Theory of Machines Formulas for GATE ME Exam | Theory of Machines (TOM) - Mechanical Engineering PDF Download

 Page 1


Theory of Machines and Vibrations – Short Notes 
Instantaneous Centre of Velocity (I-centre) 
• The instantaneous centre of velocity can be defined as a point which has no velocity 
with respect to the fixed link. 
Centro 
• Instantaneous centre is also called centro 
• Primary Centro One which can be easily located by a mere observation of the 
mechanism. 
• Secondary Centro Centros that cannot be easily located 
Aronhold-Kennedy Theorem of Three Centre 
• It state that if three bodies are in relative motion with respect to one another, the 
three relative instantaneous centers of velocity ar collinear. 
 
Number of Centros in a Mechanism 
• For a mechanism of n links, the number of centros (Instantaneous centre) N is 
 
Linkages are the basic building blocks of all mechanisms 
• Links: rigid member having nodes. 
• Node: attachment points. 
• Binary link: 2 nodes 
• Ternary link: 3 nodes 
• Quaternary link: 4 nodes 
• Joint: connection between two or more links (at their nodes) which allows motion; 
(Joints also called kinematic pairs) 
D’Alembert’s Principle and Inertia Forces 
Page 2


Theory of Machines and Vibrations – Short Notes 
Instantaneous Centre of Velocity (I-centre) 
• The instantaneous centre of velocity can be defined as a point which has no velocity 
with respect to the fixed link. 
Centro 
• Instantaneous centre is also called centro 
• Primary Centro One which can be easily located by a mere observation of the 
mechanism. 
• Secondary Centro Centros that cannot be easily located 
Aronhold-Kennedy Theorem of Three Centre 
• It state that if three bodies are in relative motion with respect to one another, the 
three relative instantaneous centers of velocity ar collinear. 
 
Number of Centros in a Mechanism 
• For a mechanism of n links, the number of centros (Instantaneous centre) N is 
 
Linkages are the basic building blocks of all mechanisms 
• Links: rigid member having nodes. 
• Node: attachment points. 
• Binary link: 2 nodes 
• Ternary link: 3 nodes 
• Quaternary link: 4 nodes 
• Joint: connection between two or more links (at their nodes) which allows motion; 
(Joints also called kinematic pairs) 
D’Alembert’s Principle and Inertia Forces 
• D’Alembert’s principle states that the reverse-effective forces and torques and the 
external forces and torques on a body together give statical equilibrium 
 
Gear Terminology 
 
Circular Pitch (p):  
• It is a distance measured along the circumference of the pitch circle from a point on 
one tooth to the corresponding point on the adjacent tooth. 
 
Diametrical Pitch (P) 
• It is the number of teeth per unit length of the pitch circle diameter in inches. 
 
Module (m) 
• It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI 
units in place of diametrical pitch. 
 
Gear Ratio (G) 
• It is the ratio of the number of teeth on the gear to that on the pinion. 
Page 3


Theory of Machines and Vibrations – Short Notes 
Instantaneous Centre of Velocity (I-centre) 
• The instantaneous centre of velocity can be defined as a point which has no velocity 
with respect to the fixed link. 
Centro 
• Instantaneous centre is also called centro 
• Primary Centro One which can be easily located by a mere observation of the 
mechanism. 
• Secondary Centro Centros that cannot be easily located 
Aronhold-Kennedy Theorem of Three Centre 
• It state that if three bodies are in relative motion with respect to one another, the 
three relative instantaneous centers of velocity ar collinear. 
 
Number of Centros in a Mechanism 
• For a mechanism of n links, the number of centros (Instantaneous centre) N is 
 
Linkages are the basic building blocks of all mechanisms 
• Links: rigid member having nodes. 
• Node: attachment points. 
• Binary link: 2 nodes 
• Ternary link: 3 nodes 
• Quaternary link: 4 nodes 
• Joint: connection between two or more links (at their nodes) which allows motion; 
(Joints also called kinematic pairs) 
D’Alembert’s Principle and Inertia Forces 
• D’Alembert’s principle states that the reverse-effective forces and torques and the 
external forces and torques on a body together give statical equilibrium 
 
Gear Terminology 
 
Circular Pitch (p):  
• It is a distance measured along the circumference of the pitch circle from a point on 
one tooth to the corresponding point on the adjacent tooth. 
 
Diametrical Pitch (P) 
• It is the number of teeth per unit length of the pitch circle diameter in inches. 
 
Module (m) 
• It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI 
units in place of diametrical pitch. 
 
Gear Ratio (G) 
• It is the ratio of the number of teeth on the gear to that on the pinion. 
 
 
where, T = number of teeth on the gear 
t = number of teeth on the pinion 
Velocity Ratio 
• The velocity ratio is defined as the ratio of the angular velocity of the follower to the 
angular velocity of the driver gear 
 
Gear Train 
• A gear train is a combination of gears used to transmit motion from one shaft to 
another. Gear trains are used to speed up or stepped down the speed of driven 
shaft. The following are main types of gear trains. 
Simple Gear Train 
• Series of gears, capable of receiving and transmitting motion from one gear to 
another is called a simple gear train. 
Train value 
 
Speed ratio  
 
Gears-and-gear-trains  
• The intermediate gears have no effect on the speed ratio and therefore they are 
known as idlers. 
Compound Gear Train 
Page 4


Theory of Machines and Vibrations – Short Notes 
Instantaneous Centre of Velocity (I-centre) 
• The instantaneous centre of velocity can be defined as a point which has no velocity 
with respect to the fixed link. 
Centro 
• Instantaneous centre is also called centro 
• Primary Centro One which can be easily located by a mere observation of the 
mechanism. 
• Secondary Centro Centros that cannot be easily located 
Aronhold-Kennedy Theorem of Three Centre 
• It state that if three bodies are in relative motion with respect to one another, the 
three relative instantaneous centers of velocity ar collinear. 
 
Number of Centros in a Mechanism 
• For a mechanism of n links, the number of centros (Instantaneous centre) N is 
 
Linkages are the basic building blocks of all mechanisms 
• Links: rigid member having nodes. 
• Node: attachment points. 
• Binary link: 2 nodes 
• Ternary link: 3 nodes 
• Quaternary link: 4 nodes 
• Joint: connection between two or more links (at their nodes) which allows motion; 
(Joints also called kinematic pairs) 
D’Alembert’s Principle and Inertia Forces 
• D’Alembert’s principle states that the reverse-effective forces and torques and the 
external forces and torques on a body together give statical equilibrium 
 
Gear Terminology 
 
Circular Pitch (p):  
• It is a distance measured along the circumference of the pitch circle from a point on 
one tooth to the corresponding point on the adjacent tooth. 
 
Diametrical Pitch (P) 
• It is the number of teeth per unit length of the pitch circle diameter in inches. 
 
Module (m) 
• It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI 
units in place of diametrical pitch. 
 
Gear Ratio (G) 
• It is the ratio of the number of teeth on the gear to that on the pinion. 
 
 
where, T = number of teeth on the gear 
t = number of teeth on the pinion 
Velocity Ratio 
• The velocity ratio is defined as the ratio of the angular velocity of the follower to the 
angular velocity of the driver gear 
 
Gear Train 
• A gear train is a combination of gears used to transmit motion from one shaft to 
another. Gear trains are used to speed up or stepped down the speed of driven 
shaft. The following are main types of gear trains. 
Simple Gear Train 
• Series of gears, capable of receiving and transmitting motion from one gear to 
another is called a simple gear train. 
Train value 
 
Speed ratio  
 
Gears-and-gear-trains  
• The intermediate gears have no effect on the speed ratio and therefore they are 
known as idlers. 
Compound Gear Train 
• When a series of gears are connected in such a way that two or more gears rotate 
about an axis with the same angular velocity. 
 
Train value 
 
Planetary or Epicyclic Gear Train 
• A gear train having a relative motion of axes is called a planetary or an epicyclic gear 
train. In an epicyclic train, the axis of at least one of the gears also moves relative to 
the frame. 
• If the arm a is fixed the wheels S and P constitute a simple train. However if the 
wheel S is fixed so that arm a can rotate about the axis of S. The P would be moved 
around S therefore it is an epicyclic train 
Flywheel 
• A flywheel is used to control the variations in speed during each cycle of an 
operation. A flywheel acts as a reservoir of energy which stores energy during the 
period when the supply of energy is more than the requirement and releases the 
energy during the period when the supply energy is less than the requirement. 
Maximum fluctuation of energy (e), 
 
where, 
?max and ?min are the maximum and minimum angular speed respectively. 
Page 5


Theory of Machines and Vibrations – Short Notes 
Instantaneous Centre of Velocity (I-centre) 
• The instantaneous centre of velocity can be defined as a point which has no velocity 
with respect to the fixed link. 
Centro 
• Instantaneous centre is also called centro 
• Primary Centro One which can be easily located by a mere observation of the 
mechanism. 
• Secondary Centro Centros that cannot be easily located 
Aronhold-Kennedy Theorem of Three Centre 
• It state that if three bodies are in relative motion with respect to one another, the 
three relative instantaneous centers of velocity ar collinear. 
 
Number of Centros in a Mechanism 
• For a mechanism of n links, the number of centros (Instantaneous centre) N is 
 
Linkages are the basic building blocks of all mechanisms 
• Links: rigid member having nodes. 
• Node: attachment points. 
• Binary link: 2 nodes 
• Ternary link: 3 nodes 
• Quaternary link: 4 nodes 
• Joint: connection between two or more links (at their nodes) which allows motion; 
(Joints also called kinematic pairs) 
D’Alembert’s Principle and Inertia Forces 
• D’Alembert’s principle states that the reverse-effective forces and torques and the 
external forces and torques on a body together give statical equilibrium 
 
Gear Terminology 
 
Circular Pitch (p):  
• It is a distance measured along the circumference of the pitch circle from a point on 
one tooth to the corresponding point on the adjacent tooth. 
 
Diametrical Pitch (P) 
• It is the number of teeth per unit length of the pitch circle diameter in inches. 
 
Module (m) 
• It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI 
units in place of diametrical pitch. 
 
Gear Ratio (G) 
• It is the ratio of the number of teeth on the gear to that on the pinion. 
 
 
where, T = number of teeth on the gear 
t = number of teeth on the pinion 
Velocity Ratio 
• The velocity ratio is defined as the ratio of the angular velocity of the follower to the 
angular velocity of the driver gear 
 
Gear Train 
• A gear train is a combination of gears used to transmit motion from one shaft to 
another. Gear trains are used to speed up or stepped down the speed of driven 
shaft. The following are main types of gear trains. 
Simple Gear Train 
• Series of gears, capable of receiving and transmitting motion from one gear to 
another is called a simple gear train. 
Train value 
 
Speed ratio  
 
Gears-and-gear-trains  
• The intermediate gears have no effect on the speed ratio and therefore they are 
known as idlers. 
Compound Gear Train 
• When a series of gears are connected in such a way that two or more gears rotate 
about an axis with the same angular velocity. 
 
Train value 
 
Planetary or Epicyclic Gear Train 
• A gear train having a relative motion of axes is called a planetary or an epicyclic gear 
train. In an epicyclic train, the axis of at least one of the gears also moves relative to 
the frame. 
• If the arm a is fixed the wheels S and P constitute a simple train. However if the 
wheel S is fixed so that arm a can rotate about the axis of S. The P would be moved 
around S therefore it is an epicyclic train 
Flywheel 
• A flywheel is used to control the variations in speed during each cycle of an 
operation. A flywheel acts as a reservoir of energy which stores energy during the 
period when the supply of energy is more than the requirement and releases the 
energy during the period when the supply energy is less than the requirement. 
Maximum fluctuation of energy (e), 
 
where, 
?max and ?min are the maximum and minimum angular speed respectively. 
E = kinematic energy of the flywheel at mean speed. 
Flywheel in Punching Press 
• Generally, flywheel is used to reduce fluctuation of speed where the load on the 
crank shaft constant while the applied torque varies. 
• However, the flywheel can also be used to reduce fluctuation of speed when the 
torque is constant but load varies during the cycle e.g., in punching press in riveting 
machine. 
• Let E be energy required for one punch energy supplied to crank shaft from the 
motor during punching 
 
Governors 
The function of a governor is to maintain or regulate the speed of an engine within specified 
limits whenever there is variation of load. 
Types of Governors 
The broadly classification of the governors are given below. 
Centrifugal Governor 
• In this type of governor, the action of governor depends upon the centrifugal effects 
produced by the masses of two balls. 
Inertia Governor 
• In this type of governor, positions of the balls are effected by the forces set up by an 
angular acceleration or deceleration of the given spindle in addition to centrifugal 
forces on the balls. 
Pendulum Type Watt Governor 
 
height of each bal