Capitolo 0. INTRODUCTION 3.1
Mechanical transmission: state space model
• Mechanical transmission:
• POG dynamic model of the considered mechanical transmission:
F -
? 1 s
? 1 M1
?
x˙1
-
b
6
6 - -
-
6 1 s 6
K
6
Fe
-
-
? 1 s
? 1 M2j
?
x˙2
-
b2
6
6
- -- 1 R -
ω2
1
R -
Kj
s
6
6
τ
-
-
? 1 s
? 1 J1
ω?1
-
d1
6
6 - -
τd
• State vector (output power variables):
x =
x˙1 Fe x˙2 τ ω1
T
• The force F is the system control input. The torque τd is an external disturbance input.
• The output variable is the angular velocity ω1 of the last rotational element.
Zanasi Roberto - System Theory. A.A. 2015/2016
Capitolo 3. DYNAMIC MODELING 3.2
• The equivalent mass M2j can be expressed as follows:
M2j = M2 + J2
R2
• State space equations:
M1 x¨1 1 K
F˙e M2j x¨2
1 Kj τ˙ J1 ω˙1
| {z } L ˙x
=
−b1−1 0 0 0 1 0 −1 0 0 0 1−b2−R1 0 0 0 1
R 0 −1
0 0 0 1−d1
| {z }
A
x˙1
Fe x˙2
τ ω1
| {z } x
+
1 0 0 0 0 0 0 0 0 −1
| {z }
B
F τd
|{z}u
y =
0 0 0 0 1
| {z }
C
x that is
( L ˙x = A x + B u y = C x
⇓
( x = L˙ −1A x + L−1B u y = C x
dove L =
M1 0 0 0 0 0 1
K 0 0 0
0 0 M2j 0 0 0 0 0 1
Kj 0 0 0 0 0 J1
• The symmetric part of matrix A is a function of the dissipative parameters b1, b1, R and d1. The skew-symmetric part of matrix A is a function only of the “connection” coefficients.
• Possible values for the parameters to be used for simulations in Ma- tlab/Simulink environment:
M1 = 0.6*Kg; % First mass
b1 = 2*N/(40*m/sec); % Linear friction coefficient of the first mass K = 100*N/(1*cm); % Stiffness of the first spring
M2 = 1*Kg; % Second mass
b2 = 1*N/(50*m/sec); % Linear friction coefficient of the first mass R = 10*cm; % Radius of the wheel
J2 = 150*gr*(12*cm)^2; % Inerzia of element J2
Kj = 100*N/(0.1*rad); % Stiffness of the torsional spring J1 = 190*gr*(10*cm)^2; % Inerzia of element J1
d1 = 10*N*m/(100*rad/sec); % Linear friction coefficient of element J1 M2j = M2+J2/(R^2); % Equivalent translational mass
Zanasi Roberto - System Theory. A.A. 2015/2016
