Implementation and Validation of the Fokker-Planck Equation for Narrow-band excitation

The weak form of the FPE is written as follows, denoting the domain by \(\mathcal{D}\) and using a trial function \(g:\mathcal{D}\to\mathcal{H}\) (with \(\mathcal{H}\) satisfying Dirichlet B.C.s),

where \(\partial \mathcal{D}\) denotes the boundary and \(n_i\) denotes the relevant boundary normal.

• The symbol \(\int_{\mathcal{D}} g p\) denotes some inner product defined over the domain \(\mathcal{D}\).

• \(\lambda_1=0\): Corresponds to a Stationary Process at steady-state..
• \(\Re\{\lambda_1\}=0,\,\Im\{\lambda_1\}\neq0\): Corresponds to a Periodic random process at steady-state.

• MMS:

  \begin{align*} \dfrac{d}{dt} \begin{bmatrix} q\\\\\beta \end{bmatrix} = \left(\textcolor{red}{\epsilon} \begin{bmatrix} -\zeta \omega_0 q - \dfrac{F\sin \beta}{2\omega_0}\\\\-(\Omega-\omega_0)-\dfrac{F\cos \beta}{2\omega_0q} \end{bmatrix} + \textcolor{red}{\epsilon^2} \begin{bmatrix}\dfrac{F\zeta \cos \beta}{4\omega_0} + (\Omega-\omega_0)\dfrac{F\sin \beta}{4\omega_0^2}\\\\ -\dfrac{\zeta^2\omega_0}{2} - \dfrac{F\zeta\sin \beta}{4\omega_0q} + (\Omega-\omega_0)\dfrac{F\cos \beta}{4\omega_0^2q} \end{bmatrix}\right) + \nu \left(\textcolor{red}{\epsilon} \begin{bmatrix}0\\\\ -1\end{bmatrix} + \textcolor{red}{\epsilon^2} \begin{bmatrix}\dfrac{F\sin \beta}{4\omega_0^2}\\\\ \dfrac{F\cos \beta}{4\omega_0^2q}\end{bmatrix}\right) \dot{W}(t) - \cancel{\left(\textcolor{red}{\epsilon^2} \frac{d\sigma}{dT_2}\right)} \end{align*}
  • Phase: \[ \sigma = \omega_0 t + \textcolor{red}{\epsilon}(\Omega-\omega_0)t + \textcolor{red}{\epsilon} \nu W(t) \]

• EMMS:

  \begin{align*} \dfrac{d}{dt} \begin{bmatrix} q\\\\\beta \end{bmatrix} = \left(\textcolor{red}{p} \begin{bmatrix} -\zeta \omega_0q - \dfrac{F\sin \beta}{2\Omega}\\\\ -\dfrac{\Omega^2-\omega_0^2}{2\Omega}-\dfrac{F\cos \beta}{2\Omega q}\end{bmatrix} + \textcolor{red}{p^2} \begin{bmatrix} \dfrac{F\zeta \omega_0\cos \beta}{4\Omega^2}-(\Omega^2-\omega_0^2)\dfrac{F\sin \beta}{8\Omega^3}\\\\ -\dfrac{(\Omega^2-\omega_0^2)^2}{8\Omega^3}-\dfrac{\zeta^2\omega_0^2}{2\Omega}-\dfrac{F\zeta\omega_0\sin \beta}{4\Omega^2q}-(\Omega^2-\omega_0^2)\dfrac{F\cos \beta}{8\Omega^3q} \end{bmatrix}\right) + \left( \textcolor{red}{p} \begin{bmatrix} 0\\\\ -1\end{bmatrix} + \textcolor{red}{p^2} \begin{bmatrix} \dfrac{F\sin \beta}{4\Omega^2}\\\\ \dfrac{F\cos \beta}{4\Omega^2q}\end{bmatrix} \right) \dot{W}(t) - \cancel{\left(\textcolor{red}{p^2} \dfrac{d\sigma}{dT_2}\right)} \end{align*}
  • Phase: \[ \sigma = \Omega t + \textcolor{red}{p} \nu W(t) \]

Here is a summary plot. Deterministically, the MMS approaches seem to match the

Looking closer seems to show that although the others track the true envelope closer, they seem to have some bias, while CXA does not. The MMS approaches, however, show better performance in standard deviation.

Here are the Eigenspectra of the FPK pencils. (Note that the analytical solution is written as \(p = \eta_i \phi_i e^{-\lambda_i t}\)).

Used EPMC to compute the backbones first

This is next used to obtain the PDF of the slow flow system using the four different formulations. Here is a summary.

Used EPMC to compute the backbones first

This is next used to obtain the PDF of the slow flow system using the four different formulations. Here is a summary.

• PSD: \[ \Phi_{FF}(\omega) = \frac{\lambda^2\overline{\nu}^2}{4\pi} \frac{\overline{\Omega}^2+\frac{\overline{\nu}^4}{4}+\omega^2}{(\Omega^2+\frac{\overline{\nu}^4}{4}-\omega^2)^2+\omega^2\overline{\nu}^4}\text{, with } \overline{\nu}=\frac{\nu}{\sqrt{F_s}}. \]

• PDF:

  \begin{equation*} p_F(f) = \begin{cases} \frac{1}{\pi\sqrt{(\Omega^2-f^2)}} & |f|< F\\ 0 & \text{otherwise} \end{cases} \end{equation*}

I am retrying with higher damping since transient effects may still be present (see deterministic simulation below):

Here I am comparing this with the four slow-flow approaches

Here's a summary of the measured channels

Here are the mode shapes we got through multisine testing

Here's the force-voltage transfer function at the shaker head