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Wednesday, August 9, 2017

Kinematics of Machine Notes


A mechanism is a combination of rigid or restraining bodies so shaped and connected that they move upon each other with a definite relative motion. A simple example of this is the slider crank mechanism used in an internal combustion or reciprocating air compressor.


A machine is a mechanism or a collection of mechanisms which transmits force from the source of power to the resistance to be overcome,and thus perform a mechanical work.
Kinematic Pairs

A mechanism has been defined as a combination so connected that each moves with respect to each other.A clue to the behaviour lies in in the nature of connections,known as kinetic pairs.
The degree of freedom of a kinetic pair is given by the number independent coordinates required to completely specify the relative movement.

Lower Pairs

A pair is said to be a lower pair when the connection between two elementsis through the area of contact.Its 6 types are :
Double Slider Crank mechanism

It has four binary links, two revolute pairs, two sliding pairs.Its various types are :

Scotch Yoke mechanism:

Here the constant rotation of the crank produces harmonic translation of the yoke.Its four binary links are :
  1. Fixed Link 
  2. Crank 
  3. Sliding Block 
  4. Yoke 
The four kinematic pairs are :
  • revolute pair (between 1 & 2) 
  • revolute pair (between 2 & 3) 
  • prismatic pair (between 3 & 4) 
  • prismatic pair (between 4 & 1) 
Oldhams Coupling:

It is used for transmitting anbgular velocity between two parallel but eccentric shafts

Elliptical Trammel:

Here link 4 is fixed. Any point on the link 2 describes an ellipse as it moves.The mid-point of the link 2 will obiviously describe a circle.
Very often a mechanism with higher pair can be replaced by an equivalent mechanism with lower pair.This equivalence is valid for studying only the instantaneous characteristics.The equivalent lower-pairmechanism facilitates analysis as a certain amount of sliding takes place between connecting links in a higher-pair mechanism.
Another example of an equivalent lower-pair mechanism for a cam-follower system is shown.The sliding block is the additional link and thebhigher pair is replaced by two lower pairs, one revoluteand other prismatic. C is the center of curvature of the cam surface at the point of contact between the cam and the follower


Let n be the no. of links in a mechanism out of which, one is fixed, and let j be the no. of simple hinges(ie, those connect two links.) Now, as the (n-1) links move in a plane, in the absence of any connections, each has 3 degree of freedom; 2 coordinates are required to specify the location of any reference point on the link and 1 to specify the orientation of the link. Once we connect the linmks there cannot be anyrelative translation betweenthem and only one coordinate is necessary to specify their relative orientation.Thus, 2 degrees of freedom (translation) are lost, and only one degree of freedom (rotational) is left. So, no. of degrees of freedom is:
Most mechanisms are constrained, ie F=1. Therefore the above relation becomes,
,this is called Grubler's Criterion.
Failure of Grubler's criterion
A higher pair has 2 degrees of freedom .Following the same argument as before, The degrees of freedom of a mechanism having higher pairs can be written as,
Often some mechanisms have a redundant degree of freedom. If a link can move without causing any movement in the rest of the mechanism, then the link is said to have a redundant degree of freedom.
Example of redundant degree of freedom 
The objective of kinematic analysis is to determine the kinematic quantities such as displacements, velocities, and accelerations of the elements of a mechanism when the input motion is given. It establishes the relationship between the motions of various components of the linkage


When the kinematic dimensions and the configurations of the input link of a mechanism are prescribed, the configurations of all the other links are determined by displacement analysis.
  1. Graphical Method
  2. Analytical Method


In a graphical method of displacement analysis, the mechanism is drawn to a convenient scale and the desired unknown quantities are determined through suitable geometrical constructions and calculations.
  1. The configurations of a rigid body in plane motion are completely defined by the locations of any two points on it.
  2. Two intersecting circles have two points of intersection and one has to be careful, when necessary, to choose the correct point for the purpose in hand.
  3. The use of tracing paper, as an overlay, is very convenient and very often provides an unambiguous and quick solution.
  4. The graphical method fails if no closed loop with four links exists in the mechanism.


Figure 1
An analytical method of displacement analysis, is preferred whenever 
  1. high level of accuracy is required
  2. a large number of configurations have to be solved
  3. The graphical method fails.
In this method every link is represented by two dimensional are represented by two dimensional vectors expressed through complex notation. Considering each closed loop in the mechanism, a vector equation is established. Separating the real and imaginary parts , sufficient number of nonlinear algebraic equations are obtained to solve for the unknown quantities.
Let us consider a 4R linkage of given link lengths, viz., i=1, 2, 3, and 4. The configuration of the input link (2) is also prescribed by the angle θ2, and we have to determine the configurations of the other two links, namely, the coupler and the follower, expressed by the angles θ3 and θ4.


Figure 3
For a 4R linkage, the transmission angle ( μ ) is defined as the acute angle between the coupler (AB) and the follower ).4B), as indicated in Fig. 2.11. If ( - ABO4) is acute (Fig.2.11), then μ =- ABO4. On the other hand, if - ABO4 is obtuse, then μ =Π-- ABO4. As explained in this figure, if μ = Π /2, then the entire coupler force is utilized to drive the follower. For good transmission quality, the minimum value of μ(μmin)>300. For a crank-rocker mechanism, the minimum value of μ occurs when the crank becomes collinear with the frame, i.e.,inlinesymbol. If the swing angle (inlinesymbol) of the rocker is increased maintaining the same quick-return ratio, then the maximum possible value of μ min decreases. If the forward and return strokes of the rocker take equal time, then (μmin)max is restricted to inlinesymbol (see Problem 2.6). Therefore, such a crank rocker will have a poor transmission quality if inlinesymbol.


The concept of velocity and acceleration images is used extensively in the kinematic analysis of mechanisms having ternary, quaternary, and higher-order links. If the velocities and accelerations of any two points on a link are known, then, with the help of images the velocity and acceleration of any other point on the link can be easily determined. An example is illustrated below:
A rigid link BCDE having four hinges is sown in figure. Let the angular velocity and acceleration of this be ω and α. The absolute velocity vectors of the E, B, C and D are shown in the figure as VE, VB, VC, and VD respectively. The velocity difference vectors are
And their magnitudes are, respectively,
Hence the velocity diagram bcde is a scale drawing of the link BCDE. The figure bcde is called the velocity image of the link BCDE. The velocity image is rotated through 90o in the direction ω, as all the velocity difference vectors are perpendicular to the corresponding lined. The scale of the image is determined by and therefore the scale will be different for each link of a mechanism. The letters identifying the end points of the image are in the same sequence as that in the link diagram BCDE. The absolute velocity any point X on the link is obtained by joining the image of X(x) with the pole of the velocity diagram o.


  1. Instantaneous Centre Method
  2. Relative Velocity Method


  1. First determine the number of instantaneous centers (N) by using the relation
  2. Make a list of all the instantaneous centers in the mechanism.
  3. Locate the fixed and permanent instantaneous centers by inspection.
  4. Locate the remaining neither fixed nor permanent instantaneous centers by Kennedy’s theorem. This can be done by circle diagram. Mark points on a circle equal to the number of links in a mechanism.
  5. Join the points by solid lines to show that these circles are already found. In the lines indicate the instantaneous centers corresponding to those particular two points.
  6. In order to find the remaining instantaneous centers, join two such points that the line joining them forms two adjacent triangles in the circle diagram. The line which is responsible for completing two triangles should be a common side to the two triangles. 


The relative velocity method is based upon the velocity of the various points of the link.
Consider two points A and B on a link. Let the absolute velocity of the point A i.e. VA is known in magnitude and direction and the absolute velocity of the point B i.e. VB is known in direction only. Then the velocity of B may be determined by drawing the velocity diagram as shown.
Figure 5
  1. Take some convenient point o, known as the pole.
  2. Through o, draw oa parallel and equal to VA, to some convenient scale.
  3. Through a, draw a line perpendicular to AB. This line will represent the velocity of B with respect to A, i.e.
  4. Through o, draw a line parallel to VB intersecting the line of VBA at b.
  5. Measure ob, which gives the required velocity of point B to the scale.


Cams come under higher pair mechanisms. As we already know that in higher pair the contact between the two elements is either point or line contact, instead of area in the case of lower pairs.
Figure 1
In CAMs, the driving member is called the cam and the driven member is referred to as the follower. CAM is used to impart desired motion to the follower by direct contact. Generally the CAM is a rotating or reciprocating element, where as the follower may de rotating, reciprocating or oscillating element. Using CAMs we can generate complex, coordinate movements that are very difficult with other mechanisms. And also CAM mechanisms are relatively compact and easy it design. Cams are widely used in automatic machines, internal combustion engines, machine tools, printing control mechanisms and so on. Along with cam and follower one frame also will be there with will supports the cam and guides the follower.


A follower can be classified in three ways
  1. According to the motion of the follower.
  2. According to the nature of contact.
  3. According to the path of motion of the follower.


According to the motion of the follower
  1. Reciprocating or Translating follower
    : When the follower reciprocates in guides as the can rotates uniformly, it is known as reciprocating or translating follower.
  2. Oscillating or Rotating follower
    : When the uniform rotary motion of the cam is converted into predetermined oscillatory motion of the follower, it is called oscillating or rotating follower.


According to the nature of contact:
  1. The Knife-Edge follower
    : When contacting end of the follower has a sharp knife edge, it is called a knife edge follower. This cam follower mechanism is rarely used because of excessive wear due to small area of contact. In this follower a considerable thrust exists between the follower and guide.
  2. The Flat-Face follower
    : When contacting end of the follower is perfectly flat faced, it is called a flat faced follower. The thrust at the bearing exerted is less as compared to other followers. The only side thrust is due to friction between the contact surfaces of the follower and the cam. The thrust can be further reduced by properly offsetting the follower from the axis of rotation of cam so that when the cam rotates, the follower also rotates about its axis. These are commonly used in automobiles.
  3. The Roller follower
    : When contacting end of the follower is a roller, it is called a roller follower. Wear rate is greatly reduced because of rolling motion between contacting surfaces. In roller followers also there is side thrust present between follower and the guide. Roller followers are commonly used where more space is available such as large stationary gas or oil engines and aircraft engines.
  4. The Spherical-Faced follower
    : When contacting end of the follower is of spherical shape, it is called a spherical faced follower. In flat faced follower’s high surface stress are produced. To minimize these stresses the follower is machined to spherical shape.
According to the path of motion of the follower:
  1. Radial follower
    : When the motion of the follower is along an axis passing through the centre of the cam, it is known as radial follower.
  2. Off-set follower
    : When the motion of the follower is along an axis away from the axis of the cam centre, it is called off-set follower.
A Cam can be classified in two ways:
  1. Radial or Disc cam
    : In radial cams, the follower reciprocates or oscillates in a direction perpendicular to the cam axis.
  2. Cylindrical cam
    : In cylindrical cams, the follower reciprocates or oscillates in a direction parallel to the cam axis. The follower rides in a groove at its cylindrical surface.


Figure 2
The various terms we will very frequently use to describe the geometry of a radial camare defined as fallows.
  1. Base Circle
    : It is the smallest circle, keeping the center at the camcenter, drawn tangential to cam profile. The base circle decides the overall size of the cam and thus is fundamental feature.
  2. Trace Point
    : It is a point on the follower, and it is used to generate the pitch curve. Its motion describing the movement of the follower. For a knife-edge follower, the trace point is at knife-edge. For a roller follower the trace point is at the roller center, and for a flat-face follower, it is a t the point of contact between the follower and the cam surface when the contact is along the base circle of the cam. It should be note that the trace point is not necessarily the point of contact for all other positions of the cam
[The figure on the right shows a radial cam with radial-translating roller follower.]


The various terms we will very frequently use to describe the geometry of a radial cam are defined as fallows.
  1. Pitch Curve
    : It is the curve drawn by the trace point assuming that the cam is fixed, and the trace point of the follower rotates around the cam, i.e. if we hold the cam fixed and rotate the follower in a direction opposite to that of the cam, then the curve generated by the locus of the trace point is called pitch curve.
    For a knife-edge follower, the pitch curve and the cam profile are same where as for a roller follower they are separated by the radius of the roller.
  2. Pressure Angle
    : It is the measure of steepness of the cam profile. The angle between the direction of the follower movement and the normal to the pitch curve at any point is called pressure angle. Pressure angle varies from maximum to minimum during complete rotation. Higher the pressure angle higher is side thrust and higher the chances of jamming the translating follower in its guide ways. The pressure angle should be as small as possible within the limits of design. The pressure angle should be less than 450 for low speed cam mechanisms with oscillating followers, whereas it should not exceed 300 in case of cams with translating followers. The pressure angle can be reduced by increasing the cam size or by adjusting the offset.
  3. Pitch Point
    : The point corresponds to the point of maximum pressure angle is called pitch point, and a circle drawn with its centre at the cam centre, to pass through the pitch point, is known as the pitch circle.
  4. Prime Circle
    : The prime circle is the smallest circle that can be drawn so as to be tangential to the pitch curve, with its centre at the cam centre. For a roller follower, the radius of the prime circle will be equal to the radius of the base circle plus that of the roller where as for knife-edge follower the prime circle will coincides with the base circle.


A gear is a toothed element commonly used for transmitting rotary motion from one shaft to another if the centre to center distance is relatively less. It is a higher-pair mechanism. The power transmitting capacity for gears is high when compared to the other power transmitting elements. Gears are used widely in our day to day life, automobile industry to meet different speed requirements for the same power.


When two parallel shafts are connected to each other by a pair of toothed wheels (gears), the number of teeth from each gear passing through the engagement zone in a given period of time is equal. If this number be N for a period of 1 second and the number of teeth on the two gears 1 an 2 be N1 and N2, respectively, then gars 1 and 2 make (N / N1) and (N / N2) revolutions, respectively. Hence, the average angular velocities of these two gears can be written as

The ratio of these average angular velocities in
However, the ratio of the instantaneous angular speeds in general is not a constant and takes the form
Where P(t) is a periodic function of time with zero average. This relation suggests that, for a given constant driving speed, the rotation of the driven gear is not uniform rotation. This results in harmful effects, and proper gearing action requires the tooth profile to be so chosen that, for a constant droving speed, the driven shat also rotates uniformly. Thus, proper gearing action implies
To attain the above condition the gear pair must satisfy the condition “The common normal AB to the involutes at the point of contact Q (called the line of action) meets the line of centres O1O2 at the (fixed) pitch point P. this is the condition required for maintaining a constant angular velocity ratio and is known as the fundamental law of gearing


Figure 3
Here we are going in to the quantitative analysis of some important aspects relating to involute gears.

Contact Ratio
To transit rotational motion continuously, there must be at least one pair of teeth in contact at all times. In the actual case, certain amount of overlap exists between the actions of two consecutive pairs of teeth. The term contact ratio is used to provide a quantitative measure of the amount of this overlap. Figure shows a pair of teeth at the beginning and end of contact. So, at the point E ( on the line of action ,i.e., the common tangent to the base circles of the two gears), the contact begins and at F the contact ends. The points S and T are the points on a tooth of gear2 (on its base circle) at the start and end of contact, respectively. Since E and F are two points on two involutes with the same base circle, the length EF (called the length of action) will be equal to ST. now, the ratio (ST/ base pitch ) is a measure of the contact ratiomc. So,
The base pitch of gear 2 can be expressed as inlinesymbol, where rb2 is the base circle radius of gear 2 and N2 is the number of teeth on this gear. The above figure shows the essential dimensions of the gears in contact. From those figure, the length of action.
If ro1 and roOfc are the outer circle (same as the addendum circle ) radii, then this relation yields
Where C= rp1 + rp2 ( = centre distance) and rb1 is the base circle radius of gear 1, rp1 and rpfc being the pitch circle radii with α as the pressure angle. So, the expression for contact ratio becomes
Normally, the contact ratio is not a whole number. If the ratio is 1.3 it means that there alternately, one pair and two pairs of teeth in contact and the time average is 1.3. In practice, a value of 1.4 is recommended for m c for smooth and good performance. It cam de seen from figure (3) that by increasing the addendum the 
length of the path of contact EF can de increased , resulting in a larger value of the contact ratio. But increasing the addendum can lead to some problems.