## Abstract

The paper deals with analytically predicting the effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam performing bending oscillations. Two cases are considered: (i) large transverse deflections, where nonlinear (true) curvature, nonlinear material and nonlinear inertia owing to longitudinal motions of the beam are taken into account, and (ii) mid-plane stretching nonlinearity. A novel approach is employed, the method of varying amplitudes. As a result, the isolated as well as combined effects of the considered sources of nonlinearities are revealed. It is shown that nonlinear inertia has the most substantial impact on the dispersion relation of a non-uniform beam by removing all frequency band-gaps. Explanations of the revealed effects are suggested, and validated by experiments and numerical simulation.

## 1. Introduction

The analysis of the behaviour of linear periodic structures can be traced back over 300 years to Sir Isaac Newton [1], but until Rayleigh's work [2] the systems considered were lumped masses joined by massless springs. Much attention was given to this topic in the twentieth century (e.g. [1,3,4]). In recent years, the topic has experienced rising interest [5–9]. An essential feature of periodic structures is the presence of frequency band-gaps, i.e. frequency ranges in which waves cannot propagate. The determination of band-gaps and the corresponding attenuation levels is an important practical problem [3–9]. A large variety of analytical methods have been developed to solve this problem, most of them based on Floquet theory [1]; this holds, for example, for the classical Hill's method of infinite determinants [10,11] and the method of space harmonics [12]. However, application of these for nonlinear problems is impossible or cumbersome, because Floquet theory is applicable for linear systems only. Thus, the nonlinear effects arising in periodic structures have not yet been fully discovered, while, at the same time, applications may require the effects of nonlinearity on the structural response to be accounted for. Only a few papers have been devoted to this topic, most of them either considering lumped-parameter models, i.e. discrete periodic mass-spring chains [13–15], or implying a certain discretization of continuous periodic structures, e.g. by the Galerkin weighted residuals approach [16] or finite elements [17]. Khajehtourian & Hussein [18] consider longitudinal wave motion in a one-dimensional elastic metamaterial, which is a thin uniform rod with periodically attached local resonators, whereas Abedinnasab & Hussein [19] studied the dispersion relations for a uniform rod and Bernoulli–Euler beam under finite deformation, with the effects of nonlinearities accounted for. The latter problem represents a special case of the one covered in this paper, in the limit of zero modulation.

This article deals with analytically predicting the dynamic responses for a nonlinear continuous elastic periodic structure without employing system discretization. Specifically, the effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam performing bending oscillations are analysed. Various sources of nonlinearity are considered: nonlinear (true) curvature, nonlinear inertia owing to longitudinal beam motions, nonlinear material and the nonlinearity associated with mid-plane stretching. A novel analytical approach is employed, the *method of varying amplitudes* (MVA) [20,21]. This approach is inspired by the method of direct separation of motion (MDSM) [22,23], and may be considered a natural continuation of the classical methods of harmonic balance [11] and averaging [24–26]. It implies a solution in the form of a harmonic series with varying amplitudes; however, in contrast to the asymptotic methods, the amplitudes are not required to vary slowly. Thus, the approach does not imply separation of variables into slow and fast, which is the key assumption of the MDSM. It is also strongly related to Hill's method of infinite determinants [1,10,11], and to the method of space harmonics [12].

Possible sources of nonlinearities for a Bernoulli–Euler beam performing bending oscillations have been discussed in many works; see, for example, [27] and the classical monograph [11]. In [11], the main sources were identified as nonlinear stiffness and nonlinear inertia. It was noted that the character of the nonlinearity strongly depends on the specific boundary conditions applied to the beam. For example, when there is no restriction on the longitudinal motion of the beam ends, large deflections are possible, so that nonlinear (true) curvature and nonlinear inertia owing to longitudinal motion of the beam should be taken into account. The effects of nonlinear material may also be of significance in this case. If both ends of the beam are restricted from moving in the longitudinal direction, then another source of nonlinearity becomes important, namely mid-plane stretching. This nonlinearity appears to be much stronger than the others [28,29] and influences the beam response at relatively small deflections.

Because, in real structures, boundary conditions affect the character of the nonlinearity, *finite* structures are to be considered. On the other hand, the analysis of dispersion relations and frequency band-gaps, in its conventional formulation [1,12], implies considering *infinite* structures. The transition from infinite to finite structures and the discussion of the importance of dispersion relations and band-gaps for finite structures are given in many studies [1,5,7,12,30]. Dispersion relations are of utmost importance for finite structures, e.g. because they facilitate determining natural frequencies and mode shapes. Frequency band-gaps, in turn, are key features of sufficiently long structures for which waves from band-gap ranges attenuate strongly before reaching the boundaries.

Section 2 is concerned with the formulation of the governing equations of transverse motions of the beam and their brief analysis. In §3, the equations are solved by the MVA; §4 presents the obtained dispersion relations and reveals the effects of nonlinearities on the frequency band-gaps. Section 5 is concerned with the discussion as well as with the experimental and numerical validation of the results.

## 2. Governing equations

### (a) Case A: beam unrestricted longitudinally; relatively large deflections possible

Consider the case with no restriction on the longitudinal motions of the beam, when relatively large transverse deflections are possible. This case corresponds to, for example, the clamp-free beam schematically shown in figure 1*a*, with the kinematics of the beam element also presented. The internal bending moment of a Bernoulli–Euler beam with spatially varying properties is defined by

where *I* is the moment of inertia of the cross section, *E* is Young's modulus of the beam material, *Θ* is the period of modulation, and
*χ*_{A}<1, 0≤*χ*_{I}<1, *k*=2*π*/*Θ*, and in the simplest case of constant *ρ*, *E* and *r*, we have *χ*_{A}=*χ*_{I}. Here, a more general case is considered, where the modulation amplitudes *χ*_{A} and *χ*_{I} are not required to be equal, though modulations of the beam mass per unit length and stiffness have the same phase shift *ϕ*. According to [1], the approximation similar to (2.11) is valid for predicting at least the lowest two band-gaps of a periodic structure.

The governing equation of transverse motions of the beam is [11,28]

Introducing non-dimensional variables *π*/*k* propagating in the corresponding uniform beam, (2.13) can be rewritten in dimensionless form as
*x*.

Because *k*, which is the spatial frequency of modulations, so that for large *k*, i.e. for rapidly varying cross section, the effect can be significant even for relatively small physical deflections of the beam.

Solutions of (2.14) are sought in the form of a series, with over-bars denoting complex conjugation
*Θ* of modulation are considered, so high-frequency oscillations are outside the scope of this article. In addition, dimensionless deflections *w* are assumed to be finite, but not very large, so that nonlinearities can be considered weak, permitting only the fundamental harmonic in (2.16) to be included. This assumption, in particular, implies that for rapidly varying cross section, i.e. large *k*, only very small physical deflections

Substituting (2.16) and (2.15) into (2.14) and balancing the terms at the fundamental harmonic *ω*, one obtains the governing ordinary differential equation for *φ*(*x*),
*Θ* of modulation are considered, and owing to the choice of the non-dimensional variables, we have *ω*=*O*(1) (which comprises also the case *ω*≪1).

The integral term in (2.17) represents nonlinear inertia, whereas the term with *β*_{n} is related to nonlinear material, and the remaining nonlinear terms are due to the true measure of curvature (2.2). Note that, although *k*.

### (b) Case B: mid-plane stretching

Now consider the case when both ends of the beam are restricted from moving in the longitudinal direction, and mid-plane stretching occurs. This case corresponds to, for example, the clamped–clamped beam schematically shown in figure 1*b*. To transversely deform such a beam considerably more energy needs to be supplied, because bending is coupled with axial stretching of the beam. Consequently, one can expect transverse deformations to be much smaller than in case A (§2a), so that the linear measure of curvature and material stress–strain relation can be adopted, and nonlinear inertia can be neglected [28]. In the absence of external axial forces, the assumption regarding longitudinal inertia implies that
*η* describes a small initial stretch of the beam, and

Inserting (2.23) into (2.12), adopting the linearized curvature and neglecting the possible effects of nonlinear material, one obtains
*μ*=*A*_{0}/(*I*_{0}*k*^{2})=(*kr*_{0})^{−2}, where *r*_{0} is the radius of gyration of the corresponding uniform beam, and here again primes denote derivatives with respect to *x*.

Employing the Bernoulli–Euler theory and considering waves of length much larger than the height of the beam implies that
*μ*≫1, and the nonlinear term in (2.25) is much larger than the nonlinear terms in (2.14), illustrating why mid-plane stretching nonlinearity is much stronger than all other nonlinearities considered in §2a.

Searching for a solution to (2.25) in the form (2.16) with only the first harmonic taken into account, which is valid for weak nonlinearity, we obtain equation (2.28) for the new variable *φ*(*x*)
*l* and phase *ϕ* are here present in the governing equation for *φ*(*x*), so that the effect of nonlinearity may depend on these parameters. However, it is expected that for relatively large *l*, allowing attenuation of waves from band-gap ranges before reaching the boundaries, this dependency should vanish.

## 3. Solution by the method of varying amplitudes

### (a) Case A: beam unrestricted longitudinally; relatively large deflections possible

Conventional methods for analysing spatially periodic structures, e.g. the classical Hill method of infinite determinants [10,11] and the method of space harmonics [12], are not applicable for the problem considered here because they are based on Floquet theory, which is valid for linear systems only. In addition, the governing equations (2.17) and (2.28) are nonlinear integrodifferential equations involving strong parametric excitation, and solving such equations by standard asymptotic methods, e.g. the multiple scales perturbation method, is impossible or very cumbersome [26]. Consequently, a novel approach, the MVA [20,21], is employed. Following this approach, a solution to (2.17) is sought in the form of a series of spatial harmonics with varying amplitudes
*b*_{j}(*x*) are not required to vary slowly in comparison with *b*_{j}(*x*). The solution ansatz implied in the MVA, i.e. the choice of harmonics in (3.1), depends on the parameters of modulation in the equation considered. For the present problem, the modulation is

The shift from the original dependent variable *φ*(*x*) to the new variables *b*_{j}(*x*) implies that additional constraints on these variables should be imposed. With the MVA, the constraints are introduced in the following way: substitute (3.1) into the governing equation (2.17) and require the coefficients of the spatial harmonics involved to vanish identically. As a result, one obtains the following infinite set of differential equations for the amplitudes *b*_{j}(*x*)
*N*_{j}(**b**) are nonlinear functions which are rather lengthy, and thus not given here. When composing equations (3.2), the relation
*G* and *g* are related by

The approximation of the method is concerned with truncation of the solution series (3.1) and neglecting the corresponding higher-order harmonic terms, similar to the method of harmonic balance [11,33]. This approximation is valid if the neglected terms in (3.2) are small in comparison with those kept, leading to the requirement that the number of terms in the solution series employed is high enough. For the present problem, truncation of the *m*th harmonic in (3.1) is adequate if
*ω*=*O*(1) and 0≤*χ*_{A}<1, an additional restriction *χ*_{I}≪1 should be imposed to satisfy (3.5) (in fact, it is sufficient to require *χ*_{I}≤0.5; see §5). So only comparatively small modulations of the beam stiffness can be considered by the means of the method; modulations of the beam mass per unit length, however, can be large.

Equations (3.2) are nonlinear differential equations in **b**(*x*). They allow a multitude of solutions, in particular those that can be written in the form
*N*_{j}(**b**)=0, equations (3.2) have solutions *only* of the form (3.6), with −*iκ* being a root of the characteristic equation of the system (3.2), and **b**_{c} the associated vector. Taking into account (2.16), (3.1) and (3.6), the solution of the linear counterpart of the initial dimensionless equation (2.14) may be written as
*cc* denotes complex conjugate terms. This solution obeys Floquet theory [1], because *F*(*x*) has the same period as the cross-section modulation, and is similar to the one implied in the method of space harmonics [12]. It describes a ‘compound wave’ [1] or a ‘wave package’ [12] propagating (or attenuating) in the beam with dimensionless frequency *ω* and wavenumber *κ*, with the relation between *ω* and *κ* defining the dispersion relation of the considered periodic structure, and with real values of *κ* corresponding to propagating waves and complex values to attenuating waves [1,12].

From (3.7), for the propagation constant *p* [1,12], describing how a travelling wave changes when passing through a single periodic cell, one obtains
*κ*_{B} is the Bloch parameter [1,12], which by (3.8) becomes
*κ* correspond to propagating waves, and complex values to attenuating waves.

Our aim is to examine the effect of nonlinearities on the beam dispersion relation and frequency band-gaps. This implies that we are interested in solutions to (3.2) *only* of the form (3.6), so that the corresponding solution of the initial dimensionless equation (2.14) takes the form (3.7) describing the propagating (or attenuating) wave with dimensionless frequency *ω* and wavenumber *κ*. Introducing
*θ*_{j} are real-valued constants, and substituting into (3.2) and their complex conjugates, gives
*θ* can take arbitrary values without affecting the resulting algebraic equations for *κ*, *ω*, *χ*_{I}, *χ*_{A} and *β*_{n}. The effect of nonlinearities on the dispersion relation and frequency band-gaps depends on the magnitude of transverse deflections *w* as given by expression (3.7). Taking into account (3.10) and (3.11), we obtain that the spatially averaged amplitude of the beam transverse deflections *w* is given by
*κ* as functions of the amplitude *B* and parameters *ω*, *χ*_{I}, *χ*_{A} and *β*_{n}. Thus, the dispersion relation *κ*=*κ*(*ω*) of the considered nonlinear periodic beam is obtained for various values of the parameters, as will be illustrated in §4, and discussed and validated in §5.

### (b) Case B: mid-plane stretching

Employing again the MVA and searching for a solution of (2.28) in the form (2.29), one obtains the following infinite set of differential equations for the new variables *b*_{j}(*x*):
*χ*_{I}≪1, is imposed for truncation of the solution series (3.1) to be adequate. Substituting the solution form (3.6) into terms multiplied by *S*_{3} in (3.14), one obtains expressions of the form *κ* and frequency *ω* as the primary one, but propagating in the *opposite* direction. Thus, the requirement for this additional wave to be negligibly weak and not affect the primary wave should be imposed for the analysis of the beam dispersion relation to be valid, leading to the condition
*S*_{3} in (3.14) are much smaller than the leading terms, so that the solution form (3.6) can be employed.

Considering *S*_{1} and *S*_{2}, it is found that for a relatively long beam *l*≫1, and with the solution form (3.6) and the real-valued constants *θ*_{j} introduced according to (3.10), these can be approximated as
*θ*_{j} satisfy relations (3.11). As can be seen (3.19) do not involve the length of the beam *l* and phase *ϕ* so that, as predicted (see §2b), the dispersion relation of the considered nonlinear beam does not depend on these parameters. The effect of the nonlinearity on this relation is governed by the term *S*_{2}, present in (3.19). Comparing expressions (3.17), (3.18) for *S*_{1} and *S*_{2}, it appears they differ only at the position of the initial pre-stretching coefficient *η*; hence, the nonlinearity is equivalent to an additional stretching of the beam
*w*; for propagating waves (real values of *κ*) *η*_{n}>0.

Thus, the approximate solution of the initial dimensionless equation (2.28) is obtained in the form (3.7), describing a propagating (or attenuating) wave with dimensionless frequency *ω* and wavenumber *κ*. As for case A, the amplitude *B* is introduced by (3.13) to define the magnitudes of *κ* as functions of the amplitude *B* and parameters *ω*, *χ*_{I}, *χ*_{A}, *μ* and *η*.

## 4. Dispersion relations and frequency band-gaps

### (a) Effects of nonlinear (true) curvature and nonlinear material

In [21], it was shown that the linear dispersion relation of the considered periodic beam is symmetric about axis *ω* and periodic with respect to the wavenumber *κ*, which agrees well with the results obtained in the classical works [1,12]. For pure modulation *χ*_{A} of the beam mass per unit length, the linear dispersion relation features two distinct band-gaps in the considered frequency range (at *ω*≈0.25 and *ω*≈1), whereas pure modulation *χ*_{I} of the beam stiffness gives one band-gap (at *ω*≈0.25), and modulations with equal amplitudes *χ*_{A}=*χ*_{I} also one band-gap (at *ω*≈1).

To simplify the analysis of the effects of nonlinearities on the dispersion relation, we first consider each source of nonlinearity separately: in this section, nonlinear (true) curvature and nonlinear material. The effects of nonlinear inertia are not taken into account, but will be studied in §4c.

As follows from the obtained solution (3.7), the structure of the beam dispersion relation does not change owing to the nonlinearities, i.e. the relation remains symmetric and periodic with respect to *κ*. The pure effect of nonlinear (true) curvature lies in shifting the dispersion relation to higher frequencies, as is illustrated in figure 2*a* for *B*=0.45, *χ*_{A}=0, *χ*_{I}=0.5 and *β*_{n}=0, with the linear dispersion relation shown for comparison (dotted line). The pure effect of nonlinear material is opposite to the effect of nonlinear curvature, i.e. the nonlinear dispersion relation is shifted to lower frequencies (figure 2*b*). Consequently, these sources of nonlinearity can compensate for the effect of each other, as is illustrated in figure 2*c*, where solid and dashed lines almost coincide. The effects of nonlinearities appear to be more pronounced for higher frequencies and the corresponding band-gaps.

According to the phase closure principle [30], the frequencies corresponding to the boundaries of band-gap regions for linear periodic structures are those where an integer number *n* of compound half-waves fits exactly into a unit cell of the structure, i.e. they correspond to the wavenumbers
*ω*_{c}, determining boundaries of the band-gaps, can be obtained by letting *κ*=*n*/2, *n*=±1, ±2, ±3,…, in (3.12). The effect of nonlinearities appears to be the same for different values of the modulation amplitudes *χ*_{A} and *χ*_{I}, and is more pronounced for higher band-gaps. The widths of the band-gaps appear to be relatively insensitive to (weak) nonlinearities. As an illustration, figure 3*a*,*b* shows the dependencies of *ω*_{c} corresponding to the first (*n*=1) and the second (*n*=2) band-gap on the amplitude *B* for *β*_{n}=0 and *χ*_{A}=0.5, *χ*_{I}=0. Figure 3*c*,*d* represents these dependencies for the second band-gap with only the effect of nonlinear material taken into account for different modulation amplitudes *χ*_{A} and *χ*_{I}. Figure 3*e* corresponds to the case of combined nonlinearities; as is seen the nonlinearities compensate for the effect of each other. It is interesting to determine the critical value of the parameter *β*_{n} at which there will be *complete* compensation of the nonlinearities, i.e. the band-gap range will not change with increasing amplitude *B*. Taking into account that the effect of nonlinearities depends weakly on the modulation amplitudes *χ*_{A} and *χ*_{I}, we get
*n* is the number of the band-gap. As can be seen, it is possible to achieve complete compensation for the nonlinearities only for one of the frequency band-gaps, e.g. the second one, *n*=2, in figure 3*e* for

### (b) Effects of initial pre-stretching and mid-plane stretching nonlinearity

Next, consider the case when both ends of the beam are restricted from moving longitudinally (case B). First, we analyse the *linear* dispersion relation and the effects of initial pre-stretching *η* of the beam. It appears that positive pre-stretching shifts the band-gaps to higher frequencies, and negative pre-stretching to lower frequencies, the effect being most pronounced for low frequencies. This is illustrated in figure 4*a*,*b*, with the dispersion relation without pre-stretching being shown for comparison by dotted lines. In particular, it is possible to shift one of the boundaries of the lowest band-gap to zero frequency, with the width of the band-gap being considerably increased (figure 4*c*). In this case, the beam serves as a high-pass filter, where all waves with frequencies lower than a certain critical one are attenuated. Also, for *χ*_{A}=*χ*_{A}, when the unstretched beam does not feature a frequency band-gap at *ω*≈0.25, the pre-stretched beam does (figure 4*d*).

The width of the second band-gap is relatively weakly affected by pre-stretching, though it can be effectively shifted to a higher- or lower-frequency range. The width of the first band-gap, by contrast, is strongly affected by pre-stretching, as is illustrated in figure 5, where the dependencies of the critical frequencies *ω*_{c}, determining the boundaries of the first band-gap, on the value of the initial pre-stretching *η* are shown. For pure modulation of the beam mass per unit length, positive pre-stretching increases the band-gap, whereas negative pre-stretching decreases it, as seen in figure 5*a*, and at a certain value of *η* the width of the band-gap essentially vanishes. In the case of pure modulation of the beam stiffness (figure 5*b*), the effect of pre-stretching is the opposite, and it is even possible to obtain a large band-gap with zero frequency as the lower boundary. If modulations with equal amplitudes are imposed (figure 5*c*), then negative as well as positive pre-stretching increases the band-gap.

Considering the isolated effect of mid-plane stretching nonlinearity, it is found that it is similar to that of nonlinear curvature: the band-gaps are shifted to higher frequencies, whereas the width of the band-gaps is changed only slightly, as illustrated by figure 6. However, this source of nonlinearity is much stronger than the nonlinear curvature, being pronounced already at very small values, *B*∼10^{−2}, of transverse beam deflections.

### (c) Effects of nonlinear inertia

Now consider the isolated effect of nonlinear inertia, governed by the term (*Nw*′)′ in (2.14), on the dispersion relation. This nonlinearity is involved in (2.14) along with nonlinear curvature and nonlinear material; however, we discuss it separately, because the effects it causes differ considerably from those already described in §4a.

Substituting the obtained solution *w*_{l} for the *linear* beam problem into (*Nw*′)′, one finds that for integer values of 2*κ* and *χ*_{A}≠0 or *χ*_{I}≠0 the term tends to infinity for arbitrarily small values of the amplitude *B*. Thus, the dispersion relation for the non-uniform beam should change considerably for wavenumbers close to *κ*=0,

Dispersion relations, relating a certain frequency to a certain wavenumber, as well as frequency band-gaps are relevant for linear or weakly nonlinear wave motion only. For strongly nonlinear waves, comprising many components with different frequencies, these notions are of little use [1,12]. For example, it is impossible or rather cumbersome to achieve attenuation of all components of such a wave by periodicity effects. In addition, when solving the initial equations by the MVA, the involved nonlinearities were assumed to be weak, so that only weakly nonlinear waves are captured. A 1 : 3 ratio between the maximum absolute values of the nonlinear term (*Nw*′)′ in (2.14) and the linear term

The dispersion relation of the beam does not feature frequency band-gaps owing to the nonlinear inertia. Instead of the band-gaps, relatively narrow frequency ranges arise in which the wave motion is strongly nonlinear. As an illustration, figure 7 shows the dispersion relation of the considered beam with the isolated effect of nonlinear inertia taken into account, with the linear dispersion relation shown for comparison by the dotted lines. Regions in which the wave motion is strongly nonlinear are bounded by dashed lines; in these regions, nonlinear dispersion relations are not shown, because the method employed, the MVA, captures only weakly nonlinear waves. Figure 7*a* corresponds to the case of pure modulation of the beam mass per unit length, figure 7*b* to pure modulation of the beam stiffness, and figure 7*c* to modulations with equal non-zero amplitudes. As can be seen, frequency ranges with strongly nonlinear wave motion arise instead of the band-gaps and correspond to wavenumbers slightly shifted from *κ*=0, *B*: the larger the amplitude, the larger the shift; compare, for example, figure 7*e* for *B*=0.4 with figure 7*f* for *B*=0.1. As appears from figure 7*g*,*h*, the effect of nonlinear inertia is strong even for very small beam deflections, *B*∼10^{−2}, so that the dispersion relation does not feature frequency band-gaps.

A frequency range implying a strongly nonlinear wave motion arises near *κ*=±1/2 also in the case of modulations with equal amplitudes, *χ*_{A}=*χ*_{I}, when there is no band-gap in the linear dispersion relation at *κ*=±1/2 (figure 7*h*). The effects described above are present for the non-uniform beam only; for the uniform beam, the influence of nonlinear inertia on the dispersion relation is much weaker (figure 7*d*). For example, for the wave motion to be strongly nonlinear, the beam deflections should be much larger than those considered previously, e.g. *B*∼1.

The obtained results clearly indicate that nonlinear inertia has a substantial impact on the non-uniform beam dispersion relation. It appears to remove all the band-gaps, with the frequency ranges implying a strongly nonlinear wave motion arising instead. The effects of nonlinear inertia seem to prevail over other nonlinearities considered in §4a, as illustrated by figure 8, which shows the dispersion relation with all three sources of nonlinearity taken into account. The effects of nonlinear inertia can be pronounced also in case B when both ends of the beam are restricted from moving in the longitudinal direction, because these effects are strong even for very small beam deflections resulting in vanishing of all the band-gaps.

From the results obtained, it follows that real periodic beam structures with *continuous* modulations of parameters performing bending oscillations should not feature frequency band-gaps. In the case of *piecewise constant* modulations, however, the effects of nonlinear inertia can be much weaker, as is suggested by the results obtained for the uniform beam. So such beams *can* feature frequency band-gaps, which has also been shown by laboratory experiments [12].

## 5. Validation of the results

### (a) Numerical validation

As appears from §§2 and 3, the obtained analytical solution involves two approximations. The first one is concerned with the truncation of the series (2.16), and the second is implied in the MVA. Both of them imply certain nonlinear high-order harmonic terms in the equations considered to be discarded. This simplification is valid under the condition that the involved nonlinearities are weak, which is the key assumption of this analysis, and has been carefully checked for all results presented.

For the isolated effects of the nonlinear curvature and nonlinear material, this condition appears to be satisfied if
*κ*=*O*(1) and *β*_{n}=*O*(1). Relation (5.1) was considered to be fulfilled for *B*^{2}≤1/3. Mid-plane stretching nonlinearity appears to be weak if the following condition is satisfied:

By contrast to the other sources of nonlinearity, nonlinear inertia implies strongly nonlinear wave motion in certain, relatively narrow, frequency ranges for any, even very small, *B*∼10^{−3}, beam deflections (see §4c). A 1 : 3 ratio between the maximum absolute values of the nonlinear term (*Nw*′)′ in (2.14) and the linear term

The MVA implies also that certain *linear* terms are discarded; as is shown, this is valid under condition (3.5), which leads to the requirement for modulations of the beam stiffness to be small, *χ*_{I}≪1. To further validate the results, a series of numerical experiments were conducted. The initial non-dimensional governing equation (2.14) (or (2.25)) was numerically integrated directly using Wolfram Mathematica v. 7.0 (NDSolve), with periodic boundary conditions and the following initial conditions imposed:
*θ*=0 and the first five terms taken into account in series (3.1). Consequently, in accordance with the theoretical predictions, at such initial conditions, the beam should oscillate with frequency *ω*. This allows the obtained dispersion relations between frequency *ω* and wavenumber *κ*, as well as the solution (3.7) itself, to be validated.

Typical results of the numerical experiments are shown in figure 9, where solid lines are values of the frequency *ω* obtained analytically (see §4), and filled circles represent numerical data for various values of the modulation amplitudes *χ*_{I}, *χ*_{A} and other parameters.

Figure 9*a* illustrates the pure effect of nonlinear curvature for *B*=0.5, *χ*_{A}=0.5, *χ*_{I}=0, and figure 9*b* the pure effect of nonlinear material for *β*_{n}=0.25, *B*=0.5, *χ*_{A}=*χ*_{I}=0.5. The discrepancy between the numerical and analytical values of the frequency *ω* is less than 0.3% for all values of *κ*, although in case (b) for *κ*=1 it rises to 1%. Additional (high-)frequency components are present in the beam response; these are due to the nonlinearity, and at the parameter values considered do not exceed 3% of the total response amplitude; the larger the frequency *ω*, the more pronounced these components become.

Figure 9*c* illustrates the effect of initial pre-stretching of the *linear* beam for *μ*=100, *η*=−0.002, *χ*_{A}=0, *χ*_{I}=0.5. Here, the discrepancy between the numerical and analytical values of the frequency *ω* is even smaller, around 0.2%, and no additional frequency components are present in the beam response. Figure 9*d* represents mid-plane stretching nonlinearity for *μ*=100, *B*=0.06, *η*=0.001, *χ*_{A}=*χ*_{I}=0.5. It should be noted that Wolfram Mathematica, as well as other similar software packages, is not able to handle nonlinear partial integrodifferential equations. Thus, when solving numerically the considered equation, the integral term was calculated using the obtained analytical solution. The resulting discrepancy between the numerical and analytical values of the frequency *ω* is less than 0.5%, and the additional frequency components in the beam response do not exceed 5% of the total response amplitude. To validate the simplification that we employed, the integral term was calculated using the obtained numerical solution and compared with the analytical one; the resulting discrepancy between them was less than 0.6%.

Figure 9*e*,*g* illustrates the isolated effect of nonlinear inertia for *B*=0.4, *χ*_{A}=0.5, *χ*_{I}=0. Here the discrepancy between numerical and analytical values of the frequency *ω* is again very small, around 0.2%. However, the additional (high-)frequency components in the beam response are about 10% for *κ* near 0.5 and 1, and the closer the wavenumber *κ* to the regions in which the wave motion is strongly nonlinear, the more pronounced these components become. Similar results were obtained for the case when all three sources of the nonlinearity were taken into account: figure 9*f*,*h* for *B*=0.4 and (*f*) *β*_{n}=0.2, *χ*_{A}=0, *χ*_{I}=0.5; (*h*) *β*_{n}=0.3, *χ*_{A}=*χ*_{I}=0.5.

In summary, good agreement between the numerical and analytical results for all the considered cases can be noted, and, thus, the effects revealed in §4 are validated numerically.

### (b) Experimental validation

To further validate the results obtained, laboratory experiments [34] were conducted. Two 1 m long steel beams with rectangular cross section of a constant height *h* and varying width *b* were impacted by an instrumented impact hammer, and their frequency response functions (FRFs) thus determined and band-gaps identified. The height of the first beam was *h*=5 mm and that of the second beam was *h*=15 mm. The width of both beams varied piecewise linearly (so that the considered modulation was *continuous*), as is shown in figure 10*a*, with

A sketch of the experimental set-up is shown in figure 10*b*. The beam hangs in soft rubber bands which simulate free–free boundary conditions. An accelerometer at beam point 6 monitors acceleration in the *y*-direction (figure 10*c*). FRFs were obtained by exciting the structure at points 1–5 by the impact hammer and measuring the output at point 6. Each of the measurements was repeated three times, checking for acceptable measures of coherence (close to unity in the frequency range of relevance except at (anti-)resonance).

The frequency spectrum of the measured force at perfect impact is constant, providing excitation energy at all frequencies. A real hammer hit, however, features a finite frequency range of excitation, which depends on the stiffness of the tip of the hammer and the excited structure; in the experiments, the excited frequency range was tailored to be below about 5 kHz.

The results of the experiments are shown in figure 11; (*a*) corresponds to the beam with *h*=5 mm and (*b*) to the beam with *h*=15 mm. No frequency band-gaps can be detected from the FRF diagrams; this supports the conclusions of §4, where it was shown that real periodic beam structures with *continuous* modulations of parameters performing bending oscillations should not feature frequency band-gaps owing to nonlinear inertia. The linear theory predicts band-gaps in the considered frequency range, e.g. for the first beam the first band-gap should be near 1.1 kHz, and for the second beam this band-gap is from 3.3 to 3.4 kHz, indicating that this theory does not suffice to correctly describe the beams' response and nonlinearities should be accounted for, as is done in the theoretical part of this paper.

The results presented in this section may be considered as a first step in the experimental validation of the theoretical predictions of the paper. Further in-depth experimental testing is relevant, e.g. considering periodic beams with wider frequency band-gaps predicted by the linear theory.

## 6. Conclusion

The effects of weak nonlinearity on the dispersion relation and frequency band-gaps of a periodic Bernoulli–Euler beam performing bending oscillations are analysed. Two cases are considered: (i) relatively large transverse deflections, where nonlinear (true) curvature, nonlinear material and nonlinear inertia owing to longitudinal motions of the beam are taken into account, and (ii) mid-plane stretching nonlinearity. As a result, several notable effects are revealed by means of the method of varying amplitudes; in particular, a shift of the band-gaps to a higher frequency owing to nonlinear curvature, whereas the effect of nonlinear material is the opposite. The width of the band-gaps appears to be relatively insensitive to these nonlinearities. It is shown that initial pre-stretching of the beam considerably affects the dispersion relation: it is possible for new band-gaps to emerge and the band-gaps can be shifted to a higher or lower frequency, their width being considerably changed. The isolated effects of mid-plane stretching nonlinearity are similar to those of nonlinear curvature, though mid-plane stretching nonlinearity is pronounced already at much smaller beam deflections.

It has been shown that, of the four sources of nonlinearity considered, nonlinear inertia has the most substantial impact on the dispersion relation of a non-uniform beam with continuous modulations of cross-section parameters. It appears to remove all the band-gaps, with the frequency ranges implying a strongly nonlinear wave motion arising instead. The results obtained are validated by experiments and numerical simulation, and explanations of the revealed effects are suggested.

## Data accessibility

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## Authors' contributions

V.S.S. derived the governing equations and performed their solution by the MVA, found the nonlinear dispersion relations and conducted numerical validation of the obtained results. J.J.T. supervised the research and established the logical organization of the paper.

## Competing interests

We declare we have no competing interests.

## Funding

The work was carried out with financial support from the Danish Council for Independent Research and FP7 Marie Curie Actions—COFUND: DFF—1337-00026.

## Acknowledgements

The authors are grateful to Prof. J. S. Jensen and Corresponding Member of the Russian Academy of Science D. A. Indeitsev for valuable comments on the paper, and to Solveig Dadadottir for her efforts in conducting the presented laboratory experiments.

- Received October 29, 2015.
- Accepted January 13, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.