Mechanism & Machines - Mechanical Engineering SSC JE (Technical) PDF Download

Kinematic Link/Element

Every part of a machine which has some relative motion with respect to another part is called a kinematic link or kinematic element. Links connect together to form mechanisms; each link may be rigid, flexible, or fluid depending on its behaviour under load.

Types of Links

• Rigid Link: Deformation is negligible under the working loads; the link can be treated as an undeformable body for kinematic analysis.
• Flexible Link: Deformation occurs but remains within permissible limits for the intended application; elastic effects may be considered when necessary.
• Fluid Link: Motion or power is transmitted by a fluid medium (for example, in hydraulic transmissions) where the fluid serves as the connecting element between parts.

Mechanism and Machine

• Mechanism: An assembly of kinematic links arranged so that the motion of one element causes constrained and predictable motion of the others.
• Machine: A practical application or combination of mechanisms that not only imparts definite motion to its parts but also transmits and converts mechanical energy into useful work.

Types of Constrained Motion

(I) Completely constrained motion

When the relative motion between two elements of a pair is confined to a definite direction independent of the direction of the applied force, the motion is called completely constrained. Example: a slider moving in a guide (sliding pair).

(II) Successfully constrained motion

When relative motion between two elements could occur in more than one direction but is made to occur only in the desired direction by the use of external means (pins, guides, cams, bearings), it is called successfully constrained. Example: a piston in an internal combustion engine is constrained to reciprocate by the piston pin, cylinder walls and connecting mechanism.

(III) Incompletely constrained motion

When the relative motion between the elements of a pair may occur in more than one direction and the actual motion depends on the direction of the applied forces, the motion is called incompletely constrained. Example: a cylindrical shaft in a loosely fitting round bearing where rotation and small lateral displacements may both occur depending on loading.

Rigid Body and Resistant Body

• Rigid Body: A body that does not suffer any appreciable distortion under the action of applied forces (ideal assumption for many kinematic analyses).
• Resistant Body: A body that is effectively rigid for the purpose it serves, e.g. a belt used primarily under tensile loading behaves as a resistant element in belt drives.

Kinematic Pair and Their Classification

A kinematic pair (or simply a pair) is a joint of two links that permits relative motion between them. Classification follows several criteria.

(I) According to the nature of contact

• Lower Pair: Members have surface (area) contact. Examples: shaft rotating in a bearing, nut turning on a screw, many journal bearings and typical turning pairs.
• Higher Pair: Members have point or line contact and typically dissimilar contact surfaces. Examples: cam and follower, wheel rolling on a surface, gear tooth contact, ball and roller bearings.

(II) According to the nature of mechanical constraint

• Closed Pair: When the elements are geometrically complementary (one solid and one hollow) and the contact cannot be broken without destroying at least one member. Most lower pairs are closed pairs.
• Unclosed Pair: When the two links remain in contact due to external forces such as gravity, springs, or applied loads and the contact can be separated without destruction.

(III) According to the nature of relative motion

• Sliding Pair: Relative motion is pure translation (sliding) between contacting surfaces. Example: a rectangular rod sliding in a rectangular guide (prismatic pair).
• Turning (Revolving) Pair: Relative motion is rotation about an axis. Example: a pin joint or shaft in a bearing in a typical four-bar linkage.
• Rolling Pair: Relative motion is pure rolling. Example: a wheel rolling on a flat surface, roller bearings.
• Screw (Helical) Pair: Relative motion combines rotation and translation along the same axis. Example: lead-screw and nut in a lathe feed mechanism.
• Spherical Pair: One member is spherical and turns within a socket, allowing rotation about multiple axes. Example: a ball-and-socket joint.

Degree of Freedom

The degree of freedom (D.O.F.) of a system is the minimum number of independent variables required to uniquely define the position or motion of the system. For a body in space the maximum D.O.F. is 6 (three translations and three rotations). A constraint removes one or more of these independent motions.

General spatial relation: Degree of freedom = 6 - (number of restraints).

Degree of Freedom of a Mechanism (General Spatial)

Let:

• F = Degree of freedom (D.O.F.)
• N or L = Total number of links in the mechanism (including the frame or fixed link)
• P1 = Number of pairs having one D.O.F. (binary 1-DOF joints)
• P2 = Number of pairs having two D.O.F.
• P3 = Number of pairs having three D.O.F.
• P4 = Number of pairs having four D.O.F.
• P5 = Number of pairs having five D.O.F.

If one link is fixed, the number of movable links is N - 1 and the total unconstrained degrees of freedom of these movable links is 6(N - 1). Each pair reduces the D.O.F. of the mechanism by the number of restraints it provides. Hence the general expression:

F = 6 (N - 1) - 5P1 - 4P2 - 3P3 - 2P4 - 1P5

Degree of Freedom of Plane (2D) Mechanism - Grübler / Kutzbach Criterion

Most planar mechanisms permit motion in two directions (x and y) and rotation about an axis perpendicular to the plane. Such a rigid body in plane has 3 degrees of freedom. Using similar restraint counting for planar mechanisms:

F = 3 (N - 1) - 2P1 - 1P2

Here P1 is the number of binary (1-DOF) joints (for planar mechanisms these are usual turning or sliding pairs), and P2 are joints with 2 DOF (rare in planar chains).

Kutzbach's equation (alternate form)

F = 3 (L - 1) - 2 j - h

where L is number of links, j is number of binary joints (1-DOF joints), and h is number of higher pairs (each higher pair usually contributes one restraint less).

Grübler's equation (for single degree of freedom mechanisms with no higher pairs)

3 L - 2 j - 4 = 0

Here L is the number of links and j is the number of binary joints. This is the condition for a planar mechanism with single degree of freedom and zero higher pairs.

Degree of Freedom for different frames

• F = 0 : (statically determinate) frame
• F < 0 : redundant frame (over-constrained)
• F > 0 : constrained / partially constrained / movable frame

Simple Mechanism

Mechanisms with four links form the simplest interesting kinematic chains. Mechanisms with more than four links are termed compound mechanisms. Three common simple mechanisms are:

• Four-bar mechanism: Four links joined by four turning pairs (revolute joints).
• Slider-crank mechanism: Four links with three turning pairs and one sliding (prismatic) pair.
• Double slider-crank mechanism: Four links with two turning pairs and two sliding pairs.

Four-bar Mechanism

• In a four-bar linkage, the link that can rotate fully is called the crank. The link opposite the fixed link is the coupler. The remaining link may oscillate (a rocker) or rotate fully (a crank).
• Grashof's law: For continuous relative rotation of one link with respect to another in a planar four-bar linkage, the sum of the shortest and longest link lengths must be less than or equal to the sum of the remaining two link lengths.

If the sum of the shortest and longest link lengths is less than the sum of the other two, the linkage is a Class I (Grashof) four-bar linkage; this permits at least one link to make a complete revolution relative to the frame. Depending on which link is fixed, the types obtained are:

• Crank-rocker (crank-lever): One link rotates fully (crank) and the opposite oscillates (rocker).
• Double-crank (crank-crank): Both adjacent links to the fixed link can rotate fully - a rotary-rotary converter.
• Double-rocker (rocker-rocker): Both links adjacent to the fixed link oscillate - an oscillating-oscillating converter.

When the sum of the largest and smallest links is greater than the sum of the other two links, the linkage is a Class II four-bar and continuous rotation of any link is not possible; the mechanism behaves as a rocker-crank or double-rocker as per inversions.

Inversion of 4-bar Mechanism

• Coupling rod of locomotive - example of a crank-crank inversion.
• Beam engine - crank-rocker inversion.
• Watt's indicator - rocker-rocker inversion.

Note:

• Approximate straight line mechanisms: Watt's straight-line linkage, modified Scott-Russell mechanism, Grasshopper mechanism.
• Exact straight line mechanisms: Peaucellier-Lipkin linkage, Hart mechanism, Scott-Russell mechanism (exact forms).

Inversion of Slider-Crank Mechanism

A slider-crank chain has four links: crank, connecting rod (coupler), slider, and frame. By fixing different links, different inversions are obtained with practical applications.

• Fixing the frame and letting the crank rotate with the slider sliding gives the classic reciprocating engine or reciprocating compressor arrangement.
• Fixing the crank gives the Whitworth quick-return mechanism (used in shapers) and other rotary engines.
• Fixing the coupler or other links can produce oscillating cylinder engines, crank-and-slotted-lever mechanisms, and other useful conversions.

Crank and Slotted Lever Mechanism

• This is a form of quick-return mechanism commonly used in slotting and shaping machines. The mechanism combines rotating and sliding motions and is used to obtain different velocities in forward and return strokes.
• Given the crank angular speed and geometry, one can compute the instantaneous linear velocity of the slider and the absolute velocity of other points using kinematic relations.

Let:

• r = length of crank (OP)
• l = length of slotted lever (AR)
• c = distance between fixed centres (AO)
• ω = angular velocity of the crank

During the cutting stroke the instantaneous kinematic configuration can be represented and analysed using the geometric construction of the mechanism.

• If link 4 of the slider-crank mechanism is fixed, the mechanism becomes that of a hand pump.

Inversion of Double Slider-Crank Mechanism

A double slider-crank chain is a four-link chain having two turning pairs and two sliding pairs, arranged so that pairs of the same kind are adjacent. Typical inversions:

First inversion - Elliptical trammel

• Elliptical trammel produces exact elliptical motion. The geometric relation gives the equation of an ellipse for the traced point.

Let AC and BC be the semi-axes; then the position (x, y) of the tracing point satisfies

x²/(BC)² + y²/(AC)² = 1

When AC = BC, the locus becomes x² + y² = AC², a circle of radius AC.

Second inversion - Scotch yoke

Fixing one slide block (from the trammel) and allowing the other to move gives the scotch yoke mechanism. This converts rotary motion into simple harmonic translation and is a common method for generating reciprocation with sinusoidal velocity variation.

Third inversion - Oldham coupling

Oldham coupling: Used to connect two parallel shafts with a small offset between axes. It transmits torque while accommodating small misalignment. The relative sliding within the coupling has a maximum dependent on the shaft offset and rotational speed.

Maximum sliding velocity approximately equals the peripheral velocity corresponding to the offset, i.e. proportional to the angular speed multiplied by the distance between shaft centres.

Mechanical Advantage

• Mechanical Advantage (MA): The ratio of the output force or torque to the input force or torque at any instant. For torque-driven links, MA = output torque / input torque.
• If friction and inertial effects are neglected and input torque T2 drives the output link 4 with resisting torque T4, power balance gives:

T₂ ω₂ = T₄ ω₄

Therefore,

T₄ / T₂ = ω₂ / ω₄

Thus the mechanical advantage is the reciprocal of the instantaneous velocity ratio between the driving and driven links.

Solved Numericals

Q1: A five-bar mechanism is shown in the figure. What will be the degrees of freedom of this plane mechanism? 

(a) 3

(b) 1

(c) 2

(d) 0

Ans: (c)

Sol:

Kutzbach equation for DOF is given by

DOF = 3 (n - 1) - 2 j - h

where n = Number of links, j = Number of joints, h = Number of higher pairs.

There are 5 links and 5 binary joints.

Therefore, L = 5, j = 5, h = 0.

DOF = 3 (5 - 1) - 2 (5) - 0

DOF = 12 - 10 = 2

Q2: The degree of freedom of the mechanism shown in the figure is

(a) two

(b) zero

(c) one

(d) negative one

Ans: (c)

Sol:

Given:

Number of links N = 4.

Number of 1-DOF pairs P₁ = 3.

Number of 2-DOF pairs P₂ = 1.

F_r = 1 (redundant kinematic pair - the roller follower is used to reduce friction and does not change the kinematic function; its motion is dictated by cam rotation).

Use planar DOF formula with redundancy:

F = 3 (N - 1) - 2 P₁ - 1 P₂ - F_r

F = 3 (4 - 1) - 2 (3) - 1 (1) - 1

F = 9 - 6 - 1 - 1 = 1

Hence, the degree of freedom is 1.

Q3: The degrees of freedom of a plane mechanism as shown in the figure is:

(a) 3

(b) 4

(c) 2

(d) 1

Ans: (d)

Sol:

Number of links, N = 8.

Number of binary joints j = 10.

Number of higher pairs h = 0.

Use Kutzbach formula:

F = 3 (N - 1) - 2 j - h

F = 3 (8 - 1) - 2 × 10 - 0

F = 21 - 20 = 1

Q4: The number of degrees of freedom of the linkage shown in the figure is

(a) -3

(b) 0

(c) 1

(d) 2

Ans: (c)

Sol:

Number of links, N = 6.

Total number of binary joints j = 7.

Number of higher pairs h = 0.

Using planar formula:

F = 3 (N - 1) - 2 j

F = 3 (6 - 1) - 2 (7)

F = 15 - 14 = 1

Q5: Of the kinematic linkage below, the number of degrees of freedom (F) is:

(a) 2

(b) 3

(c) 4

(d) 1

Ans: (d)

Sol:

Given: n = 8 (links), j = 10 (binary joints), h = 0 (no higher pairs).

Use Kutzbach formula for planar mechanisms:

DOF = 3 (n - 1) - 2 j - h

DOF = 3 (8 - 1) - 2 (10) - 0

DOF = 21 - 20 = 1