. For example in the figure below when one of the link of a 3 link chain is fixed and the second link is rotated the whole linkage breaks, incase when the number of links are four and we fix one link and rotate any other link, we see a definite motion of other links whereas, when the number of links are five and we fix one link and rotate any other link we find that there are infinte number of ways in which the other links can move.Thus we can call a three link chain as a structure, a four link chain as a mechanism (when one of the link is fixed) and the  motion from the 5 link chain is not useful as it is not constrained.

1.9 Types of Joints

(i) Binary Joint (J): If two Links are joined at the same connection

(ii) Ternary Joint(T): If three links are joined by same connection.

J=T(n-1), where J = no. of binary joint, T = no. of ternary joints, n = no. of links in a joint. For any ternary joint J = 1(3-1) =2, Therefore one ternary joint is equivalent to 2 binary joints.

(iii) Quaternary Joint (Q): If four links are joined
J = Q(n-1) where Q = no. of quaternary joint, n = no. of links in a joint. For any quarternary joint J = 1(4-1) = 3, Therefore one quaternary joint is equivalent to 3 binary joints.

If "n" numbers of links are connected at a joint it is equivalent to (n-1) binary joints.

1.10 Degrees of freedom (DOF)

DOF also called the mobility of the device is one of the first concerns in either design or the analysis of the mechanism. The mobility of a mechanism is the number of input parameters which must be controlled independently in order to bring the device into a particular position. The mobility of the mechanism can be determined from the number of links and the no of joints which it includes.

An unconstrained rigid body moving in space can describe the following independent motions

• Translational motion along any three mutually perpendicular axes x, y and z.
• Rotational motions about these axes

Thus a rigid body possesses six degrees of freedom.

1.11 Grubler's condition for a mechanism

Let n be the no. of links in a mechanism out of which, one is fixed and let j be the number of joints. Now, as the (n-1) links move in a plane in the absence of any connection, each has a 3 DOF, 2 co-ordinates are required to specify the location of any reference point in a link and 1 to specify the orientation of the link. When the links are connected, there cannot be any relative translation between them and only one co-ordinate is necessary to specify their relative orientation. Thus two translational DOF are lost and only one rotational DOF is left.

Fig 1.6 Degree of Freedom

So, the no. of DOF is

$F=3(n-1)-2j$

This equation is known as Kutzbach equation. Most mechanisms are constrained i.e. F = 1. Therefore the above relation becomes

$1=3(n-1)-2j => 3n-2j-4 =0$

This is called Grubler ‘s Criterion for planar mechanism. Thus Grubler’s criterion is applied to a mechanism with single DOF. The planar mechanism satisfying the condition cannot have odd number of links. The simplest possible mechanism of this type is a four bar mechanism and a slider crank mechanism.

1.12 Four Bar Chain and Grashoff’s Law

A very important consideration when designing a mechanism to be driven by a motor, is to ensure that the input crank can make a complete revolution. Mechanism in which no link makes a complete revolution would not be useful in such application.

Grashof’s law states that for planar four bar linkage, the sum of the shortest and the longest link lengths cannot be greater than the sum of the remaining two link lengths if there is to be continuous relative motion between two members.

Fig 1.7.a Four Bar Chain

Let $l_1,l_2,l_3,l_4$ be the length of the link where 2 and 3 represents the longest and shortest link then if the inequality $l_2+l_3 <= l_1+l_4$ is not satisfied no link will make a complete revolution relative to other.

Test for Four Bar mechanism by Grashof’s law

1.7.b Test of Four bar chain by Grashof's law

1.13 Inversions

 If the number of links in a kinematic chain is $L$, then in general,$L$  different mechanisms may be obtained by fixing each of the links in turn. The mechanism obtained in this way may be very diferent in appearance and the purposes for which they are used. Each mechanism is termed as inversion of the original kinematic chain. Thus for a four bar chain depending on different type of kinematic pairs, it can be classified into three categories :

1. 4R – Kinematic chain which has all four kinematics pairs as revolute pairs
2. 3R-1P - Kinematic chain which has three revolute pairs and one prismatic pair. This is also called as single slider crank chain.
3. 2R-2P – Kinematic chain which has two revolute pairs and two prismatic pairs. This is also called as double slider crank chain.

 1.13.1 Inversions of 4R- Kinematic chain

Kinematically speaking, all four inversions of 4R-kinematic chain are identical. However, by suitably altering the proportions of the length of the link several mechanisms are obtained. Thus the different inversion that can be obtained from the 4R-kinematic chain are as below:                      

• Crank-lever mechanism or Crank-rocker mechanism
• Double-lever mechanism or Rocker-Rocker mechanism
• Double Crank mechanism 
  • Parallel crank mechanism
  • Drag link mechanism

1. Crank-lever mechanism or Crank-rocker mechanism

Crank-and-Rocker mechanism (Crank-Lever mechanism): This mechanism is shown in Figure below. Link 2 is the crank and link 4 is the rocker or lever.

   

              Fig 1.8.a Crank-Lever Mechanism            

Link proportions for this case are:

$L_1+L_2 <L_3+L_4$, and  $L_2+L_3<L_1+L_4$

Double-lever mechanism or Rocker-Rocker mechanism

 In this mechanism, both the links 2 and 4 can only oscillate.

Fig 1.8.b Double-Lever Mechanism

Link $O_2A$ oscillates between positions $O_2A_1$ and $O_2A_2$ whereas $O_4B$ oscillates between positions $O_4B_1$ and $O_4B_2$. Position $O_4B_2$ is obtained when $O_2A$ and AB are along straight line and position $O_2A_1$is obtained when AB and $O_4B$ are along straight line. This mechanism must satisfy the following relations.

$L_3+L_4<L_1+L_2$ or $L_2+L_3<L_1+L_4$

It may be observed that link AB has shorter length as compared to other links. If links 2 and 4 are of equal lengths and $L_1>L_3$, this mechanism forms automobile steering gear.

Double Crank mechanism

     The links 2 and 4 of the double crank mechanism makes complete revolutions There are two forms of this mechanism

Parallel crank mechanism

In this mechanism the length of the links 2 and 4 are same. It is a four-bar mechanism having cranks of equal length and a coupler with length equal to that of the fixed link (frame).

         

                             Fig 1.8.c Locomotive Coupling Rod                  

 A familiar example is coupling of the locomotive wheels where wheels act as cranks of equal length and length of the coupling rod is equal to center distance between the two coupled wheels.

Drag Link Mechanism

In this case, the ground link is the shortest one and the coupler should be longer than that. Both the input and output links perform complete 360◦ revolutions.

Fig 1.8.d Drag Link Mechanism

 The proportions of this mechanism are

$L_3>L_1; L_4>L_2;L_3>(L_1+L_4-L_2),and , L_3<(L_2+L_4-L_1)$

Length of link 1 is smaller as compared to other links.

 1.13.2 Inversions of 3R-1P kinematic chain:

The slider crank chain consists of three turning pairs and one sliding pair. Different inversions are

(a) Reciprocating Engine Mechanism, reciprocating compressor:

Fig 1.9.a Single Slider Crank Chain

(b) Oscillating Cylinder Engine Mechanism:

It is derived from the slider-crank chain by fixing the link CP which in the reciprocating engine mechanism forms the connecting rod. As the crank OC revolves about the axis through C, the slotted link OQ slides over the block which is pivoted to the fixed link at P.

Fig 1.9.b Oscillating Cylinder Engine

(c) Whitworth quick-return Mechanism:   Link 3 is the crank here. Slider 4 drives link 1. Link 1 drives a cutting tool through the connecting rod. And link 2 is Fixed. The forward stroke starts with link 3 in position AQ and ends at AP through AS. The return motion is faster from AP through AR to AQ. These linkages are most useful in saving time, since the return stroke, which is an idle stroke, is faster than a useful forward stroke when metal is removed. Since the crank link MN rotates at uniform angular velocity therefore time taken during the cutting stroke (or forward stroke) is more than the time taken during the return stroke. In other words, the mean speed of the ram during cutting stroke is less than the mean speed during the return stroke. The ratio between the time taken during the cutting and return strokes is given by

${Time of cutting stroke \over Time of return stroke} = { \alpha\over \beta}$

Fig 1.9.c Quick Return Motion Mechanism

Another example is rotary engine.

Fig 1.9.d Rotary Engine

(d) Pendulum pump or Bull engine Mechanism: It is obtained by fixing the die block. Here the link OQ to reciprocate along a vertical straight line. At the same time OC will rotate and the line CP will oscillate about the pin P attached to the fixed block.

Fig 1.9.e Pendulum Pump

(e) Crank and slotted lever quick return motion mechanism:

The motion of link 4 in crank and slotted lever quick return motion mechanism can be taped through link 5 and can be transferred to ram. O₂A₁ and O₂A₂ are two positions of crank when link 4 will be tangential to the crank circle and corresponding to which ram will have extreme positions. When crank travels from position O₂A₁ and O₂A₂ forward stroke takes place. When crank moves further from position O₂A₂ to O₂A₁ return stroke takes place. Therefore, for constant angular velocity for crank ‘ω'.

$Quick return ratio ={ Time of forward stroke\over time of return stroke} = {2\pi - \theta\over\theta}$

 

Fig 1.9.f Crank and slotted lever mechanism

 

 1.13.3 Inversion of 2R-2P kinematic chain

This four bar kinematic chain has two revolute or turning pairs . the inversion of this type of kinematic chains are as below:

Scotch Yoke : By fixing link 1 the mechanism we get is called Scotch Yoke. The link 1 is a slider similar to link 3. Link 2 works as a crank. Link 4 is a slotted link. When link 2 rotates, link 4 has simple harmonic motion for angle ‘θ’ of link 2, the displacement of link 4 is given by $x = OA* Cos\theta$.

Fig 1.10.a Skotch  Yoke Mechanism

Fixing link 2 we get a mechanism called Oldham’s coupling. This coupling is used to connect two shafts which have eccentricity ‘ε’. The axes of the two shafts are parallel but displaced by distance “ε” . The link 4 slides in the two slots provided in links 3 and 1. The centre of this link will move on a circle with diameter equal to eccentricity.

Oldham’s Coupling:    An oldham's coupling is used for connecting two parallel shafts whose axes are at a small distance apart. The shafts are coupled in such a way that if one shaft rotates, the other shaft also rotates at the same speed. The flanges have diametrical slots cut in their inner faces. The intermediate piece (link 4) which is a circular disc, have two tongues (i.e. diametrical projections) T₁ and T₂ on each face at right angles to each other. The tongues on the link 4 closely fit into the slots in the two flanges (link 1 and link 3).

Fig 1.10.b Oldham Coupling

The link 4 can slide or reciprocate in the slots in the flanges. When the driving shaft A is rotated, the flange C (link 1) causes the intermediate piece (link 4) to rotate at the same angle through which the flange has rotated, and it further rotates the flange D (link 3) at the same angle and thus the shaft B rotates. Hence links 1, 3 and 4 have the same angular velocity at every instant. A little consideration will show, that there is a sliding motion between the link 4 and each of the other links 1 and 3. If the distance between the axes of the shafts is constant, the center of intermediate piece will describe a circle of radius equal to the distance between the axes of the two shafts. Therefore, the maximum sliding speed of each tongue along its slot is equal to the peripheral velocity of the centre of the disc along its circular path.

Let, $\omega =$anular velocity of each shaft in rad/s, and $r=$distancce between the axes of shaft in meters, then the maximum slidin speed of each togue is $V=\omega*r$.

Elliptical Trammels:  It is an instrument used for drawing ellipses. This inversion is obtained by fixing the slotted plate (link 4), as shown in Figure. The fixed plate or link 4 has two straight grooves cut in it, at right angles to each other. The link 1 and link 3, are known as sliders and form sliding pairs with link 4. The link A B (link 2) is a bar which forms turning pair with links 1 and 3.

Fig 1.10.c Elliptical Trammel

When the links 1 and 3 slide along their respective grooves, any point on the link 2 such as P traces out an ellipse on the surface of link 4, as shown in Figure. A little consideration will show that AP and BP are the semi-major axis and semi-minor axis of the ellipse respectively. This can be proved as follows: Let us take O X and O Y as horizontal and vertical axes and let the link B A is inclined at an angle θ with the horizontal, as shown in Figure. Now the co-ordinates of the point P on the link BA will be

$x = PQ =AP*Cos\theta; y = PR = BP*Sin\theta$, or

$({x\over AP})^2 +( {y\over BP})^2=Sin^2\theta +Cos^2\theta$, or 

$({x\over AP})^2 +( {y\over BP})^2= 1$

This is the equation of an ellipse. Hence the path traced by point P is an ellipse whose semi major axis is AP and semi-minor axis is BP. 

1.14 Conclusion

We saw that for a quadric cycle chain various useful mechanism can be generated by changing the fixed link and by changing the type of joint between them. Similarly one can  explore multi-link chains and make a systematic study to identify possible new mechanisms. However, in the present topic we will limit the study to quadric cycle chain only.