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Physical Modeling (Guitar) \[ \overbrace{\underbrace{\frac{\partial^2…

- - - - E
    - - I
- - - - Guitar String Properties
        
        \(\sigma_{\tiny{TLM}} = \frac{T}{\mu} = \frac{T L}{M}\)
        
        Tension \(T\)
        
        Mass \(M\)
        
        Length \(L\)
        
        Linear density \(\mu = \frac{M}{L}\)
    - - Ideal String
        
        Vibrates only in one direction
        
        No friction and no other losses
        
        Perfectly flexible (rubber band)
- - - - \[ \left. \begin{align} \delta_{t+} &:= \frac{1}{\Delta t}\ \left( e_{t+}-1 \right) \\ \\ \delta_{t-} &:= \frac{1}{\Delta t}\ \left( 1-e_{t-} \right) \\ \\ \delta_{t} &:= \frac{1}{2 \ \Delta t}\ \left( e_{t+}-e_{t-} \right) \end{align} \right\} \ \approx \ \frac{d}{dt} \]
        
        \[ \begin{align} \delta_{t+} \, u_{\left[ n \right]} &= \ \frac{1}{\Delta t}\ \left( u_{\left[ n+1 \right]} - u_{\left[ n \right]} \right) \\ \\ \delta_{t-} \, u_{\left[ n \right]} &= \ \frac{1}{\Delta t}\ \left( u_{\left[ n \right]} - u_{\left[ n-1 \right]} \right) \\ \\ \delta_{t} \; u_{\left[ n \right]} &= \ \frac{1}{2 \Delta t}\ \left( u_{\left[ n+1 \right]} - u_{\left[ n-1 \right]} \right) \end{align} \]
      - \[ \left. \begin{align} \mu_{t+} &:= \frac{1}{2}\ \left( e_{t+}+1 \right) \\ \\ \mu_{t-} &:= \frac{1}{2}\ \left( 1+e_{t-} \right) \\ \\ \mu_{t} &:= \frac{1}{2}\ \left( e_{t+}+e_{t-} \right) \end{align} \right\} \ \approx \ 1 \]
      - Identity
        \[ 1 \; u_{\left[ n \right]} = \ u_{\left[ n \right]} \]
      - \[ \delta_{t t} \ := \ \delta_{t+} \ \delta_{t-} \ = \ \frac{1}{\Delta t^2} \left( e_{t+} - 2 + e_{t-} \right) \ \approx \ \frac{d^2}{dt^2} \]
      - \[ \delta_{t t t t} \ := \ \delta_{t t} \ \delta_{t t} \ = \ \frac{1}{\Delta t^4} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]
      - Shift
        \[ \begin{align} e_{t+} \, u_{\left[ n \right]} &= \ u_{\left[ n+1 \right]}\\ \\ e_{t-} \, u_{\left[ n \right]} &= \ u_{\left[ n-1 \right]} \end{align} \]
      - \[ \delta_{t-} \delta_{x x} \ = \ \frac{1}{2 \ \Delta t \ \Delta x^2} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]
- - - - \[ \begin{align} \frac{1}{\Delta t^2} \left( y_{\left( x , \ t - \Delta t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x , \ t + \Delta t \right)} \right) \quad =& \quad \tfrac{T}{\mu} \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \right) \qquad \ldots \\ \\ & \quad - \quad 2 \ \sigma_0 \frac{1}{2 \ \Delta t} \left( y_{\left( x , \ t + \Delta t \right)} \ - \ y_{\left( x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad \sigma_1 \frac{1}{\Delta t} \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \ - \ y_{\left( x - \Delta x , \ t - \Delta t \right)} \ + \ 2 \ y_{\left( x , \ t - \Delta t \right)} \ - \ y_{\left( x + \Delta x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad E \ \ I \ \ \frac{1}{\Delta x^4} \left( y_{\left( x + 2 \Delta x , \ t \right)} -\ 4 \ y_{\left( x + \Delta x, \ t \right)} +\ 6 \ y_{\left( x , \ t \right)} \ - \ 4 \ y_{\left( x - \Delta x, \ t \right)} \ + \ y_{\left( x - 2 \Delta x , \ t \right)} \right) \end{align} \]
        
        \[ \begin{align} y_{\left( x , \ t + \Delta t \right)} \quad = \quad \frac{1}{1 + \sigma_0 \ \Delta t} & \Bigg( 2 \ y_{\left( x , \ t \right)} \ \ - \ \ y_{\left( x , \ t - \Delta t \right)} \quad \ldots \\ & \quad +\quad \tfrac{T}{\mu} \frac{\Delta t^2}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad \sigma_0 \ \Delta t \ y_{\left( x , \ t - \Delta t \right)} \qquad \ldots \\ \\ & \quad + \quad \sigma_1 \frac{\Delta t}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \ - \ y_{\left( x - \Delta x , \ t - \Delta t \right)} \ + \ 2 \ y_{\left( x , \ t - \Delta t \right)} \ - \ y_{\left( x + \Delta x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad E \ \ I \ \ \frac{\Delta t^2}{\Delta x^4} \left( y_{\left( x + 2 \Delta x , \ t \right)} -\ 4 \ y_{\left( x + \Delta x, \ t \right)} +\ 6 \ y_{\left( x , \ t \right)} \ - \ 4 \ y_{\left( x - \Delta x, \ t \right)} \ + \ y_{\left( x - 2 \Delta x , \ t \right)} \right) \Bigg) \end{align} \]
        
        \[ y_{\left( x, t + \Delta t \right)} \quad = \quad \left( \frac{1}{1+\sigma_0 \Delta t} \right) \; \underbrace{ \begin{bmatrix} ~ \\ 1 && \dfrac{T}{\mu} \dfrac{\Delta t^2}{\Delta x^2} && \sigma_0 \ \Delta t && \sigma_1 \ \dfrac{\Delta t}{\Delta x^2} && E \ I \ \dfrac{\Delta t^2}{\Delta x^4} \\ ~ \end{bmatrix} }_{\cdots} \; \begin{bmatrix} 2 & & 1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 0 & & 1 & & 1 & & 0 & & 0 & & 0 & & 0 \\ \\ 0 & & 1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 2 & & 1 & & 1 & & -1 & & -1 & & 0 & & 0 \\ \\ 6 & & 0 & & -4 & & -4 & & 0 & & 0 & & 1 & & 1 \end{bmatrix} \; \begin{bmatrix} y_{\left( x \ , \ t \right)} \\ \\ y_{\left( x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x - \Delta x \ , \ t \right)} \\ \\ y_{\left( x +\Delta x \ , \ t \right)} \\ \\ y_{\left( x - \Delta x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x +\Delta x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x - 2 \Delta x \ , \ t \right)} \\ \\ y_{\left( x + 2 \Delta x \ , \ t \right)} \end{bmatrix} \]
        
        \[ y^{(1 \ldots 6)}_{\left( x, t + \Delta t \right)} \ldots = \quad \left( \frac{1}{1+\sigma_0 \Delta t} \right) \; \overbrace{ \begin{bmatrix} ~ \\ 1 && \dfrac{T}{\mu} && \sigma_0 && \sigma_1 && E \ I \\ ~ \end{bmatrix} \; \begin{bmatrix} 1 && 0 && 0 && 0 && 0 \\ \\ 0 && \dfrac{\Delta t^2}{\Delta x^2} && 0 && 0 && 0 \\ \\ 0 && 0 && \Delta t && 0 && 0 \\ \\ 0 && 0 && 0 && \dfrac{\Delta t}{\Delta x^2} && 0 \\ \\ 0 && 0 && 0 && 0 && \dfrac{\Delta t^2}{\Delta x^4} \end{bmatrix} }^{\cdots} \; \begin{bmatrix} 2 & & 1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 0 & & 1 & & 1 & & 0 & & 0 & & 0 & & 0 \\ \\ 0 & & 1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 2 & & 1 & & 1 & & -1 & & -1 & & 0 & & 0 \\ \\ 6 & & 0 & & -4 & & -4 & & 0 & & 0 & & 1 & & 1 \end{bmatrix} \; \begin{bmatrix} y_{\left( x \ , \ t \right)} \\ \\ y_{\left( x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x - \Delta x \ , \ t \right)} \\ \\ y_{\left( x +\Delta x \ , \ t \right)} \\ \\ y_{\left( x - \Delta x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x +\Delta x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x - 2 \Delta x \ , \ t \right)} \\ \\ y_{\left( x + 2 \Delta x \ , \ t \right)} \end{bmatrix} \]
        
        generalised for 6 strings
        
        1 more item...