Review Outline for System Dynamics

Rectangular form and polar form
- \(z=a+bj\) (a,b) are the coordinate in complex plane
- \(z=|z|e^{j\theta},|z|=\sqrt{a^2+b^2},\theta=arctan(\frac{b}{a})\)
Euler’s Formula
- \(e^{j\theta}=cos(\theta)+jsin(\theta)\)
Complex operations
- Complex conjugate: \(z^*=a-bj=|z|e^{-j\theta}\)
- Addition and Subtraction
- Multiplication: \(z_1z_2=|z_1||z_2|e^{j(\theta_1+\theta_2)}\)
- Division: \(\frac{z_1}{z_2}=\frac{|z_1|}{|z_2|}e^{j(\theta_1-\theta_2)}\)
- Inverse: \(z^{-1}=|z|^{-1}e^{-j\theta}\)
Complex variable and complex function
- \(s=\sigma+j\omega,~\sigma~and~\omega\in R\)
- \(F(s)=F_x+jF_y\), \(F_x\) and \(F_y\) are real functions
zeros and poles of a complex function
- \(F(s)=\frac{k(s+z_1)(s+z_2)...(s+z_m)}{(s+p_1)(s+p_2)...(s+p_n)}\)
- zeros: \(-z_1,-z_2,...,-z_m\)
- poles: \(-p_1,-p_2,...,-p_n\)
- In practice, the number of zeros is always less than or equal to the number of poles, \(m\le n\)

Definition of Laplace Transform (L.T.)
Existence criteria
Unit step function
- Used to transform the signal into L.T. applicable form
Laplace Transform table
Time delay and damping effect
Superposition of Laplace Transform
- Used to decompose the general input signal into typical forms easier for L.T.
Solving Differential Equation
- From time domain to Laplace domain, back to time domain
Final Value Theorem
- Used to compute the steady-state value of the output

\[ X(s)=\frac{b_0+b_1s+...+b_ms^m}{a_0+a_1s+...+a_ns^n}=\frac{B(s)}{A(s)} \]

Work by hand
- \(m\lt n\)
  - Distinct poles
  - Repeated poles
- \(m\ge n\)
  - Distinct poles
  - Repeated poles
Matlab
- Matlab Convention
  - The power of polynomials are in descending form and the power of poles are in ascending form

Two aspects of the construction of response
- Transient repsonse (due to initial condition and input) + steady-state response (due to input)
- Free vibration (dut to initial condition) + Forced vibration (due to input)
Find the steady-state response using Final Value Theorem

Definition of state-space model
- \(\dot{x}=Ax+Bu,~y=Cx+Du\)
- Applicable to nonlinear, time-varying differential equation
Example of a 2 DOF mechanical system
Block diagram of state-space model
Transfer function of state-space model
- \(G(s)=C(sI-A)^{-1}B+D=\frac{Y(s)}{U(s)}\)

Reynold Number
- Inertial force vs Viscous force
- Turbulent vs Laminar
Valve equation (Relationship between head H and flow rate Q)
- Laminar
- Turbulent
Capacitance
- Conservation of Volume
Example: Tank system
- Equation 1: conservation of volume
- Equation 2: valve equation

First order ODE system \(a\ddot{x}+b\dot{x}=0\)
- Time constant T: \(T=\frac{a}{b}\)
- Typical values
  - 0.37 at T, 0.83 at 2T, 0.95 at 3T, 0.98 at 4T
Second order ODE system
- only for underdamped system
- Time constant for the Exponential Decay Envelop
  - \(T=\frac{1}{\xi \omega_n}\)
- Experimentally determine the dample natural frequency and damping ration
  - \(T=\frac{2\pi}{\omega_d},~ln(\frac{x_1}{x_2})=\frac{2\pi\xi}{\sqrt{1-\xi^2}}\)
- Some terminology
  - Settling time-converge to 2% or 5%
  - Overshoot \(\%M_p\frac{x_{peak}-x_{ss}}{x_{ss}}\)
  - Peak time-rise to peak
  - Delay time-reach 50% of the steady-state value
  - Rise time-from 10% to 90% of the steady-state value

Steady-state response of a system to a sinusoidal input
- \(u(t)=Psin(\omega t),~x_{ss}=Xsin(\omega t+\phi)\)
- \(X\) and \(\phi\) are determined by \(\omega\)
Forced vibration without damping
Forced vibration with damping
Sinusoidal transfer function (Frequency response function)
- step 1: obtain transfer function \(G(s)\)
- step 2: let \(s\) equal to \(jw\)
- step 3: \(\frac{x_0}{P}=|G(jw)|,~\phi=\angle G(jw)\)

Condition for zero vibration in main system

\(\sqrt{\frac{k_a}{m_a}}=\omega_n\)