Nonlinear free vibrations of beams in space due to internal resonance

Abstract

The geometrically nonlinear free vibrations of beams with rectangular cross section are investigated using a p-version finite element method. The beams may vibrate in space, hence they may experience longitudinal, torsional and non-planar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The geometrical nonlinearity is taken into account by considering Green’s nonlinear strain tensor. Isotropic and elastic beams are investigated and generalised Hooke’s law is used. The equation of motion is derived by the principle of virtual work. Mostly clamped–clamped beams are investigated, although other boundary conditions are considered for validation purposes. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. One constant term, odd and even harmonics are assumed in the Fourier series and convergence with the number of harmonics is analysed. The variation of the amplitude of vibration with the frequency of vibration is determined and presented in the form of backbone curves. Coupling between modes is investigated, internal resonances are found and the ensuing multimodal oscillations are described. Some of the couplings discovered lead from planar oscillations to oscillations in the three dimensional space.

Introduction

The knowledge of the modes of vibration allows one to understand the dynamic characteristics of a structure and can be employed in the prediction of the response to external excitations. Whilst in a linear conservative system, it is possible to define a mode of vibration by its natural frequency and mode shape, which do not change with the vibration amplitude, in a system experiencing geometrically nonlinear vibrations this is not true. In the latter case, periodic, free, oscillations can still be found, but the shape assumed by the system along a vibration period is not constant and the oscillations can significantly deviate from harmonic. A few studies have been carried out on the variation of the shape of vibration and of the vibration period with the vibration amplitude [1], [2], [3], [4].

The definition of mode of vibration in nonlinear systems is not unequivocal and has been a matter of discussion [5], [6], [7], [8]. Rosenberg [6] suggested that a nonlinear natural mode of a discrete system could be interpreted as a motion where the masses execute periodic, but not necessarily harmonic, vibrations, all masses vibrating with the same period, achieving the maximum amplitude displacement and the static equilibrium points simultaneously. In these conditions, the coordinates of the system are determined by the position of any of the masses, as in linear modes, but the ratio between the vibration amplitudes of any two points is not constant. This concept can be extended to continuous systems, by considering an infinite number of infinitesimal masses.

The natural modes of discrete (or discretized) linear systems are computed by solving eigenvalue problems. In discrete models of nonlinear systems an eigenvalue problem related with steady-state free vibration can also be defined, but the stiffness matrix depends upon the unknown eigenvector. Hence, the eigenvalues and the eigenvectors are, generally, amplitude dependent. We defend that these define frequencies and shapes of vibration that depend on the amplitude of vibration displacement and that approximately represent the modes of vibration of the nonlinear system. Rather conveniently, these “nonlinear modes” tend to the linear modes when the vibration amplitude decreases.

Due to the variation of the natural frequencies, two or more natural frequencies may become commensurable, creating conditions for the strong interaction of the modes involved. As a result of this phenomenon, known as internal resonance, energy imparted initially to one of the modes involved in the internal resonance will be continuously exchanged among all the modes involved in that internal resonance and multimodal vibration appears [3], [4], [9], [10].

The study of undamped nonlinear free vibrations is important for understanding the system’s vibratory response in the nonlinear regime. One reason why the computation and investigation of nonlinear free, undamped oscillations is valuable is that resonances in the forced responses occur in the neighbourhood of these free vibration oscillations [11], this fact is also indicated by many analytical and numerical investigations [5], [9], [10]. Hence, a free vibration study provides information on the dynamic characteristics of the system, information that is of use when the system is excited by practically any force. During the fundamental resonance of forced vibration, the shape of vibration is identical to that of the neighbouring nonlinear free response; moreover a bifurcation of the free response can considerably affect the resonances of a forced system [12]. Since bifurcations in free vibrations give rise to new types of dynamic behaviour, they have significant effects on the dynamics of forced systems. Hence, the knowledge of the backbone curves and the bifurcations offers important understanding on the dynamics of the structure. The study of free vibrations of nonlinear systems, with definition of shapes, natural frequencies and bifurcation, is the natural development of studies of normal modes in linear systems.

Nonlinear and non-planar vibrations in space appear in various engineering problems. At the macro-scale one could refer to wind turbine and helicopter blades, airplane wings, robot arms and shafts [13]. Another example is provided by bridges, as occurred in the famous case of Tacoma Narrows bridge [14]. More recently, non-planar dynamics of beams have also been referred to at the nano-scale, with experimental evidence [15], [16].

Bending–torsion coupling may arise in linear oscillations of beams due to anisotropy in the elastic properties of the material, as investigated in [17], [18] or due to non-coincident shear and mass centres [19], [20]. Moreover, it has been shown that bending–torsion coupling also appears in isotropic materials with coincident mass and shear centres, due to the nonlinear effects. But we could not find a study on free vibrations of beams in space with bending–torsion coupling. In fact, to the authors’ knowledge, in the works published so far on this issue there is either a motion imposed on one of the beam boundaries – the so called base or support – or an external force. For example, in [21] the nonlinear vibration of a cantilever beam subjected to parametric excitations is analysed, using a combination of the methods of multiple scales and Galerkin, unstable planar motion is found and non-planar motion described. An analysis using a similar model and solution methods is carried out in [22], but to investigate non-planar responses of a cantilever beam subjected to lateral harmonic base excitation. In [23] it was shown experimentally that planar motions can give rise to coupled bending–torsion oscillations even if the bending and torsional modes are uncoupled in the linear model. Under a bifurcation, the planar motion looses stability and non-planar motion occurs. Nonlinear flexural–torsional coupling was also investigated in [24]. The beam was excited by a periodic force with frequency near the bending and the torsional natural frequencies and the nonlinear differential equations were solved by the method of multiple scales. A nonlinear dynamic model for slender isotropic beams was formulated in [25] using Cosserat theory and considering flexure about both principal axes, extension, shear and torsion. The bending–torsion motion occurred under an external force in both transverse directions. Lateral buckling analysis of beams of arbitrary cross section was presented in [26]. The model considers large displacements in both transverse planes and large angles of twist. A p-version finite element for bending–torsional displacements of beams with rectangular cross section was presented in [27]. Forced vibrations were considered and different models were compared, the most appropriate for bending–torsional problems was chosen.

As written in the former lines, although there are publications that addressed the geometrically nonlinear free vibrations of beams in a plane, e.g. [1], [2], [3], [4], studies on the interaction of vibration modes with bending and torsion in the free regime apparently do not exist. The current paper studies free periodic vibrations of beams, involving bending modes from different planes and torsion, i.e. the vibrations are not restricted to a particular bending plane. Isotropic beams with rectangular cross section are analysed, therefore the shear centre coincides with the centroid of the cross section. An accurate model developed and validated in [27] is employed; it is based on Timoshenko’s theory for bending and Saint-Venant’s theory for torsion, includes a warping function and considers geometrical nonlinearity. The differential equation of motion is transformed to an algebraic nonlinear system by the harmonic balance method. A continuation method is employed and the solutions of the nonlinear free vibration problem are presented in the frequency domain. The major aim of the paper is to show that due to bifurcations, bending vibrations in one plane can couple with bending vibrations in another plane, leading to oscillations in three dimensional space which include torsion. Also, secondary branches which involve not only odd but also even harmonics are presented.