Amplitude-modulated Motions Research Articles

Vibration Control of Horizontally Excited Structures Utilizing Internal Resonance of Liquid Sloshing in Nearly Square Tanks

Passive control of vibrations in an elastic structure subjected to horizontal, harmonic excitation by utilizing a nearly square liquid tank is investigated. When the natural frequency ratio 1:1:1 is satisfied among the natural frequencies of the structure and the two predominant sloshing modes (1,0) and (0,1), the performance of a nearly square tank as a tuned liquid damper (TLD) is expected to be superior to rectangular TLDs due to internal resonance. In the theoretical analysis, Galerkin's method is used to determine the modal equations of motion for liquid sloshing considering the nonlinearity of sloshing. Then, van der Pol's method is used to obtain the expressions for the frequency response curves for the structure and sloshing modes. Frequency response curves and bifurcation set diagrams are shown to investigate the influences of the aspect ratio of the tank cross section and the tank installation angle on the system response. From the theoretical results, the optimal values of the system parameters can be determined in order to achieve maximum efficiency of vibration suppression for the structure. Hopf bifurcations occur and amplitude modulated motions (AMMs) may appear depending on the values of the system parameters. Experiments were also conducted, and the theoretical results agreed well with the experimental data.

Vibration Control of Two-Degree-of-Freedom Structures Utilizing Sloshing in Nearly Square Tanks

This paper investigates the vibration control of a towerlike structure with degrees of freedom utilizing a square or nearly square tuned liquid damper (TLD) when the structure is subjected to horizontal, harmonic excitation. In the theoretical analysis, when the two natural frequencies of the two-degree-of-freedom (2DOF) structure nearly equal those of the two predominant sloshing modes, the tuning condition, 1:1:1:1, is nearly satisfied. Galerkin's method is used to derive the modal equations of motion for sloshing. The nonlinearity of the hydrodynamic force due to sloshing is considered in the equations of motion for the 2DOF structure. Linear viscous damping terms are incorporated into the modal equations to consider the damping effect of sloshing. Van der Pol's method is employed to determine the expressions for the frequency response curves. The influences of the excitation frequency, the tank installation angle, and the aspect ratio of the tank cross section on the response curves are examined. The theoretical results show that whirling motions and amplitude-modulated motions (AMMs), including chaotic motions, may occur in the structure because swirl motions and Hopf bifurcations, followed by AMMs, appear in the tank. It is also found that a square TLD works more effectively than a conventional rectangular TLD, and its performance is further improved when the tank width is slightly increased and the installation angle is equal to zero. Experiments were conducted in order to confirm the validity of the theoretical results.

Nonlinear responses of spherical pendulum vibration absorbers in towerlike 2DOF structures

This paper investigates the vibration control of a flexible towerlike structure using a single spherical pendulum vibration absorber (SPVA). The SPVA is attached to the structure, which is modeled as a two-degree-of-freedom system, subjected to horizontal, harmonic excitation. In the theoretical analysis, the nonlinearity of the restoring moment of the SPVA is considered, and new coordinates are implemented to avoid the non-integrability of the equations of motion due to a singularity. Van der Pol’s method is used to determine the frequency response curves for the structure and SPVA. When the excitation direction \(\alpha \) is \(0^{\circ }\), the structure is only excited in the x-direction; however, the structure vibrates in both the x- and y-directions due to nonlinear coupling between the structure and SPVA, known as autoparametric resonance. Clockwise and counterclockwise whirling motions appear in the structure and SPVA depending on the excitation frequency. On part of the branches of the response curves, Hopf bifurcations occur followed by amplitude-modulated motions, and combination resonances are also observed in the numerical simulations. When \(\alpha =45^\circ \), the structure is equally excited in the x- and y-directions; however, the vibrations in these directions appear at different amplitudes in the frequency response curves. Such phenomena also occur in the SPVA. The influence of the spring constant of the structure in the y-direction on the frequency response curves is also investigated. Experiments were conducted, and their results confirmed the validity of the theoretical analysis.

Localization phenomena in annular-type oscillator arrays

This paper investigates intrinsic localized modes (ILMs) in a system where N nonlinear oscillators are arranged in a circle and connected by weak springs when the system is subjected to sinusoidal, harmonic excitation torque. These oscillators consist of point masses at the tip of pinned, massless rigid rods, nonlinear springs, and dashpots. In the theoretical analysis, van der Pol’s method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves for N=3 are shown in the cases of both soft-spring and hard-spring nonlinear arrays and compared with the results of the numerical simulations. As a result, in the case of the soft-spring nonlinear arrays, eight stable oscillation patterns may appear when the connecting spring constants are comparatively small, and ILMs appear in six patterns depending on the initial conditions and the excitation frequency. The number of the oscillation patterns decreases as the connecting spring constants are increased. Comparatively, small values of the connecting spring constants may cause Hopf bifurcation to appear followed by amplitude modulated motions (AMMs), including regular ILMs, moving ILMs, and chaotic vibrations. In the case of the hard-spring nonlinear arrays, similar phenomena are observed except for moving ILMs. ILMs do not appear when the values of connecting springs exceed specific values in the both cases.

Internal resonance of nonlinear sloshing in rectangular liquid tanks subjected to obliquely horizontal excitation

Nonlinear sloshing in rectangular tanks subjected to obliquely horizontal, harmonic excitation is investigated when the internal resonance condition 1:1 is satisfied between the natural frequencies of predominate modes (1, 0) and (0, 2). Galerkin’s method is employed to derive the nonlinear modal equations of motion for sloshing, considering nine sloshing modes. Then, van der Pol’s method is applied in order to obtain the expressions of the frequency response curves for amplitudes and phase angles of the predominate modes. The frequency response curves are calculated and reveal that (0, 2) mode may occur even though it is not directly excited because it is nonlinearly coupled with (1, 0) mode due to the autoparametric terms. In the numerical simulations, it is found that planar motions of (1, 0) mode, clockwise and counter-clockwise swirl motions, and translational motions may appear. Furthermore, Hopf bifurcation occurs, and amplitude modulated motions (AMMs), including chaotic motions, may appear depending on the value of the excitation frequency. Three-dimensional distribution charts of the maximum liquid surface elevation are calculated to show the risk of liquid overspill. The influence of the difference between the horizontal excitation direction and the tank side on the frequency response curves is also examined. Bifurcation sets are calculated to clarify this influence. Experimental data confirmed the validity of the theoretical results.

Intrinsic Localized Modes of Principal Parametric Resonances in Pendulum Arrays Subjected to Vertical Excitation

Intrinsic localized modes (ILMs) are investigated in an N-pendulum array subjected to vertical harmonic excitation. The pendula behave nonlinearly and are coupled with each other because they are connected by torsional, weak, linear springs. In the theoretical analysis, van der Pol's method is employed to determine the expressions for frequency response curves for the principal parametric resonance, considering the nonlinear restoring moment of the pendula. In the numerical results, frequency response curves for N = 2 and 3 are shown to examine the patterns of ILMs, and demonstrate the influences of the connecting spring constants and the imperfections of the pendula. Bifurcation sets are also calculated to show the excitation frequency range and the conditions for the occurrence of ILMs. Increasing the connecting spring constants results in the appearance of Hopf bifurcations. The numerical simulations reveal the occurrence of ILMs with amplitude modulated motions (AMMs), including chaotic motions. ILMs were observed in experiments, and the experimental data were compared with the theoretical results. The validity of the theoretical analysis was confirmed by the experimental data.

Intrinsic localized modes of 1/2-order subharmonic oscillations in nonlinear oscillator arrays

1/2-order subharmonic oscillations in an array with \(N\) nonlinear, identical oscillators are theoretically investigated. Each oscillator is connected by weak, linear springs and subjected to sinusoidal excitation. Van der Pol’s method is employed to determine the frequency response curves for 1/2-order subharmonic oscillations when \(N=2\) and 3. Frequency response curves for hard- and soft-type nonlinearities of the oscillators are examined and compared with each other. All patterns of oscillations are classified depending on the results of the response curves, and it is determined in which pattern intrinsic localized modes (ILMs) appear. When disturbances are input in order to observe ILMs, basins of attraction demonstrate which pattern of ILMs may occur. Bifurcation sets are also calculated to examine the influence of the connecting spring constants on the response curves and the occurrence of Hopf bifurcations and amplitude-modulated motions (AMMs), including chaotic vibrations. Furthermore, the influence of oscillator imperfections on the response curves is investigated by slightly changing the value of the spring constant. The numerical simulations for \(N=2\) and 3 confirm the validity of the frequency response curves and the occurrence of AMMs. ILMs were also observed in the numerical simulations conducted for \(N=10\).

Intrinsic Localized Modes of Harmonic Oscillations in Pendulum Arrays Subjected to Horizontal Excitation

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant above which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.

Intrinsic Localized Modes of Harmonic Oscillations in Nonlinear Oscillator Arrays

Intrinsic localized modes (ILMs) are investigated in an array with N Duffing oscillators that are weakly coupled with each other when each oscillator is subjected to sinusoidal excitation. The purpose of this study is to investigate the behavior of ILMs in nonlinear multi-degree-of-freedom (MDOF) systems. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are shown for N = 2 and 3 and compared with the results of the numerical simulations. Basins of attraction are shown for a two-oscillator array with hard-type nonlinearities to examine the possibility of appearance of ILMs when an oscillator is disturbed. The influences of the connecting springs for both hard- and soft-type nonlinearities on the appearance of the ILMs are examined. Increasing the values of the connecting spring constants may cause Hopf bifurcation followed by amplitude modulated motion (AMM) including chaotic vibrations. The influence of the imperfection of an oscillator is also investigated. Bifurcation sets are calculated to show the influence of the system parameters on the excitation frequency range of ILMs. Furthermore, time histories are shown for the case of N = 10, and many patterns of ILMs may appear depending on the initial conditions.

Nonlinear Dynamic Responses of Elastic Structures With Two Rectangular Liquid Tanks Subjected to Horizontal Excitation

Nonlinear vibrations of an elastic structure with two partially filled liquid tanks subjected to horizontal harmonic excitation are investigated. The natural frequencies of the structure and sloshing satisfy the tuning condition 1:1:1 when tuned liquid dampers are used. The equations of motion for the structure and the modal equations of motion for the first, second, and third sloshing modes are derived by using Galerkin’s method, taking into account the nonlinearity of the sloshing. Then, van der Pol’s method is employed to determine the frequency response curves. It is found in calculating the frequency response curves that pitchfork bifurcation can occur followed by “localization phenomenon” for a specific excitation frequency range. During this range, sloshing occurs at different amplitudes in the two tanks, even if the dimensions of both tanks are identical. Furthermore, Hopf bifurcation may occur followed by amplitude- and phase-modulated motions including chaotic vibrations. In addition, Lyapunov exponents are calculated to prove the occurrence of both amplitude-modulated motions and chaotic vibrations. Bifurcation sets are also calculated to show the influence of the system parameters on the frequency response. Experiments were conducted to confirm the validity of the theoretical results. It was found that the theoretical results were in good agreement with the experimental data.

Non-linear dynamic responses of elastic two-story structures with partially filled liquid tanks

Abstract This paper deals with the non-linear vibrations of an elastic two-story structure with two liquid tanks installed under horizontal harmonic excitation. The influence of the configuration of the two rectangular tanks on the response of the structure is investigated. In the theoretical analysis, Galerkin's method is applied to derive the equations of motion for the structure and the modal equations for sloshing, while considering the non-linear liquid forces. Then, van der Pol's method is used to determine the frequency response curves. Three cases are investigated: In the first case two tanks are installed, one on the top and one on the second story of the structure, in the second case one tank is installed on top, and in the third case two tanks are installed on top. The theoretical results of the first case are compared with those of the second and third cases. In the numerical calculations, it is found that Hopf bifurcations occur near the tuning frequency and then amplitude modulated motion appears in both the first and third cases. It is thus concluded that multiple tanks yield less effectiveness in suppressing the vibrations of the structure. The experimental data confirm the validity of the theoretical results for the first and third cases.

Autoparametric interaction of a liquid surface in a rectangular tank with an elastic support structure under 1:1 internal resonance

Autoparametric interaction of a liquid free surface in a rectangular tank with an elastic support structure, which is subjected to vertical excitation, is investigated. When the natural frequency of the structure is equal to the lowest natural frequency of liquid sloshing, this system is categorized as an autoparametric system with an internal resonance ratio 1:1. The structure is elastically supported so there is a higher possibility that the 1:1 internal resonance can be observed. The nonlinear theoretical analysis is conducted for a fluid assumed to be perfect in a tank with a finite liquid depth. The equations of motion for the first three sloshing modes are derived employing Galerkin’s technique and considering both the nonlinearity of the fluid motion, and the viscous damping effect. Then the theoretical frequency response curves for the harmonic oscillations of the structure and sloshing are determined using van der Pol’s method. The frequency response curves show that high amplitudes of the structure’s vibrations facilitate the liquid sloshing. Furthermore, the influence of the internal detuning parameter is investigated by showing the frequency response curves and bifurcation sets. Hopf bifurcations may occur followed by amplitude-modulated motions. The theoretical results are in quantitative agreement with the experimental data.

Nonlinear Vibrations of Elastic Structures Containing a Cylindrical Liquid Tank under Vertical Excitation

This study investigates the nonlinear vibrations of an elastic structure, with a liquid-filled cylindrical tank, which is subjected to a vertical sinusoidal excitation. This structure-tank system behaves as an autoparametric system. Modal equations governing the coupled motions of the structure and liquid sloshing are derived when the natural frequency of the structure is equal to twice the natural frequency of the first axisymmetric sloshing mode. Van der Pol's method is applied to the modal equations to determine the theoretical resonance curves. The theoretical results can be concluded as follows: (1) As the liquid level decreases, the resonance curve for the liquid sloshing changes from a soft spring type to a hard spring type. (2) The structure's resonance curve flattens out at small amplitude when the liquid level is appropriate. (3) Amplitude-modulated motions appear for the negative and positive values of the internal resonance ratio's deviation (the detuning parameter) in the high and low liquid levels, respectively. (4) Furthermore, the results of the bifurcation analysis, Poincare maps and Lyapunov exponents reveal that amplitude-modulated motions and chaotic oscillations can occur in the system. In experiments, the theoretical resonance curves were quantitatively in agreement with the experimental data.

BIFURCATION AND AMPLITUDE MODULATED MOTIONS IN A PARAMETRICALLY EXCITED TWO-DEGREE-OF-FREEDOM NON-LINEAR SYSTEM

The non-linear response of a T-shaped beam–mass structure is investigated theoretically and experimentally for the case of one-to-two internal resonance and principal parametric resonance of the lower mode. The method of multiple scales is used to determine four first order amplitude- and phase-modulation equations. The non-trivial steady state solutions are obtained from trivial solutions through pitchfork bifurcation. The Melnikov's method is used to predict the critical parameter at which the dynamical system possesses a Smale horseshoe type of chaos. To verify the analytical results, experiments were performed on the T-shaped beam–mass structure. The periodically amplitude-modulated motions and chaotically amplitude-modulated motions were observed during experiments. The results of the experiment showed good qualitative agreement with the theoretical predictions.

Amplitude modulated motions in a two degree-of-freedom system with quadratic nonlinearities under parametric excitation: experimental investigation

We conducted an experimental investigation of amplitude modulated response of a two degree-of-freedom mechanical structure with quadratic nonlinearities to parametric excitation. The linear natural frequencies of the system were tuned so that they were approximately in the ratio of two-to-one, and the excitation frequency was in principal parametric resonance with the first mode. We observed periodically amplitude-modulated motions and chaotically amplitude-modulated motions.

Amplitude modulated dynamics and bifurcations in the resonant response of a structure with cyclic symmetry

Periodic structures with cyclic symmetry are often used as idealized models of physical systems and one such model structure is considered. It consists ofn identical particles, arranged in a ring, interconnected by extensional springs with nonlinear stiffness characteristics, and hinged to the ground individually by nonlinear torsional springs. These cyclic structures that, in their linear approximations, are known to possess pairwise double degenerate natural frequencies with orthogonal normal modes, are studied for their forced response when nonlinearities are taken into account. The method of averaging is used to study the nonlinear interactions between the pairs of modes with identical natural frequencies. The external harmonic excitation is spatially distributed like one of the two modes and is orthogonal to the other mode. A careful bifurcation analysis of the amplitude equations is undertaken in the case of resonant forcing. The response of the structure is dependent on the amplitude of forcing, the excitation frequency, and the damping present. For sufficiently large forcing, the response does not remain restricted to the directly excited mode, as both the directly excited and the orthogonal modes participate in it. These coupled-mode responses arise due to pitchfork bifurcations from the single-mode responses and represent traveling wave solutions for the structure. Depending on the amount of damping, the coupled-mode responses can undergo Hopf bifurcations leading to complicated amplitude-modulated motions of the structure. The amplitude-modulated motions exhibit period-doubling bifurcations to chaotic amplitude-modulations, multiple chaotic attractors as well as “crisis”. The existence of chaotic amplitude dynamics is related to the presence of Sil'nikov-type conditions for the averaged equations.

Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam

The nonlinear dynamics of a clamped-clamped/sliding inextensional elastic beam subject to a harmonic axial load is investigated. The Galerkin method is used on the coupled bending-bending-torsional nonlinear equations with inertial and geometric nonlinearities and the resulting two second order ordinary differential equations are studied by the method of multiple time seales and by direct numerical integration. The amplitude equations are analyzed for steady and Hopf bifurcations. Depending on the amplitude of excitation, the damping and the ratio of principal flexural rigidities, various qualitatively distinct frequency response diagrams are uncovered and limit cycles and chaotic motions are found. In the truncated two-degree-of-freedom system the transition from periodic to chaotic amplitude-modulated motions is via the process of torus doubling and subsequent destruction of the torus.

Complex in-line and whirling response of structures to oscillatory flow

A two-degree-of-freedom model of offshore structures is studied for its dynamic response in a steady-current and regular-wave environment. The two degrees of freedom represent motions in-line with, and orthogonal to, the current direction. The hydrodynamic forces on the structure are modeled by the modified Morison equation and, additionally, a non-linear symmetric hardening spring is included in the model. For “large inertia” structures, both the cases of coincident and non-coincident current and wave directions are investigated. By using the shooting technique, the steady state periodic response is numerically obtained over a wide range of wave frequencies. Floquet theory is utilized to evaluate the stability of these solutions. Non-periodic steady state solutions, arising due to instabilities of the periodic responses, are obtained by direct time integration. For large damping and drag coefficients and for coincident current and wave directions, the only response possible is in the direction of the current. As the wave direction deviates from the current direction, the response becomes bidirectional, and both the in-line and the orthogonal components exhibit primary and superharmonic resonances. When the wave is orthogonal to the current, the primary resonance in the current direction is completely eliminated. For lower damping and drag coefficients, a bidirectional primary response is excited even for the case of coincident wave and current directions. As the damping is decreased the periodic bidirectional motion is shown to further bifurcate to periodic, and then chaotic, amplitude-modulated motions. The chaotic motion is, over an interval in wave frequencies, found to be destroyed by heteroclinic bifurcations.

Torus doublings and chaotic amplitude modulations in a two degree-of-freedom resonantly forced mechanical system

This work studies the response of a weakly non-linear vibratory system with two degrees-of-freedom when the system is excited near resonance. The two linear modes are in 1:3 internal resonance. The asymptotic method of averaging and direct numerical integration are used to obtain the response of the system. Over a range of excitation frequencies and modal damping, the averaged equations in slow time are found to possess limit cycle solutions. These solutions undergo period doubling bifurcations to chaotic solutions. The averaging theory then implies the existence of amplitude modulated motions, the exact nature of modulations not being well defined. Numerical simulation of the original vibratory two degree-of-freedom system shows that the system does undergo amplitude modulated motions. For sufficiently large damping, only periodic modulations arise in the form of a 2-torus. For lower damping, the 2-torus can undergo doubling and ultimate destruction to result in a chaotic attractor. Poincare sections of steady state solutions are used to characterize the various types of amplitude modulated motions.

Asymptotic techniques and complex dynamics in weakly non-linear forced mechanical systems

The applicability of asymptotic techniques and their ability to predict complex dynamical motions in weakly non-linear forced mechanical systems is investigated. The basic theorems in the method of averaging and integral manifolds are reviewed. A two degrees-of-freedom system, representing one mode truncation of general non-planar equations of a harmonically excited string, is used as the example. It is shown that the averaged equations, for small enough damping, possess non-planar constant solutions which become unstable and give rise to limit cycles, period-doublings, and isolated periodic solutions, as well as chaotic attractors. The truncated string equations are also directly integrated for small forcing amplitudes. There are non-planar periodic responses that bifurcate into amplitude-modulated motions on a two-torus. Changes in damping and frequency detuning result in torus-doubling, coexisting torus branches, and merging as well as destruction of the torus, leading to chaotic amplitude modulations. The bifurcation values of parameters are found to exhibit a scaling behavior and the results of averaged equations are found to be in qualitative agreement with the actual response.