o Qk1 Qk2 Qk3 C1 C2 C3 TB TA R1a R12 R2a R23 R3B R3a RBa Problem 1: In the thermal system shown below, Qk1, Qk2, Qk3 and TB are the inputs. Thermal capacities of the rooms are given as Ci. Thermal resistances between the rooms and the outdoor are also given as Ri. Draw the equivalent electric circuit of the thermal system. - + VB C1 R1a R12 C2 R2a R23 C3 R3a R3B RBa VA Thermal Elec. Ri Ri Ci Ci Ti Vi Qi

7. Modeling of Electromechanical Systems Example 7.1 System including DC Motor Motor + - Vk Jm , Bm Ra , La Ki , Kb 1 2 By K2 z2 z1 JL 3 4 (Rigit shaft) For shaft 2: K2 : Rotational rigidity of shaft 2 Ra : Motor winding resistance La : Motor winding inductance Jm : Motor mass moment of inertia l: Length of the rotor windings Bm : Motor rotational damping constant Ki : Motor torque constant B r Kb : Counter-electromotive force constant Vk : Motor voltage F : Motor armature current JL : Mass moment of inertia of the load By : Rotational damping constant of bearing Counter-electromotive voltage Motor torque

; Generalized variables (Outputs): qa, θm, θL Motor + - Vk Jm , Bm Ra , La Ki , Kb 1 2 By K2 z2 z1 JL 3 4 (Rigit shaft) Shaft 2: Energy expressions : Input: Vk ; Generalized variables (Outputs): qa, θm, θL

Motor + - Vk Jm , Bm Ra , La Ki , Kb 1 2 By K2 z2 z1 JL 3 4

Example 7.2 Capacitor with moving plate x(t) Inputs: Vk(t) and fa(t) Generalized variables: q(t) and x(t) b/2 k/2 R C Fixed Movable, m Vk + - fa(t) System of Nonlinear Differential Equations: Solve with Runga Kutta method

x(t) Problem 2: m Vk R L(x) - + The core of the inductor moves with the x coordinate and the variation of the inductance is given by L(x). Find the mathematical model of the system. x(t) and q(t) are the generalized variables and Vk() is the input

R2 Problem 3: k1 m2 fk L(x1) Vk R1 - + x1 m1 C(x2) x2 fk(t) and Vk(t) are the inputs of the electro-mechanical system shown in the figure. q(t), x1(t) and x2(t) are the generalized variables. Find the mathematical model of the system.