lab 6 torsional oscillations_v3
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8/18/2019 Lab 6 Torsional Oscillations_v3
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ME 413: System Dynamics & ControlME 413: System Dynamics & ControlME 413: System Dynamics & ControlME 413: System Dynamics & Control
MechanicalMechanicalMechanicalMechanical Systems (Systems (Systems (Systems (3333))))Torsional OscillationsTorsional OscillationsTorsional OscillationsTorsional Oscillations
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8/18/2019 Lab 6 Torsional Oscillations_v3
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MECHANICAL SYSTEMS (3)
TORSIONAL OSCILLATIONS
OBJECTIVES
1. To find the mathematical model of a coupled discs assembly in torsion.2. To determine the response of a coupled discs assembly for given step an
impulsive loading.
Part 2: Assignment: Simulation
In the system shown in the figure below, the two shafts are assumed to be flexible
with stiffness constants1
k and2
k . The two disks, with moments of inertia1
J and2
J
are supported by bearings whose friction is negligible compared to the viscous
friction element denoted by the coefficients1
B and2
B . The reference position for1
θ
and2
θ are the positions of the reference marks on the rims of the disks when the
system contains no stored energy.
Figure 1.
1. Draw the FBD of the system and obtain the system differential equations ofmotion.
2. For zero initial conditions obtain the transfer functions ( )1G s and ( )2G s
given by ( ) ( )
( )1
1
a
sG s
sτ
Θ= and ( )
( )
( )2
2
a
sG s
sτ
Θ=
where ( )1 sΘ and ( )2 sΘ are the Laplace transforms of ( )1 t θ and ( )2 t θ ,
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8/18/2019 Lab 6 Torsional Oscillations_v3
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respectively; and ( )a sτ is the Laplace transforms of ( )a t τ . It is clear that
the transfer functions ( )1G s and ( )2G s should be given in terms of the
system parameters1
k ,2
k ,1
B ,2
B and1
J ,2
J .
3. If in SI units1
J =2
J =1.0 and1
B =2
B =0.7 and1
k =2
k =10; use MATLAB to
plot the responses ( )1 t θ and ( )2 t θ if ( )a t τ is :
•
a step input of magnitude 10 N.m.• an impulsive torque of magnitude 10 N.m
4.
From the above expression of ( )2G s , write the explicit expression of
( )2 t θ if ( )a sτ is a step input of magnitude 10 N.m. as in the previous case.
5.
Use the final value theorem to find the steady state values of ( )1 t θ and( )2 t θ .
References
[1] C. M Close, D. H. Frederick and J. C. Newell, Modeling and Analysis of DynamicSystems, Third Edition, 2002, John Wiley & Sons.