Coupled-mode Response Research Articles

On experiments in harmonically excited cantilever plates with 1:2 internal resonance

This work presents some experimental results for resonant nonlinear response of hyperelastic plates for 1:2 internal resonance. Previously developed topology optimization methods are used to design and fabricate candidate resonant plates using 3-D printing. One such plate is subjected to harmonic transverse excitation with increasing amplitudes in a frequency range where 1:2 internal resonances are expected to be activated. While the fabricated structure exhibits coupled mode internal resonance activated response when subjected to higher levels of excitation, the plate also displays other interesting nonlinear behavior. These include nonlinear periodic as well as amplitude modulated motions of the directly excited mode and these motions super-imposed on the coupled mode response.

Nonlinearity of Finite-Amplitude Sloshing in Rectangular Containers

This paper investigates the influence exerted by small surface tension on the nonlinear normal sloshing modes of a two-dimensional irrotational, incompressible fluid in a rectangular container. To this end, the influence of surface tension on the modal frequencies is investigated by assuming pure slipping at the contact line and a 90 deg contact angle between the fluid surface and the walls. The regions of possible nonlinear internal resonances up to the fifth mode are highlighted. Away from the highlighted regions, the influence of surface tension on the effective nonlinearity of the lowest four modes is studied and used to shed light onto its effect on the softening/hardening behavior of the uncoupled nonlinear modes. Subsequently, the response of the sloshing waves near two-to-one internal resonances is studied. It is shown that, in the vicinity of such internal resonance, the steady-state sloshing response can either contain a contribution from the two interacting modes (coupled-mode response) or only the high-frequency mode (high-frequency uncoupled mode response). The regions where the coupled mode uniquely exists are shown to depend on the surface tension. Moreover, it is demonstrated that such regions may be underestimated considerably when neglecting the influence of the cubic nonlinearities.

Structural optimization for nonlinear dynamic response.

Much is known about the nonlinear resonant response of mechanical systems, but methods for the systematic design of structures that optimize aspects of these responses have received little attention. Progress in this area is particularly important in the area of micro-systems, where nonlinear resonant behaviour is being used for a variety of applications in sensing and signal conditioning. In this work, we describe a computational method that provides a systematic means for manipulating and optimizing features of nonlinear resonant responses of mechanical structures that are described by a single vibrating mode, or by a pair of internally resonant modes. The approach combines techniques from nonlinear dynamics, computational mechanics and optimization, and it allows one to relate the geometric and material properties of structural elements to terms in the normal form for a given resonance condition, thereby providing a means for tailoring its nonlinear response. The method is applied to the fundamental nonlinear resonance of a clamped-clamped beam and to the coupled mode response of a frame structure, and the results show that one can modify essential normal form coefficients by an order of magnitude by relatively simple changes in the shape of these elements. We expect the proposed approach, and its extensions, to be useful for the design of systems used for fundamental studies of nonlinear behaviour as well as for the development of commercial devices that exploit nonlinear behaviour.

Vibration Dynamics of an Inclined Cable Excited Near Its Second Natural Frequency

Inclined cables are essential structural elements that are used most prominently in cable-stayed bridges. When the bridge deck oscillates due to an external force, such as passing traffic, cable vibrations can arise not only in the plane of excitation, but also in the perpendicular plane. This undesirable phenomenon can be modeled as an auto-parametric resonance between the in-plane and out-of-plane modes of vibration of the cable. In this paper, we consider a three-mode model, capturing the second in-plane, and first and second out-of-plane modes, and use it to study the response of an inclined cable that is vertically excited at its lower (deck) support at a frequency close to the second natural frequency of the cable. Averaging is applied to the model and then the solutions and bifurcations of the resulting averaged differential equations are investigated and mapped out with numerical continuation. In this way, we present a detailed bifurcation study of the different possible responses of the cable. We first consider the equilibria of the averaged model, of which there are four types that are distinguished by whether each of the two out-of-plane modes is present or not in the cable response. Each type of equilibrium is computed and represented as a surface over the plane of amplitude and frequency of the forcing. The stability of the equilibria changes and different surfaces meet along curves of bifurcations, which are continued directly. Overall, we present a comprehensive geometric picture of the two-parameter bifurcation diagram of the constant-amplitude coupled-mode response of the cable. We then focused on bifurcating periodic orbits, which correspond to cable dynamics with varying amplitudes of the participating second in-plane and second out-of-plane modes. The range of excitation amplitude and frequency is determined where such whirling cable motion can occur. Further bifurcations — period-doubling cascades and a Shilnikov homoclinic bifurcation — are found that lead to a chaotic cable response. Whirling and chaotic cable dynamics are confirmed by time-step simulations of the full three-mode model. The different cable responses are characterized, and can be distinguished clearly, by their motion at the quarter-span and by their frequency spectra.

Finite element based coupled mode initial post-buckling analysis of a composite cylindrical shell

This paper presents a finite element formulation of Koiter's initial post-buckling theory using a multi-mode approach. Initial post-buckling theory provides direct information about the imperfection sensitivity of a structure under compressive loading, and is also the basis of a nonlinear reduced order model. The objective of the present work is to illustrate the capability of the implementation for buckling analysis of shell structures including modal interaction. A coupled mode initial post-buckling analysis for a composite cylindrical shell under axial compression, including the effect of a nonlinear pre-buckling state, has been carried out using a small number of representative modes. For small imperfection amplitudes the limit-point buckling loads obtained with the reduced order model compare reasonably well with full model nonlinear analysis, illustrating that a fast prediction of the coupled mode response of imperfect shells is possible using the approach proposed.

On the coupling of non-linear normal modes

The phenomenon of internal resonance is known as the exchange of energy between the modes and the existence of coupled-mode response under a single-mode excitation. This phenomenon is observed whenever a non-linear normal mode loses its stability, called the modal coupling. The details of modal coupling are formulated in the free vibrations of two-degree-of-freedom systems, and compared with internal resonance. The theory is based on the structural change in Poincaré map due to the stability change of normal modes. It is shown that every change in stability of normal modes gives rise to a pitchfork or a period-doubling bifurcation. The functional form is derived to compute the coupled modes by the method of harmonic balance. Examples are given to describe the procedure of stability analysis of non-linear normal modes, to compute the coupled modes, and then to demonstrate that results of internal resonances can be derived by model coupling. Other examples are given to demonstrate that the results of some modal couplings cannot be obtained by internal resonances.

A novel elastic suspension system for wind tunnel section model studies

A novel three-degree-of-freedom (DOF) suspension system has been developed for the wind-tunnel section model study of wind-excited vibrations of flexible structures. This system enables simultaneous vertical, horizontal, and torsional motion of the suspended model and captures the effect of coupling between different degrees of freedom of a flexible structure immersed in a dynamic flow field. It utilizes pneumatic bushings that glide along polished steel shafts in the vertical and horizontal directions of motion. Torsional assemblies are used to generate torsional motion of the model. Force measurement is accomplished with compact strain-gage type force transducers that allow precise placement for accurate measurements. Static measurements are also possible by restraining the desired DOF. Vibration frequencies of the system are tuned with combinations of springs. System damping is low due to the low-friction pneumatic bushings, restriction of coil spring wire to small diameters, and highly polished stainless steel guide shafts. Relatively large displacements can be generated by the suspension system, if required. Results from two experiments on aeroelastic studies are presented to demonstrate the functioning of the suspension system. The effect of including the lateral motion on the coupled-mode response is illustrated.

Non-stationary flexural vibration behaviour of a cylindrical shell

An analytical/numerical method involving a small number of generalized coordinates is presented for the analysis of the non-linear vibration and dynamic stability behaviour of imperfect anisotropic cylindrical shells. The small amplitude vibration mode is characterized by m axial half-waves and ℓ circumferential full waves. The radial excitation is assumed to have the same spatial distribution as the small amplitude vibration mode. For large excitations, a coupled mode response can occur, consisting of the directly excited ‘driven mode’ and the so-called ‘companion mode’. The latter has the same spatial form as the driven mode, but is circumferentially 90° out-of-phase. Donnell-type governing equations are used and classical laminate theory is employed. The assumed modes approximately satisfy ‘simply supported’ boundary conditions. A formulation is used which can account for a possible skewedness of the asymmetric modes. Certain axisymmetric modes which are essential for a satisfactory description of the non-linear behaviour are also included in the assumed deflection function. The effect of axial and torsional inertia, including the inertia effect of a ring or disk at the loaded end of the shell, can be taken into account approximately. Viscous modal damping is included in the analysis. The shell is statically loaded by axial compression, radial pressure, and torsion. A two-mode imperfection model, consisting of an axisymmetric and an asymmetric mode, is used. The static state response is assumed to be affine to the given imperfection. In order to find approximate solutions for the dynamic state equations, Hamilton's principle is applied to derive a set of modal amplitude equations. The dynamic response is obtained via numerical time integration of the set of non-linear ordinary differential equations. The coupled mode non-linear vibration behaviour of an axially loaded isotropic shell is simulated using this approach. Non-stationary vibrations, where the response drifts between single mode and different types of coupled mode solutions, are observed in a small frequency region near resonance.

A model for non-linear viscoelastic coupled mode response of an elastomeric bushing

An elastomeric bushing is a device used in automotive suspension systems to cushion the loads transmitted from the wheel to the frame of the vehicle. A bushing is essentially an elastomeric hollow cylinder which is bonded to a solid metal shaft at its inner surface and a metal sleeve at its outer surface. The shaft is connected to the suspension and the sleeve is connected to the frame. The elastomeric cylinder provides the cushion when it deforms due to relative motion between the shaft and sleeve. The relation between the force or moment applied to the shaft or sleeve and the relative displacements or rotations is non-linear and exhibits features of viscoelasticity. An explicit force–displacement relation for elastomeric bushings is important for multi-body dynamics numerical simulations. A boundary value problem for the bushing response leads to an implicit relation which requires extensive computation time to implement and is hence unsuitable. In the present work, an explicit relation for coupled axial and torsional mode response is introduced and studied. A boundary value problem is formulated for coupled axial and torsional mode bushing response. A constitutive model is defined in which the axial force and torsional moment are each expressed explicitly in terms of axial displacement and rotation. Each relation contains a relaxation function which depends on the axial displacement and rotation. The relaxation functions are constructed using numerical results obtained by solving the boundary value problem. Numerical solutions of the boundary value problem also allow for comparison between the exact coupled mode response and that predicted by the proposed model. It is shown that the predictions of the proposed moment–rotation relation are in very good agreement with the exact results.

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The “locked pendulum” mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.

Non-linear dynamics of a shallow arch under periodic excitation —I.1:2 internal resonance

The weakly non-linear resonance response of a two-degree-of-freedom shallow arch subjected to simple harmonic excitation is examined in detail for the case of 1:2 internal resonance. The method of averaging is used to yield a set of autonomous equations of the first-order approximations to the response of the system. The averaged equations are then examined to determine their bifurcation behavior. Our analysis indicates that by varying the detuning para- meters from the exact external and internal resonance conditions, the coupled mode response can undergo a Hopf bifurcation to limit cycle motion. It is also shown that the limit cycles quickly undergo period-doubling bifurcation, giving rise to chaos. In order to study the global bifurcation behavior, the Melnikov method is used to determine the analytical results for the critical parameter at which the dynamical system possesses a Smale horseshoe type of chaos.