The document summarizes mechanical translational and rotational systems. It describes three types of forces that resist translational motion: inertia force, damping force, and spring force. It provides the equations of motion for each type. Similarly, it describes three types of torques that resist rotational motion: inertia torque, damping torque, and spring torque. It establishes the translational-rotational counterparts and provides examples applying D'Alembert's principle to mechanical systems. The document also provides an example problem drawing free body diagrams and deriving the differential equations of motion for a given multi-body system.

1. MECHANICAL SYSTEM
TRANSLATIONAL SYSTEM: The motion takes place along a strong
line is known as translational motion. There are three types of
forces that resists motion.
INERTIA FORCE: consider a body of mass ‘M’ and acceleration
‘a’, then according to Newton’s law of motion
FM(t)=Ma(t)
If v(t) is the velocity and x(t) is the displacement then
dt
tdv
MtFM
)(
)(  2
2
)(
dt
txd
M
SYED HASAN SAEED
M
)(tx
)(tF

2. DAMPING FORCE: For viscous friction we assume that the
damping force is proportional to the velocity.
FD(t)=B v(t)
Where B= Damping Coefficient in N/m/sec.
We can represent ‘B’ by a dashpot consists of piston and
cylinder.
SYED HASAN SAEED
dt
tdx
B
)(

)(tFD
)(tx

3. SPRING FORCE:A spring stores the potential energy.
The restoring force of a spring is proportional to
the displacement.
FK(t)=αx(t)=K x(t)
Where ‘K’ is the spring constant or stiffness (N/m)
The stiffness of the spring can be defined as
restoring force per unit displacement
SYED HASAN SAEED
 dttvKtFK )()(

4. ROTATIONAL SYSTEM: The rotational motion of a
body can be defined as the motion of a body
about a fixed axis. There are three types of
torques resists the rotational motion.
1. Inertia Torque: Inertia(J) is the property of an
element that stores the kinetic energy of
rotational motion. The inertia torque TI is the
product of moment of inertia J and angular
acceleration α(t).
Where ω(t) is the angular velocity and θ(t) is the angular
displacement.
SYED HASAN SAEED
2
2
)()(
)()(
dt
td
J
dt
td
JtJtTI

 

5. SYED HASAN SAEED

6. 2. Damping torque: The damping torque TD(t) is the
product of damping coefficient B and angular
velocity ω. Mathematically
3. Spring torque: Spring torque Tθ(t) is the product
of torsional stiffness and angular displacement.
Unit of ‘K’ is N-m/rad
SYED HASAN SAEED
dt
td
BtBtTD
)(
)()(

 
)()( tKtT  

7. D’ALEMBERT PRINCIPLE
This principle states that “for any body, the
algebraic sum of externally applied forces
and the forces resisting motion in any given
direction is zero”
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8. D’ALEMBERT PRINCIPLE contd……
External Force: F(t)
Resisting Forces :
1. Inertia Force:
2. Damping Force:
3. Spring Force:
SYED HASAN SAEED
2
2
)(
)(
dt
txd
MtFM 
dt
tdx
BtFD
)(
)( 
)()( tKxtFK 
M
F(t)
X(t)

9. According to D’Alembert Principle
SYED HASAN SAEED
0)(
)()(
)( 2
2
 tKx
dt
tdx
B
dt
txd
MtF
)(
)()(
)( 2
2
tKx
dt
tdx
B
dt
txd
MtF 

10. Consider rotational system:
External torque: T(t)
Resisting Torque:
(i) Inertia Torque:
(ii) Damping Torque:
(iii) Spring Torque:
According to D’Alembert Principle:
SYED HASAN SAEED
dt
td
JtTI
)(
)(


dt
td
BtTD
)(
)(


)()( tKtTK 
0)()()()(  tTtTtTtT KDI

11. SYED HASAN SAEED
0)(
)()(
)(  tK
dt
td
B
dt
td
JtT 

)(
)()(
)( tK
dt
td
B
dt
td
JtT 


D'Alembert Principle for rotational motion states
that
“For anybody, the algebraic sum of externally
applied torques and the torque resisting rotation
about any axis is zero.”

12. TANSLATIONAL-ROTATIONAL COUNTERPARTS
S.NO. TRANSLATIONAL ROTATIONAL
1. Force, F Torque, T
2. Acceleration, a Angular acceleration, α
3. Velocity, v Angular velocity, ω
4. Displacement, x Angular displacement, θ
5. Mass, M Moment of inertia, J
6. Damping coefficient, B Rotational damping
coefficient, B
7. Stiffness Torsional stiffness
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13. EXAMPLE: Draw the free
body diagram and write
the differential equation
of the given system
shown in fig.
Solution:
Free body diagrams are
shown in next slide
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14. SYED HASAN SAEED
2
1
2
1
dt
xd
M
)( 211 xxK 
dt
xxd
B
)( 21
1

)(tF
1M 2M
)( 211 xxK 
dt
xxd
B
)( 21
1

2
2
2
2
dt
xd
M
dt
dx
B 2
2 22xK

15. 22
2
22
2
2
2
21
1211
211
21
12
1
2
1
211
21
1
2
1
2
1
)(
)(
)(
)(
)(
)(
)(
xK
dt
dx
B
dt
xd
M
dt
xxd
BxxK
similarly
xxK
dt
xxd
B
dt
xd
MtF
xxKF
dt
xxd
BF
dt
xd
MF
K
D
M










SYED HASAN SAEED

16. Figure
Write the differential equations describing the
dynamics of the systems shown in figure and
find the ratio
SYED HASAN SAEED
)(
)(2
sF
sX

17. FREE BODY DAIGRAMS
SYED HASAN SAEED
1M
2
1
2
1
dt
xd
M
)( 211 xxK 
)(tF
2M
2
2
2
dt
xd
M
22xK
)( 211 xxK 

18. Differential equations are
SYED HASAN SAEED
2
2
2
222211
2112
1
2
1
)(
)()(
dt
xd
MxKxxK
xxK
dt
xd
MtF


LaplaceTransform of aboveequations
)()()()(
)()()()(
2
2
2222111
21111
2
1
sXsMsXKsXKsXK
sXKsXKsXsMsF


Solve equationsabove for
2
11
2
1212
2
12
))(()(
)(
KMsKKKMs
K
sF
sX


)(
)(2
sF
sX

19. THANK YOU
SYED HASAN SAEED