THEORY OF MACHINE

DIPLOMA IN MECHANICAL ENGINEERING

3^rd SEMESTER

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Dr. Durjyodhan Sethi

PROFESSOR

GANDHI INSTITUTE FOR EDUCATION AND TECHNOLOGY, BANIATANGI, BHUBANESWAR

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Kinematics of Machinery - Unit - I

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KINEMATICS OF MECHINERY

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KINEMATICS OF MECHINERY

• Unit – I-Basics of Mechanism
• Unit – II – Kinematics
• Unit – III – Cams
• Unit – IV – Gears
• Unit – V - Friction

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Unit – I-Basics of Mechanism

1.Terminologies

Machine

Mechanism

Kinematic Pair

Links

Kinematic Chain

2.DOF

Kutzhback Equation

Grubler Equation

3.Grashoff Law

4.Mechanism

Four Bar

Single-Slider Crank

Double-Slider

5.Inversion of Mechanism

6.Mechanical Advantage

7.Transmission Angle

8.Design of Mechanism

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MECHANISM

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Mechanism – Part of a machine, which transmit motion and power from input point to output point

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Example for Mechanism

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Example for Mechanism

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PLANAR MECHANISMS

When all the links of a mechanism have plane motion, it is called as a planar mechanism. All the links in a planar mechanism move in planes parallel to the reference plane.

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MACHINE

A machine is a mechanism or collection of mechanisms, which transmit force from the source of power to the resistance to be overcome.

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Though all machines are mechanisms, all mechanisms are not machines

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KINEMATICS

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RELEVANCE OF KINEMATIC STUDY

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• Motion requirements
• Design requirements

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MOTION STUDY

Study of position, displacement, velocity and acceleration of different elements of mechanism

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Given input Desired output

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Motion requirement

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DESIGN REQUIREMENTS

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Design: determination of shape and size

1. Requires knowledge of material
2. Requires knowledge of stress

Requires knowledge of load acting

(i) static load

(ii) dynamic/inertia load

DYNAMIC/INERTIA LOAD

Inertia load require acceleration

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LINK OR ELEMENT

Any body (normally rigid) which has motion relative to another

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• Binary link
• Ternary link
• Quaternary link

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Examples of rigid links

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PAIRING ELEMENTS

Pairing elements: the geometrical forms by which two members of a mechanism are joined together, so that the relative motion between these two is consistent. Such a pair of links is called Kinematic Pair.

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PAIRING ELEMENTS

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PAIRING ELEMENTS

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KINEMATIC PAIRS

• A mechanism has been defined as a combination so connected that each moves with respect to each other. A clue to the behavior 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.�

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TYPES OF KINEMATIC PAIRS

Based on nature of contact between elements

• (i) Lower pair : The joint by which two members are connected has surface contact. A pair is said to be a lower pair when the connection between two elements are through the area of contact. Its 6 types are

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Revolute(Or)TurningPair �

Prismatic(Or)SlidingPair

�Screw(Or)HelicalPair

�CylindricalPair �

Spherical(Or)GlobularPair

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Flat(or)PlanarPair

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(ii) Higher pair: The contact between the pairing elements takes place at a point or along a line.

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Based on relative motion between pairing elements

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(a) Siding pair [DOF = 1]

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(b) Turning pair (revolute pair)

[DOF = 1]

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Based on relative motion between pairing elements

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(c) Cylindrical pair [DOF = 2]

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(d) Rolling pair

[DOF = 1]

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Based on relative motion between pairing elements

(e) Spherical pair [DOF = 3]

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(f) Helical pair or screw pair [DOF = 1]

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Based on the nature of mechanical constraint

(a) Closed pair

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(b) Unclosed or force closed pair

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CONSTRAINED MOTION

one element has got only one definite motion relative to the other

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(a) Completely constrained motion

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(b) Successfully constrained motion

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(c) Incompletely constrained motion

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KINEMATIC CHAIN

Group of links either joined together or arranged in a manner that permits them to move relative to one another.

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Kinematic Chain

Relation between Links, Pairs and Joints

L=2P-4

J=(3/2) L – 2

L => No of Links

P => No of Pairs

J => No of Joints

L.H.S > R.H.S => Locked chain

L.H.S = R.H.S => Constrained Kinematic Chain

L.H.S < R.H.S => Unconstrained Kinematic Chain

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LOCKED CHAIN (Or) STRUCTURE

Links connected in such a way that no relative motion is possible.

L=3, J=3, P=3 L.H.S>R.H.S

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Kinematic Chain Mechanism

Slider crank and four bar mechanisms

L=4, J=4, P=4

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L.H.S=R.H.S

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Working of slider crank mechanism

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Unconstrained kinematic chain��L=5,P=5,J=5 �� L.H.S < R.H.S

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DEGREES OF FREEDOM (DOF):

It is the number of independent coordinates required to describe the position of a body.

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Degrees of freedom/mobility of a mechanism

It is the number of inputs (number of independent coordinates) required to describe the configuration or position of all the links of the mechanism, with respect to the fixed link at any given instant.

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GRUBLER’S CRITERION

Number of degrees of freedom of a mechanism is given by

F = 3(n-1)-2l-h. Where,

• F = Degrees of freedom
• n = Number of links in the mechanism.
• l = Number of lower pairs, which is obtained by counting the number of joints. If more than two links are joined together at any point, then, one additional lower pair is to be considered for every additional link.
• h = Number of higher pairs

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Examples - DOF

• F = 3(n-1)-2l-h
• Here, n = 4, l = 4 & h = 0.
• F = 3(4-1)-2(4) = 1
• I.e., one input to any one link will result in definite motion of all the links.

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Examples - DOF

• F = 3(n-1)-2l-h
• Here, n = 5, l = 5 and h = 0.
• F = 3(5-1)-2(5) = 2
• I.e., two inputs to any two links are required to yield definite motions in all the links.

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Examples - DOF

• F = 3(n-1)-2l-h
• Here, n = 6, l = 7 and h = 0.
• F = 3(6-1)-2(7) = 1
• I.e., one input to any one link will result in definite motion of all the links.

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Examples - DOF

• F = 3(n-1)-2l-h
• Here, n = 6, l = 7 (at the intersection of 2, 3 and 4, two lower pairs are to be considered) and h = 0.
• F = 3(6-1)-2(7) = 1

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Examples - DOF

• F = 3(n-1)-2l-h
• Here, n = 11, l = 15 (two lower pairs at the intersection of 3, 4, 6; 2, 4, 5; 5, 7, 8; 8, 10, 11) and h = 0.
• F = 3(11-1)-2(15) = 0

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Examples - DOF

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Determining DOF and Pairs

• N_b=No of Binary Links
• N_t=No of Ternary Links
• N_o=No of Other Links
• N=Total No of Links
• L=No of Loops

• P=No of Pairs
• M=Mobility or DOF

P=N+L-1

M=N-(2L+1)

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Determining DOF and Pairs

P=N+L-1

M=N-(2L+1)

N_b = 4,N_t=2, N₀=0

N=6, L=2

Sol:

P=6+2-1=7

M=6-(2x2 +1)=1

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Determining DOF and Pairs

P=N+L-1

M=N-(2L+1)

N_b = 5,N_t=1, N₀=0

N=6, L=2

Sol:

P=6+2-1=7

M=6-(2x2 +1)=1

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Determining DOF and Pairs

P=N+L-1

M=N-(2L+1)

N_b = 9,N_t=0, N₀ =2

N=11, L=5

Sol:

P=11+5-1=15

M=11-(2x5 +1)=0

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Grashoff Law

• The sum of the shortest and longest link length should not exceed the sum of the other two link lengths.

s+l < p+q

(e.x) (1+2) < (3+4)

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INVERSIONS OF MECHANISM

A mechanism is one in which one of the links of a kinematic chain is fixed. Different mechanisms can be obtained by fixing different links of the same kinematic chain. These are called as inversions of the mechanism.

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INVERSIONS OF MECHANISM

• 1.Four Bar Chain

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• 2.Single Slider Crank

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• 3.Double Slider Crank

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1. FOUR BAR CHAIN

• (link 1) frame
• (link 2) crank
• (link 3) coupler
• (link 4) rocker

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INVERSIONS OF FOUR BAR CHAIN

Fix link 1& 3. Crank-rocker

or Crank-Lever

mechanism

Fix link 2. Drag link

or Double Crank

mechanism

Fix link 4. Double rocker

mechanism

Pantograph

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APPLICATION�link-1 fixed-� CRANK-ROCKER MECHANISM OSCILLATORY MOTION

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CRANK-ROCKER MECHANISM

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Link 2 Fixed- DRAG LINK MECHANISM

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Locomotive Wheel - DOUBLE CRANK MECHANISM

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��2.SLIDER CRANK CHAIN�Link1=Ground�Link2=Crank�Link3=ConnectingRod �Link4=Slider� �

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lnversions of slider crank chain

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1. crank fixed (b) connecting rod fixed (c) slider fixed

Link 2 fixed Link 3 fixed Link 4 fixed

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Application�Inversion II – Link 2 Crank fixed �Whitworth quick return motion mechanism

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Quick return motion mechanisms

Drag link mechanism

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Rotary engine– II inversion of slider crank mechanism. (crank fixed)

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Application�Inversion III -Link 3 Connecting rod fixed �Crank and slotted lever quick return mechanism

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Crank and slotted lever quick return motion mechanism

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Crank and slotted lever quick return motion mechanism

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Application of Crank and slotted lever quick return motion mechanism

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Oscillating cylinder engine–III inversion of slider crank mechanism (connecting rod fixed)

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Application�Inversion IV – Link 4 Slider fixed �Pendulum pump or bull engine

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3. DOUBLE SLIDER CRANK CHAIN

It is a kinematic chain consisting of two turning pairs and two sliding pairs.

Link 1 Frame

Link 2 Slider -I

Link 3 Coupler

Link 4 Slider - II

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Inversion I – Frame Fixed�Double slider crank mechanism

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Elliptical trammel

AC = p and BC = q, then,

x = q.cosθ and y = p.sinθ.

Rearranging,

Inversion II – Slider - I Fixed �SCOTCH –YOKE MECHANISM

Turning pairs –1&2, 2&3; Sliding pairs – 3&4, 4&1

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Inversion III – Coupler Fixed �OLDHAM COUPLING

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Other Mechanisms�1.Straight line motion mechanisms

Condition for perfect steering

Locus of pt.C will be a straight line, ┴ to AE if,

is constant.

Proof:

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1.a) Peaucellier mechanism

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1.b) Robert’s mechanism

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1.c) Pantograph

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2.Indexing Mechanism�

Geneva wheel mechanism

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3.Ratchets and Escapements

Ratchet and pawl mechanism

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Application of Ratchet Pawl mechanism

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4. Toggle mechanism

Considering the equilibrium condition of slider 6,

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For small angles of α, F is much smaller than P.

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5.Hooke’s joint

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Hooke’s joint

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6.Steering gear mechanism

Condition for perfect steering

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Ackermann steering gear mechanism

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Mechanical �Advantage

• Mechanical Advantage of the Mechanism at angle a2 = 0⁰ or 180⁰

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• Extreme position of the linkage is known as toggle positions.

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Transmission �Angle

θ = a1=Crank Angle

γ = a2 =Angle between crank and Coupler

μ = a3 =Transmission angle

Cosine Law

a² + d² -2ad cos θ =

b² + c² -2 bc cos μ

Where a=AD, b=CD,

c=BC, d=AB

Determine μ.

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Design of �Mechanism

• 1.Slider – Crank Mechanism

Link Lengths, Stroke Length, Crank Angle specified.

• 2.Offset Quick Return Mechanism

Link Lengths, Stroke Length, Crank Angle, Time Ratio specified.

• 3.Four Bar Mechanism – Crank Rocker Mechanism

Link Lengths and Rocker angle Specified.

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ALL THE BEST

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