Transcript 投影片 1

REVIEW NETWORK MODELING OF PHYSICAL SYSTEMS EXAMPLE:
VIBRATION IN A CABLE HOIST (CONTINUED)
Bond graphs of the cable hoist models help to develop insight about how
the electrical R-C filter affects the mechanical system dynamics.
Equivalent mechanical system:
velocity source (equivalent to switch
voltage)
viscous damper (equivalent to
electrical
mass equivalent of capacitor
spring (cable compliance)
mass of elevator cage
force source (elevator weight)
Modeling and Simulation of Dynamic Systems
Cable Hoist Example, continued
"switch
voltage
"
"resistor"
"capacitor"
cable
complianc
e
elevator
mass
elevato
r
weight
page 1
Resonant oscillation:
due to out-of-phase motion of the two masses opposed by the spring
Coniser the two massess and the spring in isolation
No external forces
—net momentum = zero
—the masses move in opposite directions
m v = –m v
1
1
2
2
(subscripts as indicated in the diagram)
v = –m
v
2
1
1
m2
Kinetic (co-) energy:
E* = m
k
v12
1
2
+m
2
v22
2
=m
1
v12
2
m2
+ 1
m2
Modeling and Simulation of Dynamic Systems
v 12
=m
1
2
1+
m1
v12
m2
2
Cable Hoist Example, continued
page 2
Potential energy:
Undamped natural frequency:
Thus the undamped natural frequency will be increased by the
factor
Modeling and Simulation of Dynamic Systems
Cable Hoist Example, continued
page 3
Check the numbers:
The parameters used in the MATLAB simulations were as follows:
R = 10 ohms
C = 0.1 farads
K_motor = 0.03 Newtonmeters/amp n_gear = 0.02
r_drum = 0.05 meters
k_cable = 200000 Newton/meter
m_cage = 200 kilograms
Undamped natural frequency without the R-C filter:
= 31.6 radian/second = 5
Hertz
This agrees
with the numerical simulation.
Modeling and Simulation of Dynamic Systems
Cable Hoist Example, continued
page 4
The mass equivalent of the capacitor is
C = 90 kilograms !!!
Undamped natural frequency with the R-C filter:
= 56.8 radians/second = 9
Hertz
This agrees quite well with the numerical simulation.
Modeling and Simulation of Dynamic Systems
Cable Hoist Example, continued
page 5
Decay time constant:
both masses move in unison, opposed by the damper
The viscous damping equivalent of the resistor is
= 90 Newton-seconds/meter
If the equivalent mass and equivalent damper were isolated, the decay
time constant would be
t
isolated
= m /b = RC = 1 second
2
which is the time constant of the electrical filter—as it should be.
In the coupled electro-mechanical system, both masses interact with the
damper and the decay time constant is
t = (m + m )/b = 3.2 seconds
This also agrees quite well with the numerical simulation.
coupled
1
2
Modeling and Simulation of Dynamic Systems
Cable Hoist Example, continued
page 6
Less can be better
The designer’s original objective may be achieved without increasing the
frequency of oscillation by eliminating the capacitor. The system becomes
second order.
Modeling and Simulation of Dynamic Systems
Cable Hoist Example, continued
page 7