Course Instructor: Dr. M. Mohsin Khan (mohsinkhan@nitsri.ac.in)
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PLANER MECHANISM
LINK
A link is defined as a single part which can be a resistant body or a combination of resistant bodies having inflexible connections and having a relative motion with respect to other parts of the machine. A link is also known as kinematic link or element. Links should not be confused with the parts of the mechanism. Different parts of the mechanism can be considered as single link if there is no relative motion between them.
Example: The frame of any machine is considered as single link as there is no relative motion between the various parts of the frame. As shown in slider crank mechanism shown below, the frame is considered as one link (link 1) as there is no relative motion in frame itself. The crank here is link 2 & connecting rod is again single link (link 3). The slider or piston is link 4 as there is no relative motion it. In this way, many complex mechanisms can be describe by simple configuration diagram by considering the definition of a link.
Fig: 2.1 Slider crank mechanism
TYPES OF LINKS: Links can be classified into Binary, Ternary, and Quaternary etc. depending upon its ends on which revolute or turning pairs can be placed.
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Fig.2.2 Types of links
The links can also be classified into Rigid, Flexible, Fluid according to its nature such as
Rigid link is the link which do not deform while transmitting the motion
Flexible link is the link which deform while transmitting the motion but does not affect its function of transmitting motion such as belts, chains etc.
Fluid link is the link which uses the fluid pressure to transmit the motion such as hydraulics jack, brakes and lifts.
RIGID BODY
A rigid body is a body in which the distance between the two points on the body remains constant or it does not deform under the action of applied force. In actual practice no body is perfectly rigid but we assume it to be rigid to simplify our analysis.
RESISITANT BODY
A Resistant body is a body which is not a rigid body but acts like a rigid body whiles its functioning in the machine. In actual practice, no body is the rigid body as there is always some kind of deformation while transmitting motion or force. So, the body should be resistant one to transmit motion or force.
Examples: The cycle chain is the resistant body as it acts like rigid body while transmitting motion to the rear wheel of the cycle, Belt in belt and pulley arrangement.
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KINEMATIC PAIR OR PAIR
A kinematic pair is a connection between rigid bodies, which permits relative motion between them. When the links are supposed to be rigid in kinematics, then, there cannot be any change in the relative positions of any two chosen points on the selected link. In other words, the relative position of any two points does not change and it is treated as one link. Due to this rigidness, many complex shaped links can be replaced with simple schematic diagrams for the kinematic and synthesis analysis of mechanism.
CLASSIFICATION OF PAIRS
Kinematic pairs can be classified according to
a) Type of contact between elements
Lower Pairs: A pair of links having surface or area contact between the members is known as a lower pair. The surfaces in contact of the two links are similar.
Examples: Nut turning on a screw, shaft rotating in a bearing, universal joint, etc.
Fig. 2.3 Nut and screw (lower pair)
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Higher Pair: When a pair has a point or line joint contact between the links, it is known as a higher pair. The contact surfaces of the two links are dissimilar.
Examples: Wheel rolling on a surface, cam and follower pair, tooth gears, ball and roller bearings, etc.
Fig. 2.4 ball and roller bearing (higher pair)
b) Type of relative motion
Sliding Pair: When two pairs have sliding motion relative to each other.
Examples: piston and cylinder, rectangular rod in rectangular hole.
Turning Pair: When one element revolves around another element it forms a turning pair.
Examples: shaft in bearing, rotating crank at crank pin.
Screw Pair: This is also known as helical pair. In this type of pair two mating elements have threads on it or its relative motion takes place along a helical curve.
Examples: Nut and screw pair as shown in figure 2.4, Screw jack
Rolling Pair: When one element is free to roll over the other one.
Examples: Ball and rolling as shown in figure 2.5, motion of wheel on flat surface
Spherical pair: When one element move relative to the other along a spherical surface.
Examples: Ball and socket joint
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Explanation
In slider crank mechanism (Fig.2.6), crank (link 2) rotates relative to ground (link 1) and form a turning pair. Similarly, crank (link 2), connecting rod (link 3) and connecting rod (link 3), slider (link 4) also form turning pairs. Slider (link 4) reciprocates relative to ground (link 1) and form a sliding pair.
Fig.2.5 Slider crank mechanism
It should be noted here that the slider crank mechanism showed here is useful only in kinematic analysis and synthesis of the mechanism as actual physical appearance will be different and more complex than showed here. For designing the machine component the different approach will be followed.
c) Nature of Constraint or Type of closure
Closed pair: One element is completely surrounded by the other.
Examples: Nut and screw pair
Open Pair: When there is some external mean has been applied to prevent them from separation.
Examples: cam and follower pair
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DEGREE OF FREEDOM
DEGREE OF FREEDOM
An object in space has six degrees of freedom. Translatory motion along X, Y, and Z axis (3 D.O.F.)
Rotary motion about X, Y, and Z axis (3 D.O.F)
Fig.2.6 Degree of freedom
The rigid body has 6 DOF in space but due to formation of linkage one or more DOF is lost due to the presence of constraint on the body. The total number constraints cannot be zero as the body has to be fixed at some place to make the linkage possible. Thus the degree of freedom is given by
DOF= 6- (Numbers of Restraints)
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Fig.2.7 Pairs having varying degree of freedom
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S. No. | Geometrical Shapes involved | Restraints on | Degree of freedom | Total restraints | |
Translatory motion | Rotary motion | | | ||
(a) | Rigid | 0 | 0 | 0 | 6 |
(b) | Prismatic | 2 | 3 | 1 | 5 |
(c) | Revolute | 3 | 2 | 1 | 5 |
(d) | Parallel cylinders | 2 | 2 | 2 | 4 |
(e) | Cylindrical | 2 | 2 | 2 | 4 |
(f) | Spherical | 3 | 0 | 3 | 3 |
(g) | Planer | 1 | 2 | 3 | 3 |
(h) | Edge slider | 1 | 1 | 4 | 2 |
(i) | Cylindrical slider | 1 | 1 | 4 | 2 |
(j) | Point slider | 1 | 0 | 5 | 1 |
(k) | Spherical slider | 1 | 0 | 5 | 1 |
(l) | Crossed cylinder | 1 | 0 | 5 | 1 |
Table 2.1
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Figure | Explanation for DOF |
2.9 a | (0) As there is no motion hence DOF is zero |
2.9 b | (1) As movement is possible only in Z direction. |
2.9 c | (1) As it can revolve around Y axis |
2.9 d | (2) As one element can move in Z axis & also revolve around Z axis |
2.9 e | (2) As element inside can revolve around Z axis and also move in Z axis |
2.9 f | (3) As element can revolve around X,Y&Z axis |
2.9 g | (3) As element can revolve around Y axis & can move in Z & X axis |
2.9 h | (4) As element can revolve around Z & Y axis & can move in Y axis |
2.9 i | (4) As element can revolve around Z & Y axis & can move in Z & X axis |
2.9 j | (5) As an element can revolve around X,Y&Z axis & can move in X & Z axis |
2.9 k | (5) As element can revolve around X,Y & Z axis & can move in X & Z axis |
2.9 l | (5) As element can revolve around X,Y & Z axis & can move in X & Z axis |
Table 2.2
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KINEMATIC CHAIN
a) Kinematic chain: A kinematic chain is an assembly of links which are interconnected through joints or pairs, in which the relative motions between the links is possible and the motion of each link relative to the other is definite.
a. kinematic chain
b. non kinematic chain
Fig.2.8 kinematic chains
c. redundant chain
DEGREE OF FREEDOM IN A MECHANISM
Degrees of freedom of a mechanism in space can be explained as follows: Let
N = total number of links in a mechanism
F = degrees of freedom
J1 = number of pairs having one degree of freedom
J2 = number of pairs having two degree of freedom and so on.
When one of the links is fixed in a mechanism
Then, the number of the movable links are
= N - 1
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Degrees of freedom of (N- 1) movable links = 6(N-1)
(Because each movable link has six degree of freedom) Each pair having one degree of freedom imposes 5 restraints on the mechanism reducing its degrees of freedom by 5J1 this is because of the fact that the restraint on any of the link is common to the mechanism as well. Other pairs having 2, 3, 4 and 5 degrees of freedom reduce the degree of freedom of the mechanism by putting constraints on the mechanism as well.
Then, the DOF can be given by
F = 6(N-1) - 5J1 - 4J2 - 3J3 - 2J4 - 1J5
Most of the mechanism we generally study are two dimensional in nature, such as slider-crank mechanism in which translatory motion is possible along two axes(one restraint) and rotary motion about only one axis(two restraints).Thus there are three general restraints in a two dimensional mechanism. This can be shown with the help of figure 2.10 that a link has three degree of freedom in two dimensions.
Fig.2.9 a line in a plane has three DOF: x, y, θ
Therefore, for plane mechanism, the following relation can be used for degrees of freedom,
F = 3 (N-1) - 2J1 - 1J2
This equation is known as Gruebler’s criterion for degrees of freedom of plane mechanism. It should be noted here that gruebler’s criterion does not take care of geometry of the mechanism so it can give wrong prediction. So, inspection should be done in certain cases to find the degrees of freedom.
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Example: 2.1 Find the degree of freedom of the mechanism given below.
Fig.2.10
Solution:
Number of links=8
Numbers pairs having one degrees of freedom=10 by counting How to calculate pairs
Pair 1
Pair 2
Pair 3
Pair 4
Pair 5
Pair 6
Pair 7
Pair 8
Pair 9
Pair 10
Link 1 (ground) and link 2 constitute a single turning pair Link 2 and link 3 constitute a single turning pair
Link 3 and link 5 constitute a single turning pair
Link 4 and link 5 constitute a single turning pair Link 5 and link 6 constitute a single turning pair Link 6 and ground (link 1) constitute a turning pair Link 5 and link 7 constitute a turning pair
Link 7 and link 8 constitute a turning pair
Link 8 and ground (link 1) constitute a sliding pair Link 4 and ground (link 1) constitute a turning pair
As all the pair calculated have one degree of freedom so there is only term J1 is used as it denotes the pair having single degree of freedom.
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J1 = 10 (as all pairs have one degree of freedom)
F = 3 (N-1) - 2J1 - 1J2
DOF=3(8-1)-2×10=1
The degree of freedom is one for this mechanism.
Example 2.2 Find the degree of freedom of the mechanism given below.
Fig. 2.11
6
7
7
Solution:
Number of links = Number of Pairs =
J1 =
DOF=3(6-1)-2×7=1
The degree of freedom is one.
(six turning pairs and one sliding pair)
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Example 2.3: Find the mobility or degree of freedom of the following mechanism.
Fig. 2.12
Solution:
Number of links = Number of Pairs = J1 =
J2 =
7
8
(six turning pairs and one sliding pair) (Fork joint is two DOF joint)
7
1
DOF=3(7-1)-2×7-1×1=3
The degree of freedom is one.
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FOUR BAR MECHANISM
The four bar linkage, as shown in figure 2.13 below, is a basic mechanism which is quite common. Further, the vast majority of planar one degree-of-freedom (DOF) mechanisms have "equivalent" four bar mechanisms. The four bar has two rotating links (2 and 4)) which have fixed pivots. One of the levers would be an input rotation, while the other would be the output rotation. The two levers have their fixed pivots with the ground link (1) and are connected by the coupler link (3).
Fig.2.13 four bar mechanism
Crank (2) - a ground pivoted link which is continuously rotatable.
Rocker (4) - a ground pivoted link that is only capable of oscillating between two limit positions and cannot rotate continuously.
Coupler (3) - a link opposite to the fixed link.
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Fig.2.14 crank rocker
In fig 2.14 the links adjacent to the shortest link b is fixed. The mechanism such obtained is known as crank-lever or crank rocker mechanism.
If the shortest link b is fixed, the mechanism obtained is crank-crank or double crank mechanism.
Fig.2.15 double crank
If the link opposite to the shortest link is fixed then the mechanism is know as double-rocker or double lever mechanism.
Fig. 2.16 double rocker
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INVERSIONS OF SLIDER CRANK MECHANISM
Different mechanisms are obtained when we fixed different links of a Kinematic chain and the phenomenon is known as inversion of mechanism. A slider crank mechanism has the following inversions.
First Inversion
This inversion is obtained when link 1 is fixed (as shown in fig 2.17) and links 2 and 4 are made the crank and the slider respectively.
Application: This mechanism is commonly used in I.C. engines, steam engines and reciprocating compressor mechanism.
Fig.2.17 reciprocating engine
Second Inversion
By fixing link 2 of a slider mechanism gives second inversion. Rotary engine mechanism or gnome engine is the application of second inversion. It is a rotary cylinder V – type internal combustion engine used as an aero engine. The rotary engine has generally seven cylinders in one plane. The crank (link 2) is fixed and all the connecting rods from the pistons are connected to this link. In this mechanism when the pistons reciprocate in the cylinders, the whole assembly of cylinders, pistons and connecting rods rotate about the axis O, where the entire mechanical power developed, is obtained in the form of rotation of the crank shaft.
Application: Rotary engine mechanism or gnome engine
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Fig.2.18 rotary engine
Third Inversion
By fixing the link 3(connecting rod) of the slider crank mechanism we can obtain third inversion (as shown in fig.2.19). It is used in hoisting engine mechanism and also in toys. In hoisting purposes, its main advantages lie in its compactness of construction as it allows simple method of supplying steam to the cylinder.
Application: It is used in hoisting engine mechanism and also in toys
Fig.2.19 oscillating cylinder engine
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Fourth Inversion
By fixing the link 4 of the slider crank mechanism we can obtain the fourth inversion of slider crank. Fixing the slider means that the slider should be fixed in position and also should be fixed in respect to rotation. In this case, the cylinder will have to be slotted to give passage to piston pin of connecting rod as the cylinder slides over the piston. Due to this difficulty, the shapes of the cylinder and piston are exchanged as shown in figure below.
Fig.2.20 pendulum pump
Application: hand pump