Passage through a resonance for a mechanical system, having time-varying parameters and possessing a single trapped mode. The principal term of the resonant solution

TL;DR

This paper analyzes the passage through resonance in an infinite string system with time-varying parameters and a trapped mode, deriving an asymptotic solution and validating it with numerical simulations.

Contribution

It provides the first asymptotic expression for the resonant response of a string with a trapped mode and time-varying parameters, verified by numerical methods.

Findings

Asymptotic principal term of the resonant solution derived.

Good agreement between analytical and numerical results near resonance.

Validated the theoretical results with finite difference numerical simulations.

Abstract

We consider a forced oscillation and passage through resonance for an infinite-length system, having time-varying parameters and possessing a single trapped mode. The system is a string, lying on the Winkler foundation and equipped with a discrete linear mass-spring oscillator of time-varying stiffness. We obtain the principal term of the asymptotic expansion for the resonant solution describing the motion of the inclusion (i.e., the mass-spring oscillator). The obtained result was verified by independent numerical calculations based on solution of the corresponding partial differential equation by means of the method of finite differences. The comparison demonstrates a good agreement in a neighbourhood of the instant of resonance.

Equations

$$u^{\prime \prime }-\ddot{u}-u=-P(\tau )\delta (\xi ),$$

$$P(\tau )=-M\ddot{u}(0,\tau )-K(ϵ\tau )u(0,\tau )+p(\tau ).$$

$$u\big{|}_{\tau<0}\equiv 0.$$

$$p(\tau )=H(\tau )\exp (-\mathrm{i}\hat{\Omega }\tau ),$$

$$\big{[}u^{\prime}(0,\tau)\big{]}=-{P(\tau)}=M\ddot{u}(0,\tau)+K(\epsilon\tau)u(0,\tau)-p(\tau).$$

$$u^{\prime \prime }-\ddot{u}-u=0(\xi >0)$$

$$u^{\prime }(0,\tau )=\frac{M\ddot{u}(0,\tau )+K(ϵ\tau )u(0,\tau )-p(\tau )}{2}.$$

$$u=\hat{u}(\xi )\exp (-i\Omega \tau ).$$

$$0<|\Omega |<1.$$

$${\int }_{-\infty }^{+\infty }{\hat{u}}^2\mathrm{d}\xi <\infty ,{\int }_{-\infty }^{+\infty }{\hat{u}}^{\prime }{}^2\mathrm{d}\xi <\infty .$$

$${\hat{u}}^{\prime \prime }-(1-{\Omega }^2)\hat{u}=(K-M{\Omega }^2)\hat{u}(0)\delta (\xi ).$$

$$\hat{u}=-{(K-M\Omega^{2})\hat{u}(0)}\frac{\exp\big{(}-\sqrt{1-\Omega^{2}}|\xi|\big{)}}{2\sqrt{1-\Omega^{2}}}.$$

$$2\sqrt{1-{\Omega }_0^2}=M{\Omega }_0^2-K.$$

$$-2<K<M$$

$${\Omega }_0^2=\frac{2}{M^2}\left(\sqrt{M^2-MK+1}+\frac{MK}{2}-1\right),$$

$${\Omega }_0^2=1-\frac{K^2}{4}.$$

$${\Omega }_0{}_K^{\prime }>0.$$

$$\underset{K=-2+0}{\lim }{\Omega }_0(K)=+0,\underset{K=M-0}{\lim }{\Omega }_0(K)=1-0.$$

$$\mathscr{A}=\frac{C_{0}\left(1-{\Omega}^{2}_{0}\right)^{1/4}}{\Omega_{0}^{1/2}\Big{(}1+M\sqrt{1-\Omega_{0}^{2}}\Big{)}^{1/2}},$$

$$1\ll K<M$$

$$\mathcal{A}=\frac{C_1}{{\Omega }_0^{1/2}}$$

$$\begin{gathered}\dot{K}>0,\\ \Omega_{0}\big{(}K(0)\big{)}>0,\quad\Omega_{0}\big{(}K(\epsilon\tau_{0})\big{)}=\hat{\Omega},\quad\Omega_{0}\big{(}K(\epsilon\tau_{\ast})\big{)}<1,\end{gathered}$$

$$T=\sqrt{ϵ}(\tau -{\tau }_0),X=\sqrt{ϵ}\xi .$$

$$K(\tau )=K_0+ϵK_1(\tau -{\tau }_0)+o(ϵ),$$

$$K(T)=K_0+\sqrt{ϵ}{K}_{1}T+o(\sqrt{ϵ}).$$

$${\Omega }_0(\tau )=\hat{\Omega }+ϵ{\Omega }_{01}(\tau -{\tau }_0)+o(ϵ),$$

$${\Omega }_0(T)=\hat{\Omega }+\sqrt{ϵ}{\Omega }_{01}T+o(\sqrt{ϵ}).$$

$${{\Omega }_0^{\prime }}_T={\Omega }_{01}\sqrt{ϵ}+o(\sqrt{ϵ}),$$

$$u(\xi ,\tau )=W(X,T)\exp \phi (\xi ,\tau ),$$

$${\phi }^{\prime }=\mathrm{i}\omega (X,T),\dot{\phi }=-\mathrm{i}\Omega (X,T),$$

$$W(X,T)=\frac{W_0(X,T)}{\sqrt{ϵ}}+W_1(X,T)+O(\sqrt{ϵ}).$$

$$\begin{array}{c}\dot{(\cdot )}=-\mathrm{i}\Omega {\partial }_{\phi }+\sqrt{ϵ}{\partial }_T+O(ϵ),\\ \ddot{(\cdot )}=-{\Omega }^2{\partial }_{\phi \phi }^2-2\mathrm{i}\sqrt{ϵ}\Omega {\partial }_{\phi T}^2-\mathrm{i}\sqrt{ϵ}{{\Omega }^{\prime }}_T{\partial }_{\phi }+O(ϵ),\\ (\cdot {)}^{\prime }=\mathrm{i}\omega {\partial }_{\phi }+\sqrt{ϵ}{\partial }_X+O(ϵ),\\ (\cdot {)}^{\prime \prime }=-{\omega }^2{\partial }_{\phi \phi }^2+2\mathrm{i}\sqrt{ϵ}\omega {\partial }_{\phi X}^2+\mathrm{i}\sqrt{ϵ}{\omega }_X^{\prime }{\partial }_{\phi }+O(ϵ).\end{array}$$

Full text

††journal: Journal of Sound and Vibration

1 Introduction

In the paper, we consider a forced oscillation of an infinite-length mechanical system having time-varying parameters and possessing a single trapped mode [1] characterized by a frequency $\Omega_{0}(\epsilon\tau)$ (here and in what follows, $\epsilon$ is a small parameter, $\tau$ is the time). The system is a string, lying on the Winkler foundation, and equipped with a discrete linear mass-spring oscillator of time-varying stiffness. In the case of a constant spring stiffness, the spectrum of natural oscillations for such a mechanical system may contain unique (positive) eigenvalue, which is less than the lowest frequency for the string on the uniform foundation (the cut-off frequency). This special natural frequency corresponds to a trapped mode of oscillation with eigenform localized near the spring. The phenomenon of trapped modes was discovered in the theory of surface water waves [2].

The discrete oscillator is subjected to a harmonic external force with constant frequency $\hat{\Omega}$. In the case of the passage through the resonance, we obtain the principal term of the asymptotic expansion describing the motion of the inclusion (i.e., the mass-spring oscillator). To do this, we use a combination of two asymptotic approaches.

The first approach was suggested in [3] to describe a free localized oscillation in systems possessing a single trapped mode. It was successfully used to investigate various free localized oscillation of spatially non-uniform infinite length systems with time-varying properties [4, 5, 6, 7, 8, 9, 1]. In particular, in our recent study [1] we used this approach to study a free localized oscillation in the system considered in this paper. The extensive bibliography on trapped modes and localized waves in infinite-length linear systems can be found in recent papers [8, 9] and monograph [10]. The second approach was used in [11, 12] to describe a forced oscillation and passage through a resonance in a single degree of freedom system (a linear oscillator).

The paper by Fowler & Lock [13] was probably the first one (see [14]), where the resonant excitation of a linear system of ODE \colorblack (ordinary differential equations) with slowly varying coefficients was considered from the asymptotic point of view. In the series of studies by Feschenko, Shkil & Nikolenko summarized in monograph [15], the authors obtained an asymptotically simplified system of ODE by describing the passage through a resonance in a system with several degrees of freedom (the same approach is discussed in [14]). Kevorkian in [11] obtained, using the method of multiple scales, the matched asymptotic expansion that is uniformly valid both in the resonant case (“the inner expansion”) and in the non-resonant case (before and after the resonance, “the outer expansions”). In [16] Ablowitz, Funk & Newel demonstrated that the Kevorkian’s solution for times greater than the instant of resonance contained an error: the oscillation of order $1/\sqrt{\epsilon}$ emerged near the resonance should never vanish after the resonance. This fact is in contradiction with results of [11]. The error was also pointed out in [17], where the problem was solved by a modified WKB \colorblack (the Wentzel–Kramers–Brillouin) approach. The error was resolved in erratum [12]. Skinner [18] applied the stationary phase method to obtain a uniformly valid solution of the linear problem. Modification of the Kevorkian procedure for the case of a weakly nonlinear system with slowly varying properties are considered in many studies, see e.g. [16, 19, 20, 21, 22] and many other references.

\color

black Our work have at least two important motivations. First, we can discover trapped modes in many systems used as simple analytical models for engineering applications. For instance, an infinite continuous system with discrete inertial moving loads under certain conditions can possess trapped modes [3, 23, 24, 25, 26, 27, 28], though this fact is not always properly recognized in the corresponding studies. The trapped mode frequency is a resonance frequency for the infinite mechanical system. It is not very difficult to imagine a real engineering problem, where it is necessary to consider the passage through resonance for such a system. For example, such a problem naturally emerges if we want to consider a non-uniformly moving discrete oscillator subjected to a harmonic force (some kind of extension of the problem considered in recent paper [28]).

The second motivation is a more theoretical one. Let us imagine a mechanical system with several identical distant discrete inclusions and corresponding trapped modes with frequencies close to each other [29]. We can naturally transform such a system to a system with slowly time-varying parameters if we additionally assume that inclusions can slowly move along the continuous system. In the latter case, we obtain a linear system with internal resonances of a fundamentally new type (the resonances between different weakly coupled trapped modes), which has not been considered, as far as we know, in the literature before. The problem considered in this paper is our very initial attempt to describe such resonances analytically.111\colorblack Note that our initial interest was related with an unfinished (for the time being) attempt to take into account the existence of a trapped mode [26] in analytical treatment of the problem for a semi-infinite system with a single moving inclusion considered in [24, 25]. The problem for the latter semi-infinite system, in fact, can be equivalently transformed to a problem for an infinite system with two inclusions and two coupled trapped modes. Accordingly, we have chosen a model problem (a string equipped with an oscillator of time-varying stiffness) to prepare the mathematical tricks, which we plan to use in future for more complicated and better motivated (by engineering or physical practice) problems.

With regard to the following problems related to

free oscillation for a linear oscillator with time-varying stiffness; 2. 2.

resonant forced oscillation for a linear oscillator with time-varying stiffness; 3. 3.

free oscillation for a system, having time-varying parameters and possessing a single trapped mode; 4. 4.

resonant forced oscillation for the latter system;

we claim that they are quite different from the asymptotic point of view. For problems 1 & 2 this difference is well-known (see e.g. [11, 12, 14, 15]). In our previous studies [1, 3, 4, 5, 6, 7, 8, 9] we have demonstrated that problems of the third type are much more difficult and complicated than problem 1. And now, in the current manuscript, we deal with problem 4. To the best of our knowledge, this paper is the first study, where passage through a resonance for such a system is considered.

The paper is organized as follows. In Section 2 we consider the formulation of the problem. In Section 3 we present a summary of some known results related with free oscillation in the system under consideration. These results are necessary to consider the forced oscillation. In Section 4 the forced oscillation and passage through the resonance are considered. We obtain the principal singular term of the “inner expansion”. In the particular case of a discrete oscillator with sufficiently large stiffness and mass, the obtained formulas transform into formulas obtained in [11, 12]. The particular cases of an increasing and a decreasing stiffness are considered in Sections 4.1 and 4.2, respectively. In Section 5 we verify the obtained analytical results by independent numerical calculations based on solution of the corresponding \colorblack partial differential equation by means of the method of finite differences. The comparison of the analytical and numerical solutions demonstrates a good agreement in a neighbourhood of the instant of resonance. In Conclusion (Section 6) we discuss the basic results of the paper, future plans and possible applications.

2 Mathematical formulation

The governing equations for the system in a dimensionless form are

[TABLE]

Here, and in what follows, we denote by prime the derivative with respect to the spatial coordinate $\xi$ and by overdot the derivative with respect to the time $\tau$, $u(\xi,\tau)$ is the displacements, $p(\tau)$ is the given external force on the discrete oscillator, $P(\tau)$ is the unknown force on the string from the discrete oscillator, constant $M\geq 0$ is the mass in the discrete oscillator, $K(\epsilon\tau)$ is the spring stiffness for the discrete oscillator (a given slowly varying function of time), $\epsilon$ is a formal small parameter. We do not assume that $K>0$ (hence, the spring stiffness can be negative [6, 8, 7, 9] or zero). The initial conditions for Eq. (1) can be formulated in the following form [30]:

[TABLE]

In order to consider forced oscillation, we put

[TABLE]

where $\hat{\Omega}=\operatorname{const}$ is the resonant frequency, $H({\tau})$ is the Heaviside function.

The problem under consideration (1), (2), (3) is symmetric with respect to $\xi=0$. Integrating (1) over $\xi=0$ results in the following condition:222It is only the highest spatial derivative term in the left-hand side of Eq. (1) that can involve a delta-like term [30]

[TABLE]

Here, and in what follows, $[\mu]\equiv\mu(\xi+0)-\mu(\xi-0)$ for any arbitrary quantity $\mu$. Expression (5) is the equation for balance of momentum formulated for the mass-spring oscillator attached to the string. In fact, this is the Rankine-Hugoniot jump condition [31] formulated at the fixed position $\xi=0$. It also can be obtained from the first principles (see, e.g., [32, 33, 34], where the more complicated case of geometrically nonlinear model of the string with a point moving load is considered). Due to the symmetry, one has $[u^{\prime}]=2u^{\prime}(\xi+0)$. Thus, the problem for the infinite string can be equivalently reformulated as the problem for the homogeneous equation

[TABLE]

with the corresponding boundary condition at $\xi=0$

[TABLE]

The formulation (6), (7), (3) is equivalent to the initial one (1), (2), (3) and is used for numerical calculations (see Section 5).

Note that dropping out Eq. (1) and putting the left-hand side of Eq. (2) to zero yields the equation describing a forced oscillation and passage through resonance for a single degree of freedom system [11].

3 Free localized oscillation (some known results)

Put $p=0$, $K=\operatorname{const}$ and consider the steady-state problem concerning the natural oscillations of the system described by Eqs. (1)–(2). Take

[TABLE]

Let us show that such a system possesses a mixed spectrum of natural frequencies. There exists a continuous spectrum of frequencies, which lies higher than the cut-off (or boundary) frequency: $|\Omega|\geq\Omega_{\ast}\equiv 1$. The modes corresponding to the frequencies from the continuous spectrum are harmonic waves. Trapped modes correspond to the frequencies from the discrete part of the spectrum, which lies lower than the cut-off frequency:

[TABLE]

These are modes with finite energy, therefore, we require

[TABLE]

Now we substitute Eq. (8) into Eq. (1). This yields

[TABLE]

The solution of Eq. (11), which satisfies (10), is

[TABLE]

Calculating the right-hand side of Eq. (12) at $\xi=0$ yields the equation for the trapped mode frequency:

[TABLE]

Provided that restriction

[TABLE]

is true, there exists a unique trapped mode (see [1, 9, 7, 29]). The corresponding squared frequency $\Omega_{0}^{2}$ is

[TABLE]

where $M\neq 0$. In the special case $M=0$ (an inertialess discrete oscillator, i.e. a spring with a negative stiffness, see Eq. (14)), one gets

[TABLE]

Inside the interval (14), one has

[TABLE]

The boundary limiting values of $\Omega_{0}$ are

[TABLE]

The free localized oscillation in the case $K=K(\epsilon\tau)$ is considered in [1, 7]. It is shown that the amplitude of the free localized oscillation is proportional to the following quantity

[TABLE]

where $C_{0}$ is a constant.

Remark 1*.*

In the limiting case

[TABLE]

Eq. (19) transforms into classical formula

[TABLE]

for a single degree of freedom system [1], where $\Omega_{0}^{2}=K(\epsilon\tau)/M$, $C_{1}=C_{0}/M$. Thus, the greater the mass $M$, the closer the distributed system is to a discrete mass-spring oscillator. The opposite (most different) limiting case is $M=0$, $K<0$.

4 Passage through resonance

The aim of this paper is to use the method of multiple scales to obtain a singular principal term of the asymptotic expansion of $u(0,t)$ in a resonant case [11, 15]. In terminology of [11] we look for the principal term of the inner expansion. We take a finite time interval $[0,\tau_{\ast}]$, and the frequency of external excitation $\hat{\Omega}$ such that $0<\hat{\Omega}<1$.

4.1 The case of an increasing stiffness

We assume (see Eqs. (17), (18)) that $K(\epsilon\tau)$ is a smooth function of $\epsilon\tau$ uniformly bounded with all its derivatives such that

[TABLE]

where $\tau_{0}\in[0,\tau_{\ast}]$ is the instant of resonance. We introduce slowly varying variables

[TABLE]

We represent the spring stiffness in the form of the following expansion

[TABLE]

or, equivalently,

[TABLE]

Accordingly to Eqs. (15), (22), $\Omega_{0}(K)$ is a smooth function in interval $[K(0),K(\epsilon\tau_{\ast})]$, thus,

[TABLE]

or

[TABLE]

Under the accepted assumptions, it follows from Eq. (27) that

[TABLE]

where $\Omega_{01}>0$ due to Eqs. (17), (22).

We represent the solution in the form of the following ansatz:

[TABLE]

The variables $X$, $T$, ${\varphi}$ are assumed to be independent. We use the following representations for the differential operators:

[TABLE]

Following [1, 3, 5, 9], we require that wavenumber $\omega(X,T)$ and frequency $\Omega(X,T)$ satisfy the dispersion relation

[TABLE]

and the equation

[TABLE]

which follows from Eq. (30). We assume that

[TABLE]

Additionally, we require that

[TABLE]

The phase $\varphi(\xi,\tau)$ should be defined by the formula

[TABLE]

Applying differential operators (32) to $u(\xi,\tau)$, given by Eq. (29), one obtains:

[TABLE]

Here the wavenumber $\hat{\omega}=\pm\mathrm{i}\sqrt{1-{\hat{\Omega}}^{2}}$ corresponds to the frequency $\hat{\Omega}$ due to dispersion relation (33). Taking into account Eqs. (31), we get:

[TABLE]

Substituting the above expressions into Eq. (5) and equating coefficients of like powers $\epsilon$, one obtains that the term of order $\epsilon^{-1/2}$ equals zero identically due to frequency equation (13). The zeroth order term is

[TABLE]

Note that putting the left-hand side of the last equation to zero yields the equation describing passage through the resonance for a single degree of freedom system [11].

The unknown quantity $\left[\frac{\partial W_{0}}{\partial X}\right]$ in the left-hand side of Eq. (40) can be defined by consideration of Eq. (1) at $\xi=\pm 0$. To do this, we substitute ansatz (29)–(31) and representations (32) into Eq. (1) and equate coefficients of like powers $\epsilon$. Taking into account dispersion relation (33), one obtains

[TABLE]

Now we equate the right-hand sides of Eqs. (40), (41), and get the equation for $\mathscr{W}_{0}(T)\equiv W_{0}(0,T)$:

[TABLE]

Here

[TABLE]

Substituting expressions (25), (27) into frequency equation (13) and equating coefficients of like powers, one can demonstrate that

[TABLE]

Hence, Eq. (42) can be written as follows:

[TABLE]

The case of passage through the resonance for a single degree of freedom system [11] can be formally obtained by the choice

[TABLE]

Remark 2*.*

In the limiting case (20), one gets

[TABLE]

(see also Section 3 and [1]). Therefore, in the case (20), the principal term of the asymptotic solution for the problem under consideration transforms into the principal term of the solution for the corresponding problem for a linear oscillator. Thus, the greater the mass $M$, the closer the distributed system is to a discrete mass-spring oscillator. The opposite (most different) limiting case is $M=0$, $K<0$. The same conclusions are true for a free localized oscillation in the system under consideration (see Remark 21).

According to Eqs. (22), (44), one has $\Omega_{01}>0$. To solve Eq. (45), following to [11], and taking into account the last inequality, we introduce the new variable $\eta$

[TABLE]

We can now rewrite Eq. (45) as follows:

[TABLE]

Here and in what follows superscript “$-$” corresponds to the case $T<0$, and “$+$” corresponds to $T>0$. We search a solution of Eq. (49) in the following form:

[TABLE]

where $a^{\pm}$ are unknown constants,

[TABLE]

One has

[TABLE]

[TABLE]

where $\operatorname{erf}(\cdot)$ is the error function [35], $C(\cdot)$, $S(\cdot)$ are the normalized Fresnel integrals [35].

One can put [12]

[TABLE]

Since the solution must be continuous at the instant $T=0$, we require

[TABLE]

By virtue of Eq. (51), one gets:

[TABLE]

Hence,

[TABLE]

Thus, using Eq. (48) for $\eta$, we get:

[TABLE]

where $f(\hat{\Omega})$ is defined by Eq. (43). Note that taking here $f(\hat{\Omega})$ in the form of Eq. (46) yields the classical result for a single degree of freedom system [11, 12]. To obtain the physical displacements that correspond to the real part of the right-hand side of Eq. (4), one need to take the real part of Eq. (62).

4.2 The case of a decreasing stiffness

Consider now the case, when

[TABLE]

In this case, the quantity $K_{1}$ in Eq. (25) is such that

[TABLE]

Hence, due to Eq. (44), the quantity $\Omega_{01}$ in Eq. (27) is also negative:

[TABLE]

Taking into account the last inequality, and substituting Eq. (48) into Eq. (45), analogously to the case $K_{1}>0$ one gets

[TABLE]

Here $\overline{\mathscr{W}_{0}^{-}}$, $\overline{\mathscr{W}_{0}^{+}}$ denote complex conjugates of functions $\mathscr{W}_{0}^{-}$, $\mathscr{W}_{0}^{+}$ (given by Eqs. (60), and (61)), respectively.

5 Numerics

Unlike the case of a single degree of freedom system, the applicability of the suggested asymptotic approach to the non-stationary problem for the considered distributed system can be verified only numerically. One of the reasons for this is the presence of contributions of all frequencies in a non-stationary solution. Such a verification is the main aim of calculations presented in this section. For numerical study we use SciPy software. The discretization of PDE (6) is defined in accordance with the following scheme:

[TABLE]

where integers $i,\ j$ ($0\leq j\leq N,\ 1\leq i$) are such that

[TABLE]

This scheme conserves [36, 37] the discrete energy for a nonlinear Klein-Gordon equation with constant coefficients. Numeric boundary conditions that correspond to (7) are taken in the form [38]

[TABLE]

where $i>1$,

[TABLE]

At the right end of the string, we use the condition

[TABLE]

\color

black that corresponds to the physical boundary condition $u^{\prime}=0$. Actually, the specific form of this boundary condition is not very important in our calculations, since we consider the discrete model of the string with sufficiently large length such that the wave reflections at the right end do not occur. Numerical initial conditions are

[TABLE]

All numerical results in what follows are obtained for the choice

[TABLE]

In Figure 1 we compare the analytical and numerical results in the case of inertialess oscillator $M=0$ with an increasing stiffness (see Remark 47, which clarifies the motivation of such a choice). In Figure 2 we compare the results for the same system in the case of a decreasing stiffness. In Figure 3 we compare the results for the system with a massive oscillator, where $M=2$ in the case of an increasing stiffness.

One can see that the comparison demonstrates a good mutual agreement in a neighbourhood of the instant of resonance, and suggested asymptotic approach is really applicable to the problem under consideration. One can observe also that the influence of the term of order $O(1)$ is more noticeable in the case of a massive oscillator.

6 Conclusion

The most important result of the paper is the expressions (see Eqs. (60), (61), (62), and (66)) for the principal singular term of order $O(1/\sqrt{\epsilon})$ of the resonant solution describing a forced oscillation of an infinite-length mechanical system possessing a single trapped mode. The system is a string, lying on the Winkler foundation and equipped with a discrete linear mass-spring oscillator of time-varying stiffness, see Eqs. (1), (2). According to Eq. (23), the diameter of the neighbourhood, where the resonant solution is applicable, has an order $O(1/\sqrt{\epsilon})$. The results can be generalized for other mechanical systems possessing a single trapped mode.

\color

black We have to emphasize that the structure of the first approximation equation (40) for the problem under consideration is quite different comparing to the problem on passage through resonance for a linear oscillator. The latter case corresponds to Eq. (40), where the left-hand side equals zero. Assuming additionally that we deal with the special case $M=0$ (discussed in Remarks 21, 47), we get the first approximation equation333\colorblack In the form of Eq. (40), where two corresponding terms equal zero, which is, clearly, senseless. Thus, the presented in this paper analytical solution cannot be obtained straightforwardly using known solutions [11, 12] for a linear oscillator.

Nevertheless, the obtained final solution has a similar structure with the corresponding solution for a linear oscillator [11, 12]. The difference is in the structure of function $f(\hat{\Omega})$ (see Eqs. (43), (46)). Note that in the limiting case (20) the solution transforms into the corresponding solution for a linear oscillator, see Remark 47. The obtained analytical solution was verified by independent numerical calculations and a good agreement in a neighbourhood of the instant of resonance was demonstrated.

In order to obtain a uniformly valid asymptotic solution for both non-resonant and resonant cases, we definitely need to obtain the next term (of order $O(1)$) for the resonant solution. This, probably, will allow us to match the corresponding terms in the resonant and non-resonant solutions. Note that the problem to determine the next term of the “inner expansion” seems to be much more complicated than the one considered in this paper. On the other hand, the expression for the next term of the resonant solution will contain unknown constants, which should be determined as a result of matching [11, 12]. Thus, the expression for the next term cannot be verified numerically before matching is done.

\color

black Note that for the time being we do not know how to take into account properly the possible presence of a viscous friction (i.e., a dissipation) considering non-stationary oscillations in the system under consideration. The presence of a dissipation makes the problem more sophisticated. It may be a subject of a separate future study.

The results obtained in the paper can be useful, in particular, for investigation of the internal resonances in a linear infinite-length system, having time-varying parameters and possessing several trapped modes with corresponding frequencies close to each other (e.g., a string on the Winkler foundation with several distant moving discrete inclusions, see [24, 26]).

Acknowledgements

The authors are grateful to D.A. Indeitsev for useful and stimulating discussions.

Declaration of interest

None