1 Introduction

Sensors are utilized in all kinds of applications to measure a multi-physical system’s transient and steady states. The design and analysis of dynamics are under investigation and utilized in many different areas ranging from classical engineering [19, 33, 37] over physics [7] up to environmental and biological engineering [41].

For successful measurements, it is crucial for the sensor that its own dynamics do not distort the measurement. This is achieved by designing fast dynamics of the sensor. The quality factor limits the speed of the sensor response dynamics. A high-quality factor Q is needed to allow for a high sensor sensitivity [20, 25]. However, high Q leads simultaneously to a low damping and slow sensor transients. The measurement system must wait for these sensor transients’ decay to gain a reliable reading of the system dynamics under investigation, not the sensor dynamics. This contribution highlights a methodology for speeding up the transient time of a sensor while keeping the advantageous high-quality factor [26].

In the scope of this investigation, we focus on time-periodic systems. Time-periodic systems have been investigated for decades. A time-periodicity can be unavoidable and undesired, e.g. the base excitation of a motor in a ship subject to an external excitation by the sea [16]. A time-periodicity can also be implemented in the context of sensors for exploiting a parametric resonance that amplifies the sensor sensitivity, e.g. the [23, 32].

Time-periodic systems, even linear ones, cannot be solved analytically in closed form. However, several analytical techniques exist which allow for an approximate solution. The most commonly applied methods are singular perturbation methods like the multiple scales method [24], the method of averaging [40] or complexification [18], and several textbooks exist that apply them interchangeably [5, 35, 38]. Both methods rely on the assumption of small bookkeeping parameter \(\varepsilon \). It can be shown that for an approximation of first order in \(\varepsilon \) both methods lead to the same result on the time scale of \(1/\varepsilon \). The method of averaging [39] is followed here, specifically in the general case. The benefit of this method was outlined in more detail very recently in [40]. Introducing multiple parametric frequencies in the present work results in a quasi-periodic parametric excitation for which the method of averaging in the general case remains applicable, and the findings in [15] can be built upon.

Time-periodic systems exhibit parametric combination resonances for [9] at which the system vibrations increase, depending on the strength, frequency of the parametric excitation and the system damping. The parameter combinations for which an increase of vibrations is observed lie within the famous Arnould tongues [28] or instability tongues that are visualized in the Ince-Strutt stability diagram [8]. For a multiple-degree-of-freedom-system, instability tongues may appear close to the principal parametric resonance frequencies \(2\Omega _i/n\) or at parametric combination resonance frequencies \(\vert \Omega _i\mp \Omega _j\vert /n\) with \(n\in \mathbb {N}\) and \(\Omega _i\) being the natural frequency of the underlying system without parametric excitation [43].

The numerical stability analysis of time-periodic systems needs particular attention, and several methods were developed in the time domain (Monodromy matrix [27]) and the frequency domain (Hill matrix [4]). Recent developments for single as well as coupled Mathieu equations can be found in [8], also utilizing symplectic properties [27, 28]. The quasi-periodic Mathieu equation, the simplest time-periodic system with multiple frequency excitation, was investigated in [31, 45], showing very rich dynamics. In the present contribution dealing with a set of coupled Mathieu equations, such rich dynamics were not observed, which is why the treatment by singular perturbation leads to still compact expressions. The numerical analysis of more complex systems with quasi-periodic parametric excitation can be analyzed by following [42].

Under certain conditions, parametric combination resonances can show a stabilizing behaviour observed initially in [36] wherein the combination between a destabilizing self-excitation and a parametric combination resonance led to an overall stable system. This phenomenon was then termed parametric anti-resonance to underline the stabilizing behaviour. This observation was investigated in a series of works analytically and numerically [10,11,12, 14]. These studies showed that not the interaction between self-excitation and parametric excitation leads to vibration mitigation but solely a properly tuned parametric combination resonance ([10]). This triggered the possibility that any vibrating system with at least two degrees of freedom and small damping has the potential to exploit the vibration mitigation by a parametric combination resonance. This finding was confirmed experimentally for several discrete and continuous systems in [13].

MEMS (micro electromechanical systems) are utilized in several applications [2, 22] and allow for a simple electrical implementation of a parametric excitation [1, 21, 26]. During operation, MEMS can be driven to linear or non-linear regimes using different actuation mechanisms [44]. MEMS based on resonators can be composed of coupled structures (mechanically or electrically) in which coupling and energy transfer between vibration modes are analyzed for a specific purpose [17]. This coupling involves various vibration modes of a structure and exploits an intermodal coupling or the internal resonance phenomenon. These represent a nonlinear mechanism for transferring energy from one vibration mode to another [3, 6]. Furthermore, these structures can be excited in various ways, in their first mode or higher modes of vibration [34]. MEMS resonators possess diverse dynamics and its potential applications lie in energy harvesting, frequency stabilization in oscillators and synchronization [17]. The present work focuses exclusively on the linear dynamics and achieves modal coupling by a parametric excitation. Introducing a specific parametric excitation leads to a selective modal coupling and consequently to a selective energy transfer between vibration modes as originally shown in [11, 14]. Doing so we can trigger a selective modal coupling between the first mode and the remaining modes of the system. The usage of a single frequent parametric excitation for vibration mitigation was investigated theoretically in [29, 30] and verified for a MEMS sensor in [25]. Motivated by the results for a MEMS with 2DOF, the configuration is extended here to multiple DOFs, always leading to linearly coupled Mathieu equations. The concept is then generalized to the more complex case of a quasi-periodic parametric excitation at several parametric anti-resonance frequencies. Analytical conditions for a confined pulses of parametric excitations are given for achieving a mitigation of transient vibrations. The first order approximation of the slow flow dynamics is derived by applying the averaging method in the general case [40]. It shown that driving a linear MEMS at different parametric anti-resonance frequencies triggers a continuous energy transfer between selected vibration modes. The benefit for reducing the settling time for MEMS with three, four and more beams is highlighted.

In summary, the averaging technique is applied to a multi-parametric excited NDOF system at the anti-parametric frequency \(\Omega _p = \vert \Omega _i\ - \Omega _j\vert \); at this frequency, the averaged system accurately describes the system’s dynamics under the influence of anti-parametric excitation. The transfer of energy vibrations between the system modes is achieved when the parametric excitation is tuned at \(\Omega _p\). Using these dynamics behaviour, a technique for time-settling reduction is proposed, leading to a 90.39%, 93.13%, 94.44% and 96.79% time-settling reduction for 2DOF, 3DOF, 4DOF and 10DOF, respectively.

Fig. 1
figure 1

MEMS schematic diagram base on [26]

Full size image

2 MEMS

The MEMS considered is sketched in Fig. 1 consisting of two flexible beams that are fixed on both sides to the same frame as presented in [26]. The beam at the bottom is the sensor. The external excitation is indicated by the external forcing \(f_1(t)\). The beam on the top is manufactured the same way but is an entirely passive part of the system. The parametric excitation is applied to the sensing beam solely. Due to the physical coupling via the frame, the parametric excitation is also acting on the other beam. Approximating the dynamics of each beam by a single mode and its corresponding natural frequency \(\Omega _i\), the equations of motion can be written as

$$\begin{aligned} \ddot{x}_{1} +d_{1}\dot{x}_{1} +\Omega _{1}^{2} x_{1} +b_{11} p(t) x_1+ b_{12} p(t) x_{2}&=f_{1}(t) \nonumber \\ \ddot{x}_{2} +d_{2}\dot{x}_{2} + \Omega _{2}^{2}x_{2} + b_{21}p(t)x_1 +b_{22} p(t)x_2&=0 \end{aligned}$$
(1)

where the parametric excitation \(p(t)=\cos \Omega _{p}t\) couples the individual systems. Figure 1 depicts the conceptual configuration of the MEMS resonators according to [25, 29].

3 Applying averaging method

The coupled Mathieu equations in Eq. (1) without external excitation, i.e. after the measurement time, can be rewritten in a more general form as

$$\begin{aligned} \ddot{\textbf{x}}\left( t\right) +\textbf{D}\dot{\textbf{x}}\left( t\right) +\left( \varvec{\Omega }^{2}+\textbf{B}\cos \left( \Omega _{p}t\right) \right) \textbf{x}\left( t\right) =0 \end{aligned}$$
(2)

where \(\textbf{x}(t)\) is an n vector, \(\textbf{D}\), \(\textbf{B }\) and \(\textbf{Q}\) are \(n\times n\) constant matrices. Assuming distinct natural frequencies, we can write the matrix \(\mathbf {\Omega }^{2}=\text {diag}\left\{ \Omega _{1}^{2},\Omega _{2}^{2},\ldots ,\Omega _{n}^{2}\right\} \) being diagonal incorporating the natural frequencies of the undamped system. Adopting the Einstein summation as a notation, where repeated indices are implicitly summed over, Eq. (2) leads to the comprehensive index form

$$\begin{aligned} \ddot{x}_{i}\left( t\right) +\Omega _{i}^{2}x_{i}\left( t\right) =-\varepsilon \left[ \hat{d}_{ij}\dot{x}_{i}-\hat{b}_{ij}x_{i}\cos \left( \Omega _{p}t\right) \right] \end{aligned}$$
(3)

with \(i,j=1,2,\ldots \), and where \(\varepsilon \) is assumed to be a small parameter needed for the singular perturbation that rescales the damping and coupling coefficients in Eq. (2) according to by \(d_{ij} = \varepsilon \hat{d}_{ij}\) and \(b_{ij} = \varepsilon \hat{b}_{ij}\). Rescaling the time to normalize the frequency by

$$\begin{aligned} \tau =\Omega _{p}t \end{aligned}$$
(4)

leads to

$$\begin{aligned} x_{i}^{\prime \prime } + \omega _{i}^{2}x_{i} =-\frac{\varepsilon }{\Omega _{p}^{2}}\left( \Omega _{p}\hat{d}_{ij}x_{i}^\prime \left( \tau \right) +\hat{b}_{ij}x_{i}\left( \tau \right) \cos \tau \right) \end{aligned}$$
(5)

where \(\left( \cdot \right) ^{\prime }\) is the derivative with respect to \( \tau \) and \(\omega _{i}=\Omega _{i}/\Omega _{p}\). Defining the amplitude-phase coordinate transformation

$$\begin{aligned} x_{i}(\tau )&=u_{i}(\tau )c_{i}+v_{i}(\tau )s_{i}\nonumber \\ x_{i}^{\prime }(\tau )&=-u_{i}(\tau ){\omega _{i}}s_{i}+v_{i}(\tau ) {\omega _{i} } c_{i} \end{aligned}$$
(6)

using the abbreviations \(s_{i}=\sin (\omega _{i}\tau )\), \(c_{i}=\cos (\omega _{i}\tau )\), the equations of motion in Eq. (5) can be rewritten into the so-called quasi-normal form [39]. For the sake of simplicity, the explicit time dependence on \(\tau \) is not written. Applying this transformation yields

$$\begin{aligned} -u_{i}^{\prime }\omega _{i}s_{i}+v_{i}^{\prime }\omega _{i}c_{i}=\frac{ \varepsilon }{\Omega _{p}^{2}}\Gamma _{i}\left( u_{i},v_{i},\tau \right) \end{aligned}$$
(7)

where the scaled right-hand side reads

$$\begin{aligned} \Gamma _{i} =&-\Omega _{p}\sum \limits _{j}\hat{d}_{ij}\left( -u_{j}\Omega _{j}s_{j}+v_{j}\nu _{j}c_{j}\right) \\&-\sum \limits _{j}\hat{b}_{ij}\left( u_jc_{j}+v_{j}s_{j}\right) \cos \tau \end{aligned}$$

Respecting Eq. (7), Eqs. (5) finally become

$$\begin{aligned} u_{i}^{\prime }= & {} -\frac{\varepsilon }{\Omega _{p}\Omega _{i}}\Gamma _{i}(\textbf{ z},\tau )s_{i}=\varepsilon M_{i}^{s}(\textbf{z},\tau ) \end{aligned}$$
(8)
$$\begin{aligned} v_{i}^{\prime }= & {} \frac{\varepsilon }{\Omega _{p}\Omega _{i}}\Gamma _{i}( \textbf{z},\tau )c_{i}=\varepsilon M_{i}^{c}(\textbf{z},\tau ) \end{aligned}$$
(9)

with vector \(\textbf{z}= \begin{bmatrix} u_{1}&v_{1}&u_2&v_2&\ldots \end{bmatrix}^{T}\). The functions on the right-hand side \(M_{i}^{s}(\textbf{z},\tau )\) and \(M_{i}^{c}(\textbf{z},\tau )\) are not quasi-periodic and can be split into a finite sum of periodic terms

$$\begin{aligned} M_{i}^{s}(\textbf{z},\tau )= & {} \sum \limits _{k=1}^{N}M_{i,k}^{s}(\textbf{z},\tau ) \end{aligned}$$
(10)
$$\begin{aligned} M_{i}^{c}(\textbf{z},\tau )= & {} \sum \limits _{k=1}^{N}M_{i,k}^{c}(\textbf{z},\tau ) \end{aligned}$$
(11)

where N is fixed, \(M_{i,k}^{s}(\textbf{z},\tau )\) and \(M_{i,k}^{c}(\textbf{ z},\tau )\) are \(T_{k}\) periodic functions in \(\tau \). For this case, the averaging in the general case can be applied Eqs. (8) and (9) resulting in

$$\begin{aligned} \hat{u}_{i}^{\prime }= & {} \varepsilon \sum \limits _{k=1}^{N}\frac{1}{T_{k}} \int _{0}^{T_{k}}M_{i,k}^{s}(\textbf{z},\tau )d\tau \end{aligned}$$
(12)
$$\begin{aligned} \hat{v}_{i}^{\prime }= & {} \varepsilon \sum \limits _{k=1}^{N}\frac{1}{T_{k}} \int _{0}^{T_{k}}M_{i,k}^{c}(\textbf{z},\tau )d\tau \text {,} \end{aligned}$$
(13)

where the hat indicates the time-averaged variables and the original \( \textbf{z}\left( \tau \right) \) and averaged \(\hat{\textbf{z}}\left( \tau \right) \) solutions differ in an order of \(O\left( \varepsilon \right) \), on the timescale \(1/\varepsilon \). Using trigonometric identities, the products of the trigonometric terms can be rewritten as a sum of basic trigonometric terms of which the period \(T_{k}\) can be determined, and the integrals in Eqs. (12) and (13) calculated. Resonant terms occur at this step at frequencies

$$\begin{aligned} \Omega _{p}=\frac{2\Omega _{i}}{n}\text { }i=1,2,\ldots n\text {,} \end{aligned}$$
(14)

and

$$\begin{aligned} \Omega _{p}^{ij}=\frac{\left| \Omega _{i}\mp \Omega _{i}\right| }{n} \text { }i,j=1,2,\ldots n\text {, for }i\ne j \end{aligned}$$
(15)

4 Averaging for MEMS with 2-DOFs

Applying the averaging procedure described above to system in Eq. (2) with \(n=2\) at the parametric anti-resonance frequency \( \Omega _{p}^{21}=\left| \Omega _{2}-\Omega _{1}\right| \). Namely,

$$\begin{aligned} {\ddot{\textbf{x}}}+ \begin{bmatrix} {d}_{11} &{} {d}_{12} \\ {d}_{21} &{} {d}_{22} \end{bmatrix} {\dot{\textbf{x}}}+\left( \mathbf {\Omega }^{2}+ \begin{bmatrix} {b}_{11} &{} {b}_{12} \\ {b}_{21} &{} {b}_{22} \end{bmatrix} \cos \left( \Omega _{p}^{21} t\right) \right) \textbf{x}=0 \end{aligned}$$
(16)

where \(\mathbf {x=} \begin{bmatrix} x_{1}&x_{2} \end{bmatrix}^{T}\) and \(\mathbf {\Omega }^{2}=\text {diag}\left( \Omega _{1}^{2},\Omega _{2}^{2}\right) \), we obtain

$$\begin{aligned} {\hat{\textbf{z}}_\textbf{2}}^{~~\prime } =\hat{\textbf{A}}_\textbf{2}\hat{\textbf{z}}_{\textbf{2}} \end{aligned}$$
(17)

with the coefficient matrix

$$\begin{aligned} \hat{\textbf{A}}_\textbf{2}=\frac{1}{\Omega _{p}^{21}} \begin{bmatrix} -\frac{1}{2}d_{11} &{} 0 &{} 0 &{} \frac{1}{4\Omega _{1}}b_{12} \\ 0 &{} -\frac{1}{2}d_{11} &{} -\frac{1}{4\Omega _{1}}b_{12} &{} 0 \\ 0 &{} \frac{1}{4\Omega _{2}}b_{21} &{} -\frac{1}{2}d_{22} &{} 0 \\ -\frac{1}{4\Omega _{2}}b_{21} &{} 0 &{} 0 &{} -\frac{1}{2}d_{22} \end{bmatrix} \end{aligned}$$
(18)

Note that the damping and coupling coefficients are rescaled back according to \(d_{ij}=\varepsilon \hat{d}_{ij}\) and \(b_{ij}= \varepsilon \hat{b}_{ij}\). The difference between the original solution \(\mathbf {z_2}= \begin{bmatrix} u_{1}&v_{1}&u_{2}&v_{2} \end{bmatrix}^{T}\) and the averaged solution \(\hat{\textbf{z}}_\textbf{2}= \begin{bmatrix} \hat{u}_{1}&\hat{v}_{1}&\hat{u}_{2}&\hat{v}_{2} \end{bmatrix}^{T}\) is of order \(\varepsilon \), i.e. \(\hat{\textbf{z}}_{\textbf{2}}(\tau )-\textbf{z} _{\textbf{2}}(\tau )=\mathcal {O}(\varepsilon )\) on the time scale \(1/\varepsilon \). Tuning the parametric excitation to the parametric anti-resonance frequency \(\Omega _{p} = \Omega _{p}^{21}\), note that the direct coupling coefficients \(b_{ii}\) as well as the off-diagonal damping coefficients \(d_{ij}\) do not appear in the first order approximation in Eqs. (18). This fact was already observed in [15]. The slow flow of the system in Eq. (2) at \(\Omega _{p}=\Omega _{p}^{21}\) is entirely characterized by the coupling terms \(b_{12}\) and \(b_{21}\), and the diagonal damping coefficients \(d_{11}\) and \(d_{22}\).

Fig. 2
figure 2

Characterization of the response envelope of the system in Eq. (16) according to inequality in Eq. (19)

Full size image

The coefficient matrix \(\hat{\textbf{A}}_\textbf{2}\) possesses two repeated eigenvalues \(\hat{\lambda }_{1}^{\hat{\textbf{A}}_\textbf{2}}=\hat{\lambda }_{3}^{\hat{\textbf{A}}_\textbf{2}}\) and \( \hat{\lambda }_{2}^{\hat{\textbf{A}}_\textbf{2}}=\hat{\lambda }_{4}^{\hat{\textbf{A}}_\textbf{2}}\) given by

$$\begin{aligned} \hat{\lambda }_{1,2}^{\hat{\textbf{A}}_\textbf{2}}=-\frac{1}{4\Omega _{p}}\left( d_{11}+d_{22}\pm \sqrt{\left( d_{11}-d_{22}\right) ^{2}-\frac{b_{12}b_{21}}{\Omega _{1}\Omega _{2}}}\right) \end{aligned}$$
(19)

They represent the behaviour of the dynamics on the slow time scale which refers to the envelope of the system response in Eq.  (18) and is shown in Fig. 2. Therefore, if \(\hat{\lambda }_{12}\) are purely real, the response envelope decreases exponentially in time, as shown in Fig. 2a. Alternatively, complex conjugate eigenvalues correspond to a modulated envelope, sometimes called a beating signal as depicted in Fig. 2b. Thus, to determine this nature of the stable response \(\hat{\lambda }_{12}\in \mathbb {R} \) or \(\hat{\lambda }_{12}\in \mathbb {C} \) it is sufficient to verify the following inequality [10]

$$\begin{aligned} \left( d_{11}-d_{22}\right) ^{2}>\frac{b_{12}b_{21}}{\Omega _{1}\Omega _{2}} \ge 0\text {.} \end{aligned}$$
(20)

If this inequality is fulfilled, \(\hat{\lambda }_{12}\) are real. Otherwise, \(\hat{\lambda }_{12}\) are complex-valued in this case the imaginary part of \(\hat{\lambda }_{12}\) can be interpreted as the frequency of the envelope signal

$$\begin{aligned} \omega _{\hat{\textbf{A}}_{\textbf{2}}}=\pm \frac{1}{4\Omega _{p}^{21}} \sqrt{\frac{b_{12}b_{21}}{\Omega _{1}\Omega _{2}}-\left( d_{11}-d_{22}\right) ^{2}}\text {.} \end{aligned}$$
(21)

Moreover, the two envelope response signals for the system in Eq.  (16) are in anti-phase because there are two equal frequencies. Therefore, it is possible to determine the period of the envelope signals in anti-phase. Rescaling the time \(\tau \) according to Eq. (4) and considering \(\omega _{\hat{\textbf{A}}_{\textbf{2}}} \) as the frequency, we estimate the period of the slow motion as

$$\begin{aligned} T_{\hat{\textbf{A}}_{\textbf{2}}}=\frac{8\pi }{\sqrt{\frac{b_{12}b_{21} }{\Omega _{1}\Omega _{2}}-\left( d_{11}-d_{22}\right) ^{2}}} \end{aligned}$$
(22)
Table 1 2DOF system parameters
Full size table

To highlight the prediction quality of Eq. (22), the simulation of a specific system represented by equations of motion in Eq. (16) is performed. The following system parameter were chosen according to [25, 26] and are listed in Table 1 where the quality factors are defined as \(Q_1= \Omega _1/d_{11}\) and \(Q_2=\Omega _2/d_{22}\). The envelopes of the system responses \( Env[x_1(t)]\) and \( Env[x_2(t)]\) are shown in Fig. 3. The period identified from the numerical simulation \(T_{\hat{\textbf{A}}_{\textbf{2}}}=38.4383\) ms coincides with the analytical prediction in Eq. (22). A direct comparison between the solution \(\textbf{x}\) of the original system in Eq. (16) and the solution \(\hat{\textbf{z}}\) resulting from the averaged system is given in the appendix.

Fig. 3
figure 3

Envelope of the time responses of the system in Eq. (16) at \(\Omega _{p}^{21}=\left| \Omega _{2}-\Omega _{1}\right| \). The analytical prediction in Eq. (22) matches the observed period \(T_{\hat{\textbf{A}}_\textbf{2}}\)

Full size image

5 Pulse of parametric excitation in 2DOF

Fig. 4
figure 4

Time responses of the 2DOF-MEMS in Eq. (1) with a settling time \(t_s=100\) ms

Full size image
Fig. 5
figure 5

Time responses of the 2DOF-MEMS in Eq. (1) under the effect of a parametric excitation pulse \(t_p=9.8\) ms. The settling time is reduced to \(t_s \approx 9.8\) ms

Full size image

A continuous parametric excitation at a parametric combination resonance frequency leads to a steady energy transfer between the modes of the system, as discussed in more detail in [14]. Starting with vibration in beam 1, such a continuous excitation results in an energy transfer into beam to within the period estimated in Eq. (22). After this time however, the energy is transferred back, at least what is left after dissipation within the next period. This back and forth transfer happens continuously until all kinetic energy is eventually dissipated by the damping coefficients of both beams. In order to avoid this back-channeling of kinetic energy from beam 2 to beam 1, we apply the parametric excitation only within a short pulse with time of a single period. The time response of the 2DOF-system subject to an external excitation \(f_1(t)\) is shown in Fig. 4. The steady-state of beam 1 is reached after 100 ms. After this measurement, the excitation is switched of and the vibration level of beam 1 decays exponentially due to its damping. Beam 2 is not affected during this operation.

In contrast to this, we apply a parametric excitation right after the measurement time during the short pulse time \(t_{p_2}=T_{\hat{\textbf{A}}_\textbf{2}}/4\) given in Eq. (22). This operation is highlighted in Fig. 5 and confirms the fast decay of kinetic energy in beam 1 and increase in beam 2. The reverse energy flow, however, is eliminated by switching off the parametric excitation, which generates a single pulse of parametric excitation. The general idea was outlined already in [29].

6 Multiple parametric excitations in 3DOF

Fig. 6
figure 6

Envelope of the time responses of the 3DOF-MEMS in Eq. (23) at continuous, quasi-periodic parametric excitation \(\Omega _{p}^{21}\) and \(\Omega _{p}^{31}\). The analytical prediction in Eq. (29) matches the observed period \(T_{\hat{\textbf{A}}_\textbf{3}}\)

Full size image
Fig. 7
figure 7

Envelope of the time responses of the 3DOF-MEMS in Eq. (23) at quasi-periodic parametric excitation \(\Omega _{p}^{21}\) and \(\Omega _{p}^{31}\) pulsed during \(t_{p_3}\) predicted in Eq. (29). The settling time reduces to \(t_{s_3}\)

Full size image

The concept of reducing the settling time by a duplication of the structure is extended to more beams. It is straightforward to do this numerically. However, for a practical implementation, we need to know the time period(s) of the pulsed parametric excitations upfront. For this, we apply the averaging method to a system consisting of three beams, one sensing beam and two structural duplicates possessing similar but not exactly the same natural frequencies. With three beams, we have more than one possibility of transferring energy between the first modes of each beam, so we introduce a quasi-periodic parametric excitation of the form

$$\begin{aligned} {\ddot{\textbf{x}}}+\varvec{D}{\dot{\textbf{x}}}+\left( \varvec{\Omega } ^{2}+{\textbf{B}}\left[ \cos \left( \Omega _{p}^{21}t\right) +\cos \left( \Omega _{p}^{31}t\right) \right] \right) \textbf{x}=0 \end{aligned}$$
(23)

where \(\mathbf {x=} \begin{bmatrix} x_{1}&x_{2}&x_{3} \end{bmatrix} ^{T}\), \(\mathbf {\Omega }^{2}=\text {diag}\left\{ \Omega _{1}^{2},\Omega _{2}^{2},\Omega _{3}^{2}\right\} \) and \( \varvec{D}= \begin{bmatrix} d_{11} &{} d_{12} &{} d_{13} \\ d_{21} &{} d_{22} &{} d_{23} \\ d_{31} &{} d_{32} &{} d_{33} \end{bmatrix}\) and \(\varvec{B}= \begin{bmatrix} b_{11} &{} b_{12} &{} b_{13} \\ b_{21} &{} b_{22} &{} b_{23} \\ b_{31} &{} b_{32} &{} b_{33} \end{bmatrix} \) being fully occupied coefficient matrices for damping and time-periodic coupling. The parametric excitation consists of two periodic signals that are tuned each at a parametric anti-resonance frequency: \(\Omega _{p}^{21}=\left| \Omega _{2}-\Omega _{1}\right| \) and \(\Omega _{p}^{31}=\left| \Omega _{3}-\Omega _{1}\right| \). The system in Eq. (23) describes a dynamic system with a multiple parametric excitation, a quasi-periodic parametric excitation.

Given the identical structure of the MEMS considered, each flexible beam has similar quality factors \(Q_{i}=\Omega _{i}/d_{ii}\). To simplify the subsequent analysis, it is assumed that each beam possesses the same direct damping coefficient, i.e. \(d_{11}=d_{22}=d_{33}=d\). Applying the averaging method described in the previous section on the system in Eq. (23) with the time rescaling

$$\begin{aligned} \tau =\Omega _{p}^{21}t, \end{aligned}$$
(24)

the following slow flow is obtained

$$\begin{aligned} {\hat{\textbf{z}}_\textbf{3}}^{~~\prime }(\tau ) =\hat{\textbf{A}}_\textbf{3}\hat{\textbf{z}}_\textbf{3}(\tau ) \end{aligned}$$
(25)

with the coefficients matrix

$$\begin{aligned} \hat{\textbf{A}}_\textbf{3}=\frac{1}{\Omega _{p}^{21}} \begin{bmatrix} -\frac{d}{2} &{} 0 &{} 0 &{} \frac{b_{12}}{4\Omega _{1}} &{} 0 &{} \frac{b_{13}}{ 4\Omega _{1}} \\ 0 &{} -\frac{d}{2} &{} -\frac{b_{12}}{4\Omega _{1}} &{} 0 &{} -\frac{b_{13}}{4\Omega _{1}} &{} 0 \\ 0 &{} \frac{b_{21}}{4\Omega _{2}} &{} -\frac{d}{2} &{} 0 &{} 0 &{} 0 \\ -\frac{b_{21}}{4\Omega _{2}} &{} 0 &{} 0 &{} -\frac{d}{2} &{} 0 &{} 0 \\ 0 &{} \frac{b_{31}}{4\Omega _{3}} &{} 0 &{} 0 &{} -\frac{d}{2} &{} 0 \\ -\frac{b_{31}}{4\Omega _{3}} &{} 0 &{} 0 &{} 0 &{} 0 &{} -\frac{d}{2} \end{bmatrix} \text {.} \end{aligned}$$
(26)

The coefficient matrix for the 2DOF-system in Eq. (18) can be identified as a submatrix. Again, only the diagonal damping coefficients and the coupling terms are needed for describing the slow in this first order approximation. For the chosen parametric excitation frequencies \(\Omega _{p}^{21}\) and \(\Omega _{p}^{31}\) this results in \(b_{12}\), \(b_{21}\) and \(b_{13}\), \(b_{31}\). Since \(b_{ij}>0\) and \(d>0\) hold for this specific MEMS configuration, the coefficient matrix \(\hat{\textbf{A}}_{\textbf{3}}\) has two repeated purely real-valued eigenvalues

$$\begin{aligned} \lambda _{1}^{\hat{\textbf{A}}_\textbf{3}}=\lambda _{2}^{\hat{\textbf{A}}_\textbf{3}}=-\frac{d}{2\Omega _{p}^{21}} \end{aligned}$$
(27)

and two repeated pairs of complex conjugates

$$\lambda _{34}^{\hat{\textbf{A}}_\textbf{3}}=\lambda _{56}^{\hat{\textbf{A}}_\textbf{3}} =\frac{1}{2\Omega _{p}^{21}}\left( -d\pm j \sqrt{\frac{b_{13}b_{31}}{4\Omega _1\Omega _3}+ \frac{b_{12}b_{21}}{4\Omega _{1}\Omega _{2}}}\right) $$

For \(b_{ij}>0\) and \(d>0\), the frequency of the slow flow reads

$$\begin{aligned} \hat{\omega }_{\hat{\textbf{A}}_3}=\frac{1}{2\Omega _{p}^{21}} \sqrt{\frac{b_{12}b_{21}}{4\Omega _{1}\Omega _{2}}+\frac{b_{13}b_{31}}{4\Omega _1\Omega _3}} \end{aligned}$$
(28)

The prediction of the period of the slow flow of the system response in Eq. (23) is obtained by rescaling the time to the original physical time according to Eq. (24)

$$\begin{aligned} T_{\hat{\textbf{A}}_\textbf{3}}=4\pi \sqrt{\frac{4\Omega _{1}\Omega _{2}\Omega _{3}}{\Omega _{3}b_{12}b_{21}+\Omega _{2}b_{13}b_{31}}} = 4 t_{p_3} \end{aligned}$$
(29)

Herein, \(t_{p_3}\) is the pulse time for achieving an energy transfer from beam 1 to beam 2 and beam 3 simultanously. The envelopes of the time responses of the 3DOF-MEMS in Eq. (23) at continuous, quasi-periodic parametric excitation \(\Omega _{p}^{21}\) and \(\Omega _{p}^{31}\) are shown in Fig. 6. The period \(T_{{\hat{\textbf{A}}}_\textbf{3}}\) coincides with the analytical prediction in Eq. (29). Envelope of the very same system but activating the quasi-periodic parametric excitation only during the pulsation time \(t_{p_3}\) defined in Eq. (29) reduces the settling time to \(t_{p_3}=6.8\) ms and is shown in Fig. 7.

7 Multiple parametric excitations in 4DOF

Fig. 8
figure 8

MEMS schematic diagram with 4DOF

Full size image

Attempting to generalize the reduction of settling time presented for 2DOF-MEMS and 3DOF-MEMS, the 3DOF-MEMS in the previous section is extended by an additional beam with nominally identical parameters, see Fig. 8. By adding another beam to the structure, the system now consists of one sensing beam and three duplicates. This configuration is expected to allow for further reduction of the settling time because one additional parametric anti-resonance frequency is introduced.

For a MEMS with four beams, the equations of motion read

$$\begin{aligned} {\ddot{\textbf{x}}}+\varvec{D}{\dot{\textbf{x}}}+\left( \varvec{\Omega } ^{2}+{\textbf{B}}p(t) \right) \textbf{x}=0 \end{aligned}$$
(30)

Herein, \(\textbf{x}=\begin{bmatrix} x_{1}&x_{2}&x_3&x_{4}\end{bmatrix}^{T}\), \(\mathbf {\Omega }^2\) is the diagonal matrix of the natural frequencies of the underlying system with constant coefficients and the damping and coupling coefficient matrices are fully occupied and of size \(4\times 4\), and the parametric excitation p(t) is the sum harmonic parametric excitations at each parametric anti-resonance frequency that enables an energy transfer with beam 1. For a 4DOF-MEMS, the quasi-periodic parametric excitation consists of three harmonics

$$\begin{aligned} p(t)= \cos \left( \Omega _{p}^{21}t\right) +\cos \left( \Omega _{p}^{31}t\right) +\cos \left( \Omega _{p}^{41}t\right) \end{aligned}$$
(31)

experiencing the parametric anti-resonance frequencies \(\Omega _{p}^{21}=\left| \Omega _{2}-\Omega _{1}\right| \), \(\Omega _{p}^{31}=\left| \Omega _{3}-\Omega _{1}\right| \) and \(\Omega _{p}^{41}=\left| \Omega _{4}-\Omega _{1}\right| \). Averaging the equations of motion in Eq. (30) for the system in Fig. 8 yields

$$\begin{aligned} \hat{\textbf{z}}_\textbf{4}^{~~\prime }(\tau ) =\hat{\textbf{A}}_\textbf{4}\hat{\textbf{z}}_\textbf{4}(\tau ) \end{aligned}$$
(32)

with the coefficients matrix

$$\begin{aligned} \hat{\textbf{A}}_4=\frac{1}{\Omega _{p}^{21}} \begin{bmatrix} \frac{-d}{2} &{} 0 &{} 0 &{} \frac{b_{12}}{4\Omega _{1}} &{} 0 &{} \frac{b_{13}}{ 4\Omega _{1}} &{} 0 &{} \frac{b_{14}}{4\Omega _{1}} \\ 0 &{} \frac{-d}{2} &{} \frac{-b_{12}}{4\Omega _{1}} &{} 0 &{} \frac{-b_{13}}{4\Omega _{1}} &{} 0 &{} \frac{-b_{14}}{4\Omega _{1}} &{} 0 \\ 0 &{} \frac{b_{21}}{4\Omega _{2}} &{} \frac{-d}{2} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \frac{-b_{21}}{4\Omega _{2}} &{} 0 &{} 0 &{} \frac{-d}{2} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} \frac{b_{31}}{4\Omega _{3}} &{} 0 &{} 0 &{} \frac{-d}{2} &{} 0 &{} 0 &{} 0 \\ \frac{-b_{31}}{4\Omega _{3}} &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{-d}{2} &{} 0 &{} 0 \\ 0 &{} \frac{-b_{41}}{4\Omega _{4}} &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{-d}{2} &{} 0 \\ \frac{-b_{41}}{4\Omega _{4}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{-d}{2} \end{bmatrix} \end{aligned}$$
(33)
Fig. 9
figure 9

Envelope of the time responses of the 4DOF-MEMS in Eq. (30) at \(\Omega _{p}^{21}\), \(\Omega _{p}^{31}\) and \(\Omega _{p}^{41}\). The analytical prediction in Eq. (35) matches the observed period \(T_{\hat{\textbf{A}}_\textbf{4}}\)

Full size image
Fig. 10
figure 10

Envelope of the time responses of the 4DOF-MEMS under the pulses of excitation \(\Omega _{p}^{21}=\left| \Omega _{2}-\Omega _{1}\right| \), \(\Omega _{p}^{31}=\left| \Omega _{3}-\Omega _{1}\right| \) and \(\Omega _{p}^{41}=\left| \Omega _{4}-\Omega _{1}\right| \) for \(t_{p_4}\) in Eq. (35). The settling time reduces to \(t_{s_4}\)

Full size image

The time was rescaled by applying \(\tau =\Omega _{p}^{21}t\) similarly to Eq. (4) and the direct damping coefficients were assumed to be identical \(d_{11}=d_{22}=d_{33}=d_{44}=d\). The coefficient matrices for the 2DOF-system in Eq. (17) and for the 3DOF-system in Eq. (26) can be identified as submatrices. Similarly to the last sections, the only coefficients that describe the dynamics of the slow flow in a first order approximation of the quasi-periodic parametric excitation are the direct damping coefficients d and the coupling coefficients \((b_{12},b_{21})\), \((b_{13},b_{31})\), \((b_{14},b_{41})\). These coupling coefficients correspond to the parametric anti-resonance frequencies \(\Omega _{p}^{21}\), \(\Omega _{p}^{31}\) and \(\Omega _{p}^{41}\), respectively, to which the harmonic components in Eq. (31) are perfectly tuned to.

The coefficient matrix \(\hat{\textbf{A}}_{\textbf{4}}\) has four repeated purely real-valued eigenvalues \(\lambda _{1234}^{\hat{\textbf{A}}_\textbf{4}}\), that are identical to Eq. (27), and two repeated pairs of complex conjugates

$$\lambda _{5678}^{\hat{\textbf{A}}_\textbf{4}}=-\frac{d}{2\Omega _{p}^{21}}\pm j \hat{\omega }_{\hat{\textbf{A}}_\textbf{4}}$$

The frequency of the slow flow reads similarly to Eq. (28)

$$\begin{aligned} \hat{\omega }_{\hat{\textbf{A}}_\textbf{4}}= \frac{1}{2\Omega _{p}^{21}} \sqrt{\frac{b_{12}b_{21}}{4\Omega _{1}\Omega _{2}}+\frac{b_{13}b_{31}}{4\Omega _1\Omega _3}+\frac{b_{14}b_{41}}{4\Omega _2\Omega _3}} \end{aligned}$$
(34)

The period of the envelope of slow dynamics becomes in physical time

$$\begin{aligned} T_{\hat{\textbf{A}}_\textbf{4}}=4\pi \sqrt{\frac{4\Omega _{1}\Omega _{2}\Omega _{3}\Omega _4}{\Omega _{3}\Omega _4b_{12}b_{21}+ \Omega _{2}\Omega _4b_{13}b_{31}+\Omega _{2}\Omega _3b_{14}b_{41}}} \end{aligned}$$
(35)

The pulse time results in \(t_{p_4}=T_{\hat{\textbf{A}}_\textbf{4}}/4\) during which an energy transfer from beam 1 to beam 2, beam 3 and beam 4 is achieved simultanously. The envelopes of the time responses of a 4DOF-MEMS at continuous, quasi-periodic parametric excitation \(\Omega _{p}^{21}\), \(\Omega _{p}^{31}\) and \(\Omega _{p}^{41}\) are shown in Fig. 9. The system parameters are listed in the first four lines in Table 2. The period \(T_{\hat{\textbf{A}}_\textbf{4}}\) coincides with the analytical prediction in Eq. (35). The Envelope of the very same system but activating the quasi-periodic parametric excitation only during the pulsation time \(t_{p_4}\) defined in Eq. (35) reduces the settling time to 5.6 ms and is shown in Fig. 10.

8 Multiple parametric excitations in NDOF

In general, for an arbitrary number of beams m, the equations of motion are equivalent to Eq. (30). Herein, \(\textbf{x}=\begin{bmatrix} x_{1}&x_{2}&\ldots&x_{m}\end{bmatrix}^{T}\), \(\mathbf {\Omega }^2\) is the diagonal matrix of the natural frequencies of the underlying system with constant coefficients and the damping and coupling coefficient matrices are fully occupied and of size \(m\times m\) and the parametric excitation is the sum harmonic parametric excitations at each parametric anti-resonance frequency \(\Omega _p^{i1}=\vert \Omega _i-\Omega _1\vert \) that enables an energy transfer with beam 1,

$$\begin{aligned} p(t)= \sum _{i=2}^m \cos (\Omega _{p}^{i1}t) \end{aligned}$$
(36)

We attempt an approximation of the period for energy transfer at this highly tuned quasi-periodic parametric excitation based on the structure observed in Eqs. (22), (29) and (35). For a MEMS design similar to the one shown in Fig. 8 consisting of m similar beams, the period for energy transfer appears to be

$$\begin{aligned} T_{\hat{\textbf{A}}_\textbf{m}}= \frac{4\pi }{\sqrt{\displaystyle \sum _{i=2}^m\frac{b_{1i}b_{i1}}{4\Omega _{1}\Omega _{i}}}} = 4 t_{pN} \end{aligned}$$
(37)
Fig. 11
figure 11

Envelope of the time responses of the NDOF- MEMS starting with 2DOF in Fig. 5, 3DOF in Fig. 7, 4DOF in Fig. 10 up to 10DOF. Parameter values are according to Table 2

Full size image
Fig. 12
figure 12

Pulse or settling time in dependency of the number of beams: derivation from numerical time integration of the equations motion according to Fig. 11 in comparison to the analytical approximation in Eq. (37)

Full size image

The analytical prediction in Eq. (37) is tested for NDOF-MEMS with one to nine beam duplicates, e.g. MEMS possessing two to ten DOF. The envelope response of the sensing beam 1 is shown in Fig. 11. The system parameters are chosen according to the values listed in Table 2. The decrease of the settling time, the time for energy transfer from beam 1 to all other beams by a pulsed quasi-periodic parametric excitation is clearly highlighted. Finally, we compare the pulse time \(t_{p_N}\), or equivalently settling time of the NDOF-MEMS, to the analytical prediction in Eq. (37) in Fig. 12. The analytical prediction fits perfectly to the settling time derived from the direct numerical integration of the equations of motion for the individual MEMS-configurations in Fig. 11. This comparison confirms that in the case of identical beams, the settling time is proportional to \(1/\sqrt{N}\). The diagram allows for an extrapolation of the number of beams N of the NDOF-MEMS and provides an answer on how many similar beams are needed for achieving a certain settling time.

9 Conclusions

Table 2 Paramaters for NDOF-system
Full size table

The parametric anti-resonance concept is generalized to a multi-frequency, pulsed parametric excitation in a multi-degree-of-freedom system in order to achieve a reduction in the settling time in a specific system. This open-loop control applies one or more intentional, confined (pulsed), harmonic parametric excitations. If properly tuned, a significant reduction of the transient time is achieved. MEMS show promising applicability of this concept due to the ability to easily introduce periodic and quasi-periodic signals tuned to a specific frequency. This concept triggers multiple simultaneous energy transfers from the sensing mode of beam 1 to several beams. The selectivity of which beams are incorporated in an energy transfer is purely controlled by the individually chosen excitation frequency and coupling terms. The concept was shown originally in [29] for two beams with a single-frequent parametric excitation at \(\Omega _{p}^{21}=\left| \Omega _2-\Omega _1 \right| \). Here we greatly enhance the applicability as well as complexity by introducing N beams and multiple pulses of parametric excitation at \(\Omega _{p}^{i1}=\left| \Omega _i-\Omega _1 \right| \) with \(i=2,3,\ldots N\). The averaging method in the general case is applied to achieve an analytical prediction at very specific excitation frequencies. The approximate slow flow dynamics can accurately predict the pulsation time needed for such a quasi-periodic parametric excitation. This allows for properly designing the necessary pulse for rapid energy transfer between system modes or individual beams, which eventually leads to mitigating the kinetic energy, and therefore settling time, of the first mode. The newly proposed method is applied to the arrays of MEMS consisting of flexible beams with high quality factors. Such beams are prone to long steady-state transition times that may be too long for certain fast, repetitive operations. The influence of the number of beams in the MEMS is evaluated in more detail and shows that the original reduction of the settling time by 90.39% for a 2DOF-MEMS is reduced to 93.13% for a 3DOF-MEMS, to 94.44% for a 4DOF-MEMS and to 96.79% for a 10DOF-MEMS. This study confirms that a designed pulse of multi-frequency parametric excitation reduces the settling time significantly.