- Captured On
- Source
- Exponential Resonse
1 Can we use the superposition principle for second order equation?
1.1 Front
Can we use the superposition principle for second order equation?
i.e. \(m\ddot{x} + b\dot{x} + kx = F_{ext}\)
1.2 Back
Yes, you can. Suppose \(x_p\) is any solution to this equation, and \(x_h\) is the solution of the homogeneous second order equation (\(m\ddot{x} + b\dot{x} + kx = 0\))
Then \(x = x_p + x_h\) is a solution
2 How can we proof the superposition principle for second order equation?
2.1 Front
How can we proof the superposition principle for second order equation?
\(m\ddot{x} + b\dot{x} + kx = F_{ext}\)
2.2 Back
\begin{align*} m\ddot{x} + b\dot{x} + kx &= m\ddot{(x_h + x_p)} + b\dot{(x_h + x_p)} + k(x_h + x_p) \\\ &= (m \ddot{x_h} + m\ddot{x_p}) + (b\dot{x_h} + b\dot{x_p}) + (kx_h + kx_p)\\\ &= (m \ddot{x_h} + b \dot{x_h} + k x_h) + (m \ddot{x_p} + b \dot{x_p} + k x_p) \\\ &= 0 + F_{ext} \end{align*}
3 Can we use the superposition principle for linear equations of any order?
3.1 Front
Can we use the superposition principle for linear equations of any order?
3.2 Back
Yes, you can. We saw it as a consequence of the method of integrating factors for first order equations
4 Write a solution guess for this second order ODE?
4.1 Front
Write a solution guess for this second order ODE?
\(\ddot{x} + 8 \dot{x} + 7 x = 9e^{2t}\)
4.2 Back
\({\displaystyle x(t) = Ae^{2t}}\)
5 What is the Exponential Response Formula (ERF) of the second order equation?
5.1 Front
What is the Exponential Response Formula (ERF) of the second order equation?
\(mx’’ + kx’ +bx = Be^{at}\)
5.2 Back
Let \(p( r) = mr^2 + kr + b\) be its characteristic polynomial
\({\displaystyle x_p(t) = \frac{B}{p(a)} e^{at}}\), as \(p(a) \neq 0\)
6 What is the form of a general solution for this second order equation?
6.1 Front
What is the form of a general solution for this second order equation?
\(mx’’ + kx’ +bx = Be^{at}\)
6.2 Back
Let \(p( r) = mr^2 + kr + b\) be its characteristic polynomial
\({\displaystyle x(t) = x_h(t) + x_p(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} + \frac{B}{p(a)}e^{at}}\)
Where \(r_1\) and \(r_2\) are root of the characteristic polynomial
7 What is the Exponential Response Formula (ERF) of this ODE?
7.1 Front
What is the Exponential Response Formula (ERF) of this ODE?
\({\displaystyle m\ddot{x} + b \dot{x} + kx = B \cos(\omega t)}\)
7.2 Back
\(z_p(t) = \frac{B}{p(i\omega)} e^{i\omega t}\)
8 What is the output amplitude knowing gain and input amplitude?
8.1 Front
What is the output amplitude knowing gain and input amplitude?
8.2 Back
\(\text{output amplitude} = \text{gain} \cross \text{input amplitude}\)
9 What is the gain from this second order ODE?
9.1 Front
What is the gain from this second order ODE?
\(m \ddot{x} + b\dot{x} + kx = B \cos (\omega t)\)
Give input and output amplitude also
9.2 Back
- Input amplitude: \(B\)
- Output amplitude: \(\frac{B}{p(i \omega)}\)
- Gain: \(\frac{1}{\abs{p(i \omega)}}\)
10 What is the general solution of simple harmonic oscillator with sinusoidal input?
10.1 Front
What is the general solution of simple harmonic oscillator with sinusoidal input?
\(m \ddot{x} + kx = Be^{i \omega t}\)
Use the natural frequency
10.2 Back
Natural frequency \(\omega_n = \sqrt{k/m}\)
Rewrite the equation: \(m(\ddot{x} + \omega_n^2 x) = Be^{i \omega t}\), which complex particular solution is
\({\displaystyle z_p(t) = \frac{B}{p(i \omega)} e^{i \omega t} = \frac{B}{m(\omega_n^2 - \omega)} e^{i\omega t}}\)
Takin the real part: \({\displaystyle \operatorname{Re} (z_p) = \frac{B}{m(\omega_n^2 - \omega^2)} \cos(\omega t)}\)
Using the homogeneous solution of the simple harmonic oscillator
\({\displaystyle x(t) = c_1 \cos(\omega_n t) + c_2 \sin(\omega_n t) + \frac{B}{m(\omega_n^2 - \omega^2)} \cos(\omega t)}\)
11 What happens when \(\omega\) approaches to \(\omega_{n}\) on simple harmonic oscillator?
11.1 Front
What happens when $\omega$ approaches to $\omega_{n}$ on simple harmonic oscillator?
\(m \ddot{x} + kx = Be^{i \omega t}\)
11.2 Back
Where \(\omega_n = \sqrt{k/m}\), the particular solution \(x_p(t) = \frac{B}{m(\omega_n^2 - \omega^2)} \cos(\omega t)\) will break down.
It’s called pure resonance, and this happens when \(p(a) = 0\)
12 What is the Resonant Response Formula (RRF) of this second order ODE?
12.1 Front
What is the Resonant Response Formula (RRF) of this second order ODE?
\(m \ddot{x} + b \dot{x} + kx = Be^{a t}\)
12.2 Back
When \(p(a) = 0\), and \(p’(a) \neq 0\)
Resonant Response Formula is: \(x(t) = \frac{B}{p’(a)}t e^{a t}\)
13 What is the Generalized Exponential Response Formula?
13.1 Front
What is the Generalized Exponential Response Formula?
Let \(p(D)\) be a polynomial operator
13.2 Back
\(p(D)\) must be a polynomial operator with constant coefficient, and \(p^{(s)}\) its s-th derivative. Then
\({\displaystyle p(D)x = Be^{at}}\), where \(a\) is real or complex
has the particular solution
- \({\displaystyle x_p(t) = \frac{Be^{at}}{p(a)}}\) if \(p(a) \neq 0\)
- \({\displaystyle x_p(t) = \frac{Bte^{at}}{p’(a)}}\) if \(p(a) = 0\) and \(p’(a) \neq 0\) (Resonant Response Formula)
- \({\displaystyle x_p(t) = \frac{Bt^2e^{at}}{p’’(a)}}\) if \(p(a) = p’(a) = 0\) and \(p’’(a) \neq 0\)
- \(\cdots\)
- \({\displaystyle x_p(t) = \frac{Bt^se^{at}}{p^{(s)}(a)}}\) if \(a\) is an \(s\text{-fold}\) zero
14 What is the difference between damping and natural angular frequency?
14.1 Front
What is the difference between damping and natural angular frequency?
Let \(m \ddot{x} + b \dot{x} + kx = Be^{at}\)
14.2 Back
- Natural angular frequency: \(\omega_n = \sqrt{k/m}\)
- It’s as the system oscillate without disturbance
- Damping angular frequency: it’s from characteristic polynomial
- \({\displaystyle \omega_d = \frac{\sqrt{\abs{b^2 - 4mk}}}{2m}}\)
- It’s a pseudo-frequency, but you need to be careful to call it because only periodic functions has frequency
15 Find \(A\) so that \(A\sin(3t)\) is a solution of…
15.1 Front
Find $A$ so that $A\sin(3t)$ is a solution of…
\(\ddot{x} + 4x = \sin(3t)\)
15.2 Back
Plug the desired solution into the differential equation to solve \(A\)
\({\displaystyle -9A \sin(3t) + 4A \sin(3t) = -5A \sin(3t) = \sin(3t) \implies A = \frac{-1}{5}}\)
16 Give the general solution to this 4th order equation?
16.1 Front
Give the general solution to this 4th order equation?
\({\displaystyle \dv[4]{x}{t} -x = \cos(2t)}\)
16.2 Back
This is a constant coefficient linear equation, where the characteristic polynomial \(p(s) = s^4 - 1\), where \(p(i2) = 15 \neq 0\)
So using ERF, the particular solution with sinusoidal input is
\({\displaystyle x_p(t) = \frac{1}{15} \cos(2t)}\)
The roots of \(p(s)\) are \(\pm 1, \pm i\), so the associated homogeneous solution is
\({\displaystyle x_h(t) = c_1 e^t + c_2 e^{-t} + c_3 \cos(t) + c_4 \sin(t)}\)
\({\displaystyle x(t) = x_p(t) + x_h(t)}\)
17 How could be find a particular solution for this ODE?
17.1 Front
How could be find a particular solution for this ODE?
\({\displaystyle \ddot{y} - 4y = \frac{1}{2} (e^{2x} + e^{-2x})}\)
17.2 Back
Using superposition, \(y_p = y_{p1} + y_{p2}\), where
- \(p(D)y_{p1} = \frac{1}{2} e^{2x}\)
- \(p(D)y_{p2} = \frac{1}{2} e^{-2x}\)
Characteristic equation: \(p(s) = s^2 - 4\)
Using the ERF, \(p(\pm 2) = 0\), but \(p’(\pm 2) = \pm 4\)
- \(y_{p1} = \frac{1/2}{4} x e^{2x} = \frac{1}{8} x e^{2x}\)
- \(y_{p2} = \frac{1/2}{-4} x e^{-2x} = - \frac{1}{8} x e^{-2x}\)
\({\displaystyle y_p(t) = \frac{x}{8} (e^{2x} - e^{-2x})}\)