TMHP51 Servomechanisms (HT2012)
Lecture 06
Block Diagram of Model
Characteristics in Frequency Domain
Magnus Sethson
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1
Block Diagram
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2
Block Diagram
8
2
P
A
B
sX
F
=
M
s
Xp
>
L
p
p
p
L
t
>
<
Q = Ap sXp + 14 Vet sPL
>
>
:
Q = K q X v K c PL
Start
Kc
FL
Xv
Kq
4 e
Vt s
Ap
1
Mt s 2 + B p s
Xp
Ap s
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3
Block Diagram
8
2
P
A
B
sX
F
=
M
s
Xp
>
L
p
p
p
L
t
>
<
Q = Ap sXp + 14 Vet sPL
>
>
:
Q = K q X v K c PL
Final
FL
Kc +
Vt s
4 e
Ap
Xv
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Kq
s
h⇣
s
!h
⌘
1
Ap
+
2 h
!h s
+1
i
Xp
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4
Transfer Function
Xp =
Kq
Ap X v
s
h
Kc
A2p
s2
2
!h
⇣
+
1+
2 h
!h s
Vt s
4 e Kc
+1
i
⌘
FL
=
Kq
Ap X v
s
!s =
h
Kc
A2p
s2
2
!h
+
⇣
1+
2 h
!h s
s
!s
+1
⌘
i
FL
f
4 e Kc
(System Sti↵ness Frequency)
Vt
“An integrator, a resonant spring-mass system
and a frequency dependent disturbance”
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5
Position transfer function amplitude
50
40
30
|Xp(s)| [dB]
20
10
0
−10
−20
−30
1
10
2
3
10
10
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10
Frequency [rad/s]
Position transfer function phase
−80
−100
−120
Phase(Xp(s))
−140
−160
−180
−200
−220
−240
−260
−280
1
10
2
3
10
10
4
10
Frequency [rad/s]
Stiffness Amplitude
155
150
145
|(FL(s)| [dB]
140
135
130
125
120
115
110
105
1
10
2
3
10
10
4
10
Frequency [rad/s]
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% This is an example of ploting a frequency transfer function
% Basic parameters
D=0.050; % Piston diameter
d=0.020; % Rod diameter
L=0.8; % Cylinder length
V0=0.002; % Dead volume
Cq=0.67; % Flow ratio coefficient
rho=790; % Oil density
Ps=210e5; % Supply pressure
betae=1.5e9; % Bulk modulus
Mt=5; % Load mass
Bp=10; % Load Damping
% Operating point
xv0=0.0005; % Valve displacement
pL0=140e5; % Load pressure
% Intermediates
wv=pi*d; % Valve area gradient
Kq=Cq*wv*sqrt((Ps-pL0)/rho); % Flow coefficent
Kc=Cq*wv*xv0/(2*sqrt(rho*(Ps-pL0))); % Pressure coefficient
Ap=0.25*pi*(D*D-d*d); % Piston area
Vt=Ap*L+V0; % Cylinder volume
ws=4*betae*Kc/Vt; % System stiffnes frequency
wh=sqrt(4*betae*Ap*Ap/(Vt*Mt)); % System resonance
dh=Kc/Ap*sqrt(betae*Mt/Vt)+0.25*Bp/Ap*sqrt(Vt/(Mt*betae)); % System damping
% frequency vector
w=10:0.5:3000;
f=w/(2*pi);
s=w*i;
Gx=(ones(size(s))*Kq/Ap)./(s.*(s.*s/(wh*wh)+2*dh*s/wh+1));
Gf=(s.*(s.*s/(wh*wh)+2*dh*s/wh+ones(size(s))))./(Kc/(Ap*Ap)*(ones(size(s))+s/ws));
% Plot
subplot(3,1,1);
semilogx(w,20*log10(abs(Gx)));
grid;
xlabel('Frequency [rad/s]');
ylabel('|Xp(s)| [dB]');
title('Position transfer function amplitude');
subplot(3,1,2);
semilogx(w,unwrap(angle(Gx)/pi*180));
grid;
xlabel('Frequency [rad/s]');
ylabel('Phase(Xp(s))');
title('Position transfer function phase');
subplot(3,1,3);
semilogx(w,20*log10(abs(Gf)));
grid;
xlabel('Frequency [rad/s]');
ylabel('|(FL(s)| [dB]');
title('Stiffness Amplitude');
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Block Diagram Communities
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Block Diagram in Simulation Programs
Transfer Function
Power-Ports
(Bond Graphs)
Sub-System
Matlab/Simulink
Flowmaster (Flowmaster
Connectors represent
information flow
Connectors represent
energy flow
Amesim (LMS
Connectors represent
physical relations
Three Different “Communities”
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Next Lecture:
13:15, Friday 2012-11-16, P34
Magnus Sethson
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