\[\frac{\partial^{2}}{\partial t^{2}} y{\left(x,t \right)} = - \kappa^{2} \frac{\partial^{4}}{\partial x^{4}} y{\left(x,t \right)} \]
where:
Approximating the derivatives with finite differences:
\[- \frac{2 y{\left(x,t \right)}}{{\Delta}t^{2}} + \frac{y{\left(x,t - {\Delta}t \right)}}{{\Delta}t^{2}} + \frac{y{\left(x,t + {\Delta}t \right)}}{{\Delta}t^{2}} = - \kappa^{2} \left(\frac{6 y{\left(x,t \right)}}{{\Delta}x^{4}} + \frac{y{\left(x - 2 {\Delta}x,t \right)}}{{\Delta}x^{4}} - \frac{4 y{\left(x - {\Delta}x,t \right)}}{{\Delta}x^{4}} - \frac{4 y{\left(x + {\Delta}x,t \right)}}{{\Delta}x^{4}} + \frac{y{\left(x + 2 {\Delta}x,t \right)}}{{\Delta}x^{4}}\right) \]
Substituting discrete indices (\({t} \rightarrow {n} {Dt}\), \({x} \rightarrow {i} {Dx}\)):
The discretized form becomes:
\[\frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{2}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}t^{2}} = - \kappa^{2} \left(- \frac{4 {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} + \frac{{y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} - \frac{4 {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} + \frac{{y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} + \frac{6 {y}_{\text{i},\text{n}}}{{\Delta}x^{4}}\right) \]
Rearranging the discretized equation to solve for \({y}_{\text{i},\text{n} + 1}\) gives the explicit update rule:
\[{y}_{\text{i},\text{n} + 1} = \frac{\kappa^{2} {\Delta}t^{2} \left(4 {y}_{\text{i} + 1,\text{n}} - {y}_{\text{i} + 2,\text{n}} + 4 {y}_{\text{i} - 1,\text{n}} - {y}_{\text{i} - 2,\text{n}} - 6 {y}_{\text{i},\text{n}}\right)}{{\Delta}x^{4}} - {y}_{\text{i},\text{n} - 1} + 2 {y}_{\text{i},\text{n}} \]
These parameters control the overall simulation setup:
These parameters are derived from the independent ones and the discretization:
The spatial step \({\Delta}x\) is constrained by the need for numerical stability. We find this constraint using Von Neumann stability analysis.
We substitute the wave-form ansatz: \(y_{i,n} = \text{Y} e^{i \left(\text{i} k {\Delta}x + \text{n} w {\Delta}t\right)}\) into the finite difference scheme.
Substituting the ansatz gives:
\[\text{Y} e^{i \left(\text{i} k {\Delta}x + w {\Delta}t \left(\text{n} + 1\right)\right)} = - \frac{\kappa^{2} \text{Y} {\Delta}t^{2} e^{2 i k {\Delta}x} e^{i \text{i} k {\Delta}x} e^{i \text{n} w {\Delta}t}}{{\Delta}x^{4}} + \frac{4 \kappa^{2} \text{Y} {\Delta}t^{2} e^{i k {\Delta}x} e^{i \text{i} k {\Delta}x} e^{i \text{n} w {\Delta}t}}{{\Delta}x^{4}} - \frac{6 \kappa^{2} \text{Y} {\Delta}t^{2} e^{i \text{i} k {\Delta}x} e^{i \text{n} w {\Delta}t}}{{\Delta}x^{4}} + \frac{4 \kappa^{2} \text{Y} {\Delta}t^{2} e^{- i k {\Delta}x} e^{i \text{i} k {\Delta}x} e^{i \text{n} w {\Delta}t}}{{\Delta}x^{4}} - \frac{\kappa^{2} \text{Y} {\Delta}t^{2} e^{- 2 i k {\Delta}x} e^{i \text{i} k {\Delta}x} e^{i \text{n} w {\Delta}t}}{{\Delta}x^{4}} + 2 \text{Y} e^{i \text{i} k {\Delta}x} e^{i \text{n} w {\Delta}t} - \text{Y} e^{- i w {\Delta}t} e^{i \text{i} k {\Delta}x} e^{i \text{n} w {\Delta}t} \]
Dividing by the common factor \(\text{Y} e^{\text{I} \left(\text{i} k {\Delta}x + \text{n} w {\Delta}t\right)}\), rearranging, and simplifying leads to the numerical dispersion relation:
\[\cos{\left(w {\Delta}t \right)} = - \frac{2 \kappa^{2} {\Delta}t^{2} \left(\cos{\left(k {\Delta}x \right)} - 1\right)^{2}}{{\Delta}x^{4}} + 1 \]
Solving for the angular frequency \(w\):
\[w = \left[ \frac{- \operatorname{acos}{\left(\frac{- 2 \kappa^{2} {\Delta}t^{2} \cos^{2}{\left(k {\Delta}x \right)} + 4 \kappa^{2} {\Delta}t^{2} \cos{\left(k {\Delta}x \right)} - 2 \kappa^{2} {\Delta}t^{2} + {\Delta}x^{4}}{{\Delta}x^{4}} \right)} + 2 \pi}{{\Delta}t}, \ \frac{\operatorname{acos}{\left(\frac{- 2 \kappa^{2} {\Delta}t^{2} \cos^{2}{\left(k {\Delta}x \right)} + 4 \kappa^{2} {\Delta}t^{2} \cos{\left(k {\Delta}x \right)} - 2 \kappa^{2} {\Delta}t^{2} + {\Delta}x^{4}}{{\Delta}x^{4}} \right)}}{{\Delta}t}\right] \]
For stable solutions, \(w\) must be real. This requires the argument of arccosine to be between -1 and 1:
\[-1 \le - \frac{2 \kappa^{2} {\Delta}t^{2} \left(\cos{\left(k {\Delta}x \right)} - 1\right)^{2}}{{\Delta}x^{4}} + 1 \le 1 \]
The right-hand side (\(- \frac{2 \kappa^{2} {\Delta}t^{2} \left(\cos{\left(k {\Delta}x \right)} - 1\right)^{2}}{{\Delta}x^{4}} + 1 \le 1\)) is always satisfied. The left-hand side (\(-1 \le - \frac{2 \kappa^{2} {\Delta}t^{2} \left(\cos{\left(k {\Delta}x \right)} - 1\right)^{2}}{{\Delta}x^{4}} + 1\)) simplifies to the stability condition:
\[1 \geq \frac{\kappa^{2} {\Delta}t^{2} \left(\cos{\left(k {\Delta}x \right)} - 1\right)^{2}}{{\Delta}x^{4}} \]
The term \((\cos{\left(k {\Delta}x \right)} - 1)^2\) has a maximum value of 4 (when \(k {\Delta}x = \pi, 3\pi, \dots\)). Substituting this maximum gives the most restrictive condition:
\[1 \geq \frac{4 \kappa^{2} {\Delta}t^{2}}{{\Delta}x^{4}} \]
Rearranging this inequality gives the stability condition for the Euler-Bernoulli beam scheme:
\[{\Delta}x \geq \sqrt{2 \kappa {\Delta}t} \]
For the simple supported ideal bar we have that:
\[f_{0} = \frac{\pi \kappa}{2 \text{L}^{2}} \]
\[\kappa = \frac{2 \text{L}^{2} f_{0}}{\pi} \]
\[\frac{\text{L}}{\text{M}} \geq \frac{2 \text{L} \sqrt{f_{0}}}{\sqrt{\pi} \sqrt{f_{s}}} \]
\[\frac{\text{L}^{2}}{\text{M}^{2}} \geq \frac{4 \text{L}^{2} f_{0}}{\pi f_{s}} \]
\[f_{0} \leq \frac{\pi f_{s}}{4 \text{M}^{2}} \]
When \(f_{s} = 44100\), \(\text{M} = 3\) and \(\text{L} = 1\):
\[f_{0} \leq 1225 \pi \]
\[f_{0} \leq 3848.4510006475 \]
