Stability in parametric resonance of axially accelerating beams constituted by Boltzmann's superposition principle

Stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated. The governing equation is derived from Newton's second law, Boltzmann's superposition principle, and the geometrical relation. When the axial speed is a constant mean speed with small harmonic variations, the governing equation can be treated as a continuous gyroscopic system with small periodically parametric excitations and a damping term. The method of multiple scales is applied directly to the governing equation without discretization. The stability conditions are obtained for combination and principal parametric resonance. Numerical examples demonstrate that the increase of the viscosity coefficient causes the lager instability threshold of speed fluctuation amplitude for given detuning parameter and smaller instability range of the detuning parameter for given speed fluctuation amplitude. The instability region is much bigger in lower order principal resonance than that in the higher order.     

The effect of the number of nodal diameters on non-linear interactions in two asymmetric vibration modes of a circular plate

In order to investigate the effect of the number of nodal diameters on non-linear interactions in asymmetric vibrations of a circular plate, a primary resonance of the plate is considered. The plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one. The response of the plate is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite, and the method of multiple scales is used. In view of the corrected solvability conditions for the responses, it has been found that in order for the modes to interact, the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one. In this study the one-to-one case, in which the modes have the same number of nodal diameters, is examined. The non-linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions. The steady state responses and their stability are determined by using this system. The result shows very complicated interactions between two modes by telling existence of non-vanishing amplitudes of the mode not directly excited.     

LARGEST LYAPUNOV EXPONENT AND ALMOST CERTAIN STABILITY ANALYSIS OF SLENDER BEAMS UNDER A LARGE LINEAR MOTION OF BASEMENT SUBJECT TO NARROWBAND PARAMETRIC EXCITATION

The first order approximate solutions of a set of non-liner differential equations, which is established by using Kane's method and governs the planar motion of beams under a large linear motion of basement, are systematically derived via the method of multiple scales. The non-linear dynamic behaviors of a simply supported beam subject to narrowband random parametric excitation, in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3:1 internal resonance between the first two modes is taken into consideration, are analyzed in detail. The largest Lyapunov exponent is numerically obtained to determine the almost certain stability or instability of the trivial response of the system and the validity of the stability is verified by direct numerical integration of the equation of motion of the system.     

Nonlinear vibration of iced cable under wind excitation using three-degree-of-freedom model

High-voltage transmission line possesses a typical suspended cable structure that produces ice in harsh weather. Moreover, transversely galloping will be excited due to the irregular structure resulting from the alternation of lift force and drag force. In this paper, the nonlinear dynamics and internal resonance of an iced cable under wind excitation are investigated.Considering the excitation caused by pulsed wind and the movement of the support, the nonlinear governing equations of motion of the iced cable are established using a three-degree-of-freedom model based on Hamilton's principle. By the Galerkin method, the partial differential equations are then discretized into ordinary differential equations. The method of multiple scales is then used to obtain the averaged equations of the iced cable, and the principal parametric resonance-1/2 subharmonic resonance and the 2:1 internal resonance are considered. The numerical simulations are performed to investigate the dynamic response of the iced cable. It is found that there exist periodic, multi-periodic, and chaotic motions of the iced cable subjected to wind excitation.     

Two-degree-of-freedom systems with quadratic non-linearities subjected to parametric and self excitation

The method of multiple scales is used to study the response of two-degree-of-freedom systems with quadratic non-linearities under the simultaneous effects of a harmonic parametric excitation and self excitation. The principal parametric resonance of the first mode and a three-to-one internal resonance is considered, followed by the case of internal and parametric resonance of the second mode. In both cases the stability of the system is also studied. Amplitude and frequency response curves are presented for both cases. The character of stability and the mode in which the system loses its stability is also discussed.     

Interaction of fundamental parametric resonances with subharmonic resonances of order one-half

The interaction of fundamental parametric resonances with subharmonic resonances of order one-half in a single-degree-of-freedom system with quadratic and cubic nonlinearities is investigated. The method of multiple scales is used to derive two first-order ordinary differential equations that describe the modulation of the amplitude and the phase of the response with the non-linearity and both resonances. These equations are used to determine the steady state solutions and their stability. Conditions are derived for the quenching or enhancement of a parametric resonance by the addition of a subharmonic resonance of order one-half. The degree of quenching or enhancement depends on the relative amplitudes and phases of the excitations. The analytical results are verified by numerically integrating the original governing differential equation.     

Autoparametric resonance in a structure containing a liquid,Part II: Three mode interaction

Autoparametric coupling of the first antisymmetric liquid sloshing mode with two orthogonal structure freedoms in a simple structure containing a liquid is investigated theoretically and experimentally. Asymptotic approximation up to the first order shows four possible conditions of internal resonance. The response of the system is obtained analytically and numerically in the neighbourhood of the internal resonance conditions. Under the principal internal resonance (i.e., when one of the normal mode frequencies is twice one of the other mode frequencies) the system possesses a steady-state response. Under the summed or differenced internal resonance (i.e., when one of the normal mode frequencies equals the sum or difference of two other natural mode frequencies) the system does not achieve a constant amplitude steady-state response.Experimental investigations confirm the possible existence of most of the internal resonance conditions considered in the analytical study; however, theoretical amplitude-frequency response curves are rather higher than the experimental results. Experimental observations showed that other kinds of instabilities occur when the liquid free surface exhibits rotational flow at a forcing frequency just above twice the liquid sloshing frequency.     

A parametric multi-body section model for modal interactions of cable-supported bridges

A parametric section model is formulated to synthetically describe the geometrically nonlinear dynamics of cable-stayed and suspended bridges through a planar elastic multi-body system. The four-degrees-of-freedom model accounts for both the flexo-torsional motion of the bridge deck and for the transversal motion of a pair of hangers or stay cables. After linearization around the pre-stressed static equilibrium configuration, the coupled equations of motion governing the global deck dynamics and the local cable motion are obtained. A multi-parameter perturbation method is employed to solve the modal problem of internally resonant systems. The perturbation-based modal solution furnishes, first, explicit formulae for the parameter combinations which realize the internal resonance conditions and, second, asymptotic approximations of the resonant frequencies and modes. Attention is focused on the triple internal resonance among a global torsional mode of the deck and two local modes of the cables, due to the relevant geometric coupling which maximizes the modal interaction. The asymptotic approximation of the modal solution is found to finely describe the multiple veering phenomenon which involves the three frequency loci under small variation of the most significant mechanical parameters, including terms of structural coupling or disorder. Moreover, the veering amplitude between any two of the three frequency loci can be expressed as an explicit parametric function. Finally, the disorder is recognized as the only parameter governing a complex phenomenon of triple modal hybridization involving all the resonant modes. The entire hybridization process is successfully described by an energy-based localization factor, presented in a new perturbation-based form, valid for internally resonant system.     

Non-linear resonances in the forced responses of plates,part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.     

Nonlinear resonance interaction between the oscillation modes of a spherical liquid layer on the surface of a melting hailstone

Evolutionary equations are derived and solved that describe the time dependence of the oscillation mode amplitudes on the surface of a charged conducting liquid layer resting on a solid core. It is assumed that the layer experiences a multimode initial deformation. The equations are solved asymptotically in the second order of smallness in the small dimensionless amplitude of capillary oscillations on the surface of the layer. Mechanisms behind internal nonlinear resonance interaction between the modes of the liquid layer oscillations and behind energy transfer between the modes both in degenerate and in secondary combination resonances are investigated. It is found that in the degenerate resonance interaction between oscillation modes, the energy may be transferred not only from lower to higher modes but also vice versa if the higher mode is excited at the zero time. This conclusion is valid not only for a liquid layer on the surface of a solid core but also for a drop.     

Investigation of velocity field of gas flow moving behind a piston in rectangular branch pipe

The two-dimensional boundary-value problem of the unsteady flow of an incompressible viscous gas moving behind the piston in a “long” rectangular branch pipe is solved. An analytic solution is constructed for two velocity components with a refining polynomial, which reduces to a system of nonlinear algebraic equations after the substitution into the governing system of equations. By virtue of the solution uniqueness of the boundary-value problem under study, the only solution is found from obtained values of the refining polynomial constants for each point of the branch pipe internal space for the velocity components in analytic form.     

Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping

A theoretical analysis is presented of the response of a lightly and nonlinearly damped mass-spring system in which the spring constant contains a small randomly fluctuating component. Damping is represented by a combination of linear and nonlinear power-law damping. System response to some initial disturbance at time zero is described by a sinusoidal wave whose amplitude and phase vary slowly and randomly with time. Leading order formulations for the equations of amplitude and phase are obtained through the application of methods of stochastic averaging of Stratonovich. The equations of amplitude and phase are given in two versions: Fokker-Planck equations for transient probability and Langevin equations for response in the time-domain. Solutions in closed-form of these equations are derived by methods of mathematical and theoretical physics involving higher transcendental functions. They are used to study the behavior of system response for ever increasing time applying asymptotic methods of analysis such as the method of steepest descent or saddle-point method. It is found that system behavior depends on the power density of the parametric excitation at twice the natural frequency and on the magnitude and form of the damping. Depending on these parameters different types of system behavior are found to be possible: response which decays exponentially to zero, response which leads to a stationary state of random behavior, and response which can either grow unboundedly or which approaches zero in a finite time.     

Nonlinear vibration of micromachined asymmetric resonators

In this paper, the nonlinear dynamics of a beam-type resonant structure due to stretching of the beam is addressed. The resonant beam is excited by attached electrostatic comb-drive actuators. This structure is modeled as a thin beam-lumped mass system, in which an initial axial force is exerted to the beam. This axial force may have different origins, e.g., residual stress due to micro-machining. The governing equations of motion are derived using the mode summation method, generalized orthogonality condition, and multiple scales method for both free and forced vibrations. The effects of the initial axial force, modal damping of the beam, the location, mass, and rotary inertia of the lumped mass on the free and forced vibration of the resonator are investigated. For the case of the forced vibration, the primary resonance of the first mode is investigated. It has been shown that there are certain combinations of the model parameters depicting a remarkable dynamic behavior, in which the second to first resonance frequencies ratio is close to three. These particular cases result in the internal resonance between the first and second modes. This phenomenon is investigated in detail.     

Amplitude reduction of parametric resonance by dynamic vibration absorber based on quadratic nonlinear coupling

The paper proposes an amplitude reduction method for parametric resonance with a new type of dynamic vibration absorber utilizing quadratic nonlinear coupling. A main system with asymmetric nonlinear restoring force and harmonic excitation causes parametric resonance in the system. In contrast with autoparametric vibration absorber, the natural frequency of the vibration absorber is tuned to be in the neighborhood of twice that of the main system. For such a vibration absorber, we investigate the effect on the amplitude reduction for a parametrically excited main system. Analytical results using the method of multiple scales show that the amplitude of parametric resonance is reduced by the effect of the vibration absorber. The experimental results by a simple apparatus indicate that the parametric resonance is stabilized by the effects of both vibration absorber and Coulomb friction of the main system. Moreover, numerical results considering the Coulomb friction of the main system show that the amplitude of parametric resonance becomes close to zero by the proposed vibration absorber.     

Parametric resonance of truncated conical shells rotating at periodically varying angular speed

Parametric resonance of a truncated conical shell rotating at periodically varying angular speed is studied in this paper. Based upon the Love?s thin shell theory and generalized differential quadrature (GDQ) method, the equations of motion of a rotating conical shell are derived. The time-dependent rotating speed is assumed to be a small and sinusoidal perturbation superimposed upon a constant speed. Considering the periodically rotating speed, the conical shell system is a parametric excited system of the Mathieu–Hill type. The improved Hill?s method is utilized for parametric instability analysis. Both the primary and combination instability regions for various natural modes and boundary conditions are obtained numerically. The effects of relative amplitude and constant part of periodically rotating speed and cone angle on the instability regions are discussed in detail. It is shown that for the natural mode with lower circumferential wavenumber, only the primary instability regions exist. With the increasing circumferential wavenumber, the instability widths are reduced significantly and the combination instability region might appear. The results for different boundary conditions are substantially similar. Increasing the constant rotating speed (or cone angle) all lead to the movements of instability regions and the appearance of combination instability region. The former will cause the instability width increasing, while the latter will reduce the instability width. The variation of length-to-radius ratio only causes the movements of instability regions.     

Principal resonance response of a stochastic elastic impact oscillator under nonlinear delayed state feedback

In this paper, the principal resonance response of a stochastically driven elastic impact(EI) system with time-delayed cubic velocity feedback is investigated. Firstly, based on the method of multiple scales, the steady-state response and its dynamic stability are analyzed in deterministic and stochastic cases, respectively. It is shown that for the case of the multivalued response with the frequency island phenomenon, only the smallest amplitude of the steady-state response is stable under a certain time delay, which is different from the case of the traditional frequency response. Then, a design criterion is proposed to suppress the jump phenomenon, which is induced by the saddle-node bifurcation. The effects of the feedback parameters on the steady-state responses, as well as the size, shape, and location of stability regions are studied. Results show that the system responses and the stability boundaries are highly dependent on these parameters. Furthermore, with the purpose of suppressing the amplitude peak and governing the resonance stability, appropriate feedback gain and time delay are derived.     

Surface effect on buckling configuration of nanobeams containing internal flowing fluid: A nonlinear analysis

In this paper, the post-buckling behavior of supported nanobeams containing internal flowing fluid with two surface layers is studied based on a nonlinear theoretical model. The nonlinear governing equation, in which the surface effect and stretching-related nonlinearity are accounted for, is analytically solved for both clamped-clamped and pinned-pinned systems. The effects of nanobeam length, bulk thickness and several dimensionless parameters on the post-buckling behavior are analyzed. It is found that, the nanobeam with low flow velocity is stable at its original static equilibrium position and then undergoes a buckling instability at a critical flow velocity, which depends on the system parameters. When buckled, in all cases, the amplitude of the resultant buckling increases with the increasing flow velocity. Typically, the surface effect is explored by considering different nanobeam lengths and bulk thicknesses. The buckling amplitude is found to be length-dependent and thickness-dependent, showing that the effect of surface layers is considerably strong.     

Numerical Simulation of Shear-Horizontal-Wave-Induced Electromagnetic Field in Borehole Surrounded by Porous Formation

Seismoelectric fieM excited by purely torsional loading applied directly to the borehole wall is considered. A brief formulation and some computed waveforms show the advantage of using shear-horizontal （SH） transverseelectric （TE） seismoelectric waves logging to measure shear velocity in a fluid-saturated porous formation. By assuming that the acoustic field is not influenced by its induced electromagnetic field due to seismoeleetric effect, the coupling governing equations for electromagnetic field are reduced to Maxwell equations with a propagation current source. It is shown that this simplification is valid and the borehole seismoelectric conversion efficient is mainly dependent on the electrokinetic coupling coefficient. The receivers to detect the conversion electromagnetic field and to obtain shear velocity can be set in the borehole fluid in the SH-TE seismoelectric wave log.