Forced Vibration


By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH

Table of Contents

In [5]:
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1. Sinusoidal Signals

$$ x(t) = x_m \cos(\omega t + \theta)$$

  • amplitude: $x_m$

  • period: T sec

  • frequency (Hz): $f = \frac{1}{T}$

  • frequency (rad/sec): $\omega = 2\pi f$

  • phase angle: $\theta$

In [6]:
f = 1;
w = 2*pi*f;
T = 1/f;
xm = 1;
theta = pi/3;

t = 0:0.01:3*T;

x = xm*cos(w*t + theta);
plot(t,x,t,zeros(size(t)),'k--'), ylim([-1.5 1.5])
Out[6]:

1.1. Two sinusoidal signals in complex plane

$$ \begin{align*} z_1 &= A e^{j \omega t} \\ z_2 &= A e^{j\left(\omega t + \phi \right)} \end{align*}$$
  • Both $z_1$ and $z_2$ ar rotating with angular velocity $\omega$, but angle $\phi$ between $z_1$ and $z_2$ is remaining constant.

1.2. Sinusoidal input and output to differential system

  • input
$$ x = x_m e^{j\omega t}$$
  • output
$$ y = \frac{dx}{dt} = j\omega \cdot x_m e^{j\omega t} = j\omega x = \omega e^{j\frac{\pi}{2}}x$$
note
1 sinusoidal $\longrightarrow$ sinusoidal
2 $\omega \longrightarrow \omega$ input and output rotate with the same $\omega$
3 $\phi = \frac{\pi}{2}$ output leads input by $\frac{\pi}{2}$
4 $A = \left\lvert \frac{y}{x} \right\rvert = \omega$ amplitude gain is a function of $\omega$
In [7]:
f = 1;
w = 2*pi*f;
T = 1/f;
xm = 1;
theta = pi/3;

t = 0:0.01:3*T;

x1 = xm*exp(1j*w*t);
x2 = xm*exp(1j*(w*t + theta));

plot(t,real(x1),t,real(x2),t,zeros(size(t)),'k--'), ylim([-1.5 1.5])
legend('input','output')
Out[7]:

1.3. Sinusoidal input to linear system

2. Forced Oscillation

  • So far, we have not taken an external force into account

  • large input $\rightarrow$ large output ?

    • not always true. why ?
    • non-linear system ?

  • Swing example

    • frequency
    • phase

  • sinusoidal input and sinusoidal output ?

    • difficult to explain

2.1. Mass + spring + damper system



  • Equations of motion
$$ \quad m \ddot{x} + c \dot{x} + kx = f(t) $$
  • When $f(t) = F_0 \cos \Omega t$
$$ \quad m \ddot{x} + c \dot{x} + kx = F_0\cos \Omega t $$
  • We could also write this equation in terms of the damping ratio, $\zeta$, and natural frequency, $\omega_n$.
$$ \quad \ddot{x} + 2\zeta\omega_n\dot{x} + \omega_n^2x = \frac{F_0}{m}\cos \Omega t = \omega_n^2 \frac{F_0}{k}\cos \Omega t$$
  • in complex number representation $x = \text{Re}(z)$
$$ \quad \ddot{z} + 2\zeta\omega_n\dot{z} + \omega_n^2 z = \omega_n^2 \frac{F_0}{k}e^{j \Omega t}$$
  • Which ones are input and ouput?
    • input = $f$
    • output = $x$
  • We know that $z$ is in the form of
$$z = A e^{j(\Omega t + \phi)}$$


$$ \begin{align*}\left( -\Omega^2 + j 2\zeta \omega_n \Omega + \omega_n^2 \right)Ae^{j\phi} e^{j\Omega t} &= \omega_n^2 \frac{F_0}{k}e^{j \Omega t} \\ Ae^{j\phi} &= \frac{ \omega_n^2 \frac{F_0}{k}}{-\Omega^2 + j 2\zeta \omega_n \Omega + \omega_n^2} = \frac{F_0}{k}\frac{1}{1-\left(\frac{\Omega}{\omega_n}\right)^2 + j 2\zeta \left(\frac{\Omega}{\omega_n}\right)} \end{align*}$$


$$ \begin{align*}A &= \frac{F_0}{k}\frac{1}{ \sqrt{ \left(1-\left(\frac{\Omega}{\omega_n}\right)^2 \right)^2 + 4\zeta^2 \left(\frac{\Omega}{\omega_n}\right)^2 }} = \frac{F_0}{k}\frac{1}{ \sqrt{ \left(1-\gamma^2 \right)^2 + 4\zeta^2 \gamma^2 }}, \quad \left(\gamma = \frac{\Omega}{\omega_n} \right)\\ \phi &= -\tan^{-1} \left( \frac{2\zeta \frac{\Omega}{\omega_n}}{1-\left(\frac{\Omega}{\omega_n}\right)^2} \right) =-\tan^{-1} \left( \frac{2\zeta \gamma}{1-\gamma^2} \right) \end{align*}$$

In [19]:
r = 0:0.1:4;

zeta = 0.1:0.2:1;
M = [];
for i = 1:length(zeta)
    M(i,:) = 1./sqrt((1-r.^2).^2 + (2*zeta(i)*r).^2); % ignore a constant of F0/k
end

plot(r,M)
xlabel('\gamma','fontsize',16)
ylabel('M','fontsize',14)
legend('0.1','0.3','0.5','0.7','0.9')
Out[19]:
In [22]:
phi = [];
for i = 1:length(zeta)
    phi(i,:) = -atan2((2*zeta(i).*r),(1-r.^2));
end

plot(r,phi*180/pi)
xlabel('\gamma','fontsize',16)
ylabel('\phi','fontsize',16)
legend('0.1','0.3','0.5','0.7','0.9')
Out[22]:

2.2. Resonance

  • Input frequency near resonance frequency
  • Resonance frequency is generally different from natural frequency, but they often are close enough
In [23]:
% in time domain
f = 1;
w = 2*pi*f;
F0 = 1;
k = 1;

zeta = 0.2;
r = 3;
M = 1./sqrt((1-r.^2).^2 + (2*zeta*r)^2); % ignore a constant of F0/k
phi = -atan2((2*zeta*r),(1-r^2));

t = 0:0.01:3;

x1 = F0*exp(1j*w*t);
x2 = F0/k*M*exp(1j*(w*t + phi));

plot(t,real(x1),t,real(x2),t,zeros(size(t)),'k--'), ylim([-4 4])
legend('input','output')
Out[23]:
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