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Modeling of Physical Systems: HW 3–due 9/20/12 Page 1
Problem 1: Show whether the mechanical and electrical systems shown below are analogous by deriving and comparing the mathematical models for each. Use V 1 , the velocity of mass m 1 , and x 2 , the displacement of spring k 2 , as the state variables for the mechanical system, and i 1 , the current of the inductor L 1 , and q 2 , the charge in capacitor C 2 , as the state variables for the electrical system.
Problem 2: The L-frame shown below is part of a ‘vintage’ electromechanical means for measuring pressure within an underwater vessel. A rotational sensor placed at the pivot point, P , is used to detect the angular rotation of the L-frame. Develop a dynamic model for this system. Your model should allow you to formulate state equations for: angular velocity, ω = θ˙, of the L-frame, deflection, xL, of the spring kL, and deflection, xp, of the pivot spring. Assume that the angular displacement of the L-frame is constrained to be small (say no larger than about 10 degrees).
R.G. Longoria, Fall 2012 ME 383Q, UT-Austin
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Modeling of Physical Systems: HW 3–due 9/20/12 Page 2
Problem 3: The figure below shows a schematic of a permanent-magnet dc (pmdc) motor. The labeling of the rate of change of flux linkage, λ˙, on the inductor and rate of change of angular momentum on the rotational inertia, h˙, as equal to zero indicates that we are interested in a steady-state model. These elements are neglected in this case.
a. Show that the torque delivered to a load attached to the output shaftas a function of the shaft speed, ωL, is TL = TL(ωL) = Ts − BeωL, where Be = Bm + r^2 m/Rm. This is the ideal torque-speed curve.
b. Derive an expression for the output power as a function of ωL, and plot the power curve on the same graph as an efficiency curve (scale as necessary or use 2 independent vertical scales), with efficiency defined, η = Pout/Pin. Assume the input voltage is 24 volts and the motor is a Maxon RE040 with specifications provided below. Note: you will need to find a suitable value for Bm since an explicit value is not shown.
c. Determine a value BL,max for a linear rotational damping load that will result in maximum power being delivered by the Maxon RE040 with a 24 V input. What is the value BL,eff for maximum efficiency?
R.G. Longoria, Fall 2012 ME 383Q, UT-Austin
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