ME 450 HOMEWORK 9 Speaker Model Develop a model of the electromagnetic speaker shown in Figure 1, and obtain the transfer function relating the diaphragm displacement x (output) to the applied voltage v (input). Magnet Speaker enclosure S 100000 Sound Waves S Coil Diaphragm Figure 1: Simplified Loudspeaker System (Cross Section) The loudspeaker system can be modeled as a coupled electrical and mechanical system: R L X + + k f= Kfi Up m C 1 i - The two systems are coupled through the magnetic force f and you can assume that the back EMF is given by Kb *. Problem 1 (15 points) Derive the governing equations for the mechanical and the electrical systems. Problem 2 (5 points) Calculate the transfer function G(s) = G(s) = ¥( X(s) V(s) 1 Thermal System with Multiple Capacitances A system with two thermal capacitances is shown in Figure 2. Heat is supplied at a rate qi(t) by an electrical heater with electrical resistance R. The current through the heater resistance is I(t). I(t) T1 C1 T2 C2 q.(t) R R R2 Ta Figure 2: System with two Thermal Capacitances Assume the enclosure is perfectly insulated except for the right-end side, at constant ambient temperature Ta · R1 and R2 are equivalent thermal resistances. Problem 3 (10 points) Assuming that the system input is the power dissipated by the electrical resistance: u(t) = qi(t), derive the ODEs for the system that describe the temperature dynamics. Problem 4 (10 points) The power dissipated by the resistor, all of which is converted to heat, is qi(t) = 12(t) . R. Update the governing equations derived in a) using the current through the resistor as the new input. Problem 5 (5 points) Write the system in nonlinear state space form: dt T1(t) = f1(T1,T2,I) d dt -- T2(t) = f2 (T1,T2, Ta) Problem 6 (10 points) Linearize the system above around a general equilibrium condition using: T1(t) = T1 + ôT1(t) T2(t) = T2 + ÔT2(t) I(t) = [ + 8I(t) Ta(t) = Ta + ôTa(t) STa(t) = 0 Problem 7 (5 points) Find the transfer function: G(s) = I(s) ST2(s) Problem 8 (5 points) Determine the steady state temperature &T2 when the system is subject to a unit step input from its equilibrium point (SI(t) is a unit step). 2 1 OC Model of an Electric Oven When buying a new range for the home, dual fuel ranges, where gas is used for the cooktop and electricity for the oven may be considered. This type of range provides the best temperature control for both the cooktop and oven. In a gas cooktop, temperature changes are immediate and some chefs can judge the temperature of a stove based on the size and appearance of the gas flames. Due to the non-negligible thermal mass of the oven air, the electric oven is the go-to choice for virtually eliminating temperature fluctuations, which are otherwise common with gas ovens. This is extremely important for baking. However, these features come with a cost and the difference between a dual fuel and its gas counterpart can vary from $1k to $2k. To design a controller for the oven, a model of the system is required. The top of the line oven can be modeled as shown below considering two thermal masses, namely the oven air and the lumped heating element. The input to the system is the heat flux q; . Oven air 290 T2 C2 * To 191 R1 Heating T1 C1 19i R2 element Figure 3: Schematic of oven and heating model Problem 9 (10 points) Write the conservation of energy for the two capacitances C, and C2 as a function of the heat fluxes q1, q1, and q .. The temperature for the heating element is T, and the temperature of the oven air is T2. Problem 10 (10 points) Write the elemental relations relating q and q. with the temperatures T1,T2, and To. The equivalent thermal resistances are R, and R2. Problem 11 (10 points) Substitute the fluxes obtained in part b into the equations for the conservation of energy obtained in part a. The final answer should only contain the temperatures of the system, the model parameters (capacitances and resistances) and the input. 3