REVIEW NETWORK MODELING OF PHYSICAL SYSTEMS
EXAMPLE: VIBRATION IN A CABLE HOIST
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:
"switch
voltage"
velocity source (equivalent to switch
voltage)
"resistor"
viscous damper (equivalent to electrical
resistor)
"capacitor"
mass equivalent of capacitor
spring (cable compliance)
cable
compliance
v3
v2
elevator
mass
mass of elevator cage
force source (elevator weight)
Modeling and Simulation of Dynamic Systems
v1
Cable Hoist Example, continued
elevator
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
m1 v1 = –m2 v2
(subscripts as indicated in the diagram)
m1
v2 = –m v1
2
Kinetic energy:
Ek* = m1
v12
2
+ m2
v22
2
= m1
v12
2
⎛
m12 v12
m1⎞⎟ v12
⎜
+ m 2 = m1 ⎜ 1 + m ⎟ 2
2
2⎠
⎝
Modeling and Simulation of Dynamic Systems
Cable Hoist Example, continued
page 2
Potential energy:
m1
∆x2 = –m ∆x1
2
∆xspring = ∆x1 – ∆x2 =
Ep =
k
2
∆x
2spring
=
⎛
⎜
⎜1
⎝
m1⎞⎟
+ m ⎟ ∆x1
2⎠
k ⎛⎜
2
m1⎞⎟2
⎜1 +
⎟ ∆x12
m
2⎠
⎝
Undamped natural frequency:
ωn2 =
⎛
m1⎞⎟2
⎜
k ⎜1 + m ⎟
2⎠
⎝
⎛
m1⎞⎟
⎜
m1 ⎜ 1 + m ⎟
2⎠
⎝
k ⎛⎜
m1⎞⎟
= m ⎜1 + m ⎟
2⎠
⎝
Thus the undamped natural frequency will be increased by the factor
m1
1+m
2
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 Newton-meters/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:
kcable
mcage = 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
m2 =
⎛ K
⎞
motor ⎟2
⎜
⎜
⎟
⎝rdrum ngear⎠
C = 90 kilograms
!!!
Undamped natural frequency with the R-C filter:
k ⎛⎜
m1⎞⎟
m ⎜⎝1 + m2⎟⎠ = 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
b=
⎛ K
⎞
motor ⎟2
⎜
⎜
⎟
⎝rdrum ngear⎠
1
R = 90 Newton-seconds/meter
If the equivalent mass and equivalent damper were isolated, the decay time
constant would be
τisolated = m2/b = RC = 1 second
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
τcoupled = (m1 + m2)/b = 3.2 seconds
This also agrees quite well with the numerical simulation.
Modeling and Simulation of Dynamic Systems
Cable Hoist Example, continued
page 6