Theory of

Presentation By Prof. N Mohan Rao JNTUK

1 Plane mechanisms

• Basic terminology: Resistant body is that which does not suffer appreciable distortion or change in physical form due to the acting on it. Element is a part of which has been manufactured without the operation of assembling. Example: bolt and nut are two elements.

2 Basic terms • Link: A link is a resistant body or assembly of resistant bodies that connects parts of a machine which have relative to it. A link may be meant for transmitting motion and for guiding other links. It may be support also. Eg . Piston, piston rod and cross head of a steam engine together form one link. • Kinematic Pair: It is a movable joint of two links which are in contact with the relative between them being completely or successfully constrained. Piston and cylinder of an I.C engine form a pair.

3 Types of Kinematic Pairs • Based on type of contact – Lower pair : surface contact while in motion Ex. Shaft revolving in a bearing, straight line motion mechanisms, automobile steering .

– Higher pair : point or line contact while in motion Ex. , rope and chain drives, ,

4 Types of Kinematic Pairs… • Based on type of relative motion – Sliding pair : Two links are connected such that one is constrained to have sliding motion relative another link. Example: Cross head and guides of a reciprocating engine bearing – Turning pair : two links are connected such that one is constrained to turn or revolve about a fixed axis of another link. Example: Crank shaft turning in a bearing

– Screw pair : one link turns about the other by means of thread, relative motion being combination of turning and sliding. Example: Bolt and nut, lead screw and nut of a lathe

– Spherical pair : one element in the form of a sphere turns about the other element Example: ball and socket joint

5 Types of Kinematic Pairs… • Based on type of closure – Closed Pair : the links of a pair are held together by their geometrical arrangement under any direction of . Example: screw pair and spherical pair – Unclosed pair : the links of a pair are not held together mechanically, but are connected by the forces exerted by external arrangement. Example: and follower are connected by the forces exerted by spring and gravity only.

6 • : when the kinematic pairs are connected such that the last link is joined to the first link to transmit constrained motion, it

is called kinematic chain. Connecting rod crank

slider : It is an assemblage of links so connected that they move relative to the other with a definite relative motion. A mechanism is formed by fixing one of the links of a kinematic chain. A mechanism is a part of machine. 7 Constrained motion • Completely constrained motion: is the one in which motion takes place in a definite direction irrespective of the direction of force. The motion is complete by its own links. Example: a rectangular bar moving in a rectangular hole, a shaft with collars at each end rotating a round hole.

Fixed element collars

shaft

8 Successfully constrained motion • Successfully constrained motion: When the constrained motion is not complete by itself but by some other external means, it is called as partially or successfully completed motion. Example: a foot step bearing and a rotor of a vertical turbine

9 Incompletely constrained motion • Incompletely constrained motion: When the links are so connected that the motion can take place in more than one direction, it is called incompletely constrained motion. Direction of motion changes with the direction of impressed force. • Example: A circular bar moving in a round hole as the bar rotate or reciprocate independently.

Fixed element

shaft

10 Degrees of freedom • The number of degrees of freedom of a mechanism indicates the number of inputs the mechanism should be given in order to fulfill a useful engineering purpose. Example: A mechanism with one degree of freedom requires one input motion at a point on the mechanism to cause constrained motion of all other links. A mechanism with two degrees of freedom requires two independent to cause definite motion of other links.

11 Degrees of freedom

θ φ θ Degrees of freedom, F=1 Degrees of freedom, F=2

12 Degrees of freedom for plane mechanisms • Let the L number of links in a mechanism be

connected by P 1 number of lower pairs and P 2 number of higher pairs. • Each link has 3 degrees of freedom before it is connected to any link. • Total number of degrees of freedom for (L -1) links is 3(L-1). • For each lower pair present in the mechanism, there is a loss of 2 degrees of freedom. • For each higher pair present in the mechanism, there is a loss of 1 degrees of freedom.

13 • Hence, the degrees of freedom of a mechanism, F = 3(L-1) – 2P 1 –P2 ……..(1) This equation is known as Gruebler’s criterion for degrees of freedom of plane mechanisms • If P 2=0 i.e. there are no higher pairs, F= 3(L-1) – 2P 1 ……..(2) • If F=1, Eq. (2) takes the form

1= 3(L -1) – 2P 1 2P 1-3L+4=0……………(3) This equation is gives the relation between the number of links and the lower pairs for plane mechanisms having one degree of freedom.

14 • If a mechanism is assumed to consist of turning pairs only, then the number of turning pairs in the mechanism is equal to number of lower pairs in the mechanism. • P1=(3/2)L-2 • In order that P1 to be whole number, L should be even. • Thus, for plane mechanisms with F=1 and all turning pairs, the number of links should be even.

15 LP1 Comments 4 4 No excess turning pairs required 6 7 One excess turning pair required 8 10 Two excess turning pairs required With all the binary links (turning pairs), it is not possible to get the excess turning pairs. Therefore, to get the excess pairs or joints, links having more than two joints are to be used like ternary or quaternary links. For the six -link chain, two ternary links are required to get one excess turning pair For eight link chain, two quaternary links (or) four ternary links (or) one quaternary link and two ternary links are required to get two excess turning pairs Ternary link Ternary link

six-link chain six-link chain 16 Four bar chain • Four bar chain is the most fundamental chain of the plane mechanisms. It is the much preferred mechanism due to its simplicity. It consists of four links connected by four turning pairs.

C

B

A D Four bar chain

17 • Grashof's theorem states that a four bar mechanism has at least one revolving link, if the sum of the lengths of the shortest and longest links is less than or equal to the sum of lengths of the other two links. L + S ≤ P+Q

Mechanisms satisfying this condition fall into the following three categories: (1)When any one of the adjacent links of the shortest link is fixed, the shortest link can have full revolution and the link opposite to it oscillates. This is known as crank- rocker mechanism.

s s

18 (2) If the link opposite to the shortest link is fixed and the shortest link is made coupler, the other two links oscillate. s This is known as rocker-rocker or double- rocker mechanism.

(3) When the shortest link is fixed, the mechanism is double-crank mechanism. s

19 Inversion of a mechanism

Different mechanisms obtained by fixing different links of a kinematic chain are known as its inversions.

20 Slider-crank chain • It is a four link chain with one sliding pair and three turning pairs. This mechanism converts the reciprocating motion of piston of a steam engine into rotary motion of its crank.

OA- Crank – link 2 AB – Connected rod – link 3 B – slider – link 4 OB- Fixed link – link 1

21 Inversions of a slider- crank chain • Different inversions of the mechanism can be obtained by 4 fixing different links one link at

a . 3 (1) Link 4 (sliding pair) is fixed 2 Example: pump, 1

Duplex pump used as feed water 1 pump to boilers (two pistons are 4 Pendulum pump fitted to link 1) (link 4 fixed)

22 Link 3 (connecting rod) fixed • 1 Examples: 2

(a) oscillating cylinder 3

engine 1 (b) Crank and slotted lever 4 type quick return Oscillating cylinder link 3 fixed motion mechanism

23 C • B Crank and slotted lever 4 1 type quick return α 2 O A motion mechanism 1 β • Used in shaper/planer 3

O2 Crank and slotted lever type quick return motion mechanism

24 Link 2 (rotating pair) fixed

• (a) Whitworth type quick return motion mechanism B 1 C O2

2 4 β A 3 O1 α

25 • Gnome engine

26 Double slider crank chain

• Consists of 2 turning pairs and 2 sliding pairs • Links 1 and 2 form turning pair • Links 2 and 3 form turning pair • Links 3 and 4 form sliding pair • Links 4 and 1 form sliding pair

27 Inversions of Double slider crank chain Oldham’s coupling – link 2 fixed

28 • Elliptical trammel – link 4 fixed

x = BC cosθ y = AC sinθ (x / BC) = cos θ

(y / AC) = sinθ O

(x 2 / BC 2) + (y 2 / AC 2) = cos 2θ + sin 2θ = 1

Equation for ellipse. Locus of point C is an ellipse with O as the centre

29 Questions

1. A four bar mechanism is made up of links of length 100, 200, 300 and 350 mm. If the 350 mm link is fixed, the number of links that can rotate fully is ______.

2. The number of degrees of freedom in a planar mechanism having n links and j simple hinge joints is (A) 3(n−3)−2j (B) 3(n−1)−2j (C) 3n−2j (D) 2j−3n+4 Answer : (B) 3(n−1)−2j 30 (3) The number of degrees of freedom of the planetary gear train shown in the figure is (A)0 (B)1 (C)2 (D)3 Answer : (C) 2

L=4, P1=3 P2=1 F= 3(4-1)-2x3-1x1=2

31 (4) The number of degrees of freedom of the shown in the figure is (A)-3 (B)0 (C)1 (D)2 Answer : (C) 1

L=6 P1=7 F=3(6-1) – 2x7=1

32 (5) For the given statements: I. Mating spur gear teeth is an example of higher pair II. A is an example of lower pair Indicate the correct answer. (A) Both I and II are false

(B) I is true and II is false

(C) I is false and II is true

(D) Both I and II are true

Answer : (D) Both I and II are true

33 (6) A 4-bar mechanism with all revolute pairs has link lengths lf = 20

mm, lin = 40 mm, lco = 50 mm and lout = 60 mm. The suffixes 'f', 'in', 'co' and 'out' denote the fixed link, the input link, the coupler and output link respectively. Which one of the following statements is true about the input and output links?

(A) Both links can execute full (B) Both links cannot execute full circular motion (C) Only the output link cannot execute full circular motion (D) Only the input link cannot execute full circular motion co Answer: A ou in

f 34 (7) A planar closed kinematic chain is formed with rigid links PQ = 2.0 m, QR = 3.0 m, RS = 2.5 m and SP = 2.7 m with all revolute joints. The link to be fixed to obtain a double rocker (rocker-rocker) mechanism is (A) PQ PQ+QR< RS+ SP

R (B) QR S

(C) RS P Q (D) SP Answer : (C) RS

35 (8) Mobility of a statically indeterminate structure is (A) ≤−1

(B) 0

(C) 1

(D) ≥2 Answer : (A) ≤−1

36 (9) Which of the following statements is INCORRECT? (A) Grashof’s rule states that for a planar crank-rocker four bar mechanism, the sum of the shortest and longest link lengths cannot be less than the sum of the remaining two link lengths. (B) Inversions of a mechanism are created by fixing different links one at a time. (C) Geneva mechanism is an intermittent motion device (D) Gruebler’s criterion assumes mobility of a planar mechanism to be one. Answer : (A) Grashof’s rule states that for a planar crank- rocker four bar mechanism, the sum of the shortest and longest link lengths cannot be less than the sum of the remaining two link lengths.

37 (10) A simple quick return mechanism is shown in the figure. The forward to return ratio of the quick return mechanism is 2:1. If the radius of the crank O 1P is 125 mm, then the distance 'd' (in mm) between the crank centre to lever pivot centre point should be

(A) 144.3 (B) 216.5 (C) 240.0 (D) 250.0

180 x1/(1+2)=60 o o O1 cos 60 =0.5=125/ O1O2 60 o d= O1O2=250 P 90 o Ans. D

O2

38 (11)Match the approaches given below to perform stated / dynamics analysis of machine

(A) P-1,Q-2,R-3,S-4

(B) P-3,Q-4,R-2,S-1

(C) P-2,Q-3,R-4,S-1

(D) P-4,Q-2,R-1,S-3

Answer : (B) P-3,Q-4,R-2,S-1

39 (12) A planner mechanism has 8 links and 10 rotary joints. The number of degrees of freedom of the mechanism, using Gruebler’s criterion, is (A) 0

(B) 1

(C) 2

(D) 3

Answer : (B) 1

F=3(8-1)-2x10=1

40 analysis • To determine the of different points on links of a mechanism for a given input motion. Determination of the motion characteristics of links in a mechanism is required for the force analysis • Velocities of links and of points of mechanism can be determined by different methods. 1. method or velocity polygon method 2. Instantaneous centre method. Velocity of link • In order to determine the relative motion of A the ends of a link, one of the ends is assumed to be moving relative to the other end. • Let a link AB rotate about A in anti-clockwise direction such that the end B rotates relative to A. B • Then the direction of relative motion of B with respect to A is perpendicular to AB . • Therefore, the direction of relative velocity of a point w.r.t any other point on a link is always along perpendicular to the straight line joining the two points. Velocity of link… A • This method is useful in drawing the velocity diagram and finding the relative velocities of points on a link. • Let the velocity of B w.r.t A be represented by ω ab. Then ab is drawn perpendicular to AB to C any convenient scale as shown in the velocity diagram. B • If the ω = of link AB about its a end A, then velocity of B w.r.t A = c b • Similarly, velocity of any intermediate point C w.r.t A = .. It is represented by the vector ac in the velocity diagram. Velocity of link…

• Hence, the point c divides the vector ab in the same ratio as the point C divides the link AB. Relative velocity method Velocity of any point on a link: A Consider a link AB where the directions VA

of motion of its ends are shown. The VB velocity vectors are shown. oa – velocity of the end A C ob - velocity of the end B B ab - velocity of B with respect to A b c ba - velocity of A with respect to B

a o Velocity of any point on a link… • Velocity of any point C on the link can be determined by dividing the vector ab at c such that

Then, the velocity of C with respect to A

Hence, the vector oc represents the velocity of point C. References

• The Theory of Machines – THOMAS BEVAN • Theory of Machines – S S RATTAN • Theory of Mechanisms &Machines – DR. JAGDISH LAL • Website links related to this subject.

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