2.141 Modeling and Simulation of Dynamic SystemsAssignment #2Problem 1Problem 2Problem 32.141 Modeling and Simulation of Dynamic Systems Assignment #2 Out: Wednesday September 20, 2006 Due: Wednesday October 4, 2006 Problem 1 The sketch shows a highly simplified diagram of a dry-dock used in ship maintenance. Once the ship enters the dock, water is drained out by positive-displacements pumps, an operation termed “pumping down the dock”. When work is complete, a gate to the harbor admits water, an operation termed “floating the dock”. Your task is to model the heave-mode oscillatory behavior (i.e. vertical motion only) of a ship in the dock. For convenience, you should assume the ship and the dock are simple block shapes as illustrated. As a further simplification, assume that any motion of the water can be described as “slug flow”; that is, its velocity is constant across any given flow cross section. Finally, neglect all frictional losses. 1. Derive a steady-state force-displacement relation for vertical motion of the ship by partial differentiation of an appropriate energy (or co-energy) function for (a) pumping down the dock with the ship still floating and (b) floating the dock. 2. Derive a relation between velocity and momentum for vertical motion of the ship by partial differentiation of an appropriate energy (or co-energy) function. 3. Develop a model suitable to describe heave motion of the ship in response to pump flow input. Express it as a bond graph and as a set of state equations (no need to simulate). 4. Assuming the sea gate is closed and the pump flow is fixed at zero, derive an expression for the frequency of small oscillations about an equilibrium height. Evaluate it for two extreme cases: (i) dock surface area only slightly larger than ship horizontal cross-section area and (ii) dock surface area very much larger than ship horizontal cross-section area.5. Given that the simplifying assumptions made above are unlikely to be true in reality, your model predictions may differ from the real system behavior. Will your model over- or under-estimate the frequency of oscillation? 6. It is often helpful to identify analogous behavior in different physical domains. Can you develop a model of an equivalent system involving mechanical translation only? Problem 2 Electro-magnetic solenoids are a very common form of linear actuator, especially in applications requiring limited linear or angular travel. (If your car has power door locks, they are almost certainly operated by solenoids.) In some applications, transient response time is especially important. The following figure shows data from a typical manufacturer’s catalog (www.electroid.com). In this product (see the cutaway rendering) the solenoid squeezes two rotating plates together, serving to brake or clutch one plate relative to the other. The bottom trace shows the braking torque vs. time in response to abruptly energizing the electromagnet (i.e., applying voltage to the coils). Of greater interest is the corresponding coil current vs. time, shown in the top trace. Note the little “blip” at t1; what causes it? A clearer picture is found in the following figure from another manufacturer’s website (www.solenoids.com) which depicts coil current vs. time in response to abruptly applying voltage to the coil. The current, instead of heading uniformly towards its steady-state value, turns back towards zero before abruptly changing direction and resuming its progress towards steady state1. These figures are obviously sketches but reflect a real phenomenon. Your task is to develop the simplest model competent to explain it. 1 The three curves in this figure obviously refer to different conditions but I could find nothing in the associated material detailing what those conditions were (though I can guess).1. What’s going on? Describe qualitatively what causes (i) the smooth reversal (between a and b in the figure above) and (ii) the abrupt reversal (e.g., at b or c in the figure above). Assume the geometry depicted in the following sketch, which is based on the design shown in the cutaway rendering. coilflux lines gap fluxstationary field assembly axis of rotationrotor & armature assembly A flat cylindrical coil is rated for a steady current of 0.35 amps at 90 VDC excitation. It has 2450 turns and an inductance of 22 henries. It is contained in a stationary housing with outside diameter 5.0 inches, wall thickness 0.3 inches; inner diameter 2.75 inches, wall thickness 0.3 inches. A disk-shaped armature, weight 2.8 lbf, may translate along the axis of rotation. It is restrained from doing so by a leaf spring characterized by where is displacement from equilibrium, F is force and c is a constant, 0.8 x 10-6 in/lbf3. At equilibrium the gap between armature and stationary field housing is 0.03 inches. The relative permeability of the armature and housing material is 5000. 3cFx =ΔxΔ 2. Develop state equations suitable to reproduce the current transient. What’s the minimum system order required? For simplicity, neglect eddy current losses, hysteresis and fringing fields. 3. As a check, develop a linearized version of your model. Can it describe (i) the smooth reversal? (ii) the abrupt reversal? 4. Develop a numerical simulation of the current transient.Problem 3 Electro-active Polymers (EAPs) have been proposed as the basis of novel actuator materials (see http://ndeaa.jpl.nasa.gov/nasa-nde/lommas/eap/EAP-web.htm). One EAP is PolyPyrrole (PPY). Immersed in an ionic solution, applying a voltage between PPY and a “counter electrode” induces an accumulation of charged ions adjacent to the material surface and a subsequent diffusion of ions into the material. Through a mechanism that is not yet fully understood2, absorption of ions induces a volumetric strain in the material, thereby providing a mechanism for electro-mechanical energy transduction. Loaded uniaxially, it serves as a translational actuator. Your task is to develop the simplest model competent to describe an experimental EAP actuator in the form of a rectangular PPY film 12 mm long (x direction) 3.5 mm wide (y direction) and 19 μm high (z direction) uniaxially loaded in the x direction. All shear stresses and strains may be ignored. Mechanically, the polymer deforms under stress as an isotropic elastic material with a non-zero Poisson’s ratio. Electrically, the polymer stores ionic charge in