The nonlinear dynamic features of compression roller batteries were investigated together with their nonlinear response to primary resonance excitation and to internal interactions between modes. Starting from a parametric nonlinear model based on a previously developed Lagrangian formulation, asymptotic treatment of the equations of motion was first performed to characterize the nonlinearity of the lowest nonlinear normal modes of the system. They were found to be characterized by a softening nonlinearity associated with the stiffness terms. Subsequently, a direct time integration of the equations of motion was performed to compute the frequency response curves (FRCs) when the system is subjected to direct harmonic excitations causing the primary resonance of the lowest skewsymmetric mode shape. The method of multiple scales was then employed to study the bifurcation behavior and deliver closedform expressions of the FRCs and of the loci of the fold bifurcation points, which provide the stability regions of the system. Furthermore, conditions for the onset of internal resonances between the lowest roller battery modes were found, and a 2:1 resonance between the third and first modes of the system was investigated in the case of harmonic excitation having a frequency close to the first mode and the third mode, respectively.
Aerial ropeways are transportation systems which are becoming increasingly popular in recent years, not only in mountain regions with ski resorts and in sightseeing areas, but also in urban environments [
On the other hand, little attention was paid to the study of the local dynamic interactions taking place between the vehicle, the cable, the roller battery, and the supporting tower; few works have investigated these aspects [
To the best of the author’s knowledge, there are no publications dealing with the characterization of the nonlinear dynamic behavior of the ropeways roller batteries or aimed at investigating their dynamic response to periodic excitations. The present work aims at filling the lack of knowledge regarding the characterization of the nonlinear dynamic features of roller battery systems and to investigate the effects of the periodic excitations induced by the moving cable that may give rise to resonance phenomena and nonlinear modal interactions. These phenomena were studied in the past in different mechanical systems, such as cranes [
By employing the method of multiple scales [
The proposed mechanical model of compression roller batteries is based on the formulation proposed, for the first time, in [
The whole modeling procedure, including all relevant details of the underlying complex mechanical behavior, such as the effects of the vehicle grip on the rollers or the periodic forcing caused by the interwire spacing on the outer layer of the monocable rope, was derived and extensively discussed in [
The kinematics are described in the plane containing the direction collinear with the cable configuration as well as the vertical direction. In particular, two fixed frames are introduced, namely, the frame
The roller battery degrees of freedom (DOFs) are the rotations
To discretize the time and spacedependent function
On the other hand, due to the multiple contact points
The unilateral contacts at points
The equations governing the motion of the roller battery system, including the interaction between the hoisting beam and the cable are derived via the EulerLagrange approach. To this end, the potential and kinetic energies of the beam and cable are first calculated together with the potential energy of the fictitious springs and the kinetic energy of the rollers to calculate the Lagrangian of the system, from which the nonlinear equations of motion of the roller battery are obtained. In particular, the potential energy of the hoisting beam and the cable can be expressed as:
On the other hand, the hoisting beam and the cable kinetic energy contributions to the dynamics of the system are given by:
The simplifications made in the model formulation proposed in [
The motion equations are then obtained via a Lagrangian approach. In particular, by introducing the Rayleigh dissipation function, and by calculating the Lagrangian of the system
The equations of motion are then nondimensionalized by adopting the characteristic time
The equations of motion reported in Equation (
Eigenvalue analysis is first performed to characterize the fundamental modes of the system. By linearizing Equation (
The
Interesting considerations emerge from the analysis of the ratios between the lowest frequencies of the system.
It is a matter of fact that the internal resonances may more easily be activated between modes having lower frequencies, such as in the case of the mechanical system investigated, shown by the first and third mode of the roller battery (see red bar and red line in
In the system investigated in this work, the cableinduced periodic excitation has a frequency dependent on the cable speed which never exceeds the nondimensional value of
The method of multiple scales [
By letting
The perturbation treatment requires a preliminary rescaling of the viscous damping force and of the excitation term; therefore, since the nonlinearities of the system are such that resonant terms are generated at the cubic order. The linear viscous damping force is assumed to appear at the third order; this, analytically implies a rescaling of the viscous damping matrix as
order
By setting
First and secondorder solutions given by Equations (
Equation (
After removing the secular terms, the solution of the inhomogeneous thirdorder problem can be calculated as:
To study the nonlinearity of the
By defining the relative phase
The function
The nonlinearity of the
The same asymptotic procedure, based on the method of multiple scales, can be used to investigate the nonlinearity of the
To study the nonlinearity of the
The solution of the firstorder problem can be written as
At the third order, the singularity generated by the resonant terms is removed when the modulation equations, having the same form of Equation (
After removing the secular terms by superimposing Equation (
To study the nonlinearity of the
The backbone curves of the lowest four nonlinear normal modes, that is, the nonlinear relationships between the modal amplitude
The results reported in
In this section the nonlinear dynamic response of roller batteries subjected to periodic excitation provided by the forces induced by the moving cable will be investigated. In particular, the focus is first devoted to the analysis of the primary resonance of the lowest mode (i.e., when
First and secondorder solutions of the hierarchical systems of equations, Equations (
To investigate the primary resonance of the
By expressing the complex amplitudes in polar form and separating the real and imaginary parts, the modulation equations in terms of the real amplitude and phase read:
By solving Equation (
The equation that allows for describing the bifurcation behavior of the system is given by
Finally, after removing the secular terms by means of Equation (
Thereafter, the nonlinear equations of motion of the system expressed by Equation (
In this section, we investigate the phenomenon of the twotoone internal resonance between two modes of the roller battery occurring when one mode of the system undergoes primary resonance due to the periodic excitation of the moving cable.
Without loss of generalization, the two resonant modes are indicated as mode
The twotoone internal resonance is caused by the quadratic nonlinearities of the system; therefore, the asymptotic treatment of Equation (
In this case, the resonant terms are generated at the quadratic order, therefore, the viscous damping force and excitation need to be rescaled so as to appear at the second perturbation order; hence, we set
order
In the case of two interacting modes (i.e., mode
To express the closeness of the two interacting frequencies, the detuning parameter
Finally, after removing the secular terms, the solution of the secondorder inhomogeneous problem can be calculated as:
In the case of primary resonance on mode
The system of four nonlinear algebraic equations (
On the other hand, in the range of low values of the excitation amplitude
For the sake of rigor, it is worth noting that the ratio between the frequency of the third and first mode of the roller battery is not exactly equal to two. In particular,
In the case of the primary resonance on mode
The force response curves shown in
The force response curves and the frequency response curves depicted in
In this work, the nonlinear dynamic behavior of roller batteries was characterized, investigated, and discussed for the first time in the literature. An accurate analytical treatment of the equations of motion was proposed to highlight the nonlinearities of the system and the method of multiple scales was employed to deliver closedform expressions of the roller battery nonlinear dynamic responses. The nonlinearity of the lowest normal modes of the system was clearly identified and showed to be of the softening type. The simplification of the asymptotic treatment, obtained through the onemode projection of the equations of motion, shed light onto the effects of the system nonlinearities and to prove that the stiffness nonlinearities govern the dynamic behavior of the roller battery.
The effects of the periodic direct excitation deriving from the cable motion were investigated by studying the primary resonance of the lowest normal mode of the roller battery. It was shown that, for moderately high values of the excitation, the nonlinear response was multistable in a wide range of excitation frequencies. The analytical results were also validated through numerical time integration of the nonlinear equations of motion. Moreover, the instability regions, whose boundaries were analytically obtained in closed form, were parametrically investigated so as to discuss the role of the structural damping, and for demonstrating that higher values of the damping coefficient reduce the region where unstable responses occur.
Furthermore, the modal characterization of the system allowed us to discover the potential scenarios of modal interactions and to investigate the phenomenon of internal resonance between two modes having frequency ratios close to two (i.e., the third and first mode) both in the case of primary resonance of the lowest mode and of the higher mode, respectively. A stability analysis of the fixed points of the modulation equations for the two mode amplitudes and phases allowed us to describe the bifurcation scenarios both for the singlemode and the coupledmode responses. It was shown that, although internal resonances were possible in the system, in order to cause largeamplitude coupledmode oscillations, the amplitude of the periodic excitation must be orders of magnitude higher and sufficient to induce large oscillations in the primary resonance of the lowest mode.
This research received no external funding.
A.A. gratefully acknowledges partial support through the Assistant Professor Sapienza Fellowship program. The project ”Dynamics of towersfeasibility of vibration absorbers” supported by POMA is gratefully acknowledged.
The author declares no conflict of interest.
(
Shapes of the lowest eight linear normal modes of the roller battery system.
Ratios of the lowest eight frequencies of the system with respect to (
(
Coefficients of the inertial (
Primary resonance: frequency response curves of the vertical displacements of points (
(
(
(
(
(
(
Lowest eight nondimensional circular frequencies.
Mode 
1  2  3  4 

0.376  0.581  0.734  0.96 
Mode 
5  6  7  8 

1.02  1.109  1.163  1.757 
Nonlinearity coefficient
All  Inertia  Velocity  Stiffness  




















