Resonance Phenomena

Resonance phenomena simulated by finite differences and biased noise inputs

In Lecture-21.nb, a second-order differencing scheme that iteratively solved  y'' + βy' + γy=0  with the specification of two initial values.
Modify this to add a little random noise to y[i] at each step and observe how this behaves---this version will store the noise added at each iteration so that it can be visualized later....

$\mathrm{GrowListGeneralNoise}\left[\text{ValuesList\_List},\text{Δ\_},\text{α\_},\text{β\_},\text{randomamp\_}\right]:=\mathrm{Module}\left[\left\{\mathrm{Minus1}=\mathrm{ValuesList}\left[\left[1,-1\right]\right],\mathrm{Minus2}=\mathrm{ValuesList}\left[\left[1,-2\right]\right],\mathrm{noise}=\mathrm{Random}\left[\mathrm{Real},\left\{-\mathrm{randomamp},\mathrm{randomamp}\right\}\right]\right\},\text{}\left\{\text{}\mathrm{Append}\left[\mathrm{ValuesList}\left[\left[1\right]\right],\text{}2*\mathrm{Minus1}-\mathrm{Minus2}+\Delta *\left(\beta *\left(\mathrm{Minus2}-\mathrm{Minus1}\right)-\alpha *\Delta *\mathrm{Minus2}\right)+\mathrm{noise}\right],\text{}\mathrm{Append}\left[\mathrm{ValuesList}\left[\left[2\right]\right],\mathrm{noise}\right]\right\}\right]$

Setting up a function that takes particular parameters and a "noise amplitude" of ${10}^{-5}$

$\mathrm{GrowListSpecificNoise}\left[\text{InitialList\_List}\right]:=\mathrm{GrowListGeneralNoise}\left[\mathrm{InitialList},\text{.001},2,0,10^\left(-5\right)\right]$

$\mathrm{Nest}\left[\mathrm{GrowListSpecificNoise},\left\{\left\{1,1\right\},\left\{0,0\right\}\right\},10\right]$

$\left\{\left\{1,1,0.9999914117795936,0.9999843172559986,0.999969270477182,0.9999428093699526,0.9999162597036201,0.9998880386207231,0.9998566204455593,0.9998204523904113,0.9997817799724843,0.9997317440858904\right\},\left\{0,0,-0.000006588220406492501,0.000003493696811337929,-0.000005952272398008523,-0.00000941435977835451,0.000001911379437887737,3.284690542412207\times {10}^{-7},-0.000001197259747345678,-0.000002750103907047068,-5.046495380609433\times {10}^{-7},-0.00000936382776220396\right\}\right\}$

$\mathrm{TheData}=\mathrm{Nest}\left[\mathrm{GrowListSpecificNoise},\left\{\left\{1,1\right\},\left\{0,0\right\}\right\},20000\right];$

$\mathrm{ListPlot}\left[\mathrm{TheData}\left[\left[1\right]\right]\right]$

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$\mathrm{ListPlot}\left[\mathrm{TheData}\left[\left[2\right]\right]\right]$

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Now suppose there is a periodic bias that tends to kick the displacement one direction more than the other:

$\mathrm{GrowListBiasedNoise}\left[\text{ValuesList\_List},\text{Δ\_},\text{α\_},\text{β\_},\text{randomamp\_},\text{lambda\_}\right]:=\mathrm{Module}\left[\left\{\mathrm{Minus1}=\mathrm{ValuesList}\left[\left[1,-1\right]\right],\mathrm{Minus2}=\mathrm{ValuesList}\left[\left[1,-2\right]\right],\mathrm{biasednoise}=0.5*\mathrm{randomamp}*\left(\mathrm{Cos}\left[2\pi \mathrm{Length}\left[\mathrm{ValuesList}\left[\left[1\right]\right]\right]/\mathrm{lambda}\right]+\mathrm{Random}\left[\mathrm{Real},\left\{-1,1\right\}\right]\right)\right\},\text{}\left\{\text{}\mathrm{Append}\left[\mathrm{ValuesList}\left[\left[1\right]\right],\text{}2*\mathrm{Minus1}-\mathrm{Minus2}+\Delta *\left(\beta *\left(\mathrm{Minus2}-\mathrm{Minus1}\right)-\alpha *\Delta *\mathrm{Minus2}\right)+\mathrm{biasednoise}\right],\text{}\mathrm{Append}\left[\mathrm{ValuesList}\left[\left[2\right]\right],\mathrm{biasednoise}\right]\right\}\right]$

$\mathrm{GrowListSpecificBiasedNoise}\left[\text{InitialList\_List}\right]:=\mathrm{GrowListBiasedNoise}\left[\mathrm{InitialList},\text{.001},2,0,10^\left(-6\right),4500\right]$

Generate the data set---this takes quite a while

$\mathrm{TheBiasedData}=\mathrm{Nest}\left[\mathrm{GrowListSpecificBiasedNoise},\left\{\left\{1,1\right\},\left\{0,0\right\}\right\},20000\right];$

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Resonance Phenomena by Solution of the ODE

The "periodic forcing function" can be any periodic function (i.e., a fourier series), but it is a bit simpler to analyze the effect of each mode separately.  Below, the forcing function will be assumed to be $F_{\mathrm{app}}$cos(${\omega }_{\mathrm{app}}$t)
Solve problems in terms of the mass and natural frequency--eliminate the spring constant in equations by defining it in terms of the mass and natural frequency.

$\mathrm{Kspring}=M{\mathrm{\omega char}}^2$

$M{\mathrm{\omega char}}^2$

Mathematica can solve the nonhomogeneous ODE with a  forcing function at with an applied frequency:

$\mathrm{yGeneralSol}=\mathrm{DSolve}\left[My\text{''}\left[t\right]+\eta y\text{'}\left[t\right]+\mathrm{Kspring}y\left[t\right]⩵\mathrm{Fapp}\mathrm{Cos}\left[\mathrm{\omega app}t\right],y\left[t\right],t\right]//\mathrm{Flatten}$

$\left\{y\left[t\right]\to e^{\frac{t\left(-\eta -\sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)}{2M}}C\left[1\right]+e^{\frac{t\left(-\eta +\sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)}{2M}}C\left[2\right]-\frac{4\left(-\mathrm{Fapp}{M}^3{\mathrm{\omega app}}^2\mathrm{Cos}\left[t\mathrm{\omega app}\right]+\mathrm{Fapp}{M}^3{\mathrm{\omega char}}^2\mathrm{Cos}\left[t\mathrm{\omega app}\right]+\mathrm{Fapp}{M}^2\eta \mathrm{\omega app}\mathrm{Sin}\left[t\mathrm{\omega app}\right]\right)}{\left({\eta }^2+2M^2{\mathrm{\omega app}}^2-2M^2{\mathrm{\omega char}}^2+\eta \sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)\left(-{\eta }^2-2M^2{\mathrm{\omega app}}^2+2M^2{\mathrm{\omega char}}^2+\eta \sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)}\right\}$

The general solution of the heterogeneous ODE is the sum of the homogeneous solution  (i.e., the one for which the right-hand-side is zero) and the particular solution (i.e., the one for the particular right-hand-side of the ODE)

$\mathrm{yParticular}\left[0\right]=\left(y\left[t\right]/.\mathrm{yGeneralSol}\right)/.\left\{C\left[1\right]\to 0,C\left[2\right]\to 0\right\}$

$-\frac{4\left(-\mathrm{Fapp}{M}^3{\mathrm{\omega app}}^2\mathrm{Cos}\left[t\mathrm{\omega app}\right]+\mathrm{Fapp}{M}^3{\mathrm{\omega char}}^2\mathrm{Cos}\left[t\mathrm{\omega app}\right]+\mathrm{Fapp}{M}^2\eta \mathrm{\omega app}\mathrm{Sin}\left[t\mathrm{\omega app}\right]\right)}{\left({\eta }^2+2M^2{\mathrm{\omega app}}^2-2M^2{\mathrm{\omega char}}^2+\eta \sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)\left(-{\eta }^2-2M^2{\mathrm{\omega app}}^2+2M^2{\mathrm{\omega char}}^2+\eta \sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)}$

$\mathrm{yParticular}\left[1\right]=\mathrm{Collect}\left[\mathrm{FullSimplify}\left[\mathrm{yParticular}\left[0\right]\right],\left\{\mathrm{Sin}\left[t\mathrm{\omega app}\right],\mathrm{Cos}\left[t\mathrm{\omega app}\right]\right\}\right]$

$\frac{\mathrm{Fapp}M\left(-{\mathrm{\omega app}}^2+{\mathrm{\omega char}}^2\right)\mathrm{Cos}\left[t\mathrm{\omega app}\right]}{{\eta }^2{\mathrm{\omega app}}^2+M^2{\left({\mathrm{\omega app}}^2-{\mathrm{\omega char}}^2\right)}^2}+\frac{\mathrm{Fapp}\eta \mathrm{\omega app}\mathrm{Sin}\left[t\mathrm{\omega app}\right]}{{\eta }^2{\mathrm{\omega app}}^2+M^2{\left({\mathrm{\omega app}}^2-{\mathrm{\omega char}}^2\right)}^2}$

The particular solutions only picks up modes from the forcing term

The homogeneous solution only has he natural frequencies

$\mathrm{yHomogenous}=\left(y\left[t\right]/.\mathrm{yGeneralSol}\right)-\mathrm{yParticular}\left[0\right]$

$e^{\frac{t\left(-\eta -\sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)}{2M}}C\left[1\right]+e^{\frac{t\left(-\eta +\sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)}{2M}}C\left[2\right]$

The General Solution is the combination of two different frequencies--this should give rise to beats if the driving frequency differs from the natural frequency

$e^{\frac{t\left(-\eta -\sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)}{2M}}C\left[1\right]+e^{\frac{t\left(-\eta +\sqrt{{\eta }^2-4M^2{\mathrm{\omega char}}^2}\right)}{2M}}C\left[2\right]$

The following shows that the solution is unphysical when ωapp → ωchar AND η→0

$\mathrm{singbehav}=\mathrm{Series}\left[\left(\mathrm{yParticular}\left[1\right]/.\left\{\eta \to \delta \mathrm{\eta 0},\mathrm{\omega app}\to \mathrm{\omega char}+ϵ\mathrm{\omega 0}\right\}\right),\left\{\delta ,0,1\right\},\left\{ϵ,0,1\right\}\right]//\mathrm{Normal}$

$\frac{\mathrm{Fapp}\mathrm{Cos}\left[t\mathrm{\omega char}\right]}{4M{\mathrm{\omega char}}^2}+\frac{-\frac{\mathrm{Fapp}\mathrm{Cos}\left[t\mathrm{\omega char}\right]}{2M\mathrm{\omega 0}\mathrm{\omega char}}+\frac{\mathrm{Fapp}t\delta \mathrm{\eta 0}\mathrm{Cos}\left[t\mathrm{\omega char}\right]}{4M^2\mathrm{\omega 0}\mathrm{\omega char}}}{ϵ}+\frac{\mathrm{Fapp}t\mathrm{Sin}\left[t\mathrm{\omega char}\right]}{2M\mathrm{\omega char}}+\frac{\mathrm{Fapp}\delta \mathrm{\eta 0}\mathrm{Sin}\left[t\mathrm{\omega char}\right]}{4M^2{ϵ}^2{\mathrm{\omega 0}}^2\mathrm{\omega char}}+\delta \left(-\frac{\mathrm{Fapp}\mathrm{\eta 0}\mathrm{Sin}\left[t\mathrm{\omega char}\right]}{16M^2{\mathrm{\omega char}}^3}-\frac{\mathrm{Fapp}{t}^2\mathrm{\eta 0}\mathrm{Sin}\left[t\mathrm{\omega char}\right]}{8M^2\mathrm{\omega char}}\right)+ϵ\left(-\frac{\mathrm{Fapp}\mathrm{\omega 0}\mathrm{Cos}\left[t\mathrm{\omega char}\right]}{8M{\mathrm{\omega char}}^3}+\frac{\mathrm{Fapp}{t}^2\mathrm{\omega 0}\mathrm{Cos}\left[t\mathrm{\omega char}\right]}{4M\mathrm{\omega char}}-\frac{\mathrm{Fapp}t\mathrm{\omega 0}\mathrm{Sin}\left[t\mathrm{\omega char}\right]}{4M{\mathrm{\omega char}}^2}+\delta \left(-\frac{\mathrm{Fapp}t\mathrm{\eta 0}\mathrm{\omega 0}\mathrm{Cos}\left[t\mathrm{\omega char}\right]}{16M^2{\mathrm{\omega char}}^3}-\frac{\mathrm{Fapp}{t}^3\mathrm{\eta 0}\mathrm{\omega 0}\mathrm{Cos}\left[t\mathrm{\omega char}\right]}{24M^2\mathrm{\omega char}}+\frac{\mathrm{Fapp}\mathrm{\eta 0}\mathrm{\omega 0}\mathrm{Sin}\left[t\mathrm{\omega char}\right]}{16M^2{\mathrm{\omega char}}^4}\right)\right)$

$\mathrm{StringTake}\text{::}\mathrm{strs}:\text{String or non-empty list of strings expected at position}1\text{in}\mathrm{StringTake}\left[\text{System`Convert`CommonDump`str},\left\{2,-2\right\}\right]\text{.}\mathrm{More\dots }$

We can determine the behavir near the singularity by looking at the effect of each term above:
    1. If the forcing frequency approaches the resonance frequency (ε→0), the solution is unbounded because there is no δ in the first part of the second term's numerator
    2. If the viscosity approaches zero (δ→0), the solution will have a linearly growing applitude because of the third term
    3. Other cases will depend on how the ratio δ/ε scales.

The following demonstrates the zero viscosity case, needs to be analyzed carefully (as above) otherwise one might miss terms:

$\mathrm{yparticularUndamped}=\mathrm{yParticular}\left[1\right]/.\eta \to 0$

$\frac{\mathrm{Fapp}\left(-{\mathrm{\omega app}}^2+{\mathrm{\omega char}}^2\right)\mathrm{Cos}\left[t\mathrm{\omega app}\right]}{M{\left({\mathrm{\omega app}}^2-{\mathrm{\omega char}}^2\right)}^2}$

Visualizing Solutions

Create a Mathematica  function that returns the solution for specified mass, viscous term, characteristic and applied frequencies

$y\left[\text{M\_},\text{η\_},\text{ωchar\_},\text{ωapp\_}\right]:=\mathrm{Chop}\left[y\left[t\right]/.\mathrm{DSolve}\left[\left\{My\text{''}\left[t\right]+\eta y\text{'}\left[t\right]+M\mathrm{\omega char}^2y\left[t\right]⩵\mathrm{Cos}\left[\mathrm{\omega app}t\right],y\left[0\right]==1,y\text{'}\left[0\right]==0\right\},y\left[t\right],t\right]//\mathrm{Flatten}\right]$

Experiment by plotting for many different values:

Undamped Resonance:

$\mathrm{Plot}\left[\mathrm{Evaluate}\left[y\left[1,0,1/2,1/2\right]\right],\left\{t,0,200\right\},\mathrm{PlotPoints}\to 200\right]$

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Undamped Near Resonance:

$\mathrm{Plot}\left[\mathrm{Evaluate}\left[y\left[1,0,1/2+0.05,1/2\right]\right],\left\{t,0,200\right\},\mathrm{PlotPoints}\to 200\right]$

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Damped Resonance:

$\mathrm{Plot}\left[\mathrm{Evaluate}\left[y\left[1,1/10,1/2,1/2\right]\right],\left\{t,0,200\right\}\right]$

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Overdamped Resonance:

$\mathrm{Plot}\left[\mathrm{Evaluate}\left[y\left[1,10,1/2,1/2\right]\right],\left\{t,0,200\right\}\right]$

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Damped Near Resonance:

$\mathrm{Plot}\left[\mathrm{Evaluate}\left[y\left[1,\text{.05},1/2+0.05,1/2\right]\right],\left\{t,0,200\right\},\mathrm{PlotPoints}\to 200\right]$

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Heavily damped Near Resonance:

$\mathrm{Plot}\left[\mathrm{Evaluate}\left[y\left[1,2.5,1/2+0.05,1/2\right]\right],\left\{t,0,200\right\},\mathrm{PlotPoints}\to 200\right]$

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Created by Mathematica  (December 18, 2005)