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Modeling of Physical Systems: HW 9–due 12/6/12 Page 1
Choose either Problem 1 or Problem 2:
Problem 1: Electro-Thermal-Mechanical System Modeling and Simulation
Consider the electro-thermo-mechanical system shown below. A piston is forced to move by the expansion of air in the cylinder. The cylinder and piston are made of steel.
Piston-Cylinder with Resistive Heater
(a) Develop a bond graph model of this system.
(b) Develop state equations for this system.
(c) Consider the following specific case. Start- ing with the air at 25 deg C and compressed enough to balance the piston, the heater is turned on. Perform a simulation of the sys- tem. What are the steady state values of critical variables? Compare response with heat trans- fer ‘off’ (insulated) and heat transfer ‘on’ (non- insulated).
Parameter data: The piston is 5 cm thick and the cylinder walls are 1 cm thick with the inner radius of the cylinder being 10 cm. The height of the cylinder is 20 cm. The ambient temperature of Ta is 25 deg Celsius. The heater coil has electrical resistance of 500 ohms and the voltage input is 110 volts 60 Hz.
NOTE: A partial solution will be provided, but you must verify and then implement the simulation.
(d) Term Project Option. Repeat steps (a)-(c) but now assume that a hole is punched in the bottom of the cylinder at the same time that the heater is turned on (t = 0). Compare the cases with and without heat transfer for holes of 2 mm and 3 mm. Fully document your new model, assumptions made, references used, etc.
R.G. Longoria, Fall 2012 ME 383Q, UT-Austin
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Modeling of Physical Systems: HW 9–due 12/6/12 Page 2
Problem 2 (Distributed-parameter modeling of a shaft-flywheel system): Study the note posted on the course log that describes the shaft-flywheel system illustrated below. Refer to that case study for all necessary data, parameter values, etc.
(a) Re-work the comparison of frequency response results derived from using a shaft model with a lumped-parameter Π-model versus a shaft modeled using a distributed- parameter element model. Use a lossless W-line for this latter case.
(b) For the Π-model, develop a simulation to solve for the response of the flywheel when the input velocity, Ω 1 (t), is a step input (starting from a value of zero stepping up to a final value to be determined).
(c) Develop a simulation of a more accurate model for this system. You have a choice here: i) either increase the complexity of your Π-model to capture at least one more ‘mode’ (the shaft would be split into two inertia elements and three spring elements), or ii) transform the ‘exact’ W-line model into the time domain. Experiment in either case to illustrate any benefits gained by increasing the model complexity.
Review these notes:
NOTE 1: If you choose to study how to transform the W-line model into time-domain, this can be approached in two different ways: i) using approximations to the sinh() and cosh() functions that allow transforming the transfer function relations into time domain, or ii) using time-difference equations. These options will be discussed in lecture.
NOTE 2: You may choose to replace this problem with an analogous system of your choice. For example, an equivalent electrical system might have a step input voltage into a transmission line with an electrical resistor-inductor load. A hydraulic system might be a pressure pulse into a long fluid line terminated by a leaking piston.
NOTE 2: You may substitute this problem with part of your term project if it includes a distributed-parameter model requirement. In this case you should explain to me what you are planning to do.
R.G. Longoria, Fall 2012 ME 383Q, UT-Austin
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