Mechanism- 2 | Theory of Machines (TOM) - Mechanical Engineering PDF Download

 Page 1


                                  
                                                                     
 
 
Steering Gear Mechanism 
 
Fig. Steering Gear Mechanism 
The steering gear mechanism is used for changing the direction of two or more of the wheel 
axles with reference to the chassis, so as move the automobile in any desired path. Usually the 
two back wheels have-a common axis, which is fixed in direction with reference to the chassis 
and the steering is done by means of the front wheels.   
cot? - cot? = c / b 
Where 
a = wheel track, 
b = wheel base, and 
c = Distance between the pivots A and B of the front axle. 
Davis Steering Gear 
Ackerman Steering Gear 
The Ackerman steering gear mechanism is much simpler than Davis gear. The difference 
between the Ackerman and Davis steering gears are: 
1.  The whole mechanism of the Ackerman steering gear is on the back of the front wheels; 
whereas in Davis steering gear, it is in front of the wheels.  
2.   The Ackerman steering gear consists of turning pairs, whereas Davis steering gear consists 
of sliding members. 
Velocity and Acceleration  
 
Page 2


                                  
                                                                     
 
 
Steering Gear Mechanism 
 
Fig. Steering Gear Mechanism 
The steering gear mechanism is used for changing the direction of two or more of the wheel 
axles with reference to the chassis, so as move the automobile in any desired path. Usually the 
two back wheels have-a common axis, which is fixed in direction with reference to the chassis 
and the steering is done by means of the front wheels.   
cot? - cot? = c / b 
Where 
a = wheel track, 
b = wheel base, and 
c = Distance between the pivots A and B of the front axle. 
Davis Steering Gear 
Ackerman Steering Gear 
The Ackerman steering gear mechanism is much simpler than Davis gear. The difference 
between the Ackerman and Davis steering gears are: 
1.  The whole mechanism of the Ackerman steering gear is on the back of the front wheels; 
whereas in Davis steering gear, it is in front of the wheels.  
2.   The Ackerman steering gear consists of turning pairs, whereas Davis steering gear consists 
of sliding members. 
Velocity and Acceleration  
 
                                  
                                                                     
 
The concept of velocity and acceleration images is used extensively in the kinematic analysis of 
mechanisms having ternary, quaternary, and higher- order links. If the velocities and 
accelerations of any two points on a link are known, then, with the help of images the velocity 
and acceleration of any other point on the link can be easily determined. An example is  
 
1. Instantaneous Centre Method 
2. Relative Velocity Method 
 
Velocity by Instantaneous Centre Method 
Instantaneous centre is one point about which the body has pure rotation. Hence for the body 
which having straight line motion, the radius of curvature of it is at infinity and hence 
instantaneous centre of this ties at infinite.   
 
Special cases of ICR 
 
2
1
I
12
   
A B
V
A
V
B
2
1
I at
12
 
 
 
 
V
A
V
B
V
C
B
A
2
C
2
2
1
I
12
 
V
A
V
B
B
2
A
I
12
1
 
 
  
Types of ICR: 
  
Page 3


                                  
                                                                     
 
 
Steering Gear Mechanism 
 
Fig. Steering Gear Mechanism 
The steering gear mechanism is used for changing the direction of two or more of the wheel 
axles with reference to the chassis, so as move the automobile in any desired path. Usually the 
two back wheels have-a common axis, which is fixed in direction with reference to the chassis 
and the steering is done by means of the front wheels.   
cot? - cot? = c / b 
Where 
a = wheel track, 
b = wheel base, and 
c = Distance between the pivots A and B of the front axle. 
Davis Steering Gear 
Ackerman Steering Gear 
The Ackerman steering gear mechanism is much simpler than Davis gear. The difference 
between the Ackerman and Davis steering gears are: 
1.  The whole mechanism of the Ackerman steering gear is on the back of the front wheels; 
whereas in Davis steering gear, it is in front of the wheels.  
2.   The Ackerman steering gear consists of turning pairs, whereas Davis steering gear consists 
of sliding members. 
Velocity and Acceleration  
 
                                  
                                                                     
 
The concept of velocity and acceleration images is used extensively in the kinematic analysis of 
mechanisms having ternary, quaternary, and higher- order links. If the velocities and 
accelerations of any two points on a link are known, then, with the help of images the velocity 
and acceleration of any other point on the link can be easily determined. An example is  
 
1. Instantaneous Centre Method 
2. Relative Velocity Method 
 
Velocity by Instantaneous Centre Method 
Instantaneous centre is one point about which the body has pure rotation. Hence for the body 
which having straight line motion, the radius of curvature of it is at infinity and hence 
instantaneous centre of this ties at infinite.   
 
Special cases of ICR 
 
2
1
I
12
   
A B
V
A
V
B
2
1
I at
12
 
 
 
 
V
A
V
B
V
C
B
A
2
C
2
2
1
I
12
 
V
A
V
B
B
2
A
I
12
1
 
 
  
Types of ICR: 
  
                                  
                                                                     
 
1 1
I
12
I
24
I
23
I
34
I
13
I
14
3
2
4
 
 
(i) Fixed ICR:  
12
I, 
14
I 
(ii)  Permanent ICR: 
23
I, 
34
I 
(iii)  Neither Fixed nor Permanent I.C: 
13
I,
24
I 
 
Three-Centre-in-line Theorem (Kennedy’s Theorem) 
Kennedy Theorem states that ?If three links have relative motion with respect to each other, 
their relative instantaneous centre lies on straight line?. 
 
V
P
2
V
P
3
P
2 3
1 1
I
12
I
13
B A
 
 
The Theorem can be proved by contradiction.  
The Kennedy Theorem states that the three IC 
12
I,
13
I, 
23
I must all lie on the same straight 
line on the line connecting two pins. 
 
Let us suppose this is not true and 
23
I is located at the point P. Then the velocity of P as a 
point on link 2 must have the direction 
2
P
V, ? to AP. Also the velocity of P as a point on link 3 
must have the direction 
3
P
V, ? to BP. The direction is inconsistent with the definition that an 
instantaneous centre must have equal absolute velocity as a part of either link. The point P 
chosen therefore, cannot be the IC 
23
I. 
This same contradiction in the direction of 
2
P
V and 
3
P
V occurs for any location chosen for point 
P, except the position of P chosen on the straight line passing through 
12
I and 
13
I. This justify 
the Kennedy Theorem. 
 
Page 4


                                  
                                                                     
 
 
Steering Gear Mechanism 
 
Fig. Steering Gear Mechanism 
The steering gear mechanism is used for changing the direction of two or more of the wheel 
axles with reference to the chassis, so as move the automobile in any desired path. Usually the 
two back wheels have-a common axis, which is fixed in direction with reference to the chassis 
and the steering is done by means of the front wheels.   
cot? - cot? = c / b 
Where 
a = wheel track, 
b = wheel base, and 
c = Distance between the pivots A and B of the front axle. 
Davis Steering Gear 
Ackerman Steering Gear 
The Ackerman steering gear mechanism is much simpler than Davis gear. The difference 
between the Ackerman and Davis steering gears are: 
1.  The whole mechanism of the Ackerman steering gear is on the back of the front wheels; 
whereas in Davis steering gear, it is in front of the wheels.  
2.   The Ackerman steering gear consists of turning pairs, whereas Davis steering gear consists 
of sliding members. 
Velocity and Acceleration  
 
                                  
                                                                     
 
The concept of velocity and acceleration images is used extensively in the kinematic analysis of 
mechanisms having ternary, quaternary, and higher- order links. If the velocities and 
accelerations of any two points on a link are known, then, with the help of images the velocity 
and acceleration of any other point on the link can be easily determined. An example is  
 
1. Instantaneous Centre Method 
2. Relative Velocity Method 
 
Velocity by Instantaneous Centre Method 
Instantaneous centre is one point about which the body has pure rotation. Hence for the body 
which having straight line motion, the radius of curvature of it is at infinity and hence 
instantaneous centre of this ties at infinite.   
 
Special cases of ICR 
 
2
1
I
12
   
A B
V
A
V
B
2
1
I at
12
 
 
 
 
V
A
V
B
V
C
B
A
2
C
2
2
1
I
12
 
V
A
V
B
B
2
A
I
12
1
 
 
  
Types of ICR: 
  
                                  
                                                                     
 
1 1
I
12
I
24
I
23
I
34
I
13
I
14
3
2
4
 
 
(i) Fixed ICR:  
12
I, 
14
I 
(ii)  Permanent ICR: 
23
I, 
34
I 
(iii)  Neither Fixed nor Permanent I.C: 
13
I,
24
I 
 
Three-Centre-in-line Theorem (Kennedy’s Theorem) 
Kennedy Theorem states that ?If three links have relative motion with respect to each other, 
their relative instantaneous centre lies on straight line?. 
 
V
P
2
V
P
3
P
2 3
1 1
I
12
I
13
B A
 
 
The Theorem can be proved by contradiction.  
The Kennedy Theorem states that the three IC 
12
I,
13
I, 
23
I must all lie on the same straight 
line on the line connecting two pins. 
 
Let us suppose this is not true and 
23
I is located at the point P. Then the velocity of P as a 
point on link 2 must have the direction 
2
P
V, ? to AP. Also the velocity of P as a point on link 3 
must have the direction 
3
P
V, ? to BP. The direction is inconsistent with the definition that an 
instantaneous centre must have equal absolute velocity as a part of either link. The point P 
chosen therefore, cannot be the IC 
23
I. 
This same contradiction in the direction of 
2
P
V and 
3
P
V occurs for any location chosen for point 
P, except the position of P chosen on the straight line passing through 
12
I and 
13
I. This justify 
the Kennedy Theorem. 
 
                                  
                                                                     
 
Properties of the IC: 
1. A rigid link rotates instantaneously relative to another link at the instantaneously 
centre for the configuration of the mechanism considered. 
 
2. The two rigid links have no linear velocity relative to each other at the instantaneous 
centre. In other words, the velocity of the IC relative to any third rigid link will be same 
whether the instantaneous centre is regarded as a point on the first rigid link or on the 
second rigid link. 
 
Number of I.C in a mechanism: 
n(n 1)
N
2
?
?
 
 N = no. of I.C. 
 n = no. of links. 
 
1. Each configuration of the link has one centre. 
 The instantaneous centre changes with alteration of configuration of mechanism. 
  
Method of locating instantaneous centre in mechanism 
 Consider a pin jointed four bar mechanism as shown in fig. The following procedure is 
adopted for locating instantaneous centre. 
  
I
14
I
24
I
34
I
23
I
13
1
4
3
2
  
1
2
3 4
 
 
1. First of all, determine the no. of IC. 
N = 
n(4 1) 4(4 1)
6
32
??
?? 
2. Make a list at all the instantaneous centre in a mechanism. 
Links 1 2 3 4
12 23 34
IC 13 24
14
??
 
3. Locate the fixed and permanent instantaneous centre by inspection. In fig 
12 14
I and I 
are fixed I.Cs and 
23
I and 
34
I are permanent instantaneous centre locate the remaining 
neither fixed nor permanent IC by Kennedy‘s Theorem. This is done by circle diagram 
Page 5


                                  
                                                                     
 
 
Steering Gear Mechanism 
 
Fig. Steering Gear Mechanism 
The steering gear mechanism is used for changing the direction of two or more of the wheel 
axles with reference to the chassis, so as move the automobile in any desired path. Usually the 
two back wheels have-a common axis, which is fixed in direction with reference to the chassis 
and the steering is done by means of the front wheels.   
cot? - cot? = c / b 
Where 
a = wheel track, 
b = wheel base, and 
c = Distance between the pivots A and B of the front axle. 
Davis Steering Gear 
Ackerman Steering Gear 
The Ackerman steering gear mechanism is much simpler than Davis gear. The difference 
between the Ackerman and Davis steering gears are: 
1.  The whole mechanism of the Ackerman steering gear is on the back of the front wheels; 
whereas in Davis steering gear, it is in front of the wheels.  
2.   The Ackerman steering gear consists of turning pairs, whereas Davis steering gear consists 
of sliding members. 
Velocity and Acceleration  
 
                                  
                                                                     
 
The concept of velocity and acceleration images is used extensively in the kinematic analysis of 
mechanisms having ternary, quaternary, and higher- order links. If the velocities and 
accelerations of any two points on a link are known, then, with the help of images the velocity 
and acceleration of any other point on the link can be easily determined. An example is  
 
1. Instantaneous Centre Method 
2. Relative Velocity Method 
 
Velocity by Instantaneous Centre Method 
Instantaneous centre is one point about which the body has pure rotation. Hence for the body 
which having straight line motion, the radius of curvature of it is at infinity and hence 
instantaneous centre of this ties at infinite.   
 
Special cases of ICR 
 
2
1
I
12
   
A B
V
A
V
B
2
1
I at
12
 
 
 
 
V
A
V
B
V
C
B
A
2
C
2
2
1
I
12
 
V
A
V
B
B
2
A
I
12
1
 
 
  
Types of ICR: 
  
                                  
                                                                     
 
1 1
I
12
I
24
I
23
I
34
I
13
I
14
3
2
4
 
 
(i) Fixed ICR:  
12
I, 
14
I 
(ii)  Permanent ICR: 
23
I, 
34
I 
(iii)  Neither Fixed nor Permanent I.C: 
13
I,
24
I 
 
Three-Centre-in-line Theorem (Kennedy’s Theorem) 
Kennedy Theorem states that ?If three links have relative motion with respect to each other, 
their relative instantaneous centre lies on straight line?. 
 
V
P
2
V
P
3
P
2 3
1 1
I
12
I
13
B A
 
 
The Theorem can be proved by contradiction.  
The Kennedy Theorem states that the three IC 
12
I,
13
I, 
23
I must all lie on the same straight 
line on the line connecting two pins. 
 
Let us suppose this is not true and 
23
I is located at the point P. Then the velocity of P as a 
point on link 2 must have the direction 
2
P
V, ? to AP. Also the velocity of P as a point on link 3 
must have the direction 
3
P
V, ? to BP. The direction is inconsistent with the definition that an 
instantaneous centre must have equal absolute velocity as a part of either link. The point P 
chosen therefore, cannot be the IC 
23
I. 
This same contradiction in the direction of 
2
P
V and 
3
P
V occurs for any location chosen for point 
P, except the position of P chosen on the straight line passing through 
12
I and 
13
I. This justify 
the Kennedy Theorem. 
 
                                  
                                                                     
 
Properties of the IC: 
1. A rigid link rotates instantaneously relative to another link at the instantaneously 
centre for the configuration of the mechanism considered. 
 
2. The two rigid links have no linear velocity relative to each other at the instantaneous 
centre. In other words, the velocity of the IC relative to any third rigid link will be same 
whether the instantaneous centre is regarded as a point on the first rigid link or on the 
second rigid link. 
 
Number of I.C in a mechanism: 
n(n 1)
N
2
?
?
 
 N = no. of I.C. 
 n = no. of links. 
 
1. Each configuration of the link has one centre. 
 The instantaneous centre changes with alteration of configuration of mechanism. 
  
Method of locating instantaneous centre in mechanism 
 Consider a pin jointed four bar mechanism as shown in fig. The following procedure is 
adopted for locating instantaneous centre. 
  
I
14
I
24
I
34
I
23
I
13
1
4
3
2
  
1
2
3 4
 
 
1. First of all, determine the no. of IC. 
N = 
n(4 1) 4(4 1)
6
32
??
?? 
2. Make a list at all the instantaneous centre in a mechanism. 
Links 1 2 3 4
12 23 34
IC 13 24
14
??
 
3. Locate the fixed and permanent instantaneous centre by inspection. In fig 
12 14
I and I 
are fixed I.Cs and 
23
I and 
34
I are permanent instantaneous centre locate the remaining 
neither fixed nor permanent IC by Kennedy‘s Theorem. This is done by circle diagram 
                                  
                                                                     
 
as shown mark the points on a circle equal to the no. of links in mechanism. In present 
case 4 links. 
 
4. Join the points by solid line to show these centres are already found. In the circle 
diagram these lines are 12, 23, 34, and 14 to indicate the ICs 
12
I, 
23
I, 
34
I and
14
I , 
 
5. In order to find the other two IC, join two such points that the line joining them forms 
two adjacent triangles in the circle diagram. The line which is responsible for 
completing two triangles should be a common side to the two triangles. In fig join 1 and 
3 to form triangle 123 and 341 and the instantaneous centre 
13
I will lie on the 
intersection of 
12 23 14 34
I , I and I I . similarly IC 
24
I is located. 
 
Angular Velocity Ratio Theorem 
According to this Theorem “the ratio of angular velocity of any two links moving in a constrained 
system is inversely proportional to the ratio of distance of their common instantaneous centre from 
their centre of rotation”. 
13 23 2
3 12 23
II
II
?
?
?
 
2 14 24
4 12 24
II
II
?
?
?
 
 
Indices of Merit (Mechanical Advantage) 
From previous concept are know that  
2 14 24
4 12 24
II
II
?
?
?
  as per angular velocity ratio Theorem. 
I
12
I
24
I
14
I
34
I
23
?
2
1 1
3
2
4
?
4
 
 
Let 
2
T represent the input torque 
4
T represent the output torque. Also consider that there is no 
friction or inertia force. 
Then  
2 2 4 4
TT ? ? ? 
? ve sign indicates that power is applied to link 2 which is negative of the power applied to link 4 by 
load. 
 ?      
4 2 14 24
2 4 12 24
T I I
T I I
?
??
?
 
 
The mechanical advantage of a mechanism is the instantaneous ratio of the output force 
(torque) to the input force (torque). From above equation we know that mechanical advantage 
is the reciprocal of the velocity ratio. 
 
Fig shows a typical position of four bar linkage in toggle, where link 2 and 3 are on the same 
straight line.