Combination Parametric Resonance Research Articles

Experimental study of parametric vibrations of arc-shaped tubes with pulsating fluid flow

The dynamics of curved tubes with fluid flow fixed on both sides was studied experimentally. Two cases were studied in which the curved tubes differed in curvature. Tests were performed with the flow without pulsation to determine the natural frequencies and with pulsation to determine the interesting dynamic phenomena occurring in the system. Two alternative measurement systems were used to study the motion of the tubes, a contact one based on accelerometers and a non-contact one based on optical technology. Acceleration and displacement time-series were analysed to determine the nature of the motion of the tubes. It was confirmed that the eigenfrequencies of the curved systems are insensitive to flow velocity which is consistent with the extensible centreline hypothesis. The interesting dynamics of tubes with pulsatile flow, the possibility of excitation of simple, combination parametric resonances, including several different combination resonances simultaneously, were identified. The vibrations in the simple resonances were polyharmonic and in the combination resonances quasi-periodic. The motion of the tubes was three-dimensional in nature and its complexity increased with increasing excitation amplitude. It was found that increasing the curvature of the tube reduced the possibility of excitation of parametric vibration.

Multi-harmonic resonance of pipes conveying fluid with pulsating flow

The parametric excitation analysis of pipes conveying pulsating fluid usually focuses on the principal parametric resonance. In this paper, the principal parametric resonance and the combination resonance under parametric excitation are compared for the first time. A dynamic model of a fluid-conveying pipe with elastic torsional constraints at both ends is established. The nonlinear partial differential-integral equation is derived for governing the transverse vibration. The natural frequencies and modes of pipe are obtained. Moreover, the analytical solutions of stable region and nonlinear amplitude-frequency responses for three kinds of parametric resonances are derived. Then, the analytical results are verified numerically. The results show that there is no differential-type combination parametric resonance for pipes supported at both ends. Increasing material damping or torsional stiffness and decreasing mean velocity can increase the stable region of the pipe. However, it is interesting to note that when combination parametric resonance occurs between symmetric and antisymmetric modes, the effect of mean velocity on stability is reduced. In addition, more than one form of resonance occurs in some regions. The sum-type combination resonance is easier to be excited than the principal parametric resonance. Among them, the combination of first-fourth mode is the most sensitive to the torsional boundary, and its response is the largest. In summary, this paper demonstrates that the combination parametric resonance, which has not been researched deeply yet, can be more destructive than the principal parametric resonance.

Numerical and Experimental Stability Investigation of a Parametrically Excited Cantilever Beam at Combination Parametric Resonance

BackgroundThe presence of parametric excitation in dynamic structures, caused by friction, crack, varying compliance, electromagnetic field, etc. may generate unbounded responses. In the literature there exist several numerical analyses of systems affected by parametric excitation, while experimental studies are less frequent.ObjectiveThe goal of the paper is to create a demonstrator of a parametrically excited system, whose stability can be modified through a controlled physical parameter. This work also investigates the applicability of the recently developed stability analysis method named Jacobian Based Approach (JBA).MethodsThis paper studies a simple experimental set-up comprising of a cantilever beam mounted on a spring with time – varying stiffness, achieved through the use of an electromagnet. The test rig allows measuring directly the magnetic force without any preknowledge of the values of electrical parameters. Results obtained from the test rig are compared with numerical results obtained from the Finite Element model. In this study, Hill’s method and JBA are employed to obtain the stability plot highlighting the regions of parametric instabilities.ResultsGood agreement is found between experimental and numerical data and the presence of unstable behavior is verified through the use of the well – known Hill’s method and the JBA. Furthermore, this study demonstrates that the stability plot, highlighting the unstable regions, computed by JBA is in complete agreement with the one obtained by Hill’s method.ConclusionsIt is shown how the parametric instability can be triggered through the regulation of a simple physical parameter, i.e. the gap between the electromagnet and the beam. The numerical model analyzed by the ad – hoc technique proposed by the authors i.e. JBA has been proven to have predictive capabilities in studying a system under parametric excitation and could be a potential substitution for state-of-the-art stability analysis techniques such Hill’s method.

Dynamic stability of graded graphene reinforced truncated conical shells under both periodic spinning speeds and axial loads considering thermal effects

This paper is concerned with dynamic stability of graded graphene reinforced truncated conical shells under both periodic spinning speeds and axial loads considering thermal effects. The volume fraction of graphene platelets (GPLs) varies continuously along the shell’s thickness direction, which induces the position-dependent effective material properties. Based upon Love’s thin shell theory and Galerkin approach, the equations of motion of the conical shells are derived by considering thermal environment, both time-variable spinning speeds and axial loads. The method of multiple scales is adopted to obtain an analytical solution on the instability boundaries under combination parametric resonances. Then, comprehensive parametric studies are conducted focusing on the instability regions, natural frequencies and critical spinning speeds of the conical shell. The sensitivities of dynamic stabilities on the thermal expansion deformation, thermal conductivity and temperature-dependent material properties are also analyzed. Results show that the conical shell system would be always instability if parametric phase is not equal to integer multiple of π. The GPL parameters, temperature variation, spinning speeds and axial loads have a significant influence on dynamic stability of the conical shell, and thermal conductivity and thermal expansion deformation are nonnegligible in the dynamic stability analysis.

Parametric Instability of Rotating Functionally Graded Graphene Reinforced Truncated Conical Shells Subjected to Both Mechanical and Thermal Loading Conditions

This article focuses on parametric instability of rotating functionally graded (FG) truncated conical shells reinforced by graphene platelets (GPLs) and subjected to both mechanical and thermal loading conditions. The GPL nanofillers are uniformly dispersed in each concentric conical layer but its weight fraction varies continuously along thickness direction, which induces the position-dependent effective material properties. Based on Love’s thin shell theory and Galerkin approach, the equations of motion for the conical shells are derived with the effects of the periodic axial loads, thermal expansion deformation, rotation-induced initial hoop tension, gyroscopic and centrifugal forces taken into account. Then, the parametric instability under combination parametric resonance for the conical shells is performed by the method of multiple scales, and the analytical solutions of both instability boundaries are obtained. A comprehensive parametric study is conducted which focuses on instability regions and vibration characteristics of the conical shell. Of particular interest in this process is the combined effect of the rotation, dynamic loads and thermal effects on dynamic stability of FG-GPL reinforced truncated conical shells.

Balancing high-speed rotors at low speed using optimized parametric excitation and tuned nonlinear feedback

The paper presents an improved slow-speed mass balancing procedure capable of reducing high-speed related imbalance. Parametric excitation is used to project the small imbalance forces onto a chosen mode of vibration, corresponding to a high critical rotation speed. The method employs two kinds of parametric excitation leading to combination and principal parametric resonances. The former projects the imbalance forces onto a selected vibration mode, and the latter significantly amplifies the dynamic response due to imbalance. In addition to the parametric excitations, specially tuned nonlinear feedback forces are applied to limit the response to desired vibration levels and achieve a pseudo-linear behavior. In situ proper tuning of the nonlinear terms, and parametric excitations, enables to increase both amplification and sensitivity to the imbalance forces without having to compromise between the two. The latter includes digital, nonlinear feedback that enhances the sensitivity to imbalance forces, while limiting the level of vibration. This method allows to identify the projection of the imbalance on any desired mode, and therefore to cancel them by adding or removing mass. An experimental realization of the method validates the theory with good accuracy.

Multiple electrically tunable parametric resonances in a capacitively coupled electromechanical resonator for broadband energy harvesting

Parametric excitation (PE) has widely been employed as a method of mechanical pre-amplification in nonlinear vibration energy harvesting systems. However, despite their advantages, most current PE systems are limited to degenerate parametric operation within a narrow frequency band around the primary instability tongue. In this paper, we simulate and experimentally demonstrate a parametrically driven capacitive electromechanical resonator having multiple electrical degrees of freedom. Multiple modes allow for several frequency bands in which the electrical resonator is driven into nondegenerate (combination) parametric resonance (PR) in addition to degenerate resonance, thereby enabling operation over a broader range of frequencies while maintaining the same mechanical footprint. These frequency bands and PR thresholds are tunable by simply changing the electrical circuit parameters and PR can be achieved in the presence of high mechanical damping making the method more adaptable than purely mechanical approaches. Experimental results are extended by simulations indicating that proper selection of operating parameters can enable the merging of instability tongues to produce a broadband region of PR for elastic wave energy harvesting thereby obtaining superior performance when compared to an equivalent single degree of freedom PE energy harvester.

Nonlinear parametric vibration of spinning pipes conveying fluid with varying spinning speed and flow velocity

In this paper, a nonlinear parameter-excited model of spinning pipes conveying fluid is proposed by considering the spinning speed and flow velocity are perturbed periodically, and the stability and nonlinear parametric vibrations of such system are studied analytically and numerically. With taking the viscoelastic material and geometrical nonlinearity due to extensible pipe axis into account, the differential equations governing two transverse vibrations of the pipe are derived by the Hamilton principle. The system stability is then analyzed via the multiple scales method, and the nonlinear responses and spatial vibration shapes are investigated using numerical simulation. The contributions of some significant parameters on the vibrations of the system are discussed in detail. It is revealed that not any other motion but combination parametric resonance with quasi-periodic motion mode can occur in the present system, and it is induced fully by the periodically perturbed fluid. A perturbed spinning motion cannot result in any parametric resonance, however it has an opposite effect on the parametric vibrations as compared to that with constant spinning speed. The flow velocity, nonlinearity and viscoelastic damping all have significant impacts on the present parametric vibrations. Moreover, the pipe performs different aperiodic whirling motions in the first two modes, and the perturbed spinning speed will lead to a gentler motion of the pipe as compared to the case of constant spinning speed.

Damping and gyroscopic effects on the stability of parametrically excited continuous rotor systems

This study is aimed at analysing damping and gyroscopic effects on the stability of parametrically excited continuous rotor systems, taking into account both external (non-rotating) and internal (rotating) damping distributions. As case-study giving rise to a set of coupled differential Mathieu–Hill equations with both damping and gyroscopic terms, a balanced shaft is considered, modelled as a spinning Timoshenko beam loaded by oscillating axial end thrust and twisting moment, with the possibility of carrying additional inertial elements like discs or flywheels. After discretization of the equations of motion into a set of coupled ordinary differential Mathieu–Hill equations, stability is studied via eigenproblem formulation, obtained by applying the harmonic balance method. The occurrence of simple and combination parametric resonances is analysed introducing the notion of characteristic circle on the complex plane and deriving analytical expressions for critical solutions, including combination parametric resonances, valid for a large class of rotors. A numerical algorithm is then developed for computing global stability thresholds in the presence of both damping and gyroscopic terms, also valid when closed-form expressions of critical solutions do not exist. The influence on stability of damping distributions and gyroscopic actions is then analysed with respect to frequency and amplitude of the external loads on stability charts in the form of Ince–Strutt diagrams.

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

In-plane vibration of a rigid body attached to a flexible rotating beam

The dynamics of rotating systems are conventionally analyzed using models that assume the axis of rotation to be stationary. However, in instances such as helicopters and gas turbines with contra-rotating rotors such models which neglect the complexity arising from the coupling between the stator and the rotor due to their comparable masses can lead to inaccurate predictions. This paper examines the coupled stator-rotor dynamics of a model comprising a rigid (but non-stationary) body which is excited with a flexible rotating beam. Equations of motion governing the model's dynamics are developed and are found to have time-periodic boundary conditions. For the axisymmetric case the system loses its time dependence; the natural frequencies and the mode shapes vary with rotation speed. Specific modes soften within certain speed regimes, while other modes stiffen. Instabilities such as divergence and flutter are observed, and the effects of parameters on the critical speeds are investigated. The non-axisymmetric case is parametrically excited at twice the rotation speed. Floquet theory is employed to develop stability charts; regimes of instability exhibit a definitive structure comprising simple and combination parametric resonances. Temporal responses are modulated, and they exhibit characteristic side-banded frequency spectra.

Radiation of a Resonant Medium Excited by a Periodically Phase-Modulated Laser in the Regime of Strong Coupling between the Field and the Matter

The radiation of a two-level resonant medium that is placed into a cavity and is excited by a periodically phase modulated laser pulse is studied theoretically. The radiation is analyzed based on the analytical and numerical solutions of the system of Maxwell–Bloch equations under conditions of a strong coupling between the field and the matter. Under these conditions, this system is similar to a polariton laser. The excitation efficiency of a polariton laser by a phase-modulated radiation pulse is shown to be high compared to a pulse without phase modulation of the carrier frequency. It is shown that the main reason for the efficient excitation of polariton modes in the medium is the appearance of difference combination parametric resonance. The obtained results open up new possibilities for the excitation of radiation of polariton lasers by low-power frequency-modulated pump laser radiation.

Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed, variable axial tension under two-frequency parametric excitations and internal resonance

This paper investigates nonlinear combined parametric transverse vibrations of a traveling viscoelastic beam. The combined parametric excitations originate from the time dependency of axial velocity as well as axial tension. Two parametric excitations are enforced into the system amid the internal resonance. Two-frequency parametric resonance is assumed to be comprised of combination parametric resonance of first two modes due to the time dependency of axial velocity, and the principal parametric resonance of first mode due to the variable tension in the axial direction in the presence of internal resonance for viscoelastic beam is considered for the first time. The higher-order integro-partial differential equation of motion is solved through direct method of multiple scales. Continuation algorithm is employed to explore the stability and various bifurcations of the nonlinear dynamic system. Focus has been made to study the effect of variations of fluctuating tension component, fluctuating velocity component independently and when combined, internal and parametric frequency detuning parameters and damping on the system response. Frequency response equilibrium curves are complex and unique in shapes which are embodied with various bifurcations. Such steady-state behavior is not seen in the existent literature. With variation in fluctuating velocity component, the number of steady-state nontrivial equilibrium curves increases to three and with variation in fluctuating axial tension, they become four. In this process, significant changes in stability, number and position of various bifurcations like supercritical and subcritical pitchfork, Hopf and saddle node are observed. Unlike the previous study, the shape, stability and bifurcations of equilibrium curves under the combined effect of axial velocity and tension closely match with the case of fluctuating axial tension component. The effect of variation in internal and parametric frequency detuning parameter is more realized for second mode compared to first mode. A comparison of the present work with a previous one where axial tension is variable reveals many qualitative and quantitative similarities and dissimilarities. But when compared with earlier work where axial velocity is constant, significant dissimilarities are surfaced. The system displays a wide ranging dynamic behavior including stable periodic, quasiperiodic and unstable chaotic behavior. The numerical computation depicts various nonlinear characteristics and oscillatory behaviors which are not found so far in the existent literature.

Parametrically Excited Nonlinear Two-Degree-of-Freedom Electromechanical Systems

This paper presents a nonlinear parametrically excited cantilever beam with electromagnets. A parametrically excited two-degree-of-freedom (2-DOF) system with linear time-varying stiffness, nonlinear cubic stiffness, nonlinear cubic parametric stiffness and nonlinear damping is considered. In previous studies the stability and bifurcation of the nonlinear parametrically excited 2-DOF were investigated through analytical, semi-analytical and numerical methods. Unlike previous studies, in this contribution the system’s response amplitude and phase at parametric resonance and parametric combination resonance are demonstrated experimentally and some novel results are discussed. Experimental and analytical amplitude-frequency plots are presented to show the stable solutions. Solutions for the system response are presented for specific values of parametric excitation frequency and the energy transfer between modes of vibrations is observed. The results presented in this paper prove that the bifurcation point and hence the bandwidth of the parametric resonance can be predicted correctly with the proposed analytical method. The proposed nonlinear parametrically excited 2-DOF can be used to design Micro ElectroMechanical Systems (MEMS) actuators and sensors. Validating the experimental results with the theory can improve the efficiency of these electrical systems.

Излучение резонансной среды, возбуждаемое лазерным излучением с периодической фазовой модуляцией в режиме сильной связи поля и вещества

In this paper, the radiation of a two-level resonant medium placed in a cavity and excited by a laser pulse with periodic phase modulation is studied theoretically. The analysis is carried out on the basis of an analytical and numerical solution of the system of Maxwell-Bloch equations under conditions when the regime of strong coupling of the field and matter is realized. Under these conditions, this system is similar to a polariton laser. A high excitation efficiency of a polariton laser by a phase-modulated radiation pulse compared with a pulse without phase modulation of the carrier frequency is shown. It is shown that the main reason for the effective excitation of polariton modes of the medium is the occurrence of a difference combination parametric resonance. The results obtained open up new possibilities in the excitation of radiation from polariton lasers by low-power pumping laser radiation with frequency modulation.

Rotor-pendulum self-synchronization of parametrical vibration generators on an isotropic elastic foundation

Significant improvement of dynamic characteristics of vibration machines on the basis of the use of resonant oscillatory systems with two or more degrees of freedom is proposed. It is shown that reasonable complication of models due to the increase in the number of degrees of freedom, taking account of nonlinearity makes it possible to increase reciprocal enhancement of vibrations of partial subsystems and to open additional opportunities in the development of new equipment and technologies. The involvement of objects of an oscillatory system in collective interaction is achieved on the basis of using compound resonances which occur in coupled systems only. Combination parametric resonance caused by pair-wise interaction of free modes of oscillations represents this kind of resonance. The results of theoretical and experimental studies of self-synchronization of parametric rotor-pendulum vibration generators installed on a common elastic isotropic foundation are presented in the paper. The dynamic model of a vibration machine is represented by a set of peer interacting nonlinear oscillators (pendulums) under resonant excitement of which the Huygens's effect, that is, pendulum clock-type self-synchronization (pendulum self-synchronization) occurs. One or several oscillators perform the functions of the working body of the vibration machine. Other oscillators of this system act as the inertial element of a rotor-pendulum vibration generator. It is shown that in the case of combined action of at least two rotors-pendulum vibration generators unbalanced rotor-type self-synchronization (rotor self-synchronization) takes place. Thus, the device under examination at the same time combines rotor and pendulum self-synchronization. The results of numerical modeling in the form of amplitude-frequency characteristics and dependences of generation frequencies on the frequency of parametric excitation are presented. It is established that the amplitude of the working body oscillations, due to resonant interactions with the generator pendulums, underreacts to an increase in damping. The sum of natural frequencies of pendulums and the working body meets the condition of combination parametric resonance over the whole range of the instability region.

Polariton condensation, superradiance and difference combination parametric resonance in mode-locked laser

The generation of the ring mode-locked laser containing resonant absorption medium in the cavity was investigated. It is shown that near the strong resonant absorption lines a condensation of polaritons arises. Intensive radiation looks like as superradiance in a medium without population inversion. We studied theoretically the microscopic mechanism of these phenomena. It was shown that in this system in absorbing medium a strong self-induced difference combination parametric resonance exists. Superradiance on polaritonic modes in the absorbing medium are due to the emergence of light-induced resonant polarization as a result of fast periodic nonadiabatic quantum jumps in the absorber.

Non-planar vibrations of slightly curved pipes conveying fluid in simple and combination parametric resonances

The paper summarises the experimental investigations and numerical simulations of non-planar parametric vibrations of a statically deformed pipe. Underpinning the theoretical analysis is a 3D dynamic model of curved pipe. The pipe motion is governed by four non-linear partial differential equations with periodically varying coefficients. The Galerkin method was applied, the shape function being that governing the beam's natural vibrations. Experiments were conducted in the range of simple and combination parametric resonances, evidencing the possibility of in-plane and out-of-plane vibrations as well as fully non-planar vibrations in the combination resonance range. It is demonstrated that sub-harmonic and quasi-periodic vibrations are likely to be excited. The method suggested allows the spatial modes to be determined basing on results registered at selected points in the pipe. Results are summarised in the form of time histories, phase trajectory plots and spectral diagrams. Dedicated video materials give us a better insight into the investigated phenomena.

Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces

Abstract This work is motivated by the self-powered component of biomedical nano-robotic device which is expected to move in arterial blood vessels. The transverse vibrations and steady-state responses of axially moving viscoelastic piezoelectric two-dimensional nanostructures are investigated based on the nonlocal viscoelasticity thin plate theory. The constitutive relations of viscoelastic piezoelectric nanoplate containing the thermal effect are performed and the governing partial differential equations of the problem model are derived using the Hamilton's principle. The natural frequencies are numerically determined via the Galerkin method, the complex mode method, as well as the finite element method for comparison. Moreover, the theoretical calculations are compared with those in previous literature to verify effects in nonlocal nanoscale framework and average speed on natural frequency. Afterwards, the instable behaviors of axially non-uniformly moving viscoelastic piezoelectric nanoplate characterized as a sine variation about the constant average speed are addressed using the method of multiple scales. The analyses are mainly focused on the boundaries of instable regions in combination parametric resonance and principal parametric resonance. The non-dimensional numerical results imply the existences of nonlocal nanoscale parameter and average speed contribute to reduce the rigidity, and further produce the coupled vibrations and flutter instabilities for complex frequencies. Additionally, the instable regions of combination and principal parametric resonances decrease with increases in the biaxial compression, change of temperature, positive electric voltage and viscoelastic coefficient. The dynamic responses of axially moving viscoelastic piezoelectric nanoplate under the coupling of thermo-electro-mechanical multi-fields are expected to play significant roles in designing biomedical nano-robots.

Nonlinear three-dimensional dynamics of flexible pipes conveying fluids

This paper presents a model which describes the non-planar vibrations of an elastic pipe induced by the pulsations of a flowing fluid. The motion of the pipe has been described by a set of four non-linear partial differential equations with periodically varying coefficients. The analysis has been conducted with the use of Galerkin method by applying functions describing the free vibrations of the beam as the shape functions. Floquet theory has been used to determine the unstable areas. The influence of the flow velocity and the pulsation frequency on the character and modes of vibrations and on the ranges of the increased vibration intensity has been examined. The possibility of the excitation of sub-harmonic and quasi-periodic vibrations in ranges of simple and combination parametric resonances has been proved. The results of experiments which confirm the occurrence of the parametric resonance phenomenon have also been presented.