2021

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]A Venkatesan

[1]PG & Research Department of Physics, Nehru Memorial College (Autonomous),
Affiliated to Bharathidasan University, Puthanampatti, Tiruchirappalli,
621007, Tamil Nadu, India

[2]Department of Nonlinear Dynamics, School of Physics, Bharathidasan
University, Tiruchirappalli, 620024, Tamil Nadu, India

# Vibrational resonance in a damped and two-frequency driven system of
particle on a rotating parabola

R Kabilan<EMAIL_ADDRESS>M Sathish Aravindh
<EMAIL_ADDRESS><EMAIL_ADDRESS>M Lakshmanan
<EMAIL_ADDRESS>*  *

###### Abstract

In the present work, we examine the role of nonlinearity in vibrational
resonance (VR) of a forced and damped form of a velocity-dependent potential
system. Many studies have focused on studying the vibrational resonance in
different potentials, like bistable potential, asymmetrically deformed
potential, and rough potential. In this connection, velocity-dependent
potential systems are very important from a physical point of view (Ex: pion-
pion interaction, cyclotrons and other electromagnetic devices influenced by
the Lorentz force, magnetrons, mass spectrometers). They also appear in
several mechanical contexts. In this paper, we consider a nonlinear dynamical
system with velocity-dependent potential along with additional damping and
driven forces, namely a particle moving on a rotating-parabola system, and
study the effect of two-frequency forcing with a wide difference in the
frequencies. We report that the system exhibits vibrational resonance in a
certain range of nonlinear strength. Using the method of separation of motions
(MSM), an analytical equation for the slow oscillations of the system is
obtained in terms of the parameters of the fast signal. The analytical
computations and the numerical studies concur well.

###### keywords:

Nonlinear systems, Vibrational Resonance, nonpolynomial oscillator, velocity-
dependent potential system

##  1 Introduction

Since its observation by Landa and McClintock landa2000vibrational , the
phenomenon of vibrational resonance (VR) has been studied analytically,
computationally, and empirically in several models of nonlinear systems,
spanning many fields of science and engineering. Vibrational resonance
primarily relies on the cooperation of two input signals with radically
different frequency values - a weak low frequency and a fast high frequency -
to improve the quality of the response of the weak frequency signal.
Theoretical studies and experimental realization of VR have been reported in
diverse fields of research.

In particular, the VR phenomenon has been studied in bistable systems
lakshmanan2003chaos ; rajasekar2016nonlinear ; baltanas2003experimental ,
multistable systems roy2017vibrational , ratchet devices du2016multiple ,
excitable systems deng2014theoretical , quintic oscillators guo2020vibrational
, coupled oscillators sarkar2019vibrational , overdamped systems, delayed
dynamical systems guo2020vibrational , asymmetric potential in Duffing
oscillators, fractional order potential oscillators yang2018vibrational , FHN
neural models ge2020vibrational , oscillatory networks qin2018vibrational ;
baysal2020effects , biological nonlinear systems ning2020vibrational ,
parametrically excited systems sarkar2019controlling , systems with nonlinear
damping laoye2019vibrational , deformed potentials vincent2018vibrational  as
well as harmonically trapped and roughed potentials abirami2017vibrational ;
laoye2019vibrational .

More significantly, experimental realizations of VR have been made,
particularly in optical system, bistable surface emitting lasers
chizhevsky2021amplification , multistable systems, arrays of hard limiters
ren2016theoretical , and Chua’s circuits usama2021vibrational . The potential
application of this novel phenomenon has been found in many areas of science,
namely energy harvester from bistable oscillator, energy detectors
ren2018generalized , signal detection, signal transmission and amplification
yao2019inhibitory ; jia2020echo , and the detection of faults in bearings
xiao2019novel , also in the design of logic gates and memory devices
venkatesh2016vibrational ; venkatesh2017implementation ; yao2022logical .

At the vibrational resonance, it is observed that the effective potential is
bifurcated. As a result many studies have focused on the effect of potentials.
Further the role played by the system parameters, delay terms, fractional
order terms, and potential deformation have been also considered on the
induction and control of VR vincent2018vibrational ; guo2020vibrational .

Despite the fact that VR is realized and observed in many systems, one can
realize that they all belong to systems whose potentials are assumed to be
polynomial functions. It will be interesting to investigate the possibility
and nature of VR in physically interesting nonpolynomial systems. From this
consideration, in the present work a mechanical system which corresponds to a
nonpolynomial velocity-dependent potential is considered for such a study.

The model under consideration is a mechanical model with a rich underlying
nonlinear dynamics venkatesan1997nonlinear ; mathews1974unique ;
nayfeh2008nonlinear . This model often depicts a particle’s motion on a
rotating parabola. This mechanical model also describes centrifugation
equipment, centrifugal forces, industrial events, and a motorbike being driven
in a revolving parabolic well in a circus.

As mentioned above, many of the works on VR are concerned with polynomial
oscillators and very few works have focused on VR in nonpolynomial
oscillators. In particular, VR in driven oscillators with position-dependent
mass roy2021vibrational , VR in the Morse oscillator rajasekar2016nonlinear ,
VR in a dual-frequency-driven Tietz-Hua quantum well olusola2020quantum ,
identification and parameter estimation of nonpolynomial forms of damping
nonlinearity in dynamic systems chintha2022identification  are few of the
works on VR in nonpolynomial oscillators. More detailed investigations on VR
in nonpolynomial oscillators are required to unravel the nature of resonance
behavior in these oscillators depending upon the various system parameters. In
particular, it will be important to explore the role of nonlinearity parameter
in contrast to the damping and forcing parameters in the onset of VR
phenomenon.

In this connection, there are numerous methods available to approximate the
solution of nonlinear equations. Methods such as the Lindstedt–Poincaré
perturbation method, the multiple scale perturbation method, the harmonic
balance method, and the averaging method are quite useful for this purpose.
Each one of the above procedures has its own set of advantages and
disadvantages, as well as ranges of validity and applicability in specific
situations. However, the method of separation of motion (MSM)
roy2021vibrational ; vincent2018vibrational  features several significant
advantages over the above methods specifically in connection with the study of
VR analytically, e.g., the simplicity in application and the transparency of
the physical interpretation. Hence, in order to investigate VR theoretically,
we used the MSM to divide the equation of motion into a slow motion and a fast
motion and obtain a set of integro-differential equations. The composite
system is completely solved by this pair of integro-differential equations,
which describe the equations of slow oscillations and fast vibrations,
respectively, and their superposition. The method of direct separation of
motion appears to be a convenient and simple tool for obtaining the equation
of a system’s slow motion. It is also applicable to solving many mechanical
problems. This MSM method is employed to study vibrational resonance in many
nonlinear systems with polynomial potential functions. In the present work, we
extend the general framework of the MSM method following the work of Roy-
Layinde et al. roy2021vibrational  to study the vibrational resonance in the
nonpolynomial oscillator with velocity-dependent potential. Also, in the
present work, the role of nonlinearity on VR in a nonpolynomial system of
damped, driven particle moving on a rotating parabola is examined. Most of the
studies on VR in the literature are concerned with role of damping in either
suppressing or increasing the resonance behaviour rajasekar2016nonlinear . But
in the present system, it is also shown that the VR occurs in an optimal range
of nonlinear strength thereby emphasizing the role of nonlinearity. It is
quite interesting and entirely different from the VR studies on other
polynomial and nonpolynomial oscillators.

It is an established fact that resonance occurs in a nonlinear oscillator
system if the matching of the natural frequency of an underdamped oscillation
of the system with an external periodic driving force causes the response
amplitude of the system to be enhanced. In recent times, it has been found
that the system amplification of response, or the term “resonance," is being
used by adjusting other system parameters, including different kinds of
forces.

As a result, many terms are being introduced, namely, stochastic resonance,
chaotic resonance, ghost resonance, auto resonance, anti resonance, and so on,
depending upon the manifestation of forces or system parameters [details are
given in ref.rajasekar2016nonlinear ; vincent2021vibrational ].

In the present work, we essentially consider a nonpolynomial nonlinear
oscillator model, namely the motion of a particle on a rotating parabola with
additional interactions, which is quite different from the generalized
Mathews-Lakshmanan oscillator model with additional interactions studied in
ref.[32]. The present oscillator is though analogous to a position-dependent
mass (PDM) system, the mass is associated with the strength of nonlinearity.
In the present paper, we mainly focus on a specific form of vibrational
resonance, which is discussed in terms of a slowly driven system’s response to
variations in the parameters associated with a high-frequency periodic force.
In the absence of nonlinearity, the model considered becomes a linear harmonic
oscillator. We have shown that no VR occurs in this system without
nonlinearity. On the other hand for a low level of nonlinearity maximum
response occurs in this model, which is the most novel feature. Thus, it is
essential and important to study the effect of nonlinearity in the system. We
have shown that the enhancement or suppression of vibrational resonance is
possible through variation of the nonlinear system parameter in the presence
of low-frequency and high-frequency forces within appropriate parameter
regimes.

Our results, while confirming the basic features of VR as identified in the
case of PDM Duffing and Morse oscillators, specifically brings out novel
features on the effect of nonlinearity. The main contribution of this paper is
that the mere presence of nonlinearity alone does not guarantee the emergence
of VR, as in the case of polynomial nonlinear oscillators. An appropriate
strength of nonlinearity region is solely responsible for the induction and
control of VR. Thus, in the present study, we have investigated the role of
nonpolynomial nonlinearity in oscillator systems for inducing vibrational
resonance. In particular, we have shown that the vibrational resonance is
realized and observed within an optimal range of nonlinear strength. From our
study, it is inferred that resonance cannot be observed at all parameter
values of nonlinearity.

The article is organized as follows : In Section 2, the dynamical model is
discussed. Analytical studies and comparison between numerical and analytical
studies based on method of separation of motions are given in Sections 3 & 4,
respectively. Finally, the paper is concluded in Section 5.

##  2 The dynamical model

In this article, we consider the nonpolynomial oscillator governed by the
equation of motion venkatesan1997nonlinear ; mathews1974unique ;
nayfeh2008nonlinear

$(1+\mu x^{2})\ddot{x}+\mu x\dot{x}^{2}+\omega_{0}^{2}x=0,$ (1)

and subjected to two additional periodic forcings and extra linear damping.

For the analysis of fundamentally nonlinear processes in a non-generic
mechanical system that correspond to a well-known model mathews1974unique ;
nayfeh2008nonlinear  is Eq.(1) with the Lagrangian,

$L=\dfrac{1}{2}\big{[}(1+\mu x^{2})\dot{x}^{2}-\omega_{0}^{2}x^{2}\big{]}.$
(2)

Figure 1: The red circle shows how a freely sliding particle moves down a
parabolic wire that rotates at a constant angular velocity $\Omega$
venkatesan1997nonlinear .

The corresponding Hamiltonian is

$H=\dfrac{1}{2}\big{[}p^{2}(1+\mu x^{2})^{-1}+\omega_{0}^{2}x^{2}\big{]},$ (3)

where the canonical conjugate momentum is

$p=\dot{x}(1+\mu x^{2}).$ (4)

We take into consideration a mechanical model that depicts how a freely moving
particle of unit mass moves down a parabolic wire that rotates around the
particle’s axis $z=\sqrt{\mu}x^{2}$ with a constant angular velocity $\Omega$
($\Omega^{2}=\Omega_{0}^{2}=-\omega_{0}^{2}+g\sqrt{\mu}$), where $\mu>0$ and
$\omega_{0}>0$ as shown in Fig.1. Here, ‘g’ stands for the acceleration
brought on by gravity, ‘$\frac{1}{\sqrt{\mu}}$’ stands for the rotating
parabola’s semi-lotus rectum, and ‘$\omega_{0}$’ stands for the initial
angular velocity venkatesan1997nonlinear ; nayfeh2008nonlinear .

Next, we investigate the nonlinear dynamics associated with Eq.(1) under the
influence of additional damping and two periodic external forcings, leading to
the equation of motion

$(1+\mu x^{2})\ddot{x}+\mu
x\dot{x}^{2}+\alpha\dot{x}+\omega_{0}^{2}x=f_{1}\cos\omega_{1}t+f_{2}\cos\omega_{2}t.$
(5)

In the above $f_{1}$ and $f_{2}$ correspond to the strength of two periodic
forces with period $2\pi/\omega_{1}$ and $2\pi/\omega_{2}$ respectively. Here
$\omega_{1}$ and $\omega_{2}$ are the strength of the low frequency and high
frequency forces, respectively.

For a single force, Eq.(5) has been well investigated and it shows the period-
doubling route to chaos, strange nonchaotic attractors, intermittency, crisis
venkatesan1997nonlinear  and prediction of extreme events using machine
learning meiyazhagan2022prediction .

We will express Eq.(5) in a way that makes our analytical process manageable,
in accordance with the work of Roy-Layinde et al. roy2021vibrational . In
particular, we divide Eq.(5) by $(1+\mu x^{2})$ throughout so that it can be
expressed as

$\displaystyle\ddot{x}+(\mu x\dot{x}^{2}+\alpha\dot{x}+\omega_{0}^{2}x)(1+\mu
x^{2})^{-1}$ $\displaystyle=(1+\mu
x^{2})^{-1}(f_{1}\cos\omega_{1}t+f_{2}\cos\omega_{2}t).$ (6)

Now apply the binomial expansion to the terms $(1+\mu x^{2})^{-1}$ and
restrict our consideration to the first three terms of the expansion only.
Eq.(6) can be expressed as,

$\displaystyle\ddot{x}+(\mu x\dot{x}^{2}+\alpha\dot{x}+\omega_{0}^{2}x)(1-\mu
x^{2}+\mu^{2}x^{4})$ $\displaystyle=(1-\mu
x^{2}+\mu^{2}x^{4})(f_{1}\cos\omega_{1}t+f_{2}\cos\omega_{2}t).$ (7)

The potential $V(x)$ associated with (7) may then be written

$\displaystyle
V(x)=\dfrac{\omega_{0}^{2}}{2}x^{2}-\dfrac{\mu\omega_{0}^{2}}{4}x^{4}+\dfrac{\mu^{2}\omega_{0}^{2}}{6}x^{6}.$
(8)

Figure 2: The system potential (8) for $\mu=0.5$ and $\omega_{0}^{2}=0.25$.

The system potential shown in Fig.2 for given parameter values,
$\omega_{0}^{2}$ and $\mu$, is computed from Eq.(8).

In connection with the analysis of the above nonlinear system, a couple of
natural questions arise: In a linear system, is it possible to detect
vibrational resonance? What happens when a linear system is subjected to a
high-frequency force? To answer these questions we consider Eq.(5) without the
nonlinear term ($\mu=0.0$). The linear equation can be easily integrated and
the exact analytical solution is given by

$x(t)=A_{1}e^{m_{1}t}+A_{2}e^{m_{2}t}+f_{\omega_{1}}\cos(\omega_{1}t+\phi_{1})+f_{\omega_{2}}\cos(\omega_{2}t+\phi_{2}),$
(9)

where

$f_{\omega_{1}}=\frac{f_{1}}{\sqrt{(\omega_{0}^{2}-\omega_{1}^{2})^{2}+\alpha^{2}\omega_{1}^{2}}},~{}~{}f_{\omega_{2}}=\frac{f_{2}}{\sqrt{(\omega_{0}^{2}-\omega_{2}^{2})^{2}+\alpha^{2}\omega_{2}^{2}}}$
(10)

From the above, it may be observed that the structure of $f_{\omega_{1}}$ is
independent of the amplitude $f_{2}$ and the frequency $\omega_{2}$.
Therefore, the amplitude $f_{\omega_{1}}$ is unaffected by the presence of the
high frequency force in a linear system rajasekar2016nonlinear . Consequently,
no VR occurs in the linear system and so it is important to study the effect
of nonlinearity via analytical and numerical methods. These are carried out in
the upcoming sections.

##  3 Theoretical analysis

The conventional method of separation of motions (MSM), which is a
perturbation method, is now used, in which it is assumed that the system
dynamics consists of a slow component $y(t)$ and a fast component $z(t,\tau)$.
The main equation of the system, Eq.(7), is solved entirely by superimposing
the solutions of the two integro-differential equations derived from each of
the components using the MSM approach. Following the factorization of like
terms and the definition $x=y+z$, the system equation (7) can be represented
as

$\displaystyle\ddot{y}+\ddot{z}+\mu[\mu^{2}y^{5}+5\mu^{2}zy^{4}+(10\mu^{2}z^{2}-\mu)y^{3}+(10\mu^{2}z^{3}-3\mu
z)y^{2}+(5\mu^{2}z^{4}-3\mu z^{2}+1)y$ $\displaystyle+\mu^{2}z^{5}-\mu
z^{3}+z](\dot{y}^{2}+2\dot{y}\dot{z}+\dot{z}^{2})+\alpha[1+\mu^{2}y^{4}+4\mu^{2}zy^{3}+(6\mu^{2}z^{2}-\mu)y^{2}$
$\displaystyle+(4\mu^{2}z^{3}-2\mu
z)y](\dot{y}+\dot{z})+\omega_{0}^{2}[\mu^{2}y^{5}+5\mu^{2}zy^{4}+(10\mu^{2}z^{2}-\mu)y^{3}+(10\mu^{2}z^{3}-3\mu
z)y^{2}$ $\displaystyle+(5\mu^{2}z^{4}-3\mu z^{4}-3\mu
z^{2}+1)y+\mu^{2}z^{5}-\mu z^{3}+z]=[1+\mu^{2}y^{4}+4\mu^{2}zy^{3}$
$\displaystyle+(6\mu^{2}z^{2}-\mu)y^{2}+(4\mu^{2}z^{3}-2\mu
z)y+\mu^{2}z^{4}-\mu z^{2}](f_{1}cos\omega_{1}t+f_{2}cos\omega_{2}t).$ (11)

Now taking the time average in a period over the fast variable $\tau$ and
considering the fact that the fast signal ’z’ is rapidly oscillating in the
term with period $\dfrac{2\pi}{\omega_{2}}$, we have

$\displaystyle\ddot{y}+\mu[\mu^{2}y^{5}+5\mu^{2}\overline{z}y^{4}+(10\mu^{2}\overline{z^{2}}-\mu)y^{3}+(10\mu^{2}\overline{z^{3}}-3\mu\overline{z})y^{2}+(5\mu^{2}\overline{z^{4}}-3\mu\overline{z^{2}}+1)y$
$\displaystyle+\mu^{2}\overline{z^{5}}-\mu\overline{z^{3}}+\overline{z}](\dot{y}^{2}+\overline{\dot{z}^{2}})+\alpha[1+\mu^{2}y^{4}+4\mu^{2}\overline{z}y^{3}+(6\mu^{2}\overline{z^{2}}-\mu)y^{2}$
$\displaystyle+(4y^{2}\overline{z^{3}}-2\mu\overline{z})y]\dot{y}+\omega_{0}^{2}[\mu^{2}y^{5}+5\mu^{2}\overline{z}y^{4}+(10\mu^{2}\overline{z^{2}}-\mu)y^{3}+(10\mu^{2}\overline{z^{3}}-3\mu\overline{z})y^{2}$
$\displaystyle+(5\mu^{2}\overline{z^{4}}-3\mu\overline{z^{2}}+1)y+\mu^{2}\overline{z^{5}}-\mu\overline{z^{3}}+\overline{z}]=[1+\mu^{2}y^{4}+4\mu^{2}\overline{z}y^{3}$
$\displaystyle+(6\mu^{2}\overline{z^{2}}-\mu)y^{2}+(4y^{2}\overline{z^{3}}-2\mu\overline{z})y+\mu^{2}\overline{z^{4}}-\mu\overline{z^{2}}+1](f_{1}cos\omega_{1}t+\overline{f_{2}cos\omega_{2}t}).~{}~{}~{}~{}~{}$
(12)

In the above, the average value of ‘z’ in relation to the fast variable
‘$\tau=\omega_{2}t$’ is given by

$\overline{z}=\dfrac{1}{2\pi}\int_{0}^{2\pi}zd\tau=0$ (13)

so that Eq.(12) becomes,

$\displaystyle\ddot{y}+\mu[\mu^{2}y^{5}+(10\mu^{2}\overline{z^{2}}-\mu)y^{3}+(10\mu^{2}\overline{z^{3}})y^{2}+(5\mu^{2}\overline{z^{4}}-3\mu\overline{z^{2}}+1)y$
$\displaystyle+\mu^{2}\overline{z^{5}}-\mu\overline{z^{3}}](\dot{y}^{2}+\overline{\dot{z}^{2}})+\alpha[1+\mu^{2}y^{4}+(6\mu^{2}\overline{z^{2}}-\mu)y^{2}+4\mu^{2}\overline{z^{3}}y]\dot{y}$
$\displaystyle+\omega_{0}^{2}[\mu^{2}y^{5}+(10\mu^{2}\overline{z^{2}}-\mu)y^{3}+10\mu^{2}\overline{z^{3}}y^{2}+(5\mu^{2}\overline{z^{4}}-3\mu\overline{z^{2}}+1)y+\mu^{2}\overline{z^{5}}-\mu\overline{z^{3}}]$
$\displaystyle=1+\mu^{2}y^{4}+(6\mu^{2}\overline{z^{2}}-\mu)y^{2}+(4\mu^{2}\overline{z^{3}}y+\mu^{2}\overline{z^{4}}-\mu\overline{z^{2}}+1)(f_{1}cos\omega_{1}t),~{}~{}~{}~{}~{}~{}$
(14)

Eq.(14) describes the slow motion of the system.

The averages in the slow motion equation can now be found using the
approximation method. For the composite system ’$x$’, this is done by first
finding the equation for the initial oscillations in ’$z$’ by deducting
Eq.(14) from Eq.(11), which is the equation for the slow component ‘$y$’.
Consequently, the equation governing the system’s fast oscillations can be
expressed as

$\displaystyle\ddot{z}+2\mu\dot{y}[\mu^{2}y^{5}+5\mu^{2}zy^{4}+(10\mu^{2}z^{2}-\mu)y^{3}+(10\mu^{2}z^{3}-3\mu
z)y^{2}+(5\mu^{2}z^{4}-3\mu z^{2}+1)y$ $\displaystyle+\mu^{2}z^{5}-\mu
z^{3}+z]\dot{z}+\mu[5\mu^{2}zy^{4}+10\mu^{2}y^{3}(z^{2}-\overline{z^{2}})+10\mu^{2}y^{2}(z^{3}-\overline{z^{3}})-3\mu
zy^{2}+$ $\displaystyle 5\mu^{2}y(z^{4}-\overline{z^{4}})-3\mu
y(z^{2}-\overline{z^{2}})+\mu^{2}(z^{5}-\overline{z^{5}})-\mu(z^{3}-\overline{z^{3}})+z]\dot{y}^{2}+[\mu^{2}y^{5}(\dot{z}^{2}-\overline{\dot{z}^{2}})$
$\displaystyle+5\mu^{2}z\dot{z}^{2}y^{4}+10\mu^{2}y^{3}(z^{2}\dot{z}^{2}-\overline{z^{2}}\overline{\dot{z}^{2}})-\mu
y^{3}(\dot{z}^{2}+\overline{\dot{z}^{2}})+10\mu^{2}y^{2}(z^{3}\dot{z}^{2}-\overline{z^{3}}\overline{\dot{z}^{2}})-3\mu
y^{2}z\dot{z}^{2}$
$\displaystyle+5\mu^{2}y(z^{4}\dot{z}^{2}-\overline{z^{4}}\overline{\dot{z}^{2}})-3\mu
y(z^{2}\dot{z}^{2}-\overline{z}^{2}\overline{\dot{z}^{2}})+y(\dot{z}^{2}-\overline{\dot{z}^{2}})+\mu^{2}(z^{5}\dot{z}^{2}-\overline{z^{5}}\overline{\dot{z}^{2}})$
$\displaystyle-\mu(z^{3}\dot{z}^{2}-\overline{z^{3}}\overline{\dot{z}^{2}})+z\dot{z}^{2}]+\alpha[4\mu^{2}zy^{3}+6\mu^{2}y^{2}(z^{2}-\overline{z^{2}})+4\mu^{2}y(z^{3}-\overline{z^{3}})$
$\displaystyle-2\mu
zy+\mu^{2}(z^{4}-\overline{z^{4}})-\mu(z^{2}-\overline{z^{2}})]\dot{y}+[\mu^{2}y^{4}+4\mu^{2}zy^{3}+(6\mu^{2}z^{2}-\mu)y^{2}$
$\displaystyle+(4\mu^{2}z^{3}-2\mu z)y+\mu^{2}z^{4}-\mu
z^{2}+1]\dot{z}+\omega_{0}^{2}[5\mu^{2}zy^{4}+10\mu^{2}y^{3}(z^{2}-\overline{z^{2}})$
$\displaystyle+10\mu^{2}y^{2}(z^{3}-\overline{z^{3}})-3\mu
y^{2}z+5\mu^{2}y(z^{4}-\overline{z^{4}})-3\mu y(z^{2}-\overline{z^{2}})$
$\displaystyle+\mu^{2}(z^{5}-\overline{z^{5}})-\mu(z^{3}-\overline{z^{3}})+z]=[4zy^{3}+b\mu^{2}(z^{2}-\overline{z^{2}})y^{2}+4\mu^{2}(z^{3}-\overline{z^{3}})y-2\mu
zy$
$\displaystyle+\mu^{2}(z^{4}-\overline{z^{4}})-\mu(z^{2}-\overline{z^{2}})](f_{1}\cos\omega_{1}t)+[\mu^{2}y^{4}+4zy^{3}+(6z^{2}\mu^{2}-\mu)y^{2}$
$\displaystyle+(4\mu^{2}z^{3}-2\mu z)y+\mu^{2}z^{4}-\mu
z^{2}+1](f_{2}\cos\omega_{2}t).~{}~{}~{}~{}~{}~{}~{}$ (15)

We observe that Eqs.(14) and (15) are a pair of integro-differential equations
that, when combined, totally solve the composite system Eq.(7) in an average
sense. They describe the equations for the slow oscillations ’$y$’ and the
fast vibrations $z$, respectively. Then, using the assumption that the
component ’$z$’ oscillates considerably faster than the slow component ’$y$’,
we apply the inertial approximation $\ddot{z}>>\dot{z}>>z>>z^{2}$ and treat
the variable ’$y$’ as constant in Eq.(15). Hence Eq.(15) is reduced to

$\ddot{z}=f_{2}cos\omega_{2}t$ (16)

which has a solution

$z=-\dfrac{f_{2}}{\omega_{2}^{2}}cos\omega_{2}t,$ (17)

leading to the mean values

$\overline{z}=\overline{z^{3}}=\overline{z^{5}}=0\mathchar
24635\relax\;~{}~{}~{}~{}~{}\overline{z^{2}}=\dfrac{f_{2}^{2}}{2\omega_{2}^{4}}\mathchar
24635\relax\;$ (18a)
$\overline{z^{4}}=\dfrac{3f_{2}^{4}}{8\omega_{2}^{8}}\mathchar
24635\relax\;~{}~{}~{}~{}~{}\overline{\dot{z}^{2}}=\dfrac{f_{2}^{2}}{2\omega_{2}^{2}}$
(18b)

Using Eq.(18b) in Eq.(14), after simplification it reads as the equation of
motion for the slow component as

$\displaystyle\ddot{y}+\mu[\mu^{2}y^{5}+(10\mu^{2}\overline{z^{2}}-\mu)y^{3}+(5\mu^{2}\overline{z^{4}}-3\mu\overline{z^{2}}+1)]\dot{y}^{2}+\alpha[\mu^{2}y^{4}+(6\mu^{2}\overline{z^{2}}-\mu)y^{2}$
$\displaystyle+\mu^{2}\overline{z^{4}}-\mu\overline{z^{2}}+1]\dot{y}+[\mu(5\mu^{2}\overline{z^{4}}-3\mu\overline{z^{2}}+1)\overline{\dot{z}^{2}}+\omega_{0}^{2}(5\mu^{2}\overline{z^{4}}-3\mu\overline{z^{2}}+1)]y$
$\displaystyle+[\mu(10\mu^{2}\overline{z^{2}}-\mu)\overline{\dot{z^{2}}}+\omega_{0}^{2}(10\mu^{2}\overline{z^{2}}-\mu)]y^{3}+(\mu^{3}\overline{\dot{z}^{2}}+\omega_{0}^{2}\mu^{2})y^{5}$
$\displaystyle=(\mu^{2}y^{4}+(6\mu^{2}\overline{z^{2}}-\mu)y^{2}+\mu^{2}\overline{z^{4}}-\mu\overline{z^{2}}+1)f_{1}\cos\omega_{1}t$
(19)

Keeping in mind the values of various mean values given in Eqs.(18b) and
redefining the quantities by the following quantities,

Figure 3: The effective potential (23) for $\mu=0.5$ and $\omega_{0}=0.25$.

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle(5\mu^{2}\overline{z^{4}}-3\mu\overline{z^{2}}+1)\mathchar
24635\relax\;$ $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle(6\mu^{2}\overline{z^{2}}-\mu)\mathchar 24635\relax\;$
$\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle(10\mu^{2}\overline{z^{2}}-\mu)\mathchar 24635\relax\;$
$\displaystyle C_{4}$ $\displaystyle=$ $\displaystyle
1+\mu^{2}\overline{z^{4}}-\mu\overline{z^{2}},$ (20) $\displaystyle\eta_{1}$
$\displaystyle=$
$\displaystyle(\mu\overline{\dot{z}^{2}}+\omega_{0}^{2})C_{1}\mathchar
24635\relax\;$ $\displaystyle\eta_{2}$ $\displaystyle=$
$\displaystyle(\mu\overline{\dot{z}^{2}}+\omega_{0}^{2})C_{3}\mathchar
24635\relax\;$ $\displaystyle\eta_{3}$ $\displaystyle=$
$\displaystyle\mu^{2}(\mu\overline{\dot{z}^{2}}+\omega_{0}^{2}),$ (21)

the slow oscillations of the system described by Eq.(14) can be written as

$\displaystyle\ddot{y}+\mu(C_{1}y+C_{3}y^{3}+\mu^{2}y^{5})\dot{y^{2}}+\alpha[C_{4}+C_{2}y^{2}+\mu^{2}y^{4}]\dot{y}+\eta_{1}y+\eta_{2}y^{3}+\eta_{3}y^{5}$
$\displaystyle=(C_{4}+C_{2}y^{2}+\mu^{2}y^{4})f_{1}\cos\omega_{1}t.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(22)

Figure 4: The relationship between $f_{2}$ and the response amplitude $Q$. The
solid line represents analytically computed response amplitude and the dotted
line represents the numerical response amplitude for the fixed value of
$\mu=0.5$, $f_{1}=0.05$ $\alpha=0.2$, $\omega_{1}=1.0$ and $\omega_{2}=4.5$.

Consequently, the system’s effective potential is given by

$V_{eff}(y)=\dfrac{\eta_{1}}{2}y^{2}+\dfrac{\eta_{2}}{4}y^{4}+\dfrac{\eta_{3}}{6}y^{6}.$
(23)

Figure 5: Phase-space trajectories for different values of frequencies of the
second periodic force. Panles (a)-(c) represent $f_{2}=5.0,9.0$ and $11.0$,
respectively, with fixed value of $\mu=0.5$, $f_{1}=0.05$ and $\alpha=0.2$.

The linearized equation of motion becomes Eq.(22) by neglecting the nonlinear
components and applying the approximation $f_{1}<<1$ such that
$\mid{y}\mid<<1$ in the long-term limit $t\rightarrow\infty$:

$\ddot{y}+\lambda\dot{y}+\omega_{r}^{2}y=F\cos\omega_{1}t,$ (24)

where the resonant frequency is $\omega_{r}=\sqrt{\eta_{1}}$, $\lambda=\alpha
C_{4}$ and $F=C_{4}f_{1}$. It is clear that the steady state solution of
equation Eq.(22) takes the form $y(t)=A_{L}\cos(\omega_{1}t+\phi)$. We define
the quantity $A_{L}/f_{1}$ as the response amplitude ‘$Q$’. The oscillation
amplitude
$A_{L}=\frac{F}{\sqrt{(\omega_{r}^{2}-\omega_{1}^{2})^{2}+\lambda^{2}\omega_{1}^{2}}}$,
$\phi=\tan^{-1}\bigg{(}\dfrac{\lambda\omega_{1}}{\omega_{1}^{2}-\omega_{r}^{2}}\bigg{)}$
and the response amplitude can be computed as

$Q_{ana}=\dfrac{A_{L}}{f_{1}}=\dfrac{C_{4}}{\sqrt{(\omega_{r}^{2}-\omega_{1}^{2})^{2}+\lambda^{2}\omega_{1}^{2}}}.$
(25)

Using the original definitions of the various quantities occurring in (25),
the response amplitude can be computed as

$\displaystyle
Q_{ana}=\dfrac{1+\frac{3\mu^{2}f_{2}^{4}}{8\omega_{2}8}-\frac{\mu
f_{2}^{2}}{2\omega_{2}4}}{[{((\omega_{0}^{2}+\frac{\mu
f_{2}^{2}}{2\omega_{2}^{2}})(\frac{15\mu^{2}f_{2}^{4}}{8\omega_{2}^{8}}-\frac{3\mu
f_{2}^{2}}{2\omega_{2}^{4}}+1)-\omega_{1}^{2}})^{2}+\alpha^{2}({1+\frac{3\mu^{2}f_{2}^{4}}{8\omega_{2}^{8}}-\frac{\mu
f_{2}^{2}}{2\omega_{2}^{4}}})^{2}\omega_{1}^{2}]^{\frac{1}{2}}}.~{}~{}~{}~{}~{}~{}~{}$
(26)

Now we analyze the VR using Eq.(26) and we verify numerically the above
theoretical results. In order to appreciate the nature of $Q_{ana}$, we
consider the quantity $S={{((\omega_{0}^{2}+\frac{\mu
f_{2}^{2}}{2\omega_{2}^{2}})(\frac{15\mu^{2}f_{2}^{4}}{8\omega_{2}^{8}}-\frac{3\mu
f_{2}^{2}}{2\omega_{2}^{4}}+1)-\omega_{1}^{2}})^{2}}{+\alpha^{2}({1+\frac{3\mu^{2}f_{2}^{4}}{8\omega_{2}^{8}}-\frac{\mu
f_{2}^{2}}{2\omega_{2}^{4}}})^{2}\omega_{1}^{2}}$ in the denominator of
equation (26) and we observe that the qualitative feature of $Q_{ana}$ can be
deduced from the nature of S. That is $Q_{ana}$ reaches the maximum value
provided the value of S is minimum. It is clear from the denominator of
equation (26), the appearance of resonance depends on the system parameters
$\omega_{0}^{2}$, $\mu$, $\alpha$, $f_{2}$, $\omega_{1}$, and $\omega_{2}$.
Now we consider the nonlinear parameter and study its effect. We compute
numerically by analyzing the cases $dS/df_{2}=0$ and $d^{2}S/df_{2}^{2}>0$ at
resonance. The results are depicted in Fig.4.

##  4 Numerical Analysis

Next, the theoretical response amplitude $Q_{ana}$ provided by Eq.(26) has
been compared with the numerical $Q_{num}$ obtained from the Fourier spectrum
of the solution of the velocity-dependent equation represented as coupled
first-order autonomous ordinary differential equations (ODEs) of the following
form:

$\displaystyle\dfrac{dx}{dt}$ $\displaystyle=$ $\displaystyle y,$
$\displaystyle\dfrac{dy}{dt}$ $\displaystyle=$ $\displaystyle(-\mu
x\dot{x}^{2}-\alpha\dot{x}-\omega_{0}^{2}x+f_{1}\cos\omega_{1}t+f_{2}\cos\omega_{2}t)\times$
(27)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1-\mu
x^{2}+\mu^{2}x^{4}).$

To substantiate our analytical study, we numerically integrate Eq.(27) using
the Runge-Kutta Fourth order (RK4) method with step size $\Delta t=0.01t$ over
a simulation of finite time interval $T_{s}=nT$, with $T=2\pi/\omega_{1}$
being the period of oscillations, where the amplitude $f_{1}$ and frequency
$\omega_{1}$ belong to the slow component and the amplitude $f_{2}$ and
frequency $\omega_{2}$ are considered as belonging to the fast component, and
n=1,2,3… is the number of complete oscillations. For our computation we fix
the system parameter values as $\mu=0.5$, $\omega_{0}^{2}=0.25$, $\alpha=0.2$,
$f_{1}=0.05$ and $\omega_{1}=1.0$. The remaining factors, $f_{2}$ and
$\omega_{2}$, are selected in a way that promotes the emergence of VR.

Figure 6: The corresponding response curve Q and the bifurcation diagram of
the displacement $x(t)$ for $\omega_{1}=1.5$. The remaining parameters are
fixed as $\mu=0.5$, $f_{1}=0.05$ and $\alpha=0.2$. Figure 7: The relationship
between $f_{2}$ and the response amplitude $Q$. The solid lines represent the
response amplitudes for different values of $\mu$ and that curves for
$\mu=0.1,0.5,1.0\text{~{}and~{}}1.5$ are represented by different colors.
Figure 8: The dependence of response amplitude $Q$ with $f_{2}$. The
analytical and numerical results are represented by the solid and dotted line
curves, respectively, for different value of $\alpha$. Figure 9: The
relationship between $f_{2}$ and the response amplitude $Q$. The solid line
represents analytically computed response amplitude and the dotted line
represents the numerical response amplitude for different values of
$\omega_{1}$ and that curves for $\omega_{1}=0.8,1.0$ and $1.5$ are
represented by different colors.

Figure 10: Two-parameter phase diagram for $\mu$ vs second force $f_{2}$ for
fixed $\omega_{1}=0.1,0.5,1.0,1.5$: the maximum response amplitude $Q$ as
indicated in the figure.

After solving numerically Eqs.(27) and using the results in the following
expression for the response function after discarding the transients, we
obtain landa2000vibrational

$Q_{num}=\dfrac{\sqrt{\rule{0.0pt}{8.61108pt}Q_{s}^{2}+Q_{c}^{2}}}{f_{1}},$
(28)

over a range of values of the forcing strength of high frequency driving force
($f_{2}$). Here the values of the quantities $Q_{s}$ and $Q_{c}$ are computed
from the Fourier spectrum of the time series of the output signal $x(t)$ as

$\displaystyle Q_{c}=\dfrac{2}{nT}\int_{0}^{nT}x(t)\cos\omega tdt,$
$\displaystyle Q_{s}=\dfrac{2}{nT}\int_{0}^{nT}x(t)\sin\omega tdt.$ (29)

The response curves of $Q$ versus selected parameters for a variety of system
parameters are superposed to compare the numerically calculated $Q$ with the
previously obtained analytical results. The maximum of each of the analytical
curves well matches with the numerically computed results and one can observe
that the VR is clearly indicated by an adequate difference between the low and
high frequencies.

We begin by examining the occurrence of VR in the response curve of the system
as given by Eq.(28) above. We fix the parameters as below: $\mu=0.5$,
$\omega_{0}^{2}=0.25$, $\alpha=0.2$, $f_{1}=0.05$, $\omega_{1}=1.0$, and
$\omega_{2}=4.5$. Fig.4 shows the response amplitude $Q$ of the system
depending on the amplitude of high-frequency drive forcing strength $f_{2}$.

It is obvious from Fig.4 that as the fast driving force $f_{2}$ increases, the
response amplitude increases and it reaches a maximum value around
$f_{2}=9.0$. Further, as we increase the value of $f_{2}$, the response
amplitude $Q$ starts decreasing. The analytically calculated response
amplitude from Eq.(26), shown as a solid line for a set of system parameter
values, and Eq.(29), which is computed directly from the system’s main
equation, is compared where the corresponding numerical response amplitude
shown by broken/dotted lines[see Fig.4 ].

To investigate further, we consider the phase trajectories of the system.
These are illustrated in Figs.5. In Fig.5(a) the phase space of the
trajectories corresponds to a one well structure. Further, on increasing the
forcing parameter to the value $f_{2}=9.0$, the attractor of the system
expands in the phase space and consequently the resonance amplitude grows
until it achieves its highest $Q$ value [see Fig.5(b)]. On further increase of
$f_{2}$ values, the expansion of the attractor decreases and the resonance
amplitude also gets decreased [see Fig.5(c)]. Consequently, it is observed in
Fig.4 that the results are consistent with theoretical and numerical studies.

By investigating the underlying dynamics and, in particular, the bifurcation
structure, we can now attempt to comprehend the mechanism of VR in the present
system. Resonance curves in nonlinear systems are well known to be closely
related to the underlying bifurcation structure. In particular, earlier
studies support the notion that symmetry-breaking (sb) and Hopf bifurcations
occur between resonances roy2016analysis ; roy2017vibrational . In this
article, we present a new dynamical mechanism linked to resonance. Figs.6(a-b)
show the response curve Q and bifurcation diagram, respectively, obtained by
increasing the forcing strength $f_{2}$. The response curve $Q$ attains the
maximum value just when the period doubling bifurcation takes place at
$f_{2}=11.766$ on increasing the value of $f_{2}$. The response curve $Q$
first increases exponentially, and then attains a maximum at the period-
doubling bifurcation point ($f_{2}=11.766$). Thus, the maximum in Q signalling
VR clearly originates from the period-doubling bifurcation, thereby we confirm
that VR is linked to the bifurcation of the attractors.

To investigate how nonlinearity affects the detected vibrational resonance,
the response amplitude $Q$ for different values of $f_{2}$ on the value of the
nonlinearity ($\mu=0.1,0.5,1.0,1.5$) of the system is presented in Fig.7. It
is interesting to note that decreasing the strength of the nonlinearity
significantly enhances the response amplitude $Q$ and it reaches a maximum
amplitude for $\mu<0.1$. Below this value of $\mu$, the Q value remains
unchanged. Enhancing the vibrational frequency in the nonlinear system
requires an understanding of the role of the damping coefficient $\alpha$. In
our study, on varying the damping term $\alpha$ ($\alpha=0.2,0.3$, and $0.5$),
our findings demonstrate that the response amplitude $Q$ diminishes as the
damping term $\alpha$ is increased (see Fig. 8).

To gain more insight on the VR phenomenon in the present mechanical model, the
response amplitude $Q$ for different ‘$\omega_{1}$’ values is investigated for
various values of the strength of high-frequency force $f_{2}$. It is observed
that on increase in the forcing strength of high-frequency causes a decrease
in the response amplitude [see Fig.9].

Fig.10 is a 3-dimensional plot which depicts the numerically computed response
as a function of the nonlinearity $\mu$ and the forcing strength $f_{2}$ of
high frequency for different values of $\omega_{1}$
($\omega_{1}=0.1,0.5,1.0,1.5$). It is clearly demonstrated in Fig.10 that the
response amplitude ‘$Q$’ is high for small values of nonlinearity $\mu$.
Increasing nonlinearity value decreases the response amplitude. Further
increase in the value of ‘$\omega_{1}$’ also suppresses the response
amplitude.

##  5 Conclusion

In this paper, we have examined the vibrational resonance (VR) exhibited by a
particle moving on a rotating parabola with additional damping force and
subjected to combined low-frequency and high-frequency driving forces using
analytical and numerical studies. It is found that nonlinearity contributes
significantly to the occurrence of VR. The origin and mechanism of vibrational
resonance in the present system are identified as corresponding to period
doubling bifurcation and it has been confirmed numerically. Usually, in many
nonlinear systems, the VR stretches along the nonlinearity, provided the other
parameters are appropriately chosen. But in the present model, nonlinearity
plays a vital role in the occurrence of VR. It is an established fact that the
presence of nonlinearity induces secondary resonances apart from primary
resonances. In most of the cases, the nonlinearity is essential for the
occurrence of VR. However, the changes in the values of the nonlinear
parameter does not make a big difference in the emergence of VR. In the
present study, we have shown that the values of the nonlinear parameter in the
present nonpolynomial system significantly contributes to the emergence of VR.
In particular, we have observed that VR occurs in an optimal range of the
nonlinear parameter.

It is established in our study that the induction or control of VR in a
nonpolynomial system is due to an appropriate range of a nonlinear system
parameter. Our results, while confirming the basic features of VR as
identified in the case of PDM Duffing roy2021vibrational  and Morse
oscillators abirami2017vibrational , specifically brings out novel features on
the effect of nonlinearity.

Also, the nonlinearity is represented by the semi-latus rectum of the rotating
parabola and is also part of the angular velocity of the parabola. Thus, the
appropriate range of the nonlinear strength in the rotating parabola
mechanical system will cause resonance in the system. From a practical point
of view, this will lead to the malfunctioning or destruction of parts of the
rotating parabola machines, like centrifugation equipment, industrial hoppers,
etc. Thus, the present analysis of VR in a particle moving on a rotating
parabola is necessary from a technological point of view also. Further, the
model is analyzed as a potential-dependent mass (PDM) system, where the mass
is defined as the strength of nonlinearity. Thus, the study of nonlinearity in
the system reveals the role played by the PDM of the system.

Acknowledgment

A.V. expresses his gratitude to the DST-SERB for funding a research project
with Grant No.EMR/2017/002813. Sincere appreciation is extended by M.S. to the
Council of Scientific & Industrial Research, India for funding his fellowship
via SRF Scheme No.08/711(0001)2K19-EMR-I. A.V. additionally thanks the DST-
FIST for funding research projects via Grant No.SR/FST/College-2018-372(C).
M.L. thanks the DST-SERB National Science Chair program for funding under
Grant No.NSC/2020/000029.

### Author contributions

RK contributed in setting general idea and implementation its analytical
analysis as well as in writing manuscript draft. MS contributed in numerical
simulations as well as in writing manuscript draft. AV and ML participated in
choosing the methods for problem treatment and presentation of the simulation
results as well as writing the paper draft.

Data availability The datasets generated during and/or analyzed during the
current study are available from the corresponding author on reasonable
request.

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