Harmonic Excitation and Response
• Steady-state response• Linear system:
– Sine in – Sine out– Excitation Frequency = Response Frequency
• Variable:– Response Amplitude– Response Phase (relative)
• Equation of Motion
• Harmonic Excitation
• Harmonic Response
mx(t) + cx(t) + kx(t) = F (t)
F (t) = F0 cos(ωt)
x(t) = X cos(ωt− φ)= X cosφ cos(ωt) +X sinφ sin(ωt)
= Xc cos(ωt) +Xs sin(ωt)
X = X2c +X
2s tanφ = Xs
Xc
−mω2Xc + cωXs + kXc = F0−mω2Xs − cωXc + kXs = 0
Xc =(k −mω2)F0
(k −mω2)2 + (cω)2
Xs =(cω)F0
(k −mω2)2 + (cω)2
X =F0p
(k −mω2)2 + (cω)2φ = tan−1
cω
k −mω2
X =X
F0/k=
kp(k −mω2)2 + (cω)2
=ω2np
(ω2n − ω2)2 + (2ζωnω)2
=1p
[1− (ω/ωn)2]2 + [2ζω/ωn]2
φ = tan−12ζωnω
ω2n − ω2
= tan−12ζω/ωn
1− (ω/ωn)2
zeta = [0;0.1;0.25;0.707;1;5];omegabar = [0:0.01:5];Xbar = 1./sqrt(ones(size(zeta))* ...
(1-omegabar.^2).^2 + (2*zeta*omegabar).^2);phi = atan2(2*zeta*omegabar, ...
ones(size(zeta))*(1-omegabar.^2));figure(1); plot(omegabar,Xbar)axis([0,3,0,10])legend('\zeta = 0','\zeta = 0.1', ...
'\zeta = 0.25‘,'\zeta = 0.707', ... '\zeta = 1','\zeta = 5')
figure(2); plot(omegabar,phi)axis([0,3,0,pi])legend('\zeta = 0','\zeta = 0.1', ...
'\zeta = 0.25‘,'\zeta = 0.707', ... '\zeta = 1','\zeta = 5')
Initial vs Harmonic
• Response Frequency: – Initial: (damped) natural frequency– Harmonic: excitation frequency
• Response Amplitude:– Initial: constant or decayed– Harmonic: increases to steady state
clear allA_undamped = [0 1;-4 0];A_damped = [0 1;-4 -0.2];B = [0;1];C = [1 0];D = [0];mysys_undamped = ss(A_undamped,B,C,D);mysys_damped = ss(A_damped,B,C,D);t = 0:0.1:50;U_resonance = cos(2*t);U_non = cos(t);lsim(mysys_undamped,mysys_damped,U_resonance,t)title('Harmonic response at Resonance')legend('excitation','undamped','damped')lsim(mysys_undamped,mysys_damped,U_non,t)title('Harmonic response at other frequencies')legend('excitation','undamped','damped')
0 10 20 30 40 50-15
-10
-5
0
5
10
15Harmonic response at Resonance
Time (sec)
Am
plitu
deexcitationundampeddamped
Harmonic response at other frequencies
Time (sec)
Am
plitu
de
0 10 20 30 40 50-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1excitationundampeddamped
Conclusions• At resonance: π/2 phase lag
– Undamped system oscillation amplitude increases linearly with time (no steady-state)
– Damped system oscillation amplitude increases linearly with time initially but then settles to the steady-state periodic response predicted by theory
• Off-resonance:– Undamped system oscillates at a constant amplitude
as predicted by the theory but the response is not harmonic because of the superposition of initial condition response (0 phase lag)
– Damped system behavior is similar to the undampedsystem except that the initial condition response decays in time so that the system finally oscillates at the steady-state response as predicted by theory (approximately 0 phase lag)
Resonance vs Instability
• Response:– Resonance: Linearly increasing in time– Instability: Exponentially increasing in time
• Cause:– Resonance: External excitation– Instability: Internal ‘excitation,’ i.e., coupling of
the subsystems where one subsystem affects the other
Fourier Series
F (t) =a02+
∞Xp=1
(ap cos pω0t+ bp sin pω0t)
ap =2
T
Z T
0
F (t) cos(pω0t)dt, p = 0, 1, 2, . . .
bp =2
T
Z T
0
F (t) sin(pω0t)dt, p = 1, 2, . . .
t
A
T
F(t)
ap =2
T
ÃZ T4
0
A cos(pω0t)dt
+
Z 3T4
T4
(−A) cos(pω0t)dt
+
Z T
3T4
A cos(pω0t)dt
!, p = 0, 1, 2, . . .
ap =2A
Tpω0
hsin(pω0t)|
T40
− sin(pω0t)|3T4T4
+ sin(pω0t)|T3T4
i, p = 0, 1, 2, . . .
ap =A
pπ
hsin(pω0t)|
T40
− sin(pω0t)|3T4T4
+ sin(pω0t)|T3T4
i, p = 0, 1, 2, . . .
For odd p:
ap =A
pπ
h³(−1) p−12 − 0
´−³−(−1) p−12 − (−1) p−12
´+³0−
³−(−1) p−12
´´i=4A
pπ(−1) p−12 , p = 1, 3, 5, . . .
ap = 0, p = 0, 2, 4, . . .For even p:
For even p:
bp =−Apπ
[(0− 1)
− (0− 0)+ (1− 0)]
= 0, p = 0, 2, 4, . . .
For odd p: bp = 0, p = 1, 3, 5, . . .
F (t) = A
∙4
πcosω0t−
4
3πcos 3ω0t+
4
5πcos 5ω0t− . . .
¸
syms tomega_0 = 2*pi;A = 1;F1 = (4*A/pi)*cos(omega_0*t);F2 = (4*A/pi)*cos(omega_0*t) ...
- (4*A/pi/3)*cos(3*omega_0*t);F10 = 0;for p = 1:2:19
F10 = F10+(-1)^((p-1)/2)*(4*A/pi/p)* ...cos(p*omega_0*t);
endplot([0,0.25,0.25,0.75,0.75,1.25, ...
1.25,1.75,1.75,2.25,2.25,2.75,2.75,3.0], ...[1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1],'k')
hold onezplot(F1,[0,3])ezplot(F2,[0,3])ezplot(F10,[0,3])hold off
Steady-State Responsex(t) =
1
k
a02
+∞Xp=1
apkXp cos(pω0t− φp)
+∞Xp=1
bpkXp sin(pω0t− φp)
Xp =1p
[1− (pω0/ωn)2]2 + [2ζpω0/ωn]2
φp = tan−1 2ζpω0/ωn1− (pω0/ωn)2
clear allsyms tomega_0 = 2*pi;A = 1;k=1;omega_n = pi;zeta = 0.1;F1 = 0;for p = 1
omegabar = p*omega_0/omega_n;Xbar_p = 1./sqrt(ones(size(zeta))* ...
(1-omegabar.^2).^2 + (2*zeta*omegabar).^2);phi_p = atan2(2*zeta*omegabar, ...
ones(size(zeta))*(1-omegabar.^2));F1 = F1+(-1)^((p-1)/2)*(4*A/pi/p)*Xbar_p/k*cos(p*omega_0*t-phi_p);
endF2 = 0;for p = 1:2:3
omegabar = p*omega_0/omega_n;Xbar_p = 1./sqrt(ones(size(zeta))* ...
(1-omegabar.^2).^2 + (2*zeta*omegabar).^2);phi_p = atan2(2*zeta*omegabar, ...
ones(size(zeta))*(1-omegabar.^2));F2 = F2+(-1)^((p-1)/2)*(4*A/pi/p)*Xbar_p/k*cos(p*omega_0*t-phi_p);
end
F10 = 0;for p = 1:2:19
omegabar = p*omega_0/omega_n;Xbar_p = 1./sqrt(ones(size(zeta))* ...
(1-omegabar.^2).^2 + (2*zeta*omegabar).^2);phi_p = atan2(2*zeta*omegabar, ...
ones(size(zeta))*(1-omegabar.^2));F10 = F10+(-1)^((p-1)/2)*(4*A/pi/p)*Xbar_p/k*cos(p*omega_0*t-
phi_p);endezplot(F1,[0,3])hold onezplot(F2,[0,3])ezplot(F10,[0,3])hold offtitle('Convergence of Periodic Response')ylabel('x(t)')