Non-linear non-planar vibrations of geometrically imperfect inextensional beams. Part II—Bifurcation analysis under base excitations

The non-linear non-planar steady-state responses of a near-square cantilevered beam (a special case of inextensional beams) with general imperfection under harmonic base excitation is investigated. By applying the combination of the multiple scales method and the Galerkin procedure to two non-linear integro-differential equations derived in part I, two modulation non-linear coupled first-order differential equations are obtained for the case of a primary resonance with a one-to-one internal resonance. The modulation equations contain linear imperfection-induced terms in addition to cubic geometric and inertial terms. Variations of the steady-state response amplitude curves with different parameters are presented. Bifurcation analyses of fixed points show that the influence of geometric imperfection on the steady-state responses can be significant to a great extent although the imperfection is small. The phenomenon of frequency island generation is also observed.     

Control of primary and subharmonic resonances of a Duffing oscillator via non-linear energy sink

The use of non-linear energy sink to passively control vibrations of a non-linear main structure under the effect of bi-frequency harmonic excitation is addressed here. The excitation is assumed to induce both 1:1 and 1:3 resonance, and the response of the system is studied after using the Multiple Scale/Harmonic Balance Method, applied to obtain amplitude modulation equations in the slow time scale. The efficiency of the non-linear energy sink to reduce or suppress vibrations of the main structure is finally discussed.     

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton's principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited. Bifurcations including the saddle-node, Hopf, perioddoubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.     

A non-linear model for the dynamics of open cross-section thin-walled beams—Part II: forced motion

The discrete equations developed in Part I are here used to analyze the non-linear dynamics of an inextensional shear indeformable beam with given end constraints. The model takes into account the non-linear effects of warping and of torsional elongation. Non-linear 3D oscillations of a beam with a cross-section having one symmetry axis is examined. Only terms of higher magnitude are retained in the equations, which exhibit quadratic, cubic and combination resonances. A harmonic load acting in the direction of the symmetry axis and in resonance with the corresponding natural frequency, is considered. Steady-state solutions and their stability are studied; in particular the effects of non-linear warping and of torsional elongation on the response are highlighted.     

Internal resonances in non-linear vibrations of a laminated circular cylindrical shell

Internal resonances in geometrically non-linear forced vibrations of laminated circular cylindrical shells are investigated by using the Amabili?CReddy higher-order shear deformation theory. A harmonic force excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are obtained by using an energy approach based on Lagrange equations that retain dissipation. Numerical results are obtained by using the pseudo-arc length continuation method and bifurcation analysis. A one-to-one-to-two internal resonance is identified, giving rise to pitchfork and Neimark?CSacher bifurcations of the non-linear response. A threshold level in the excitation has been observed in order to activate the internal resonance.     

Whirling of a forced cantilevered beam with static deflection. I: Primary resonance

The response of a slender, clastic, cantilevered beam to a transverse, vertical, harmonic excitation is investigated. The effects of nonlinear curvature, nonlinear inertia, viscous damping and static load are included. Previous work often has neglected the static deflection caused by the weight of the beam, which adds quadratic terms in the governing equations of motion. Galerkin's method is used with three modes and approximate solutions of the temporal equations are obtained by the method of multiple scales. Primary resonance is treated here, and out-of-plane motion is possible in the first and second modes when the principal moments of inertia of the beam cross-section are approximately equal. In Parts II and III, secondary resonances and nonstationary passages through various resonances are considered.     

Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlin- ear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of mul- tiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequen- cies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.     

Dynamic bifurcation and sensitivity analysis of non-linear non-planar vibrations of geometrically imperfect cantilevered beams

The non-linear non-planar dynamic responses of a near-square cantilevered (a special case of inextensional beams) geometrically imperfect (i.e., slightly curved) and perfect beam under harmonic primary resonant base excitation with a one-to-one internal resonance is investigated. The sensitivity of limit-cycles predicted by the perfect beam model to small geometric imperfections is analyzed and the importance of taking into account the small geometric imperfections is investigated. This was carried out by assuming two different geometric imperfection shapes, fixing the corresponding frequency detuning parameters and continuation of sample limit-cycles versus the imperfection parameter. The branches of periodic responses for perfect and imperfect (i.e. small geometric imperfection) beams are determined and compared. It is shown that branches of periodic solutions associated with similar limit-cycles of the imperfect and perfect beams have a frequency shift with respect to each other and may undergo different bifurcations which results in different dynamic responses. Furthermore, the imperfect beam model predicts more dynamic attractors than the perfect one. Also, it is shown that depending on the magnitude of geometric imperfection, some of the attractors predicted by the perfect beam model may collapse. Ignoring the small geometric imperfections and applying the perfect beam model is shown to contribute to erroneous results.     

On non-linear dynamics of a coupled electro-mechanical system

Electro-mechanical devices are an example of coupled multi-disciplinary weakly non-linear systems. Dynamics of such systems is described in this paper by means of two mutually coupled differential equations. The first one, describing an electrical system, is of the first order and the second one, for mechanical system, is of the second order. The governing equations are coupled via linear and weakly non-linear terms. A classical perturbation method, a method of multiple scales, is used to find a steady-state response of the electro-mechanical system exposed to a harmonic close-resonance mechanical excitation. The results are verified using a numerical model created in MATLAB Simulink environment. Effect of non-linear terms on dynamical response of the coupled system is investigated; the backbone and envelope curves are analyzed. The two phenomena, which exist in the electro-mechanical system: (a)?detuning (i.e. a natural frequency variation) and (b)?damping (i.e. a decay in the amplitude of vibration), are analyzed further. An applicability range of the mathematical model is assessed.     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

On stability of parametrically excited linear stochastic systems

The dynamic stability of a coupled two-degrees-of-freedom system subjected to parametric excitation by a harmonic action superimposed by an ergodic stochastic process is investigated. For the stability analysis, the method of moment functions is used. Explicit expressions for the stability of the second moments are obtained when the frequency of the harmonic excitation lies in the vicinity of the combination sum of the natural frequencies. Good agreement between the analytical and numerical results is obtained. As an application, the example of the flexural-torsional instability of a thin elastic beam under dynamic loading is considered     

Parametrically Excited Non-Linear Traveling Beams with and without External Forcing

The effects of parametric excitation on a traveling beam, both with and without an external harmonic excitation, have been studied including the non-linear terms. Non-linear, complex normal modes have been used for the response analysis. Detailed numerical results are presented to show the effects of non-linearity on the stability of the parametrically excited system. In the presence of both parametric and external harmonic excitations, the response characteristics are found to be similar to that of a Duffing oscillator. The results are sensitive to the relative strengths of and the phase difference between the two forms of excitations.     

Non-linear vibrations of shallow circular cylindrical panels with complex geometry. Meshless discretization with the R-functions method

Geometrically non-linear forced vibrations of a shallow circular cylindrical panel with a complex shape, clamped at the edges and subjected to a radial harmonic excitation in the spectral neighborhood of the fundamental mode, are investigated. Both Donnell and the Sanders–Koiter non-linear shell theories retaining in-plane inertia are used to calculate the elastic strain energy. The discrete model of the non-linear vibrations is build using the meshfree technique based on classic approximate functions and the R-function theory, which allows for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries; Chebyshev orthogonal polynomials are used to expand shell displacements. A two-step approach is implemented in order to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for expanding the non-linear displacements. Lagrange approach is applied to obtain a system of ordinary differential equations on both steps. Different multimodal expansions, having from 15 up to 35 generalized coordinates associated with natural modes, are used to study the convergence of the solution. The pseudo-arclength continuation method and bifurcation analysis are applied to study non-linear equations of motion. Numerical responses are obtained in the spectral neighborhood of the lowest natural frequency; results are compared to those available in the literature. Internal resonances are also detected and discussed.     

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

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.     

Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations

The principal parametric resonance of a single-degree-of-freedom system with non-linear two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations that describe to first-order the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict the existence of steady state responses. Stability is investigated. The effect of damping, magnitudes of non-linear excitation and self-excitation are analyzed.     

Non-isothermal oscillations of pseudoelastic devices

The non-linear dynamic response of a pseudoelastic oscillator embedded in a convective environment is studied taking into account the temperature variations induced, during oscillations, by the latent heat of transformation and by the heat exchange with the surroundings. The asymptotic periodic response under harmonic excitation is characterized by frequency-response curves in terms of maximum displacement, maximum and mean temperature. The periodic thermomechanical response is computed by a multi-component harmonic balance method implemented within a continuation algorithm that enables to trace out multivalued frequency-response curves. The accuracy of the results is checked by comparison with the results of the numerical integration of the basic equations governing the dynamics of the system. The response is investigated for various excitation amplitude levels and in various material parameters ranges. The resulting picture of the mechanical response shows, in some cases, features similar to other hysteretic oscillators, while, in other cases, points out peculiar behaviors. It turns out that the temperature variations induced by the phase transformations influence the mechanical response and that the results obtained under the simplifying assumption of isothermal behavior can be rather different from those obtained in a fully thermomechanical setting.     

Non-linear longitudinal vibrations of non-slender piezoceramic rods

Characteristic non-linear effects can be observed, when piezoceramics are excited using weak electric fields. In experiments with longitudinal vibrations of piezoceramic rods, the behavior of a softening Duffing-oscillator including jump phenomena and multiple stable amplitude responses at the same excitation frequency and voltage is observed. Another phenomenon is the decrease of normalized amplitude responses with increasing excitation voltages. For such small stresses and weak electric fields as applied in the experiments, piezoceramics are usually described by linear constitutive equations around an operating point in the butterfly hysteresis curve. The non-linear effects under consideration were, e.g. observed and described by Beige and Schmidt [1,2], who investigated longitudinal plate vibrations using the piezoelectric 31-effect. They modeled these non-linearities using higher order quadratic and cubic elastic and electric terms. Typical non-linear effects, e.g. dependence of the resonance frequency on the amplitude, superharmonics in spectra and a non-linear relation between excitation voltage and vibration amplitude were also observed e.g. by von Wagner et al. [3] in piezo-beam systems. In the present paper, the work is extended to longitudinal vibrations of non-slender piezoceramic rods using the piezoelectric 33-effect. The non-linearities are modeled using an extended electric enthalpy density including non-linear quadratic and cubic elastic terms, coupling terms and electric terms. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. An extended kinetic energy taking into consideration the transverse velocity is used to model the non-slender rods. The equations of motion are solved using perturbation techniques. In a second step, additional dissipative linear and non-linear terms are used in the model. The non-linear effects described in this paper may have strong influence on the relation between excitation voltage and response amplitude whenever piezoceramic actuators and structures are excited at resonance.     

Influence of Horizontal Excitations on Dynamic Stability of a Slender Beam Under Vertical Excitation

Experimental studies have been conducted to clarify the influence of horizontal harmonic excitations on the dynamic stability of a slender cantilever beam under vertical harmonic excitation. Three kinds of aluminum test beams with rectangular cross section have been used. The test beam being clamped at one end and free at the other end, was vertically stood, and was harmonically excited to both vertical and horizontal directions simultaneously. The direction of the horizontal excitation was taken parallel to one of the beam side faces, i.e. two directions were considered as X and Y directions which have the largest and smallest flexural rigidity, respectively. By varying the horizontal excitation amplitude, keeping the amplitude of excitation in the vertical direction, the influence of the horizontal excitation has been investigated on the principal instability regions in which unstable vibration of the fundamental vibration mode occurs. The excitation frequency in the vertical excitation was taken around twice the fundamental natural frequency 2f _{Y 1} in smallest rigidity direction, while that in the horizontal direction was taken around both the fundamental natural frequency f _{Y 1} and twice of it 2f _{Y 1}. Obtained experimental results present useful fundamental data for aseismatic design of structures under earthquake containing both vertical and horizontal excitation components.     

Vibration control of a cantilever beam of varying orientation

We investigate the problem of suppressing the vibrations of a non-linear system with a cantilever beam of varying orientation subject to parametric and direct excitation. It is known that the growth of the response is limited by non-linearity. Therefore, vibration control and high-amplitude response suppressions of the first mode of a cantilever beam can be performed using a simple non-linear feedback law. This control law is based on cubic velocity feedback. The method of multiples scales is used to construct first-order non-linear ordinary differential equations governing the modulation of the amplitudes and phases. The stability and effects of different system parameters are studied numerically.     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.