We employ the technique of control-based continuation to determine the response of the nonlinear oscillator to periodic forcing. Let us consider a one-degree-of-freedom forced nonlinear system as a model of the experimental rig in the form of
$$\begin{aligned} \ddot{x} + g(\dot{x}, x) = F(t), \end{aligned}$$
(1)
where the state variable x is the input voltage from the strain-gauge (IN1), the dot refers to differentiation with respect to time, the function g contains all the nonlinearities, while F(t) corresponds to the forcing provided by the shaker. Despite the fact that this model does not include the electromagnetic effects in the system explicitly, we found that the one-degree-of-freedom model can characterise the response to periodic forcing with good accuracy. Note, that, as demonstrated in [34], this is not the case for non-periodic excitation.
The forcing F(t) is composed by two parts: a periodic component and an additional control term \(F_{\mathrm{ctrl}}\)
$$\begin{aligned} F(t) = A \cos (\omega t) + B \sin (\omega t) + F_{\mathrm{ctrl}}, \end{aligned}$$
(2)
where \(\omega \) is the angular frequency of the forcing whereas A and B are constant coefficients. To capture the nonlinear response of the open-loop (uncontrolled) system, the control has to fulfil two conditions: it has to be stabilising and non-invasive, i.e. the steady-state response of the controlled system has to be equal to the steady-state response of the open-loop system. This is ensured by the proportional-derivative control law given in the form
$$\begin{aligned} F_{\mathrm{ctrl}} = k_{\mathrm{p}}(x^*- x) + k_{\mathrm{d}}(\dot{x}^*- \dot{x}), \end{aligned}$$
(3)
where \(x^*\) is the control target, while \(k_{\mathrm{p}}\) and \(k_\mathrm{d}\) are the proportional and derivative control gains. Since the derivative \(\dot{x}\) is not acquired directly in our experiment, its value at a given time \(t_i\) is obtained using a backward difference scheme
$$\begin{aligned} \dot{x} (t_i) \approx \frac{x(t_i) - x(t_{i-1})}{t_i - t_{i-1}}. \end{aligned}$$
(4)
In the control algorithm, both the desired and measured strain-gauge voltages are represented by their truncated Fourier series
$$\begin{aligned}&x(t) \approx \frac{A_0}{2} + \sum _{k = 1}^{N} \left( A_k \cos (k \omega t) + B_k \sin (k \omega t) \right) , \end{aligned}$$
(5)
$$\begin{aligned}&x^*(t) \approx \frac{A_0^*}{2} + \sum _{k = 1}^{N} \left( A_k^*\cos (k \omega t) + B_k^*\sin (k \omega t) \right) . \end{aligned}$$
(6)
In our test, the first seven harmonics were retained (\(N = 7\)).
As a result, the total forcing F(t) also can be expressed in a similar form
$$\begin{aligned} F(t) = \frac{A_{F0}}{2} + \sum _{k = 1}^{N} \left( A_{Fk} \cos (k \omega t) + B_{Fk} \sin (k \omega t) \right) , \end{aligned}$$
(7)
where the coefficients are given by
$$\begin{aligned} A_{F1}&= A+k_{\mathrm{p}}(A_1^*-A_1)-k_{\mathrm{d}} \omega (B_1^*-B_1), \end{aligned}$$
(8a)
$$\begin{aligned} B_{F1}&= B+k_\mathrm{p}(B_1^*-B_1)+k_{\mathrm{d}} \omega (A_1^*-A_1), \end{aligned}$$
(8b)
$$\begin{aligned} A_{Fk}&= k_{\mathrm{p}}(A_k^*-A_k)-k_{\mathrm{d}} k \omega (B_k^*-B_k) \; \mathrm{for} \nonumber \\ \;&k = 0,2,3,\dots ,N, \end{aligned}$$
(8c)
$$\begin{aligned} B_{Fk}&= k_{\mathrm{p}}(B_k^*-B_k)+k_{\mathrm{d}} k \omega (A_k^*-A_k) \; \mathrm{for} \nonumber \\ \;&k = 2,3,\dots ,N. \end{aligned}$$
(8d)
Studying the expressions (8a) and (8b) reveals that the fundamental harmonic component of the total forcing \(\Phi = \sqrt{A_{F1}^2+B_{F1}^2}\) is not fully determined by the open-loop forcing coefficients A and B, as it also depends on the control target \(x^*(t)\) and the response x(t). Moreover, the control introduces higher-harmonic components to the total forcing that have to be eliminated to capture the response of the open-loop system.
In our study, control-based continuation is used to generate the family of steady-state solutions of the system across a range of the forcing amplitudes while keeping the forcing frequency constant. Since, with keeping the forcing frequency \(\omega \) constant, a unique forcing amplitude corresponds to every the vibration amplitude, it is possible to trace the whole branch of solutions by a sweep in the target fundamental harmonic amplitude \(B_1^*\), with keeping \(A_1^*= 0\) to fix the phase of the response. This means that there is a linear relationship between the continuation parameter and the forcing F(t), which enables us to use a simplified version of the ‘full’ continuation algorithm (see [27] for example). Note that to retrieve the frequency response at a constant forcing amplitude the full algorithm would be required which may be less robust to noise than the simplified version. This issue is addressed by Schilder et al. [35] with techniques developed specifically to cope with noise. Another alternative is presented in [36] where the full continuation algorithm is used on a local Gaussian process regression model. Nevertheless, the frequency dependence could be equally well characterised by tracing the response curve at several frequencies as indicated by Fig. 1, where response surfaces above the forcing frequency – forcing amplitude planes, obtained by the open-loop and control-based approach, are compared. The simplified control-based continuation algorithm is briefly described below—a full description is given in [37]. It is also worth mentioning that, for parameter identification, it may be acceptable for the controller to be invasive, allowing it to overcome the issues around the Newton iterations. Our study however, along with [35, 37], specifically focuses on recovering the bifurcation diagram, while parameter identification is used as a tool to quantify the effectiveness of the open-loop and control-based approaches.
Let us assume that the experiment is running at a steady state given by the target coefficients \((A_{F1}^j, B_{F1}^j, A_0^{*j}, B_1^{*j}, A_k^{*j}, B_k^{*j})\), \(k = 2, \dots N\), with \(A_1^{*j} = 0\). Then, to find the next point in the solution branch, the fundamental harmonic coefficient of the control target is increased by the desired increment \(\Delta \) as \(B_1^{*j+1} := B_1^{*j}+ \Delta \). After waiting for the control to reach steady state, if necessary, we apply fixed point iteration to correct the higher-harmonic coefficients of the control target until coefficients corresponding to the higher harmonics of the forcing (\(A_{Fk}, \, B_{Fk}, \, k = 0,2,3, \dots N\)) are below a pre-defined tolerance. Once the higher harmonics in the forcing are eliminated, the actual state, given by \((A_0^{*j+1}, B_1^{*j+1}, A_k^{*j+1}, B_k^{*j+1})\), \(k = 2, \dots N\), is accepted as the steady-state response of the open-loop system corresponding to the forcing amplitude
$$\begin{aligned} \Phi ^{j+1} = \sqrt{\left( A_{F1}^{j+1} \right) ^2 +\left( B_{F1}^{j+1} \right) ^2 }. \end{aligned}$$
(9)
Provided that appropriately chosen control gains are used, this algorithm ensures a stable, non-invasive control, which traces the solution branch sweeping across the vibration amplitudes. A possible alternative could be to use a secant prediction to provide an initial guess for the algorithm in the direction obtained from the previous two points on the branch. This method may result in the algorithm reaching a fixed point in fewer iteration steps in an experiment with low noise; however, the amplitude sweep is more robust against noise since with this assumption, noise cannot affect the direction along the branch in which the next branch point is predicted which is an effect that can hinder progress along a branch. A further advantage is that the correction of the solution is carried out in a derivative-free way. Thus, it requires less evaluation at each iteration step, leading to faster convergence.
Note that while the control feedback given in (3) is in real time in the experiment, there is no such requirement for the continuation algorithm and the setting of new control targets for the controller. Therefore, these tasks were carried out by a PC, which was also used to process the acquired data, rather than the real-time controller.
Model of the nonlinear oscillator
We use the experimentally acquired bifurcation diagrams to identify the parameters of our model for the experimental rig, the one-degree-of-freedom nonlinear oscillator [see Eq. (1)]. We consider a linearly damped, Duffing-like oscillator with the equation of motion
$$\begin{aligned} \ddot{x} + b \dot{x} + \omega _{\mathrm{n}}^2 x + \mu x^3 + \nu x^5 + \rho x^7 = \delta _{\mathrm{st}} \omega _{\mathrm{n}}^2 \cos (\omega t), \end{aligned}$$
(10)
where \(\omega _{\mathrm{n}}\) is the linear natural angular frequency, \(\delta _{\mathrm{st}}\) is the equivalent static deflection for the forcing amplitude \(\delta _{\mathrm{st}} \omega _{\mathrm{n}}^2\) (the resulting deflection of \(\omega = 0\)), while the damping is given by the parameter b, whereas \(\mu \), \(\nu \) and \(\rho \) characterise the nonlinearities in the system. Although this model does not describe the physics-based modelling of the restoring force, our investigation indicates that considering the odd nonlinear terms up to seventh order in the Duffing-type model is satisfactory to characterise the response of the experimental rig to periodic excitation. The detailed experimental characterisation of the electromagnetic forces by means of a physics-based model of the device was performed in [6].
The fundamental harmonic component of the steady-state system response can be given as \(X \cos (\omega t + \vartheta )\) with amplitude X and phase angle \(\vartheta \). Using the method of multiple scales [38], an analytical approximate solution can be obtained for the fundamental harmonic component if the nonlinearity is weak. Based on this, for a given vibration amplitude X, the phase angle and the static deflection can be given as
$$\begin{aligned} \vartheta&= \arctan \left( \frac{{\tilde{b}} \zeta }{(\zeta ^2-1) - \frac{35}{64} X^6 {\tilde{\rho }} - \frac{5}{8} X^4 {\tilde{\nu }} - \frac{3}{4} X^2 {\tilde{\mu }}} \right) , \end{aligned}$$
(11)
$$\begin{aligned} \delta _{\mathrm{st}}&= \left|\frac{\frac{35}{64} X^7 {\tilde{\rho }} + \frac{5}{8} X^5 {\tilde{\nu }} + \frac{3}{4} X^3 {\tilde{\mu }} - X (\zeta ^2-1)}{\cos (\vartheta )}\right|, \end{aligned}$$
(12)
with \(\zeta = \omega / \omega _{\mathrm{n}}\) \({\tilde{b}} := b/\omega _\mathrm{n}\), \({\tilde{\mu }} := \mu /\omega _{\mathrm{n}}^2\), \({\tilde{\nu }} := \nu /\omega _{\mathrm{n}}^2\), \({\tilde{\rho }} := \rho /\omega _{\mathrm{n}}^2\). The derivation of these formulae is given in the appendix.
Substituting (11) into Eq. (12), we obtain the static deflection by means of the system and forcing parameters as well as the amplitude of the fundamental harmonic component of the steady-state response
$$\begin{aligned} \delta _{\mathrm{st}} = \delta _{\mathrm{st}}(X, {\tilde{\mu }}, {\tilde{\nu }}, \tilde{\rho }, {\tilde{b}}, \zeta ). \end{aligned}$$
(13)
Numerical collocation
It has to be noted that the solution presented above is only accurate for ‘weakly nonlinear’ systems where the nonlinear terms do not dominate over the underlying linear system. To check the accuracy of the approximate solution, we carried out the numerical continuation of the periodic solutions in (10). The results are compared in Fig. 6 for the parameters \(\mu = 1.499\), \(\nu = -0.3921\), \(\rho = 0.0422\), \(b = 0.3159\) and \(f_{\mathrm{n}} = 19.95 \, \mathrm{Hz}\). Both frequency and amplitude variation is checked. The results indicate that the analytical approximation provides very accurate results in the parameter range of our interest.