1. 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

2. 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

3. ; 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

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

5. 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

6. 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

7. 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.