On the stochastic parametric resonance of a rate-type viscoelastic string

Abstract

We analyze small amplitude shearing motions superimposed onto a harmonic extension of a string made of an isotropic rate-type viscoelastic material. In particular, we consider also the case in which the harmonic extension is perturbed by adding stochastic noise. We study the onset of resonance as a function of the characteristic parameters both in the absence and in the presence of noise. In the first case, we use the renormalization group (RG) method, while in the second case, we make use of a “ deterministic” approach that implies replacing the noise by high-frequency excitations. The main conclusion of our investigation is that the substitution of a classical elastic rope with a viscoelastic stress–relaxing string allows us to reproduce the parametric resonance phenomena observed by Melde. The presence of noise can change the well-known 2:1 resonance. In particular, the 2:1 resonance is reduced proportionally to the square of the amplitude ratio, frequency of the noise, and to the ratio between the characteristic viscous stress and the elastic stress.

Keywords

Melde’s experiment linear differential equation with periodic coefficients stochastic parametric resonance renormalization group effective properties

1. Introduction

Many physical systems exhibit parametric resonance [1, 2], and Melde’s experiment is a very well-known example. This experiment essentially consists of the excitation of an elastic string: a periodic change in the tension of a tight rope excites transverse waves in the string when the frequency of the oscillating tension is close to twice the natural frequency of any transverse mode. This phenomenon (usually designated as 2:1 resonance) was first reported by Melde [3]. Subsequently, Lord Rayleigh [4] investigated the phenomenon theoretically and Raman [5] provided a further experimental and theoretical analysis in 1912. Both these authors considered only the linear wave equation including damping, although they did not realize that the mathematical model was essentially Mathieu’s equation. Kidachi and Onogi [6] and Matsuda [7] investigated the nonlinear elastic case (although neglecting damping). Pucci and Saccomandi [8] showed that the parametric resonance occurs also in strings modeled as neo-Hookean materials and, in general, as hyperelastic materials. The case of nonlinear viscoelastic materials was analyzed in Pucci and Saccomandi [9].

In this paper, the vibrations of a pre-stretched string are considered in the framework of rate-type viscoelastic materials. The constitutive model we consider, in a linearized infinitesimal theory, is essentially the standard linear solid. This is a quite versatile model because it describes both creep and stress relaxation in a simple setting (see [10, 11] and the book by Rajagopal and Wineman [12]. A possible extension of this model to a general three-dimensional finite deformation context has been first put forward in Zhou [13]. However, in the general three-dimensional context, one major problem is the selection of the objective derivative required to describe stress relaxation, as illustrated, for example, in Farina et al. [14, 15].

The objectives and novelty of this study are threefold. First, to investigate the transverse waves superimposed onto an unsteady simple extension of a deformable string modeled as a rate-type viscoelastic material. We indeed point out that, although polymeric melts (i.e., viscoelastic fluids) have been deeply studied in the last several years (see, for example, [16]), relatively little attention has been devoted to solid materials. We recall the classical papers [17, 18] and also the book [19], but since then, the literature does not seem to have advanced significantly and more recent papers (e.g., [20, 21]) focus more on empirical approaches.

The second target is to investigate the stability properties of the transverse waves, which is exactly the setting of Melde’s experiment. In particular, we are interested in highlighting possible parametric resonance phenomena such as the ones occurring in Melde’s classical experiment when the rope is an elastomeric material. Indeed, when these waves are standing and the longitudinal extension is a harmonic perturbation superimposed onto a variable stretch, the mathematical model reduces to an ordinary differential equation (ODE) with periodic coefficients very similar to Mathieu’s equation.

The third target is to investigate possible stochastic parametric resonance, namely, to what extent the presence of random noise, (unavoidable) in the string pre-tension, modifies the classic 2:1 resonance. Indeed, being in a resonance framework, stochastic resonance phenomena may, in principle, occur. In this connection, there is a long history of the relevance of random noise in physical phenomena devoted to elucidate the relevance of stochastic resonance (see, for example, [22 –24]). Following the recent papers by Sorokin and Demidov [25] and Sorokin and Blekhman [26], we consider the effect of adding a stochastic noise, modeled as a high-frequency harmonic strain. This approach, usually referred to as the “ deterministic”approach, allows us to describe effectively stochastic resonance phenomena, without transforming the motion equation into the corresponding Fokker–Planck equation.

Clearly in Melde’s phenomenon, the presence of stochastic terms may not seem significant. But Melde’s string is the prototypical model of the interaction between longitudinal deformations and transverse deformations and the consequent possibility of self-resonance. There are several physical and biological systems where this phenomenon is relevant and for which stochastic perturbations are significant. This is the case of simple “ mesoscopic” model of DNA in which the binding of the RNA polymerase enzyme molecule to the promoter sequence of the DNA is included through a substrate energy term modeling the enzymatic interaction with the DNA strands. In this framework, the longitudinal degree of freedom in DNA, which may considered a principal source of parametric resonance for the strands transverse motion, is clearly affected by stochastic effects [27].

The purpose of this study is to understand to what extent the phase displacement of the stress for the current strain and strain rate (recall that we are modeling the string in the class of rate-type materials) can induce, in the framework of “ small”deformations, destabilizing phenomena. In particular, we want also to study the effect of the noise intensity on the 2:1 resonance. Indeed, it could happen that the stochastic excitations synchronize with the longitudinal oscillations of the longitudinal tension causing a sharp change of the 2:1 resonance.

The structure of this paper is as follows. In section 2, we set down, in a linearized infinitesimal theory, the constitutive equations for a rate-type viscoelastic perfectly flexible string. We then consider the transverse deformations and obtain the linear model which, for time scales much larger than the relaxation time, essentially reduces to Mathieu’s equation. In section 3, we analyze the dynamics of the rope under Melde’s experimental conditions and, exploiting the Renormalization Group (RG) method, we analyze the effect that the various parameters, which enter the model, have on stability. In section 4, the changes induced by the stochastic noise in the 2:1 resonance is studied using the deterministic approach. This is done by replacing the noise with a high-frequency excitation. Some concluding remarks are summarized in section 5.

2. Constitutive model, string deformations, and motion equations

We consider a string capable of exerting only tensile stress whose constitutive model is the standard linear solid (also known as the Zener model) which, in the Maxwell representation, is given by:

σ + \frac{ν}{E_{2}} σ_{t} = E_{1} ε + ν \frac{E_{1} + E_{2}}{E_{2}} ε_{t},

(1)

where $σ$ is the tensile stress, $ε$ is the strain, and the subscript $t$ denotes the time derivative. Basically, we are considering a spring and a spring plus a dashpot in parallel. The parameters $E_{1}$ and $E_{2}$ are Young’s moduli of the two springs and $ν$ is the viscosity of the dashpot. Henceforth, we represent model (1) following the notation introduced by Zhou [13]:

σ + \frac{1}{ϕ} σ_{t} = G ε + η ε_{t},

(2)

where $ϕ^{- 1} > 0$ is the stress relaxation time.¹ For the general 3D version of the model, we refer to Zhou [13], Nasseri [28], Phan-Thien [29], and to the abundant literature therein cited.

The string, whose relaxed (unstressed) length is $L_{r}$ , is uniformly stretched to a length $L_{o}$ , so that we have an uniform strain $ε_{o} = (L_{o} - L_{r}) / L_{r}$ . Next, assuming that the deformations occur in the $x$ , $y$ plane, we consider the following motion superimposed on the strain $ε_{o}$ (see [6]):

χ = [x (1 + ε_{o}) + u (x, t)] e_{x} + w (x, t) e_{y},

(3)

so that the total strain is:

ε = \sqrt{{(1 + ε_{o} + u_{x} (x, t))}^{2} + w_{x}^{2} (x, t)} - 1,

which, in the approximation of small deformations,² reduces to:

ε = u_{x} + ε_{o} .

Denoting by $T_{u}$ and $T_{w}$ the longitudinal and transverse components of the tension, respectively, Newton’s second law yields:

ρ A u_{tt} = {(T_{u})}_{x}, ρ A w_{tt} = {(T_{w})}_{x},

where $ρ$ is the density of the string and $A$ its cross-sectional area. In particular, being $T = A σ t$ , with:

t = \frac{\partial χ}{\partial x} {(‖ \frac{\partial χ}{\partial x} ‖)}^{- 1},

then

T_{u} = \frac{A σ (1 + ε_{o} + u_{x})}{ε + 1}, T_{w} = \frac{A σ w_{x}}{ε + 1},

and focusing only on the transverse vibrations (in the small deformations approximation), we have:

{\begin{matrix} ρ w_{tt} = {(σ w_{x})}_{x} \\ σ + \frac{1}{ϕ} σ_{t} = G ε + η ε_{t}, \\ ε = u_{x} (1 + ε_{o}) + ε_{o} . \end{matrix}

(4)

We now integrate equation (4)₂ with the initial condition ${σ |}_{t = 0} = 0$ , getting:

σ = e^{- ϕ t} \int_{0}^{t} ϕ e^{ϕ s} (G ε + η ε_{t}) ds .

We remark that if $ϕ \to \infty$ , i.e., the retardation time is much larger than all the time scales involved in the problem, then $σ$ converges toward $G ε + η ε_{t}$ . Indeed, denoting by $θ (t)$ the Heaviside function and by $δ_{D} (t)$ the Dirac distribution, we consider:

Ψ_{ϕ} (t) = \int_{- 1}^{t} ϕ e^{- ϕ s} θ (s) ds,

and observe that $lim_{ϕ \to \infty} Ψ_{ϕ} (t) = θ (t)$ . Recalling that $\overset{\cdot}{θ} (t) = δ_{D} (t)$ in the sense of distribution, we have $lim_{ϕ \to \infty} ϕ e^{- ϕ t} θ (t) = δ_{D} (t)$ , since ${\overset{\cdot}{Ψ}}_{ϕ} (t) = ϕ e^{- ϕ t} θ (t)$ . Therefore:

\begin{matrix} lim_{ϕ \to \infty} e^{- ϕ t} \int_{0}^{t} ϕ e^{ϕ s} [G ε (x, s) + η ε_{t} (x, s)] ds \\ = lim_{ϕ \to \infty} e^{- ϕ t} \int_{- 1}^{t} ϕ e^{ϕ s} [G ε (x, s) + η ε_{t} (x, s)] θ (s) ds \\ = G ε (x, t) + η ε_{t} (x, t), \end{matrix}

where the limit has to be intended in the sense of distribution.

We now exploit equation (4)₃ and obtain this expression for $σ$ :

σ = e^{- ϕ t} \int_{0}^{t} e^{ϕ s} [G (u_{x} + ε_{o}) + η (u_{xt} + {\overset{\cdot}{ε}}_{o})] ds,

which inserted in the right-hand side of equation (4)₁ gives rise to:

{(σ w_{x})}_{x} = w_{xx} ϕ e^{- ϕ t} \int_{0}^{t} e^{ϕ s} (G ε_{o} + η {\overset{\cdot}{ε}}_{o}) ds,

when the second-order terms in displacements are neglected. The motion equation (4)₁ takes the following form:

ρ w_{tt} = w_{xx} ϕ e^{- ϕ t} \int_{0}^{t} e^{ϕ s} (G ε_{o} + η {\overset{\cdot}{ε}}_{o}) ds .

(5)

We now assume:

ε_{o} = a_{o} (1 - δ \cos ω t), \Rightarrow {\overset{\cdot}{ε}}_{o} = a_{o} ω δ \sin ω t,

(6)

where $δ$ is a small parameter (not to be confused with the Dirac delta), so that:

\int_{0}^{t} e^{ϕ s} (G ε_{o} + η {\overset{\cdot}{ε}}_{o}) ds = Q + e^{ϕ t} U (t),

where

Q = a_{o} [\frac{δ (G ϕ + η ω^{2})}{ϕ^{2} + ω^{2}} - \frac{G}{ϕ}],

(7)

U (t) = a_{o} [\frac{G}{ϕ} + δ \frac{η ω (ϕ \sin ω t - ω \cos ω t) - G (ϕ \cos ω t + ω \sin ω t)}{ϕ^{2} + ω^{2}}] .

(8)

Hence, equation (5) reduces to:

ρ w_{tt} = w_{xx} ϕ [U (t) + Q e^{- ϕ t}] .

(9)

In particular, if $δ = 0$ so that $ε_{o} = a_{o}$ , we have:

w_{tt} = a_{o} \frac{G}{ρ} w_{xx} (1 - e^{- ϕ t}),

and, for $t >> 1 / ϕ$ , we end up with $w_{tt} = c_{T}^{2} w_{xx}$ , where:

c_{T}^{2} = \frac{a_{o} G}{ρ},

is the velocity of the transverse wave. In general, in the limit of $t >> 1 / ϕ$ , the term $Q e^{- ϕ t}$ can be neglected and equation (9) reduces to:

ρ w_{tt} = w_{xx} ϕ U (t) .

(10)

We now assume that only the $n th$ order transverse normal mode is excited and so we look for a solution to equation (10) of the form $w (x, t) = f (t) \sin (k_{n} x)$ with $k_{n} = n π / L_{o}$ $(n = 1, 2, 3, \dots)$ . It is convenient to rescale $t$ with a frequency $ω$ , so that $\tilde{t} = ω t$ is a dimensionless time and $x$ with $L_{o}$ , namely, $\tilde{x} = x / L_{o}$ . Equation (10) can be rewritten as follows:³

\overset{\cdot\cdot}{f} = - {(\frac{Ω_{n}}{ω})}^{2} [1 - δ (\frac{γ ω^{2} + ϕ^{2}}{ϕ^{2} + ω^{2}}) \cos t + δ (\frac{ω ϕ (γ - 1)}{ϕ^{2} + ω^{2}}) \sin t] f,

(11)

where

Ω_{n}^{2} = c_{T}^{2} k_{n}^{2} = \frac{a_{o} G}{ρ} {(\frac{n π}{L_{o}})}^{2},

(12)

is the $n th$ fundamental frequency corresponding to the $n th$ order transverse normal mode of the stretched string and where:

γ = \frac{η ϕ}{G},

(13)

is the ratio between the characteristic viscous stress and the elastic stress. Table 1 lists some values of $γ$ for materials usually modeled as a standard linear solid, i.e., rate-type viscoelastic materials.

Table 1.

Evaluated values of $γ$ for some materials usually modeled as rate-type viscoelastic.

Material	$γ$
Polybutadienes TV50 [30]	2.3
PCL/bio-active [31]	9
Collagen fibrils [31]	6
Vinyl ester [32]	4.75 $%$

We further notice that by evaluating $γ$ through relation (1), it turns out that $γ = 1 + E_{2} / E_{1} .$

We remark that equation (11) reduces to the classical Mathieu’s equation. Indeed, we have:

(\frac{γ ω^{2} + ϕ^{2}}{ϕ^{2} + ω^{2}}) \cos t - (\frac{ω ϕ (γ - 1)}{ϕ^{2} + ω^{2}}) \sin t = A \cos (t + θ),

provided

A^{2} = \frac{γ^{2} ω^{2} + ϕ^{2}}{ω^{2} + ϕ^{2}}, \tan θ = \frac{ω ϕ (γ - 1)}{γ ω^{2} + ϕ^{2}} .

(14)

Therefore, equation (11) can be rewritten as:

\overset{\cdot\cdot}{f} = - {(\frac{Ω_{n}}{ω})}^{2} [1 - A δ \cos (t + θ)] f,

(15)

which, apart from the phase shift $θ$ (irrelevant for the purposes of stability), is precisely Mathieu’s equation.

3. Stability analysis of equation (11)

An analysis of the properties of the solutions of equation (15) can be found in many standard references (see, for instance, [33, Chapter 11] or [34, 35]). The Floquet theory reveals that resonance (i.e., instability) occurs when ${(\frac{Ω_{n}}{ω})}^{2} \approx 1 / 4$ . In this section, we want to investigate how the various parameters entering the model influence the onset of resonance. To this aim, we introduce:

α_{c} = \frac{γ ω^{2} + ϕ^{2}}{8 (ϕ^{2} + ω^{2})}, α_{s} = \frac{ω ϕ (γ - 1)}{8 (ϕ^{2} + ω^{2})},

(16)

and rewrite equation (11) as:

\overset{\cdot\cdot}{f} = - {(\frac{Ω_{n}}{ω})}^{2} [1 - 8 δ α_{c} \cos t + 8 δ α_{s} \sin t] f .

(17)

We then set:

{(\frac{Ω_{n}}{ω})}^{2} = \frac{1}{4} + p_{1} δ + O (δ^{2}),

with $0 < δ << 1$ , $p_{1} = O (1)$ , which we plug into equation (17) getting:

\overset{\cdot\cdot}{f} = - \frac{f}{4} - δ (4 p_{1} - 8 α_{c} \cos t + 8 α_{s} \sin t) \frac{f}{4},

(18)

when the $O (δ^{2})$ terms are neglected. We now transform equation (18) into a first-order system as follows:

{\begin{matrix} \overset{\cdot}{f} = \frac{g}{2} \\ \overset{\cdot}{g} = - \frac{f}{2} - 4 δ (p_{1} - 2 α_{c} \cos t + 2 α_{s} \sin t) \frac{f}{2}, \end{matrix}

which, introducing $z = f + ig$ , so that $f = (z + \bar{z}) / 2$ , $g = (z - \bar{z}) / 2 i$ , can be rewritten as:

\overset{\cdot}{z} = - \frac{i}{2} z - i δ [p_{1} - α_{c} (e^{it} + e^{- it}) - i α_{s} (e^{it} - e^{- it})] (z + \bar{z}) .

(19)

We then look for a solution to equation (19) in the form of an asymptotic power series expansion in $δ$ , i.e.,

z = z_{0} + δ z_{1} + O (δ^{2}),

(20)

having as leading term (i.e., as asymptotic or perturbation limit) $z_{0}$ given by:

{\overset{\cdot}{z}}_{0} = - \frac{i}{2} z_{0}, \Rightarrow z_{0} (t) = Z e^{- \frac{i}{2} t} .

Concerning $z_{1}$ , we have the equation:

{\overset{\cdot}{z}}_{1} = - \frac{i}{2} z_{1} - i [p_{1} - α_{c} (e^{it} + e^{- it}) - i α_{s} (e^{it} - e^{- it})] \underset{Z e^{- \frac{i}{2} t} + \bar{Z} e^{\frac{i}{2} t}}{\underset{︸}{(z_{0} + {\bar{z}}_{0})}},

whose general solution is:

z_{1} = - i [Z p_{1} - (α_{c} - i α_{s}) \bar{Z}] t e^{- \frac{i}{2} t} + r (t),

where $r (t)$ is a bounded function. Thus, recalling equation (20):

z = Z e^{- \frac{i}{2} t} - i δ [Z p_{1} - (α_{c} - i α_{s}) \bar{Z}] t e^{- \frac{i}{2} t} + δ r (t) + O (δ^{2}),

where we observe the presence of a secular term which, for large values of $t$ , deteriorates the approximation. Now, we apply the RG method (see, for example, [36 –39] and the review paper [40]): we introduce an arbitrary time $τ$ , so that $t$ is decomposed as $t - τ + τ$ , and the terms containing $τ$ are absorbed into the renormalized counterpart $Z (τ) (1 + δ ζ)$ . Thus, the renormalized expansion for $z (t, τ)$ is:

\begin{matrix} z = Z (1 + δ ζ) e^{- \frac{i}{2} t} + \\ - i δ [Z (1 + δ ζ) p_{1} - (α_{c} - i α_{s}) \bar{Z} (1 + δ \bar{ζ})] (t - τ + τ) e^{- \frac{i}{2} t} + r (t) . \end{matrix}

We now select $ζ$ in order to eliminate the secular terms in $τ$ , keeping only terms in $t - τ$ . Since the parameter $τ$ is artificial, the solution must not depend on it. Hence, we require ${\frac{\partial z}{\partial τ} |}_{τ = t} = 0$ , for every $t$ , and use this condition to write the RG equation for $Z$ :

Z^{'} = - i δ [Z p_{1} - α_{c} \bar{Z}] + α_{s} δ \bar{Z} .

(21)

Setting $Z = W_{R} + i W_{I}$ , we obtain:

(\begin{matrix} {W'}_{R} \\ {W'}_{I} \end{matrix}) = - δ (\begin{matrix} - α_{s} & - p_{1} - α_{c} \\ p_{1} - α_{c} & α_{s} \end{matrix}) (\begin{matrix} W_{R} \\ W_{I} \end{matrix}) .

We thus have stability if both the eigenvalues of the matrix are imaginary, i.e., if:

S = p_{1}^{2} - (α_{s}^{2} + α_{c}^{2}) = p_{1}^{2} - \frac{γ^{2} ω^{2} + ϕ^{2}}{64 (ω^{2} + ϕ^{2})} > 0 .

Figures 1 and 2 show both stable and unstable solutions, respectively, of equation (18) while Figure 3 shows a three-dimensional view of stable/neutral/unstable regions, respectively, in the $(p_{1}, ω / ϕ, γ)$ space.

4. Stochastic noise

We now assume that the simple extension given by equation (6) is affected by the Gaussian noise $ξ (t)$ with zero mean. In this paper, we follow [25, 26, 41] and employ the “ deterministic” approach to describe the stochastic noise taking $ξ = - R \cos ω_{r} t$ , with $R$ being the noise intensity and $ω_{r}$ much larger than any other frequency characterizing the phenomenon. We thus replace equation (6) by:

ε_{o} = a_{o} [1 - δ \cos ω t \underset{ξ (t)}{] \underset{︸}{- R \cos (ω_{r} t)}} .

With this choice, equation (9) is replaced by:

ρ w_{tt} = w_{xx} ϕ [U (t) + Q e^{- ϕ t} + U_{r} (t) + Q_{r} e^{- ϕ t}],

(22)

where $Q$ and $U$ are still given by equations (7) and (8), respectively, while:

\begin{matrix} U_{r} (t) = \frac{R}{ω_{r}^{2} + ϕ^{2}} [η ω_{r} (ϕ \sin ω_{r} t - ω_{r} \cos ω_{r} t) - G (ϕ \cos ω_{r} t + ω_{r} \sin ω_{r} t)], \\ Q_{r} = R \frac{G ϕ + η ω_{r}^{2}}{ω_{r}^{2} + ϕ^{2}} . \end{matrix}

Considering, as in section 2, $t >> 1 / ϕ$ , equation (22) reduces to:

ρ w_{tt} = w_{xx} ϕ [U (t) + U_{r} (t)] .

(23)

Taking the $n th$ order transverse mode and considering the same dimensionless variables already introduced in section 2, the evolution equation for $f (t)$ , namely, equation (17), can be rewritten as:

\begin{matrix} \overset{\cdot\cdot}{f} = - {(\frac{Ω_{n}}{ω})}^{2} [1 - δ \frac{γ ω^{2} + ϕ^{2}}{ϕ^{2} + ω^{2}} \cos t + δ \frac{ω ϕ (γ - 1)}{ϕ^{2} + ω^{2}} \sin t + \\ + R (- \frac{γ ω_{r}^{2} + ϕ^{2}}{ω_{r}^{2} + ϕ^{2}} \cos \hat{ω} t + \frac{ω_{r} ϕ (γ - 1)}{ω_{r}^{2} + ϕ^{2}} \sin \hat{ω} t)] f, \end{matrix}

(24)

with $γ$ and $Ω_{n}$ given by equations (13) and (12), respectively, and $\hat{ω} = ω_{r} / ω$ . Now, $ω_{r}$ is much larger than the other frequencies involved in the problem and so $1 / ω_{r} << 1 / ϕ$ . Hence, neglecting the $O (ϕ / ω_{r})$ terms in equation (24), we obtain:

\overset{\cdot\cdot}{f} = - {(\frac{Ω_{n}}{ω})}^{2} [1 - δ \frac{γ ω^{2} + ϕ^{2}}{ϕ^{2} + ω^{2}} \cos t + δ \frac{ω ϕ (γ - 1)}{ϕ^{2} + ω^{2}} \sin t - R γ \cos \hat{ω} t] f,

which, recalling equations (14) and (15), we rewrite as:

\overset{\cdot\cdot}{f} = - {(\frac{Ω_{n}}{ω})}^{2} [1 - A δ \cos (t + θ) - R γ \cos \hat{ω} t] f,

i.e.,

\overset{\cdot\cdot}{f} + [Θ + b \cos (t + θ) + B \cos \hat{ω} t] f = 0,

(25)

with

\begin{matrix} Θ = {(\frac{Ω_{n}}{ω})}^{2}, b = - δ A {(\frac{Ω_{n}}{ω})}^{2} = - δ A Θ, \\ B = - R γ Θ = - R γ {(\frac{Ω_{n}}{ω})}^{2}, \end{matrix}

(26)

where $Θ = Ω_{n} / ω = O (1)$ , and $δ << 1$ , so that $| b | << 1$ . According to Sorokin and Blekhman [42], we look for the solution to equation (25) in the form:

f (t, τ) = F (t) + φ (t, \hat{ω} t),

(27)

and set $τ = \hat{ω} t$ to be the “ fast”timescale. In particular, we require:

〈 φ 〉 = \frac{1}{2 π} \int_{0}^{2 π} φ (t, τ) d τ = 0 .

We now plug equation (27) into equation (25) and take the average with respect to $τ$ :

\overset{\cdot\cdot}{F} + [Θ + b \cos (t + θ)] F + B 〈 φ \cos τ 〉 = 0 .

(28)

For the equation (25) to be fulfilled, we also have:

[- {\hat{ω}}^{2} + Θ + b \cos (t + θ) + \cos τ] φ + B (F \cos τ - 〈 φ \cos τ 〉) = 0 .

(29)

We now look for a solution to equation (29) in the form of Fourier series, namely:

φ = \sum_{n} C_{n} \cos (n τ) + S_{n} \sin (n τ),

(30)

since it is sufficient to determine $φ$ in an approximate way [42]. Taking into account only the first (primary) harmonic in equation (30), we have:

\begin{matrix} [- {\hat{ω}}^{2} + Θ + b \cos (t + θ) + \cos τ] (C_{1} \cos τ + S_{1} \sin τ) + \\ - \frac{B C_{1}}{2} + BF \cos τ = 0 . \end{matrix}

(31)

We now multiply equation (31) by $\cos τ$ and then by $\sin τ$ and take the average so to get:

\begin{matrix} BF + C_{1} [b \cos (t + θ) + Θ - {\hat{ω}}^{2}] = 0, \Rightarrow C_{1} = - \frac{BF}{Θ - {\hat{ω}}^{2} + b \cos (t + θ)}, \\ - {\hat{ω}}^{2} S_{1} + b S_{1} \cos (t + θ) = 0, \Rightarrow S_{1} = 0 . \end{matrix}

Hence:

φ (t, τ) \approx - \frac{BF (t) \cos τ}{Θ - {\hat{ω}}^{2} + b \cos (t + θ)},

which into equation (28) gives:

\overset{\cdot\cdot}{F} + [Θ + b \cos (t + θ)] F + \frac{B^{2} F}{2 ({\hat{ω}}^{2} - Θ - b \cos (t + θ))} = 0 .

Recalling now that $Θ = O (1)$ , $| b | << 1$ , and $\hat{ω} >> 1$ , we rewrite the previous equation as:

\overset{\cdot\cdot}{F} + [Θ + \frac{B^{2}}{2 {\hat{ω}}^{2}} (1 + \frac{Θ}{{\hat{ω}}^{2}}) + b \cos (t + θ)] F = 0,

i.e., as:

\overset{\cdot\cdot}{F} = - Θ_{eff} [1 - δ A_{eff} \cos (t + θ)] F = 0,

where

Θ_{eff} = Θ + \frac{B^{2}}{2 {\hat{ω}}^{2}} (1 + \frac{Θ}{{\hat{ω}}^{2}}),

(32)

A_{eff} = \frac{A Θ}{Θ + \frac{B^{2}}{2 {\hat{ω}}^{2}} (1 + \frac{Θ}{{\hat{ω}}^{2}})} .

(33)

Recalling equation (26), $Θ_{eff}$ can be approximated as:

Θ_{eff} = Θ + \frac{B^{2}}{2 {\hat{ω}}^{2}} = {(\frac{Ω_{n}}{ω})}^{2} + \frac{1}{2} {(\frac{R γ}{\hat{ω}})}^{2} {(\frac{Ω_{n}}{ω})}^{2} .

The so-called 2:1 resonance comes to be modified and it is affected not only by the amplitude of the stochastic noise but also by $γ$ and by the noise frequency as well. Indeed:

Θ_{eff} = \frac{1}{4}, \Rightarrow {(\frac{Ω_{n}}{ω})}^{2} [1 + \frac{1}{2} {(\frac{R γ}{\hat{ω}})}^{2}] = \frac{1}{4},

namely:

{(\frac{Ω_{n}}{ω})}^{2} = \frac{1}{4 [1 + \frac{γ^{2}}{2} {(\frac{R}{\hat{ω}})}^{2}]} .

(34)

In particular, if $\frac{R γ}{\hat{ω}} << 1$ ,

{(\frac{Ω_{n}}{ω})}^{2} \approx \frac{1}{4} - \frac{1}{8} {(\frac{R γ}{\hat{ω}})}^{2} .

Figure 4 displays ${(\frac{Ω_{n}}{ω})}^{2}$ as a function of $\frac{R}{\hat{ω}}$ for three values of $γ$ . We note that ${(\frac{Ω_{n}}{ω})}^{2}$ can deviate significantly from 0.25 as $γ$ increases. Indeed, it suffices that $γ$ is of the order of 10 for ${(\frac{Ω_{n}}{ω})}^{2}$ to reduce to $0.2$ when $(\frac{R}{\hat{ω}}) \approx 0.1$ . We, however, remark that $(\frac{R}{\hat{ω}}) \approx 0.1$ indicates a “ significant” noise intensity which is likely difficult to meet under normal experimental conditions.

5. Conclusion

In this paper, we have considered a linearized version of the constitutive model illustrated in Farina et al. [14], and applied it to model oscillations of a pre-stretched perfectly flexible string driven by periodic changes in its tension. It is well known that in the purely elastic case and for certain critical values of the various parameters, the periodic change in the longitudinal stretch generates resonance. The results we have obtained are similar to the purely elastic case. The linear model predicts the usual 2:1 resonance, i.e., the lateral vibrations of a viscoelastic rope of rate type become unstable when the frequency of the axial tension oscillations is twice the natural frequency of any transverse mode.

However, it should be stressed that our analysis does not guarantee stability to arbitrary disturbances, since we only consider a restricted class of deformations, i.e., the ones given by equation (3). A similar issue, although developed in a fluid dynamic context, has been deeply investigated in Parter and Rajagopal [43] and Rajagopal [44]. In these papers, the authors, studying the flow between rotating plates, showed that, expanding the class of possible motions, there exists an infinite number of unstable solutions (lacking symmetry) close to any stable symmetrical solution.

We have also analyzed the effect that a stochastic noise affecting the imposed strain has on the 2:1 resonance. We have used a deterministic approach, replacing the noise by a harmonic excitation whose frequency is much larger than the other frequencies characterizing the phenomenon. It is shown that the parameter $γ$ seems to play a significant role in this respect. Indeed, looking at Figure 4, one easily realizes that the deviations from the 2:1 resonance are amplified (in the sense that the ratio $Ω_{n} / ω$ is reduced) as $γ$ increases.

We finally point out that our analysis fails in describing Melde’s experiment performed with a band made of a viscoelastic rate type material in the case of finite-amplitude shearing motions.

Footnotes

Acknowledgements

The authors wish to thank the unknown referees for their precious suggestions which significantly improved the paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Angiolo Farina

Lorenzo Fusi

Fabio Rosso

Giuseppe Saccomandi

Notes

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