Short Notes: Displacement Velocity and Acceleration Analysis of Plane Mechanisms | Short Notes for Mechanical Engineering PDF Download

 Page 1


Displacement Velocity and Acceleration Analysis of Plane 
Mechanisms
Velocity Analysis in Mechanism
• Let a rigid link OA, of length r rotate about a fixed point 0 with a uniform 
angular velocity rad/s in a counter-clockwise direction OA turns through a 
small angle 56 in a small interval of time 5t. Then, A will travel along the arc 
AA’ as shown in figure.
rrrrrrrrrtrrrrrrrrn
o °
Velocity Analysis
Velocity of A relative to 0
In the limits, when
arcAA' _ r6d
68 - ~ r
v dd 
F = r — =r. 
“ A
Thus, velocity of A is wr and is perpendicular to OA. 
Velocity of Intermediate Point
• If represent the velocity of B with respect to 0, then
V i; _ ~OB _ OB
OA AO
Page 2


Displacement Velocity and Acceleration Analysis of Plane 
Mechanisms
Velocity Analysis in Mechanism
• Let a rigid link OA, of length r rotate about a fixed point 0 with a uniform 
angular velocity rad/s in a counter-clockwise direction OA turns through a 
small angle 56 in a small interval of time 5t. Then, A will travel along the arc 
AA’ as shown in figure.
rrrrrrrrrtrrrrrrrrn
o °
Velocity Analysis
Velocity of A relative to 0
In the limits, when
arcAA' _ r6d
68 - ~ r
v dd 
F = r — =r. 
“ A
Thus, velocity of A is wr and is perpendicular to OA. 
Velocity of Intermediate Point
• If represent the velocity of B with respect to 0, then
V i; _ ~OB _ OB
OA AO
b
a-*— |----------------o
Intermediate points b
i.e., b divides the velocity vector in the same ratio as B divides the link. The 
magnitude of the linear velocity of a point on the rotating body at a particular 
instant is proportional to its distance fromt the axis of rotation.
Velocity Images of Four Link Mechanism
• Figure shows a four link mechanism (quadric cycle mechanism) ABCD in 
which AD is fixed link and BC is the coupler. AB is the driver rotating at an 
angular speed of lj rad/s in the clockwise direction if it is a crank or moving 
at angular velocity c j at this instant if it is rocker.
Velocity Images of Slider-Crank Mechanism
• Consider a slider-crank mechanism in which OA is the crank moving with 
uniform angular velocity cj rad/s in the clockwise direction. At point B, a slider 
moves on the fixed guide G .
From the given configuration, the coupler AB has angular velocity in the counter­
clockwise direction. The magnitude being
BA
Velocity images ol slider-crank mechanism
Velocity of Rubbing
• Let us take two links of a turning pair, a pin is fixed to one of the links whereas 
a hole is provided in the other to fit the pin. When joined the surface of the 
hole of one link will rub on the surface of pin of the other link. The velocity of
Page 3


Displacement Velocity and Acceleration Analysis of Plane 
Mechanisms
Velocity Analysis in Mechanism
• Let a rigid link OA, of length r rotate about a fixed point 0 with a uniform 
angular velocity rad/s in a counter-clockwise direction OA turns through a 
small angle 56 in a small interval of time 5t. Then, A will travel along the arc 
AA’ as shown in figure.
rrrrrrrrrtrrrrrrrrn
o °
Velocity Analysis
Velocity of A relative to 0
In the limits, when
arcAA' _ r6d
68 - ~ r
v dd 
F = r — =r. 
“ A
Thus, velocity of A is wr and is perpendicular to OA. 
Velocity of Intermediate Point
• If represent the velocity of B with respect to 0, then
V i; _ ~OB _ OB
OA AO
b
a-*— |----------------o
Intermediate points b
i.e., b divides the velocity vector in the same ratio as B divides the link. The 
magnitude of the linear velocity of a point on the rotating body at a particular 
instant is proportional to its distance fromt the axis of rotation.
Velocity Images of Four Link Mechanism
• Figure shows a four link mechanism (quadric cycle mechanism) ABCD in 
which AD is fixed link and BC is the coupler. AB is the driver rotating at an 
angular speed of lj rad/s in the clockwise direction if it is a crank or moving 
at angular velocity c j at this instant if it is rocker.
Velocity Images of Slider-Crank Mechanism
• Consider a slider-crank mechanism in which OA is the crank moving with 
uniform angular velocity cj rad/s in the clockwise direction. At point B, a slider 
moves on the fixed guide G .
From the given configuration, the coupler AB has angular velocity in the counter­
clockwise direction. The magnitude being
BA
Velocity images ol slider-crank mechanism
Velocity of Rubbing
• Let us take two links of a turning pair, a pin is fixed to one of the links whereas 
a hole is provided in the other to fit the pin. When joined the surface of the 
hole of one link will rub on the surface of pin of the other link. The velocity of
rubbing of the two surfaces will depend upon the angular velocity of a link 
relative to the other.
Velocity of rubbing
Pin at A
• The pin at A joins links AD and AB. AD being fixed, the velocity of rubbing will 
depend upon the angular velocity of AS only.
• Velocity of rubbing = rac u
where, ra = radius of pin at A
mechanism 
Pin at B
iO b a - ojab - o j (clockwise)
> = • = i k
* BC
(counter-clockwise)
• rb = Radius of pin at B
Velocity of rubbing = rb(ouab + &W 
Pin at C
W bc = wC b (counter-clockwise)
(jJ d c - W ed (clockwise) 
rc = Radius of pin at C 
Velocity of rubbing = rc(u)bc + wc /c )
Pin at D
where, rd = radius of pin at D 
Velocity of rubbing = rd ojc d 
Instantaneous Centre of Velocity (l-centre)
• The instantaneous centre of velocity can be defined as a point which has no 
velocity with respect to the fixed link.
• Suppose there are two link 1 and link 2
Page 4


Displacement Velocity and Acceleration Analysis of Plane 
Mechanisms
Velocity Analysis in Mechanism
• Let a rigid link OA, of length r rotate about a fixed point 0 with a uniform 
angular velocity rad/s in a counter-clockwise direction OA turns through a 
small angle 56 in a small interval of time 5t. Then, A will travel along the arc 
AA’ as shown in figure.
rrrrrrrrrtrrrrrrrrn
o °
Velocity Analysis
Velocity of A relative to 0
In the limits, when
arcAA' _ r6d
68 - ~ r
v dd 
F = r — =r. 
“ A
Thus, velocity of A is wr and is perpendicular to OA. 
Velocity of Intermediate Point
• If represent the velocity of B with respect to 0, then
V i; _ ~OB _ OB
OA AO
b
a-*— |----------------o
Intermediate points b
i.e., b divides the velocity vector in the same ratio as B divides the link. The 
magnitude of the linear velocity of a point on the rotating body at a particular 
instant is proportional to its distance fromt the axis of rotation.
Velocity Images of Four Link Mechanism
• Figure shows a four link mechanism (quadric cycle mechanism) ABCD in 
which AD is fixed link and BC is the coupler. AB is the driver rotating at an 
angular speed of lj rad/s in the clockwise direction if it is a crank or moving 
at angular velocity c j at this instant if it is rocker.
Velocity Images of Slider-Crank Mechanism
• Consider a slider-crank mechanism in which OA is the crank moving with 
uniform angular velocity cj rad/s in the clockwise direction. At point B, a slider 
moves on the fixed guide G .
From the given configuration, the coupler AB has angular velocity in the counter­
clockwise direction. The magnitude being
BA
Velocity images ol slider-crank mechanism
Velocity of Rubbing
• Let us take two links of a turning pair, a pin is fixed to one of the links whereas 
a hole is provided in the other to fit the pin. When joined the surface of the 
hole of one link will rub on the surface of pin of the other link. The velocity of
rubbing of the two surfaces will depend upon the angular velocity of a link 
relative to the other.
Velocity of rubbing
Pin at A
• The pin at A joins links AD and AB. AD being fixed, the velocity of rubbing will 
depend upon the angular velocity of AS only.
• Velocity of rubbing = rac u
where, ra = radius of pin at A
mechanism 
Pin at B
iO b a - ojab - o j (clockwise)
> = • = i k
* BC
(counter-clockwise)
• rb = Radius of pin at B
Velocity of rubbing = rb(ouab + &W 
Pin at C
W bc = wC b (counter-clockwise)
(jJ d c - W ed (clockwise) 
rc = Radius of pin at C 
Velocity of rubbing = rc(u)bc + wc /c )
Pin at D
where, rd = radius of pin at D 
Velocity of rubbing = rd ojc d 
Instantaneous Centre of Velocity (l-centre)
• The instantaneous centre of velocity can be defined as a point which has no 
velocity with respect to the fixed link.
• Suppose there are two link 1 and link 2
• Link 1 may not be fixed. Rigid body 2 is shown to be in plane motion with 
respect to the link 1.
• In case of fixed link, (link 2) velocity of the point A and B are proportional to 
PA and PS respectively. So, instantaneously, the rigid body can be thought of 
as being momentarily in pure rotation about the point P. The velocity of any 
point C on the body at this instant is given by
in a direction perpendicular to Pc- This point P is called the instantaneously 
centre of velocity and its instantaneously velocity is zero.
• If both links 1 and 2 are in motion, we can define a relative instantaneous 
centre P-|2 to be a point on 2 having zero relative velocity with respect to a 
coincident point on 1. Consequently, the relative motion of 2 with respect to 1 
be appears to be pure rotation about P-|2. So P21 and P-|2 are identical.
Centro
• Instantaneous centre is also called centro. So, two coincident points 
belonging to two bodies having relative motion with the properties.
• They have the same velocities.
• They form a point in one of the rigid bodies about which the other rotates and 
vice-versa. Which is perhaps true for only an instant.
Primary Centro One which can be easily located by a mere observation of the 
mechanism.
Secondary Centro Centros that cannot be easily located.
Instantaneous Centre of Acceleration
It is defined as a point on a link having zero relative acceleration with respect 
to a coincident point on the other link and is different from the instantaneous 
centre of velocity.
Page 5


Displacement Velocity and Acceleration Analysis of Plane 
Mechanisms
Velocity Analysis in Mechanism
• Let a rigid link OA, of length r rotate about a fixed point 0 with a uniform 
angular velocity rad/s in a counter-clockwise direction OA turns through a 
small angle 56 in a small interval of time 5t. Then, A will travel along the arc 
AA’ as shown in figure.
rrrrrrrrrtrrrrrrrrn
o °
Velocity Analysis
Velocity of A relative to 0
In the limits, when
arcAA' _ r6d
68 - ~ r
v dd 
F = r — =r. 
“ A
Thus, velocity of A is wr and is perpendicular to OA. 
Velocity of Intermediate Point
• If represent the velocity of B with respect to 0, then
V i; _ ~OB _ OB
OA AO
b
a-*— |----------------o
Intermediate points b
i.e., b divides the velocity vector in the same ratio as B divides the link. The 
magnitude of the linear velocity of a point on the rotating body at a particular 
instant is proportional to its distance fromt the axis of rotation.
Velocity Images of Four Link Mechanism
• Figure shows a four link mechanism (quadric cycle mechanism) ABCD in 
which AD is fixed link and BC is the coupler. AB is the driver rotating at an 
angular speed of lj rad/s in the clockwise direction if it is a crank or moving 
at angular velocity c j at this instant if it is rocker.
Velocity Images of Slider-Crank Mechanism
• Consider a slider-crank mechanism in which OA is the crank moving with 
uniform angular velocity cj rad/s in the clockwise direction. At point B, a slider 
moves on the fixed guide G .
From the given configuration, the coupler AB has angular velocity in the counter­
clockwise direction. The magnitude being
BA
Velocity images ol slider-crank mechanism
Velocity of Rubbing
• Let us take two links of a turning pair, a pin is fixed to one of the links whereas 
a hole is provided in the other to fit the pin. When joined the surface of the 
hole of one link will rub on the surface of pin of the other link. The velocity of
rubbing of the two surfaces will depend upon the angular velocity of a link 
relative to the other.
Velocity of rubbing
Pin at A
• The pin at A joins links AD and AB. AD being fixed, the velocity of rubbing will 
depend upon the angular velocity of AS only.
• Velocity of rubbing = rac u
where, ra = radius of pin at A
mechanism 
Pin at B
iO b a - ojab - o j (clockwise)
> = • = i k
* BC
(counter-clockwise)
• rb = Radius of pin at B
Velocity of rubbing = rb(ouab + &W 
Pin at C
W bc = wC b (counter-clockwise)
(jJ d c - W ed (clockwise) 
rc = Radius of pin at C 
Velocity of rubbing = rc(u)bc + wc /c )
Pin at D
where, rd = radius of pin at D 
Velocity of rubbing = rd ojc d 
Instantaneous Centre of Velocity (l-centre)
• The instantaneous centre of velocity can be defined as a point which has no 
velocity with respect to the fixed link.
• Suppose there are two link 1 and link 2
• Link 1 may not be fixed. Rigid body 2 is shown to be in plane motion with 
respect to the link 1.
• In case of fixed link, (link 2) velocity of the point A and B are proportional to 
PA and PS respectively. So, instantaneously, the rigid body can be thought of 
as being momentarily in pure rotation about the point P. The velocity of any 
point C on the body at this instant is given by
in a direction perpendicular to Pc- This point P is called the instantaneously 
centre of velocity and its instantaneously velocity is zero.
• If both links 1 and 2 are in motion, we can define a relative instantaneous 
centre P-|2 to be a point on 2 having zero relative velocity with respect to a 
coincident point on 1. Consequently, the relative motion of 2 with respect to 1 
be appears to be pure rotation about P-|2. So P21 and P-|2 are identical.
Centro
• Instantaneous centre is also called centro. So, two coincident points 
belonging to two bodies having relative motion with the properties.
• They have the same velocities.
• They form a point in one of the rigid bodies about which the other rotates and 
vice-versa. Which is perhaps true for only an instant.
Primary Centro One which can be easily located by a mere observation of the 
mechanism.
Secondary Centro Centros that cannot be easily located.
Instantaneous Centre of Acceleration
It is defined as a point on a link having zero relative acceleration with respect 
to a coincident point on the other link and is different from the instantaneous 
centre of velocity.
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.
three centre 1. 2 and 3
P12- Instantaneous centre of fixed ground 1 and body 2.
P13 - Instantaneous centre of fixed ground 1 and body 3.
P23 - Instantaneous centre of body 2 and body 3.
Number of Centros in a Mechanism
• For a mechanism of n links, the number of centros (Instantaneous centre) N is
N =
Number of Lines of Centros
• The number of lines of centros L for a mechanism with n links is
L = i « ( tt- l) ( « - 2 ) 
6
Acceleration Analysis in Mechanism
• The rate of change of velocity with respect to time is known as acceleration 
and acts in the direction of the change in velocity. Velocity can changed by 
only changing its magnitude or its direction. Let a link OA, of length r, rotate in 
a circular path in the clockwise direction as shown in figure. It has an 
instantaneously angular velocity lj and an angular acceleration a in the same 
direction i.e., the angular velocity increases in the clockwise direction.
Tangential acceleration of A relative 0 is defined as
fL
—q r = r —
d\-
dt
Centripetal or radial acceleration of A relative to 0 is defined as