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Dynamics (((Ideal String Vibrates only in one direction No friction…

- - - - Physical Modeling (Guitar)
        
        \[ \overbrace{\underbrace{\frac{\partial^2 y}{\partial t^2}}_{\tiny\text{acceleration}} \quad =\quad \sigma_{\tiny{TLM}} \underbrace{\frac{\partial^2 y}{\partial x^2}}_{\tiny\text{curvature}} }^{\small{\text{basic wave equation}}}\ \overbrace{- \ \ 2 \sigma_{\tiny{0}} \frac{\partial y} {\partial t} \ + \ \sigma_{\tiny{1}} \frac{\partial}{\partial t} \frac{\partial^2 y}{\partial x^2} }^{\small{\text{damping}}}\ \overbrace{+ \ \ \sigma_{\tiny{E I}} \frac{\partial^4 y}{\partial x^4}}^{\small\text{stiffness}} \]
        
        Martin Shuppius - Physical modelling of guitar strings
        BILBAO, Stefan. Numerical sound synthesis: finite difference schemes and simulation in musical acoustics. John Wiley & Sons, 2009.
        
        \[ \begin{align} \frac{1}{\Delta t^2} \left( y_{\left( x , \ t - \Delta t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x , \ t + \Delta t \right)} \right) \quad =& \quad \tfrac{T}{\mu} \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \right) \qquad \ldots \\ \\ & \quad - \quad 2 \ \sigma_0 \frac{1}{2 \ \Delta t} \left( y_{\left( x , \ t + \Delta t \right)} \ - \ y_{\left( x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad \sigma_1 \frac{1}{\Delta t} \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x ,\ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \ - \ y_{\left( x - \Delta x , \ t - \Delta t \right)} \ + \ 2 \ y_{\left( x , \ t \Delta t \right)} \ - \ y_{\left( x + \Delta x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad E \ \ I \ \ \frac{1}{\Delta x^4} \left( y_{\left( x + 2 \Delta x , \ t \right)} -\ 4 \ y_{\left( x + \Delta x, \ t \right)} +\ 6 \ y_{\left( x , \ t \right)} \ - \ 4 \ y_{\left( x - \Delta x, \ t \right)} \ + \ y_{\left( x - 2 \Delta x , \ t \right)} \right) \end{align} \]
        
        \[ \begin{align} y_{\left( x , \ t + \Delta t \right)} \quad = \quad \frac{1}{1 + \sigma_0 \ \Delta t} & \Bigg( 2 \ y_{\left( x , \ t \right)} \ \ - \ \ y_{\left( x , \ t - \Delta t \right)} \quad \ldots \\ & \quad +\quad \tfrac{T}{\mu} \frac{\Delta t^2}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad \sigma_0 \ \Delta t \ y_{\left( x , \ t - \Delta t \right)} \qquad \ldots \\ \\ & \quad + \quad \sigma_1 \frac{\Delta t}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \ - \ y_{\left( x - \Delta x , \ t - \Delta t \right)} \ + \ 2 \ y_{\left( x , \ t - \Delta t \right)} \ - \ y_{\left( x + \Delta x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad E \ \ I \ \ \frac{\Delta t^2}{\Delta x^4} \left( y_{\left( x + 2 \Delta x , \ t \right)} -\ 4 \ y_{\left( x + \Delta x, \ t \right)} +\ 6 \ y_{\left( x , \ t \right)} \ - \ 4 \ y_{\left( x - \Delta x, \ t \right)} \ + \ y_{\left( x - 2 \Delta x , \ t \right)} \right) \Bigg) \end{align} \]
        
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- - - - 2-D
        
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        conservative systems
        
        \[ \begin{align} \boxed{ \boxed{ \underline{\dot{x}} = \underline{f} \left( \underline{x} \right) } \quad conservative \quad } \quad & \iff \quad \exists \; E_{\left( \cdot \right)} \; \in \; ^{conserved} _{~~quantity} \left( \mathbb{R}^n \right) \quad \implies \quad \frac{d E_{(\cdot)}}{d t} = 0 \quad \left( ^{const} _{on trajectories} \right) \\ ~ \\ ~ \\ ~ \\ \; ^{conserved} _{~~quantity} \left( \mathbb{R}^n \right) & \quad = \quad \left\{ \quad E _{\left( \right)} : \ \mathbb{R}^n \to \mathbb{R} \begin{matrix} \quad \\ \quad \end{matrix} \middle| \quad \not \exists \; \mathbb{X} \; \in \; ^{open} _{~~set} \left( \mathbb{R}^n \right) \quad . \quad \forall \underline{x} , \underline{\tilde{x}} \in \mathbb{X} \quad . \quad E \left( \underline{x} \right) = E \left( \underline{\tilde{x}} \right) = const \quad \right\} \\ ~ \\ \; ^{open} _{~~set} \left( \mathbb{R}^n \right) & \quad = \quad \end{align} \]
        
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        1-D, vectorfield on a line
        \[ \dot{x} \quad = \quad f \left( x \right) \]
        
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- - - - Video Lectures
        
        MAE5790-3 . Overdamped bead on a rotating hoop
        
        MAE5790-1 . Course introduction and overview
        
        MAE5790-4
        
        MAE5790-2 . One dimensional Systems
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    - - \[ \begin{align*} ~ & ~~~~~~~~~~\left( \begin{aligned} r \; cos(\phi) \\ ~ \\ r \; sin(\phi) \end{aligned} \right) & ~ \\ ~\\ \boxed{\boxed{ \quad \left( \begin{aligned} x \\ \\ y \end{aligned} \right) \quad }} &~~~~~~~~~~~~~~~~~~\leftrightarrows& \boxed{\boxed{ \quad \left( \begin{aligned} r \\ \\ \phi \end{aligned} \right) \quad }} \\ ~\\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ ~ \\ \text{atan2}(y,x) \end{aligned} \right) & ~ \end{align*} \]
        
        \[ \begin{align*} ~ & ~~~~~~~~~~\left( \begin{aligned} r \cdot cos(\phi) \\ ~ \\ r \cdot sin(\phi) \end{aligned} \right) & ~ \\ ~\\ \boxed{\boxed{ \quad \left( \begin{aligned} \dot{x} = f_x(x,y)\\ \\ \\ \dot{y} = f_y(x,y) \end{aligned} \right) \quad }} &~~~~~~~~~~~~~~~~~~\leftrightarrows &\boxed{\boxed{ \quad \left( \begin{aligned} \dot{r} = f_r(r,\phi) \\ \\ \\ \dot{\phi} = f_{\phi}(r, \phi) \end{aligned} \right) \quad }} \\ \\ \\ ~ & ~~~~~~~ \left( \begin{aligned} \frac{\partial \ \sqrt{ x^2 + y^2} }{\partial \ t} \quad = \quad \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ t} \quad = \quad \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot \frac{\partial \ x}{\partial t} \cdot \frac{\partial \ x}{\partial t} \end{aligned} \right) & ~ \\ ~\\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot f_x(x,y) \cdot f_y(x,y) \end{aligned} \right) & ~ \\ \\ \\ \\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot f_x(x,y) \cdot f_y(x,y) \end{aligned} \right) & ~ \end{align*} \]
        
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        {\left\downarrow\vbox to #1{}\right.\kern-\nulldelimiterspace}
        
        }
- - - - \[\normalsize \frac{d y}{dt} = f \left( t , y \right) \; , \qquad y_\left(t_0 \right) = y_0 \]
        
        \[ {\hat y}_{n+1} \quad \leftarrow \quad {\hat y}_{n} \quad + \quad \boxed{ \Delta^{(s)} {\hat y}_{n} \;\; = \; \; h \cdot{\sum _{i=1}^{s}} b_i \cdot \boxed{ k_i \;\; = \; \; f\left( \;\; \underbrace{t_n}_{n \cdot h} + c_i \cdot h \quad , \quad y_n + h \sum_{j = 1}^{i-1} a_{ij} k_j\right)\quad } }\]
        
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    - - # Finite Differences
        \[ \frac{\partial^2 y}{\partial x^2}\;\approx\; \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x\right)} -\ 2 \ y_{\left( x\right)} +\ y_{\left( x + \Delta x \right)} \right) \] differential equations approx. by difference equations
        
        operator
        
        Shift
        \[ \begin{align} e_{t+} \, u_{\left[ n \right]} &= \ u_{\left[ n+1 \right]}\\ \\ e_{t-} \, u_{\left[ n \right]} &= \ u_{\left[ n-1 \right]} \end{align} \]
        
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        \[ \left. \begin{align} \delta_{t+} &:= \frac{1}{\Delta t}\ \left( e_{t+}-1 \right) \\ \\ \delta_{t-} &:= \frac{1}{\Delta t}\ \left( 1-e_{t-} \right) \\ \\ \delta_{t} &:= \frac{1}{2 \ \Delta t}\ \left( e_{t+}-e_{t-} \right) \end{align} \right\} \ \approx \ \frac{d}{dt} \]
        
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        Identity
        \[ 1 \; u_{\left[ n \right]} = \ u_{\left[ n \right]} \]
        
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        \[ \delta_{t t} \ := \ \delta_{t+} \ \delta_{t-} \ = \ \frac{1}{\Delta t^2} \left( e_{t+} - 2 + e_{t-} \right) \ \approx \ \frac{d^2}{dt^2} \]
        
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        \[ \delta_{t t t t} \ := \ \delta_{t t} \ \delta_{t t} \ = \ \frac{1}{\Delta t^4} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]
        
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        \[ \delta_{t-} \delta_{x x} \ = \ \frac{1}{2 \ \Delta t \ \Delta x^2} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]
        
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        \[ \left. \begin{align} \mu_{t+} &:= \frac{1}{2}\ \left( e_{t+}+1 \right) \\ \\ \mu_{t-} &:= \frac{1}{2}\ \left( 1+e_{t-} \right) \\ \\ \mu_{t} &:= \frac{1}{2}\ \left( e_{t+}+e_{t-} \right) \end{align} \right\} \ \approx \ 1 \]
        
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- - - - ?
        
        ?
  - - - ?
- - - - \[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad b \cdot | z |^{2} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\cdot | z |^{2} \right)^{\tilde \ell} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
  - - - \[ \dot{z} \qquad=\qquad z \cdot \left( \alpha + \text{i} \, \omega \quad + \quad a_1 ~ | z |^{2} \;\; + \;\; a_2 ~ \varepsilon ~ | z |^{4} ~ \frac{1}{1 - \varepsilon |z|^2} \right) \qquad + \qquad c \;\; \mathcal{P} \left( \varepsilon , s \right) \; \mathcal{A} \left( \varepsilon , \bar z \right) \]
        
        \[ \boldsymbol{\dot{z}} _i \quad=\qquad \boldsymbol{z}_{\ i} \cdot \left(\qquad a_i \qquad + \qquad b_i \ | \boldsymbol{z}_{\ i} |^{2} \qquad + \qquad d_i \ \varepsilon \ | \boldsymbol{z}_{\ i} |^{4} \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\ | \boldsymbol{z}_{\ i} |^{2} \right)^{\tilde \ell} \right) \qquad+ \qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
        
        \[ \boldsymbol{\dot{z}} _i \quad=\qquad \boldsymbol{z}_{\ i} \cdot \left( \underbrace{ \left( \alpha + j \, 2 \pi \right) \ f_i }_{a_i} \qquad + \qquad \underbrace{ \left( \beta + j \, \beta' \right)f_i }_{b_i} \ | \boldsymbol{z}_{\ i} |^{2} \qquad + \qquad \underbrace{ \left( \delta + j \, \delta' \right) f_i }_{d_i} \ \frac{ \varepsilon \ | \boldsymbol{z}_{\ i} |^{4}}{1 - \varepsilon \ | \boldsymbol{z}_{\ i} |^{2}}\right) \qquad+ \qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
  - - - \[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad \sum_{ \ell = 0}^{L-1} b_{\ell+1} ~ \varepsilon^{\ell} \cdot | z |^{2(\ell+1)} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\cdot | z |^{2} \right)^{\tilde \ell} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
  - - - \[ \dot{z} \quad=\qquad z \cdot \left( \lambda_o \qquad + \qquad \sum_{ \ell = 0}^{\infty} ~ \lambda_{\ell+1} \cdot \varepsilon^{\ell} \cdot |z |^{2(\ell+1)} \right)\quad\quad\quad+\quad\quad\quad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
        
        splitting series at \( L \)
        
        \[ \dot{z} \quad=\qquad z \cdot \left( \lambda_o \qquad + \qquad \sum_{ \ell = 0}^{L-1} \lambda_{\ell+1} \varepsilon^{\ell} ~ | z |^{2(\ell+1)} \quad + \quad \sum_{ \ell = L}^{\infty} \lambda_{\ell+1} ~ \varepsilon^{\ell} ~ | z |^{2(\ell+1)} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
  - - - \[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad \sum_{ \ell = 0}^{L-1} b_{\ell+1} ~ \varepsilon^{\ell} \cdot | z |^{2(\ell+1)} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ \ell = L}^{\infty} \left( \varepsilon\cdot | z |^{2} \right)^{\ell-1} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
  - - - \[ \dot{z} \quad = \qquad z \cdot \left( \underbrace{\left( \alpha_o + j \, \omega_o \right)}_{a_o} \quad + \quad \underbrace{\left( \alpha_1 + j \, \omega_1 \right)}_{a_1} | z |^2 \right) \quad\quad\quad+\quad\quad\quad h.o.t \qquad\qquad\qquad\qquad + \qquad\qquad s\]
  - - - \[ \small \tfrac{1}{f_0} \cdot \dot{z} \quad=\quad z \left( \underbrace{\left( \beta_0 + j \, 2\pi \right)}_{b_0} \quad + \quad \underbrace{\left( \beta_1 + j \, \delta_1 \right)}_{b_1} \; | z |^2 \quad + \quad \sum_{\small k \ = 2}^{\infty} \underbrace{\left( \beta_k + j \, \delta_k \right)}_{b_k} \; ~ \varepsilon^{\, k-1} | z |^{2k} \right) \quad\quad\quad+\quad\quad\quad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]