Bond Graphs: What Are They?
2.7 Summary of Bond Direction Rules
1. R, C, and I elements have an incoming power direction (this results in positive parameters when modeling real-life components).
2. For source elements Se and Sf, the standard is outgoing as sources mostly deliver power to the rest of the system.
3. For TF and GY elements, one bond is incoming, another is outgoing.
4. For 1 and 0 junctions, some bonds could be incoming and oth-ers outgoing. The modeler has some freedom in choosing bond direction. More often than not bond directions are determined by the components that are attached to the junctions. The form of the resulting equations will change based on directions of bonds chosen.
Problems
2.1. Figure P2.1 shows a system to pump water. List all the impor-tant components in this system, identify locations where power is changing from one domain to another, and indicate what these transformations are.
2.2. Figure P2.2 shows a loudspeaker. The circuit that supplies a cur-rent through the coil is not shown in the fi gure but is a part of the system. Coil resistance and inductances are part of that cir-cuit. List all the important components in this system, identify
N S
S
Coil carrying current
Diaphragm
FIGURE P2.2
Figure for Problem 2.2, loudspeaker.
Bearing
Tank
Pump Motor
Water in Water out
FIGURE P2.1
Figure for Problem 2.1, motor driven pump used to fi ll an overhead tank.
locations where power is changing from one domain to another, and indicate what these transformations are.
2.3. Consider two mechanical springs (capacitive elements). The fi rst spring is linear, so the constitutive equations is F = k1x and the second is quadratic so the constitutive relationship is F = k2x2.
A test was run to determine the spring constants, and it was observed that for a 50 N force both the springs had a displacement of 2 cm. Determine:
The spring constants of the two springs
The displacement in each spring if the force is doubled The energy stored in each spring during the initial extension 2.4. A cantilever (Figure P2.3) is a very common feature used in many
MEMS device (the schematic shows one). This device is used both as a sensor and as an actuator in microdevices. If this microscale device is to be modeled, what are the different basic components that need to be included in the model?
2.5. Consider two electrical conductors of the same material and length. One is 0.5 mm in diameter, and the other one is 1 mm. For the same potential difference applied across both these conduc-tors, which will allow higher current?
Cantilever, mass and elasticity
Air resistance
Outer casing FIGURE P2.3
Figure for Problem 2.4, schematic of a MEMs device involving a cantilever in a casing.
2.6. Figure P2.4 is a broad overview of an automobile with some of the main items identifi ed. If a model were to be developed to study the dynamic behavior of the vehicle, determine the different subsystems, and, within each subsystem, the different components and their behaviors for which accounting is needed.
2.7. The velocity function of a moving body is given by v(t) = 5t + 7.
Compute the distance traveled over a time of 3 seconds starting from a 0 initial position. Plot both the velocity and the distance traveled as a function of time for this time period. What is the acceleration for this time period?
2.8. Figure P2.5 shows a simple system with a motor driving a fan.
Identify all the basic elements that are part of this system so that a system model can be developed.
Bearing Motor
Fan
FIGURE P2.5
Figure for Problem 2.8, schematic of a motor driven fan.
Steering
Transmission
Engine
Wheels Wind
FIGURE P2.4
Figure for Problem 2.6, schematic of a moving vehicle.
2.9. An electric heater is rated at 1500 W and is connected to a 110 V supply. What is the resistance of the heating coil? How much cur-rent is fl owing through the coil? If the supply is not a constant DC source but is a 60 Hz AC source, how does the current change with time?
2.10. In an electric car, regenerative braking is used to recharge batter-ies. The principle of regenerative braking involves converting the kinetic energy of the vehicle into energy stored in a battery. If a car is traveling at a speed of 30 km/h and is stopped completely by braking, what would be the energy stored in the battery if the mass of the car is 1000 kg and the energy conversion effi ciency is assumed to be 100%? If this braking happens at a constant rate for 5 seconds, what is the average rate of power transfer?
2.11. For mechanical translation, rotation, hydraulic, and electric domains draw a tetrahedron of state. For each case write the rele-vant variables for each domain at the four corners and write down the relationships between these variables.
2.12. Figure P2.6 shows a platform that supports a rotating machine with a mass imbalance in the rotor. The platform is supported by springs and dampers. The mass imbalance is a source of force that is peri-odic due to the rotor angular speed. For this system identify all the important components necessary to develop a system model.
F
m
B k
FIGURE P2.6
Figure for Problem 2.12, a motor mounted on suspensions with an unbalanced rotor.
2.13. The inductance of a coil is computed from:
L= N2μA
l
where L is the inductance in Henrys, N is the number of wind-ings, A is the cross-section area of the core, and μ is the perme-ability of the medium. An inductor of 2 cm core diameter and 200 turns in the coil is 6 cm long. If the current fl owing through the conductor is 0.5 A and the permeability of the medium is 1.257E-6 N /m, compute the energy stored in the coil.
2.14. The inductor energy in the previous problem is now used to charge a parallel plate capacitor for a terminal voltage of 10 V. The capacitor plates have 1 sq cm plate overlap area and the distance between the plates is 1 cm. The permittivity of the medium is 0.885E-11 F/m. Compute the charge developed across the plates.
(The capacitance of a parallel plate capacitor is given by C= εA d ) 2.15. The velocity in a system is expressed in the following
relationship:
v= 2t2m / s, 0≤ t ≤ 5 v= 50m/ s, 5 < t ≤ 20
If the damping coeffi cient B = 10 Ns/m, plot the power dissipated through this damper as a function of time. How much is the total energy dissipated over the time period of 20 seconds?
2.16. The plot in Figure P2.7 shows the voltage across an inductor of 0.5 mH. Use the data from the plot to determine the current in the inductor at t = 15μs.
2.17. The time dependent voltage across a capacitor of 50 μF is shown in Figure P2.8. Determine the current through the capacitor as a function of time using this data.
t(μs)
5 10 15 20
4
−4 V
FIGURE P2.7
Figure for Problem 2.16, voltage versus time data for Problem 2.16.
5 10 15 20
15
−15 V
t(μs)
FIGURE P2.8
Figure for Problem 2.17, voltage versus time data for Problem 2.17.
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