Non-linear modal properties of non-shallow cables

A non-linear mechanical model of non-shallow linearly elastic suspended cables is employed to investigate the non-linear modal characteristics of the free planar motions. An asymptotic analysis of the equations of motion is carried out directly on the partial-differential equations overcoming the drawbacks of a discretization process. The direct asymptotic treatment delivers the approximation of the individual non-linear normal modes. General properties about the non-linearity of the in-plane modes of different type—geometric, elasto-static and elasto-dynamic—are unfolded. The spatial corrections to the considered linear mode shape caused by the quadratic geometric forces are investigated for modes belonging to the three mentioned classes. Moreover, the convergence of Galerkin reduced-order models is discussed and the influence of passive modes is highlighted.     

Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.     

The construction of non-linear normal modes for systems with internal resonance

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.     

Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach

The non-linear free and forced vibrations of simply supported thin circular cylindrical shells are investigated using Lagrange's equations and an improved transverse displacement expansion. The purpose of this approach was to provide engineers and designers with an easy method for determining the shell non-linear mode shapes, with their corresponding amplitude dependent non-linear frequencies. The Donnell non-linear shell theory has been used and the flexural deformations at large vibration amplitudes have been taken into account. The transverse displacement expansion has been made using two terms including both the driven and the axisymmetric modes, and satisfying the simply supported boundary conditions. The non-linear dynamic variational problem obtained by applying Lagrange's equations was then transformed into a static case by adopting the harmonic balance method. Minimisation of the energy functional with respect to the basic function contribution coefficients has led to a simple non-linear multi-modal equation, the solution of which gives in the case of a single mode assumption an expression for the non-linear frequencies which is much simpler than that derived from the non-linear partial differential equation obtained previously by several authors. Quantitative results based on the present approach have been computed and compared with experimental data. The good agreement found was very satisfactory, in comparison with previous old and recent theoretical approaches, based on sophisticated numerical methods, such as the finite element method (FEM), the method of normal forms (MNF), and analytical methods, such as the perturbation method.     

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.     

Non-linear modal analysis of a rotating beam

The free non-linear vibration of a rotating beam has been considered in this paper. The von Karman strain-displacement relations are implemented. Non-linear equations of motion are obtained by Hamilton’s principle. Results are obtained by applying the method of multiple scales to a set of discretized ordinary differential equations which obtained by using the Galerkin discretization method. This set contains coupling between transverse and axial displacements as quadratic and cubic geometric non-linearities. Non-linear normal modes and non-linear natural frequencies with or without internal resonance are observed. In the internal resonance case, the internal resonance between two transverse modes and between one transverse and one axial mode are explored. Obtained results in this study are compared with those obtained from literature. The stability and some dynamic characteristics of the non-linear normal modes such as the phase portrait, Poincare section and power spectrum diagrams have been inspected. It is shown that, for the first internal resonance case, the beam has one stable or degenerate uncoupled mode and either: (a) one stable coupled mode, (b) one unstable coupled mode, (c) two stable and one unstable coupled modes, (d) three stable coupled modes, and (e) one stable coupled mode. On the other hand, for the second internal resonance case, the beam has one stable or unstable or degenerate uncoupled mode and either: (a) two stable coupled modes, (b) two unstable coupled modes, and (c) one stable coupled mode depending on the parameters.     

Non-linear vibrations of doubly curved shallow shells

Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated. Two different non-linear strain-displacement relationships, from the Donnell's and Novozhilov's shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometric imperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclength continuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio among their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under static and dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behaviour have been observed.     

Global and bifurcation analysis of a structure with cyclic symmetry

This article combines the application of a global analysis approach and the more classical continuation, bifurcation and stability analysis approach of a cyclic symmetric system. A solid disc with four blades, linearly coupled, but with an intrinsic non-linear cubic stiffness is at stake. Dynamic equations are turned into a set of non-linear algebraic equations using the harmonic balance method. Then periodic solutions are sought using a recursive application of a global analysis method for various pulsation values. This exhibits disconnected branches in both the free undamped case (non-linear normal modes, NNMs) and in a forced case which shows the link between NNMs and forced response. For each case, a full bifurcation diagram is provided and commented using tools devoted to continuation, bifurcation and stability analysis.     

Damping for large-amplitude vibrations of plates and curved panels,Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.     

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly non-linear absorber with small mass. The non-linear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of non-linear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the non-linear one. The dissipative effect of the non-linearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the non-linear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to non-linear modes) experiences localization of the vibrations in the non-linear absorber and jump phenomena.     

Non-linear modal analysis for beams subjected to axial loads: Analytical and finite-element solutions

A rigorous derivation of non-linear equations governing the dynamics of an axially loaded beam is given with a clear focus to develop robust low-dimensional models. Two important loading scenarios were considered, where a structure is subjected to a uniformly distributed axial and a thrust force. These loads are to mimic the main forces acting on an offshore riser, for which an analytical methodology has been developed and applied. In particular, non-linear normal modes (NNMs) and non-linear multi-modes (NMMs) have been constructed by using the method of multiple scales. This is to effectively analyse the transversal vibration responses by monitoring the modal responses and mode interactions. The developed analytical models have been crosschecked against the results from FEM simulation. The FEM model having 26 elements and 77 degrees-of-freedom gave similar results as the low-dimensional (one degree-of-freedom) non-linear oscillator, which was developed by constructing a so-called invariant manifold. The comparisons of the dynamical responses were made in terms of time histories, phase portraits and mode shapes.     

On the accuracy of the multiple scales method for non-linear vibrations of doubly curved shallow shells

Non-linear free and forced vibrations of doubly curved isotropic shallow shells are investigated via multi-modal Galerkin discretization and the method of multiple scales. Donnell’s non-linear shallow shell theory is used and it is assumed that the shell is simply supported with movable edges. By deriving two different forms of the stress function, the equations of motion are reduced to a system of infinite non-linear ordinary differential equations with quadratic and cubic non-linearities. A quadratic relation between the excitation and the fundamental frequency is considered and it is shown that, although in case of hardening non-linearities the results resemble those found via numerical integration or continuation softwares, in case of softening non-linearity the solution breaks down as the amplitude becomes larger than the thickness. Results reveal that, expressing the relation between the excitation and fundamental frequency in this form, which was considered by many researchers as a useful tool in analyzing strong non-linear oscillators, yields in spurious results when the non-linearity becomes of softening type.     

Non-linear dynamic analysis of tensegrity structures using a co-rotational method

A new formulation is presented for the non-linear dynamic analysis of space truss structures. The formulation is based on the dynamics of 3D co-rotational rods. In the co-rotation method, the rigid body modes are assumed to be separated from the total deformations at the local element level. In this paper a new co-rotational formulation is proposed based on the direct derivation of the inertia force vector and the tangent dynamic matrix. A closed-form equation is derived for the calculation of the inertia force, the tangent dynamic matrix, the mass matrix and the gyroscopic matrix. The new formulation is used to perform dynamic analysis of example tensegrity structures. The developed formulation is applicable to tensegrity structures with non-linear effects due to internal mechanisms or geometric non-linearities, and is applied to two numerical examples. The efficiency of the proposed approach is compared to the conventional Lagrangian method, and savings in computation of about 55%, 54% and 37% were achieved.     

Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems

The non-linear modal properties of a vibrating 2-DOF system with non-smooth (piecewise linear) characteristics are investigated; this oscillator can suitably model beams with a breathing crack or systems colliding with an elastic obstacle. The system having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of non-linear normal modes (NNMs) that are greater than the degrees of freedom. Since the non-linearities are concentrated at the origin, its non-linear frequencies are independent of the energy level and uniquely depend on the damage parameter. An analysis of the NNMs has been performed for a wide range of damage parameter by employing numerical procedures and Poincaré maps. The influence of damage on the non-linear frequencies has been investigated and bifurcations characterized by the onset of superabundant modes in internal resonance, with a significantly different shape than that of modes on fundamental branch, have been revealed.     

Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity

The non-linear normal modes (NNMs) and their bifurcation of a complex two DOF system are investigated systematically in this paper. The coupling and ground springs have both quadratic and cubic non-linearity simultaneously. The cases of ω₁:ω₂=1:1, 1:2 and 1:3 are discussed, respectively, as well as the case of no internal resonance. Approximate solutions for NNMs are computed by applying the method of multiple scales, which ensures that NNM solutions can asymtote to linear normal modes as the non-linearity disappears. According to the procedure, NNMs can be classified into coupled and uncoupled modes. It is found that coupled NNMs exist for systems with any kind of internal resonance, but uncoupled modes may appear or not appear, depending on the type of internal resonance. For systems with 1:1 internal resonance, uncoupled NNMs exist only when coefficients of cubic non-linear terms describing the ground springs are identical. For systems with 1:2 or 1:3 internal resonance, in additional to one uncoupled NNM, there exists one more uncoupled NNM when the coefficients of quadratic or cubic non-linear terms describing the ground springs are identical. The results for the case of internal resonance are consistent with ones for no internal resonance. For the case of 1:2 internal resonance, the bifurcation of the coupled NNM is not only affected by cubic but also by quadratic non-linearity besides detuning parameter although for the cases of 1:1 and 1:3 internal resonance, only cubic non-linearity operate. As a check of the analytical results, direct numerical integrations of the equations of motion are carried out.     

Large amplitude free flexural vibrations of laminated composite skew plates

Here, the large amplitude free flexural vibration behaviors of thin laminated composite skew plates are investigated using finite element approach. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman's assumptions is introduced. The non-linear governing equations obtained employing Lagrange's equations of motion are solved using the direct iteration technique. The variation of non-linear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, number of layers, fiber orientation, boundary condition and aspect ratio. The influence of higher vibration modes on the non-linear dynamic behavior of laminated skew plates is also highlighted. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and lamination parameters of the plate. Also, the degree of hardening behavior increases with the skew angle and its rate of change depends on the level of amplitude of vibration.     

Existence and stability of bushes of vibrational modes for octahedral mechanical systems with Lennard-Jones potential

The special non-linear dynamical regimes, “bushes of normal modes”, can exist in the N-particle Hamiltonian systems with discrete symmetry (Physica D 117 (1998) 43). The dimension of the bush can be essentially less than that of the whole mechanical system. One-dimensional bushes represent the similar non-linear normal modes introduced by Rosenberg. A given bush can be excited by imposing the appropriate initial conditions, and the energy of the initial excitation turns out to be trapped in this bush.In the present paper, we consider all possible vibrational bushes in the simple octahedral mechanical system and discuss their stability under assumption that the interactions between particles are described by the Lennard-Jones potential.     

Non-linear effects on the resonant frequencies of a cantilevered plate

In this paper, we address experimentally and theoretically the non-linear effects on the resonance of a periodically-forced cantilevered plate immersed in a fluid at rest. Experiments are performed with small aspect-ratio plates made of two different materials. When forced harmonically at their leading edges, these plates exhibit resonances for their first 3 structural modes. The frequencies at these resonances decrease when the forcing amplitude is increased, revealing the presence of non-linear effects. To model this phenomenon, a theoretical model is employed, which takes into account both resistive and reactive forces exerted by the fluid on the plate. By carrying out a weakly non-linear analysis, the frequencies at the resonances can then be determined. Model and experiments are in good agreement, showing that a weakly non-linear approach is suited to this kind of fluid–structure interaction and could be applied, in the future, to engineering problems such as energy harvesting with a fluttering plate or the biological problem of aquatic propulsion with a flexible fin.     

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.