2.141 Modeling and Simulation of Dynamic Systems Assignment #2

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1 2.141 Modeling and Siulation of Dynaic Systes Assignent #2 Out: Wednesday Septeber 20, 2006 Due: Wednesday October 4, 2006 Proble 1 The sketch shows a highly siplified diagra of a dry-dock used in ship aintenance. Once the ship enters the dock, water is drained out by positive-displaceents pups, an operation tered puping down the dock. When work is coplete, a gate to the harbor adits water, an operation tered floating the dock. Your task is to odel the heaveode oscillatory behavior (i.e. vertical otion only) of a ship in the dock. For convenience, you should assue the ship and the dock are siple block shapes as illustrated. As a further siplification, assue that any otion 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-displaceent relation for vertical otion of the ship by partial differentiation of an appropriate energy (or co-energy) function for (a) puping down the dock with the ship still floating and (b) floating the dock. 2. Derive a relation between velocity and oentu for vertical otion of the ship by partial differentiation of an appropriate energy (or co-energy) function.. Develop a odel suitable to describe heave otion of the ship in response to pup flow input. Express it as a bond graph and as a set of state equations (no need to siulate). 4. Assuing the sea gate is closed and the pup flow is fixed at zero, derive an expression for the frequency of sall oscillations about an equilibriu height. Evaluate it for two extree cases: (i) dock surface area only slightly larger than ship horizontal cross-section area and (ii) dock surface area very uch larger than ship horizontal cross-section area.

2 5. Given that the siplifying assuptions ade above are unlikely to be true in reality, your odel predictions ay differ fro the real syste behavior. Will your odel over- or under-estiate the frequency of oscillation? 6. It is often helpful to identify analogous behavior in different physical doains. Can you develop a odel of an equivalent syste involving echanical translation only? Proble 2 Electro-agnetic solenoids are a very coon for of linear actuator, especially in applications requiring liited linear or angular travel. (If your car has power door locks, they are alost certainly operated by solenoids.) In soe applications, transient response tie is especially iportant. The following figure shows data fro a typical anufacturer s catalog ( 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 botto trace shows the braking torque vs. tie in response to abruptly energizing the electroagnet (i.e., applying voltage to the coils). Of greater interest is the corresponding coil current vs. tie, shown in the top trace. Note the little blip at t 1 ; what causes it? A clearer picture is found in the following figure fro another anufacturer s website ( which depicts coil current vs. tie in response to abruptly applying voltage to the coil. The current, instead of heading uniforly towards its steady-state value, turns back towards zero before abruptly changing direction and resuing its progress towards steady state 1. These figures are obviously sketches but reflect a real phenoenon. Your task is to develop the siplest odel copetent to explain it. 1 The three curves in this figure obviously refer to different conditions but I could find nothing in the associated aterial detailing what those conditions were (though I can guess).

3 1. What s going on? Describe qualitatively what causes (i) the sooth reversal (between a and b in the figure above) and (ii) the abrupt reversal (e.g., at b or c in the figure above). Assue the geoetry depicted in the following sketch, which is based on the design shown in the cutaway rendering. stationary field assebly flux lines axis of rotation coil gap flux rotor & arature assebly A flat cylindrical coil is rated for a steady current of 0.5 aps at 90 VDC excitation. It has 2450 turns and an inductance of 22 henries. It is contained in a stationary housing with outside diaeter 5.0 inches, wall thickness 0. inches; inner diaeter 2.75 inches, wall thickness 0. inches. A disk-shaped arature, weight 2.8 lbf, ay translate along the axis of rotation. It is restrained fro doing so by a leaf spring characterized by Δx = cf where Δx is displaceent fro equilibriu, F is force and c is a constant, 0.8 x 10-6 in/lbf. At equilibriu the gap between arature and stationary field housing is 0.0 inches. The relative pereability of the arature and housing aterial is Develop state equations suitable to reproduce the current transient. What s the iniu syste order required? For siplicity, neglect eddy current losses, hysteresis and fringing fields.. As a check, develop a linearized version of your odel. Can it describe (i) the sooth reversal? (ii) the abrupt reversal? 4. Develop a nuerical siulation of the current transient.

4 Proble Electro-active Polyers (EAPs) have been proposed as the basis of novel actuator aterials (see One EAP is PolyPyrrole (PPY). Iersed in an ionic solution, applying a voltage between PPY and a counter electrode induces an accuulation of charged ions adjacent to the aterial surface and a subsequent diffusion of ions into the aterial. Through a echanis that is not yet fully understood 2, absorption of ions induces a voluetric strain in the aterial, thereby providing a echanis for electro-echanical energy transduction. Loaded uniaxially, it serves as a translational actuator. Your task is to develop the siplest odel copetent to describe an experiental EAP actuator in the for of a rectangular PPY fil 12 long (x direction).5 wide (y direction) and 19 μ high (z direction) uniaxially loaded in the x direction. All shear stresses and strains ay be ignored. Mechanically, the polyer defors under stress as an isotropic elastic aterial with a non-zero Poisson s ratio. Electrically, the polyer stores ionic charge in response to applied voltage, with electrical capacitance proportional to volue C = V cvv where C V (Farads) is total voluetric capacitance, c V (Farads/ ) is specific voluetric capacitance, and V ( ) is the polyer volue. In the typical operating range of ost EAPs, specific voluetric capacitance can be considered constant. 1. This EAP ay be described as ultiport network eleent. What type? How any ports? Identify appropriate power and energy variables for the echanical and electrical doains. 2. Assuing electrical linearity and sall strain, write an expression for the total energy stored in the aterial valid for general loading conditions (i.e., non-zero stress on any face of the fil). You ay find it useful to express the echanical behavior in ters voluetric and deviatoric coponents. Voluetric strain is proportional to ean stress, while deviatoric stresses defor the polyer along its principal axes without changing its volue. εv = εxx + εyy + ε σ x + σ y + σ z σ = S = σ σ x, yz, xyz,, εv ε xx, yy, = εxx, yy, 2 As far as I know.

5 where ε V is voluetric strain and ε xx, yy, are strains in the x, y and z directions, respectively; σ is ean stress, σ x, y, z are stresses in the x, y and z directions, respectively; are deviatoric stresses and ε are deviatoric strains in the x, y and z directions, respectively. S x, y, z xx, yy,. Derive constitutive equations relating port variables of this ultiport eleent. 4. Now assue uniaxial loading conditions, i.e., zero stress on the y and z faces of the fil. Revise the constitutive equations to reflect these specialized conditions. 5. If they are not already linear, linearize the constitutive equations for the uniaxial loading case. Represent the result as a network of basic bond-graph eleents (i.e., junctions, one-port passive eleents, etc.). Is the electro-echanical coupling described by a transforer or gyrator? 6. Evaluate the linearized constitutive equations about an operating point at 0.7 volts using the following EAP paraeters: x0 = y0 = z0 = ν = cv = F/ E = 0.18 GPa 8 7. Copute the no-load strain and charge added by applying a voltage of 0.7 volts to the fil. Copute the load required to double that strain and the corresponding additional charge displaced. Is transduction between the electrical and echanical doains bilaterally syetric? That is, does the echanical doain affect the electrical doain as uch as the electrical doain affects the echanical?