Theory of Machines Notes for SSC JE Mechanical: Gears, Cams, Governors and Flywheel
Rahul Kumar
SSC Exam Expert & Content Editor
Updated 15 Oct 2026
In this note
- 1.Basic Terms: Links, Pairs, Chains and Mechanisms
- 2.Inversions of the Four-Bar, Slider-Crank and Double-Slider Chains
- 3.Velocity and Acceleration in Mechanisms
- 4.Belt, Rope and Chain Drives
- 5.Gears
- 6.Cams and Followers
- 7.Governors and Flywheel
- 8.Balancing of Rotating and Reciprocating Masses
- 9.Vibrations
- 10.Friction, Clutches, Brakes and Dynamometers
- 11.Important Questions Explained
- 12.Quick Tips to Remember
- 13.Common Mistakes to Avoid
- 14.Quick Revision Points
- 15.Practice Questions With Answers
- 16.How to Practise This Topic
- 17.Related Notes and Practice
- 18.Frequently Asked Questions
PDF Notes: Theory of Machines Notes for SSC JE Mechanical: Gears, Cams, Governors and Flywheel
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Theory of Machines (TOM) is a high-weightage subject of the SSC JE Mechanical paper. Every year 6 to 8 questions come on the mechanisms, the belt and the gear drives, the cams, the governors, the flywheel, the balancing and the vibrations. These theory of machines notes for SSC JE Mechanical give you the definitions, the formulas and the standard values in simple tables, followed by 20 practice questions. Read them with the Thermodynamics Notes for SSC JE Mechanical, Strength of Materials Notes for SSC JE Mechanical and IC Engines, Refrigeration and Air Conditioning Notes for SSC JE Mechanical notes.
Basic Terms: Links, Pairs, Chains and Mechanisms
| Term | Meaning |
|---|---|
| Kinematic link | A resistant body that has the relative motion with the other bodies; it may be rigid, flexible or fluid |
| Kinematic pair | Two links in contact that have a relative motion; lower pair (surface contact: turning, sliding, screw, spherical) and higher pair (point or line contact: cam and follower, gears, ball bearing) |
| Kinematic chain | A group of the links joined so that the motion of one gives a definite motion to the others |
| Mechanism | A kinematic chain in which one link is fixed |
| Machine | A mechanism that transmits the force and does the useful work |
| Inversion | Fixing the different links of a chain one by one to get the different mechanisms |
| Degrees of freedom (mobility) | The number of the independent inputs needed to fix the position of a mechanism; Grubler's criterion for a plane mechanism: F = 3 (n - 1) - 2 j - h (n links, j lower pairs, h higher pairs) |
- A rigid body in a plane has 3 degrees of freedom; a link in space has 6.
- A mechanism with F = 1 has a constrained (definite) motion; with F = 0 it is a structure; with F less than 0 it is a super-structure.
- Grashof's law for the four-bar chain: s + l is less than or equal to p + q (s: the shortest, l: the longest link and p, q: the other links); then at least one link can make a full rotation.
Inversions of the Four-Bar, Slider-Crank and Double-Slider Chains
| Chain | Inversions and examples |
|---|---|
| Four-bar chain | Beam engine (crank and lever mechanism), coupling rod of a locomotive (double crank), Watt's indicator mechanism (a double lever) |
| Single slider crank chain | Reciprocating engine and compressor (the fixed link is the cylinder), the rotary (Gnome) engine, the crank and slotted lever quick return motion (Whitworth, used in the shaper), the oscillating cylinder engine and the pendulum pump or the Bull engine |
| Double slider crank chain | Elliptical trammel, Scotch yoke mechanism, Oldham's coupling |
- Quick return mechanism: the Whitworth quick return and the crank and slotted lever mechanism give the cutting stroke slower than the return stroke in the shaper; the quick return ratio = the cutting time / the return time.
- Steering gears: the Davis steering gear (has the sliding pairs) and the Ackermann steering gear (has the turning pairs only, so it is simple and wears less; it is used in the automobiles). The condition for the correct steering: cot (phi) - cot (theta) = b / L.
- Hooke's joint (universal joint) connects two shafts with the intersecting axes; Pantograph copies a figure to a different scale.
Velocity and Acceleration in Mechanisms
- The instantaneous centre (I-centre) is the point about which a body seems to rotate at a given instant; the number of the I-centres in a mechanism of n links = n (n - 1) / 2. A four-bar chain has 6 and a slider-crank has 6 I-centres.
- Kennedy's theorem: three bodies in a relative motion have three instantaneous centres that lie on one straight line.
- For a slider-crank mechanism (crank radius r, connecting rod length l, n = l / r, the crank angle theta from the inner dead centre): the velocity of the piston v = omega r (sin theta + sin 2 theta / (2 n)) and the acceleration a = omega squared r (cos theta + cos 2 theta / n).
- The Coriolis acceleration = 2 omega v acts when a point moves on a rotating link.
- The velocity of a point on a rotating link v = omega x r; the centripetal acceleration = omega squared r = v squared / r.
Belt, Rope and Chain Drives
| Item | Formula and fact |
|---|---|
| Velocity ratio | N1 / N2 = d2 / d1 (without the slip); with the slip: N2 / N1 = d1 / d2 x [1 - (s1 + s2) / 100] |
| Length of the open belt | L = pi (r1 + r2) + 2 x + (r1 - r2) squared / x, where x is the centre distance |
| Length of the crossed belt | L = pi (r1 + r2) + 2 x + (r1 + r2) squared / x |
| Ratio of the tensions (flat belt) | T1 / T2 = e to the power (mu theta) |
| Ratio of the tensions (V-belt) | T1 / T2 = e to the power (mu theta / sin beta), where beta is the semi-angle of the groove |
| Power transmitted | P = (T1 - T2) v |
| Centrifugal tension | Tc = m v squared (m: mass per unit length) |
| Maximum power | The power is the maximum when T1 = 3 Tc, that is, when the velocity v = square root of (T / (3 m)); T = the maximum tension of the belt = stress x area |
| Initial tension | To = (T1 + T2) / 2 (neglecting the centrifugal tension) |
- The crossed belt gives the opposite direction of rotation and more angle of contact; the open belt gives the same direction.
- The V-belt transmits more power than the flat belt (the wedge action) and has no joint; the chain drive has no slip and gives a constant velocity ratio.
- The creep is the small relative motion between the belt and the pulley due to the unequal elongation.
Gears
| Term | Meaning and formula |
|---|---|
| Types | Spur gear (parallel shafts), helical gear (parallel shafts; smooth and quiet), bevel gear (intersecting shafts), worm and worm wheel (non-parallel and non-intersecting shafts; a high speed reduction), rack and pinion |
| Module (m) | m = pitch circle diameter / number of teeth = d / T (in mm) |
| Circular pitch (pc) | pc = pi d / T = pi m |
| Diametral pitch (P) | P = T / d = 1 / m |
| Addendum and dedendum | Addendum = 1 module; dedendum = 1.25 module (the standard 20 degree full depth system); the clearance = 0.25 module |
| Pressure angle | 20 degrees is the standard (14.5 degrees was older) |
| Velocity ratio | N1 / N2 = T2 / T1 = d2 / d1 |
| Centre distance | (d1 + d2) / 2 = m (T1 + T2) / 2 |
| Law of gearing | The common normal at the point of the contact must pass through the pitch point to give a constant velocity ratio |
| Involute profile | The most common tooth profile; the centre distance variation does not change the velocity ratio |
| Interference | The tip of the tooth of one gear cuts the root of the other; the minimum teeth on the pinion to avoid the interference is 18 for the 20 degree full depth involute (32 for the 14.5 degree) |
| Contact ratio | The average number of the pairs of the teeth in contact; it should be more than 1 |
- Gear trains: simple, compound, reverted and epicyclic (planetary) gear trains. The epicyclic gear train is compact and gives a high velocity ratio (used in the automobile differential, the automatic transmission and the hoists).
- Train value = the speed of the driven / the speed of the driver = the product of the teeth of the drivers / the product of the teeth of the driven.
- The idler gear changes only the direction of the rotation and not the velocity ratio.
Cams and Followers
| Item | Types and facts |
|---|---|
| Types of cams | Disc (plate or radial) cam, cylindrical (barrel) cam, wedge cam and the translating cam |
| Types of followers | Knife-edge, roller, flat-faced (mushroom) and spherical-faced followers; the roller follower has less wear |
| Motion of the follower | Uniform velocity, simple harmonic motion (SHM), uniform acceleration and retardation (parabolic) and cycloidal motion; the cycloidal motion has no jerk and is the best for the high speeds |
| Terms | Base circle (the smallest circle), prime circle, pitch curve (the path of the trace point), trace point, pressure angle (the angle between the direction of the follower motion and the normal to the pitch curve; the maximum is about 30 degrees), lift (stroke) of the follower, dwell (the follower stays at rest) |
- The velocity of the follower in the SHM at the cam angle theta: v = (pi omega s / (2 phi)) sin (pi theta / phi); the maximum velocity = pi omega s / (2 phi), where s is the stroke and phi is the angle of the rise.
- The cam is used to open and close the valves in the IC engine (see the IC Engines, Refrigeration and Air Conditioning Notes for SSC JE Mechanical note).
- The jump of the follower happens at a high speed when the follower loses the contact with the cam.
Governors and Flywheel
| Item | Facts |
|---|---|
| Function of a governor | To control the mean speed of an engine within the limits when the load changes (by controlling the fuel supply); it is different from the flywheel |
| Function of a flywheel | To reduce the fluctuation of the speed during one cycle (the cyclic fluctuation) by storing and giving the energy |
| Centrifugal governors | Watt (a simple pendulum type), Porter (a loaded type with the central load), Proell (the balls are fixed on the extension of the links) and Hartnell (a spring-loaded type); the Hartung governor is also spring controlled |
| Terms of a governor | Sensitiveness (the ratio of the speed range to the mean speed: more is better); stability (the equilibrium speed is the one for each radius); isochronism (a constant speed for all the radii; sensitiveness is infinite); hunting (the continuous fluctuation of the engine speed due to over-sensitiveness); effort and power |
| Watt governor | Height h = 895 / N squared metres (N in rpm) |
| Coefficient of fluctuation of speed (Cs) | Cs = (N1 - N2) / N (the maximum minus the minimum speed over the mean speed) |
| Energy of the flywheel | Maximum fluctuation of the energy delta E = I omega squared Cs = m k squared omega squared Cs |
| Moment of inertia | Solid disc: m r squared / 2; thin ring: m r squared |
| Coefficient of fluctuation of energy | Ce = the maximum fluctuation of energy / the work done per cycle |
- The flywheel is used in the punching and the shearing machines (to supply the energy during the working stroke) and in the multi-cylinder engines (a smaller flywheel is enough).
- A governor is used with the steam engine and the turbine to keep the speed constant; the flywheel is used with the reciprocating engines.
Balancing of Rotating and Reciprocating Masses
- Static balancing: the resultant of all the centrifugal forces is zero (the sum of the m r vectors is zero); it is enough for the disc-type rotors in a single plane.
- Dynamic balancing: both the resultant force and the resultant couple are zero; it is needed for the long rotors (the masses in the different planes).
- A single rotating mass in a plane is balanced by the equal and opposite mass in the same plane, or by two masses in two different planes.
- Reciprocating masses: the primary unbalanced force = m omega squared r cos theta and the secondary force = m omega squared r (cos 2 theta / n). Only the partial balancing of the reciprocating mass is possible in a single cylinder engine; it causes the swaying couple, the tractive force and the hammer blow in the locomotives.
- In the in-line six-cylinder engine, the primary and the secondary forces and the couples are all balanced; in the four-cylinder in-line engine, the primary forces and the couples are balanced but the secondary forces are not.
Vibrations
| Term | Meaning and formula |
|---|---|
| Free (natural) vibration | The system vibrates under its own elastic force without an external force; the natural frequency fn = (1 / 2 pi) square root of (k / m); the time period T = 2 pi square root of (m / k) |
| Damped vibration | The amplitude decreases with the time because of the damping; the damping ratio zeta = c / c_critical; critically damped if zeta = 1, under-damped if zeta is less than 1 and over-damped if more than 1 |
| Forced vibration | Under an external periodic force; the resonance occurs when the forcing frequency is equal to the natural frequency (the amplitude becomes very large) |
| Logarithmic decrement | delta = ln (x1 / x2) = 2 pi zeta / square root of (1 - zeta squared) |
| Transmissibility | The ratio of the force transmitted to the force applied; the isolation is effective when the frequency ratio is more than square root of 2 |
| Types of the shaft vibrations | Longitudinal, transverse and torsional vibrations |
| Critical (whirling) speed of a shaft | The speed at which the shaft speed equals its natural frequency of the transverse vibration; the shaft should not run near it |
| Degrees of freedom | The number of the independent co-ordinates to describe the vibration |
Friction, Clutches, Brakes and Dynamometers
| Item | Facts and formulas |
|---|---|
| Angle of friction | tan phi = mu (the coefficient of friction) |
| Screw jack efficiency | eta = tan alpha / tan (alpha + phi) while lifting; the maximum efficiency = (1 - sin phi) / (1 + sin phi); a screw is self-locking if phi is more than alpha (efficiency less than 50 percent) |
| Single plate clutch | Uniform pressure: T = (2/3) mu W (R cubed - r cubed) / (R squared - r squared); uniform wear: T = mu W (R + r) / 2 for one friction surface; the number of the friction surfaces is 2 for a single plate clutch (both faces of the plate); the uniform wear theory is used for the design as it gives a lower torque |
| Other clutches | Multi-plate, cone and centrifugal clutches |
| Brakes | Block (shoe) brake, band brake, band and block brake, internal expanding shoe brake (the drum brake of the automobiles) and the disc brake; the brake absorbs the energy while the clutch transmits it |
| Dynamometers | Absorption type (Prony brake, rope brake, hydraulic dynamometer) measures the power by absorbing it; transmission type (belt transmission, epicyclic train and torsion dynamometers) measures the power while transmitting it |
| Brake power | BP = 2 pi N T / 60 (watts) |
- Rolling friction is less than the sliding friction; the limiting friction is the maximum friction just before the sliding.
- Lubrication: the thick film (hydrodynamic), the thin film (boundary) and the extreme pressure lubrication; the Sommerfeld number is used in the journal bearing design.
- Bearings: the sliding contact bearings (journal bearings) and the rolling contact bearings (ball and roller bearings).
Important Questions Explained
Question 1: What is Grubler's criterion for a plane mechanism?
Answer: F = 3 (n - 1) - 2 j - h, where n is the number of links, j the number of lower pairs and h the number of higher pairs.
Question 2: How many instantaneous centres does a mechanism of 4 links have?
Answer: n (n - 1) / 2 = 4 x 3 / 2 = 6.
Question 3: What is the velocity ratio of two gears of 20 and 60 teeth if the 20-teeth gear is the driver?
Answer: N1 / N2 = T2 / T1 = 60 / 20 = 3, so the driver rotates 3 times faster.
Question 4: What is the module of a gear of 100 mm pitch diameter with 25 teeth?
Answer: m = d / T = 100 / 25 = 4 mm.
Question 5: What is the standard addendum and dedendum in terms of the module?
Answer: The addendum = 1 module and the dedendum = 1.25 module.
Question 6: Which governor is a spring-loaded governor?
Answer: The Hartnell governor.
Question 7: What is the formula of the natural frequency of a spring-mass system?
Answer: fn = (1 / 2 pi) square root of (k / m).
Quick Tips to Remember
- Learn the inversions with their examples: shaper (quick return), Oldham's coupling, Scotch yoke and elliptical trammel.
- Remember the gear terms in the module: addendum m, dedendum 1.25 m and the pc = pi m.
- Remember that the governor controls the mean speed and the flywheel controls the cyclic fluctuation.
- Remember the belt formulas: T1 / T2 = e to the power mu theta; P = (T1 - T2) v and the condition T1 = 3 Tc.
- Remember the minimum teeth to avoid the interference: 18 (20 degree) and 32 (14.5 degree).
- Remember Cs = (N1 - N2) / N and the energy delta E = I omega squared Cs.
Common Mistakes to Avoid
- Mixing up the governor and the flywheel functions.
- Using the diameter in place of the radius in the belt length formula.
- Writing the wrong number of the I-centres (the formula is n (n - 1) / 2).
- Mixing the module (m = d / T) with the diametral pitch (P = T / d).
- Confusing the uniform pressure and the uniform wear theories of the clutch.
Quick Revision Points
- Grubler: F = 3 (n - 1) - 2j - h; number of I-centres n (n - 1) / 2.
- Module m = d / T; pc = pi m; addendum 1m; dedendum 1.25 m; pressure angle 20 degrees; minimum teeth 18.
- Belt: T1 / T2 = e to the power mu theta; P = (T1 - T2) v; Tc = m v squared; max power T1 = 3 Tc.
- Governor: Watt, Porter, Proell (gravity) and Hartnell (spring); sensitiveness; hunting.
- Flywheel: delta E = I omega squared Cs.
- Vibration: fn = (1 / 2 pi) root (k / m); resonance when the frequencies are equal.
- Clutch: uniform wear T = mu W (R + r) / 2.
Practice Questions With Answers
- Which pair has the surface contact?
Answer: Lower pair (Turning, sliding, screw). - Which pair has the line or point contact?
Answer: Higher pair (Cam and follower, gears). - What is Grubler's criterion?
Answer: F = 3 (n - 1) - 2j - h (Degrees of freedom). - What is the number of the instantaneous centres in a 6-link mechanism?
Answer: 15 (n (n - 1) / 2). - Which mechanism is used in a shaper?
Answer: Whitworth quick return mechanism (Slotted lever). - Which coupling is an inversion of the double slider crank chain?
Answer: Oldham's coupling (Parallel shafts). - Which steering gear has only the turning pairs?
Answer: Ackermann steering gear (Automobiles). - What is the formula of the module of a gear?
Answer: m = d / T (mm). - What is the standard pressure angle of a gear?
Answer: 20 degrees (Involute). - What is the minimum number of the teeth of a 20 degree pinion to avoid the interference?
Answer: 18 (Full depth). - What is the standard addendum of a gear?
Answer: 1 module (Dedendum 1.25 module). - Which gear is used for the large speed reduction between the non-intersecting shafts?
Answer: Worm gear (Worm and wheel). - Which cam follower motion has the least jerk?
Answer: Cycloidal motion (High speed cams). - Which governor is spring-loaded?
Answer: Hartnell governor (Centrifugal). - What is the function of a flywheel?
Answer: To reduce the cyclic speed fluctuation (Stores energy). - What is the function of a governor?
Answer: To control the mean speed against the load change (Fuel supply). - What is the formula of the coefficient of fluctuation of speed?
Answer: (N1 - N2) / N (Flywheel). - At which frequency does the resonance happen?
Answer: When the forcing frequency equals the natural frequency (Large amplitude). - What is the formula of the power transmitted by a belt?
Answer: P = (T1 - T2) v (Watts). - Which brake is used in the automobiles as the drum brake?
Answer: Internal expanding shoe brake (Brake).
How to Practise This Topic
- Make a one-page sheet of the chains and their inversions.
- Learn the gear terms and the belt formulas by solving two numerical problems each.
- Revise the governors, the flywheel and the vibration formulas the next day.
- Attempt the practice questions without looking at the answers.
- Solve the SSC practice papers and the previous year papers to practise the JE pattern.
Related Notes and Practice
- Thermodynamics Notes for SSC JE Mechanical
- Strength of Materials Notes for SSC JE Mechanical
- IC Engines, Refrigeration and Air Conditioning Notes for SSC JE Mechanical
- Mechanical Engineering (SSC JE) Notes
- Civil Engineering (SSC JE) Notes
- SSC practice papers
Frequently Asked Questions
What is the theory of machines?
The theory of machines is the branch of mechanical engineering that studies the motion and the forces in the mechanisms and the machines, such as the gears, the cams, the belts and the governors.
What is the difference between a mechanism and a machine?
A mechanism is a kinematic chain in which one link is fixed and gives a definite motion, while a machine also transmits the force and does the useful work.
What is Grubler's criterion?
Grubler's criterion gives the degrees of freedom of a plane mechanism: F = 3 (n - 1) - 2j - h.
What is the module of a gear?
The module is the ratio of the pitch circle diameter to the number of the teeth, m = d / T, in millimetres.
What is the difference between a governor and a flywheel?
A governor controls the mean speed of an engine against the load changes, while a flywheel reduces the speed fluctuation within a single cycle.
What is the law of gearing?
The law of gearing says that for a constant velocity ratio, the common normal at the point of contact of the teeth must always pass through the pitch point.
What is the interference in gears?
Interference is the rubbing of the tip of the teeth of one gear on the root of the teeth of the mating gear, which can be avoided by using at least 18 teeth on the pinion (20 degree system).
What is balancing?
Balancing is the process of making the resultant force and the couple of the rotating or reciprocating masses zero to avoid the vibrations.
What is resonance?
Resonance is the condition in which the frequency of the external force equals the natural frequency of the system and the amplitude of the vibration becomes very large.
What is the difference between a clutch and a brake?
A clutch connects and disconnects two shafts to transmit the power, while a brake absorbs the energy to slow down or stop a moving body.
How many questions come from the theory of machines in the SSC JE exam?
Usually 6 to 8 questions come from the theory of machines and the machine design in the SSC JE Mechanical paper.
How can I prepare the theory of machines for SSC JE?
Learn the definitions and the formulas in tables, solve the numerical problems, and revise the previous year questions.
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About the Author
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.