Principal Parametric Resonance Research Articles

Nonlinear and dual-parameter chaotic vibrations of lumped parameter model in blisk under combined aerodynamic force and varying rotating speed

In this paper, the nonlinear and dual-parameter chaotic vibrations are investigated for the blisk structure with the lumped parameter model under combined the aerodynamic force and varying rotating speed. The varying rotating speed and aerodynamic force are, respectively, simplified to the parametric and external excitations. The nonlinear governing equations of motion for the rotating blisk are established by using Hamilton’s principle. The free vibration and mode localization phenomena are studied for the tuning and mistuning blisks. Due to the mistuning, the periodic characteristics of the blisk structure are destroyed and uniform distribution of the energy is broken. It is found that there is a positive correlation between the mistuning variable and mode localization factor to exhibit the large vibration of the blisk in a certain region. The method of multiple scales is applied to derive four-dimensional averaged equations of the blisk under 1:1 internal and principal parametric resonances. The amplitude–frequency response curves of the blisk are studied, which illustrate the influence of different parameters on the bandwidth and vibration amplitudes of the blisk. Lyapunov exponent, bifurcation diagrams, phase portraits, waveforms and Poincare maps are depicted. The dual-parameter Lyapunov exponents and bifurcation diagrams of the blisk reveal the paths leading to the chaos. The influences of different parameters on the bifurcation and chaotic vibrations are analyzed. The numerical results demonstrate that the parametric and external excitations, rotating speed and damping determine the occurrence of the chaotic vibrations and paths leading to the chaotic vibrations in the blisk.

A Physically Consistent Model for Forced Torsional Vibrations of Automotive Driveshafts

The aim of this research was to design a physically consistent model for the forced torsional vibrations of automotive driveshafts that considered aspects of the following phenomena: excitation due to the transmission of the combustion engine through the gearbox, excitation due to the road geometry, the quasi-isometry of the automotive driveshaft, the effect of nonuniformity of the inertial moment with respect to the longitudinal axis of the tulip–tripod joint and of the bowl–balls–inner race joint, the torsional rigidity, and the torsional damping of each joint. To resolve the equations of motion describing the forced torsional nonlinear parametric vibrations of automotive driveshafts, a variational approach that involves Hamilton’s principle was used, which considers the isometric nonuniformity, where it is known that the joints of automotive driveshafts are quasi-isometric in terms of the twist angle, even if, in general, they are considered CVJs (constant velocity joints). This effect realizes the link between the terms for the torsional vibrations between the elements of the driveshaft: tripode–tulip, midshaft, and bowl–balls–inner race joint elements. The induced torsional loads (as gearbox torsional moments that enter the driveshaft through the tulip axis) can be of harmonic type, while the reactive torsional loads (as reactive torsional moments that enter the driveshaft through the bowl axis) are impulsive. These effects induce the resulting nonlinear dynamic behavior. Also considered was the effect of nonuniformity on the axial moment of inertia of the tripod–tulip element as well as on the axial moment of inertia of the bowl–balls–inner race joint element, that vary with the twist angle of each element. This effect induces parametric dynamic behavior. Moreover, the torsional rigidity was taken into consideration, as was the torsional damping for each joint of the driveshaft: tripod–joint and bowl–balls–inner race joint. This approach was used to obtain a system of equations of nonlinear partial derivatives that describes the torsional vibrations of the driveshaft as nonlinear parametric dynamic behavior. This model was used to compute variation in the natural frequencies of torsion in the global tulip (a given imposed geometry) using the angle between the tulip–midshaft for an automotive driveshaft designed for heavy-duty SUVs as well as the characteristic amplitude frequency in the region of principal parametric resonance together the method of harmonic balance for the steady-state forced torsional nonlinear vibration of the driveshaft. This model of dynamic behavior for the driveshaft can be used during the early stages of design as well in predicting the durability of automotive driveshafts. In addition, it is important that this model be added in the design algorithm for predicting the comfort elements of the automotive environment to adequately account for this kind of dynamic behavior that induces excitations in the car structure.

Nonlinear Dynamics and Bifurcation Analysis of a Hinged-Clamped Beam Under Parametric and Internal Resonances

The present study examines the stability and bifurcation of axially moving hinged-clamped beams subjected to parametric excitation originating due to speed alterations. Attention is paid to the principal parametric resonance amid 3:1 internal resonance in the subcritical speed regime. The direct method of multiple scales is applied to predict the nonlinear behavior of the beam modeled in the form of the nonlinear integro-partial differential equation. A continuation algorithm is used to solve numerical examples to illustrate the stability and bifurcations of periodic solutions for the given set of system parameters. The fixed point solution is different for different modes. Three kinds of equilibrium solution curves like trivial, two-mode, and single-mode solutions are obtained from numerical computations. The first two kinds are available for both first and second modes while the third one is available for the second mode only. The introduction of material damping converts the isolated two-mode solution into an isolated closed-loop form. The system displays saddle-node and Hopf bifurcation points in both modes while super and subcritical pitchfork bifurcation points in the second mode only. Decreasing viscous damping strengthens the effect of internal resonance. The numerically simulated results are unique, interesting, and are not available in the existing literature.

Modal Analysis for Localization Phenomena in Two-Pendulum Arrays Subjected to Vertical Excitation

When an array of two identical pendula connected by a linear, torsion spring is subjected to vertical excitation, the two pendula may oscillate with different amplitudes. This is referred to as a localization phenomenon. In this paper, modal analysis is performed to investigate the cause of the phenomenon. The equations of motion for the two pendula are derived in the physical coordinate. First, in the linearized equations of motion, the natural frequencies and the corresponding modal vectors are determined. Then, the equations of motion in the physical coordinates are transformed to those in the modal coordinates. Van der Pol’s method is employed to obtain the frequency response curves for the principal parametric resonance in the modal coordinates, and the response curves for the physical coordinates are determined using the coordinate transformation. These theoretical response curves are validated by being compared to the time histories and their FFT results. The numerical results show that as the excitation frequency decreases, a branch of the response curves for the first mode undergoes a pitchfork bifurcation, which induces the second mode owing to the nonlinear coupling with the first mode, and the two modes appear simultaneously at a specific excitation frequency range. This is the reason why localization phenomena are observed in the physical coordinates.

On Periodical Damping Ratio of a Controlled Dynamical System With Parametric Resonances

A dynamical system whose damping of oscillation or vibration is directly controlled may manifest a periodically varying damping ratio. In this letter, such a phenomenon is discovered to arise from a new type of parametric resonance, named the “zero-th order parametric resonance,” which is different from the conventional principal parametric resonance. This letter proposes an approach using the method of multiple scales to find approximate, analytical solutions of the system and thus providing an interpretation on damping variations.

Nonuniformity of Isometric Properties of Automotive Driveshafts

This paper presents an analysis of the CVJ (constant velocity joint) of automotive driveshafts from a point of view concerning the nonuniformity of isometric properties. In the automotive industry, driveshafts are considered to have constant velocity through its joints: free tripode joints and fixed ball joints, which has been proved by Mtzner’s indirect method and Orain’s direct method for tripod joint. Based on vectorial mechanics, the paper proved the quasi-isometry of velocity for polypod joints such as fixed ball joints. In the meantime, it was computed that the global nonuniformity of constant velocity joints for modern driveshafts based on the Dudita-Diaconescu homokinetic approach for the driveshafts. The nonuniformity of the velocity isometry of driveshafts was computed as a function of the input angular velocity of the driveshaft, angular inclination between the tripod–tulip axis and the midshaft axis and the angular inclination between the bowl axis and midshaft axis. The main aim of this article is how to improve the geometric and kinematic approach to add an important correction when designing the driveshaft dynamics prediction such as: forced torsional vibrations, forced bending–shearing vibrations, and coupled torsional–bending vibrations for the automotive driveshaft in the regions of specific resonances such as principal parametric resonance, internal resonance, combined resonance, and simultaneous resonances. By the way it is added, there are important corrections for the design of driveshafts, for the torsional dynamic behavior prediction, and for bending–shearing dynamic behavior of the driveshafts in the early stages of design. The results presented in the article represent a starting point for future research on dynamic phenomena in the area mentioned previously.

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.

Dynamic behavior of T-beam resonator with repulsive actuation

Electrostatic MEMS transducer driven by repulsive force is an attractive possibility and has advantages of avoiding the pull-in instability, tuning the natural frequency, and achieving high sensitivity by applying high enough voltages. In this work, a “T”-shaped beam, which is formed by attaching a secondary beam perpendicular to a primary cantilever at the tip, is introduced and its nonlinear dynamics is analyzed. A reduced-order model is derived from mode shapes formed from electromechanical coupling effects respectively. Generalized forms of forced Mathieu equation of motion are derived, and then, dynamic behaviors are investigated through the theory of multiple scales. The resonant responses, including both primary and principal parametric resonances, reveal softening behavior originating from quadratic and cubic nonlinearities in the governing equation. The behavior of the T-beam is compared with traditional cantilever structure. The resonance under repulsive force demonstrates that the T-beam has several advantages over a traditional cantilever: Lower natural frequency but higher resonant responses can improve the signal-to-noise ratio; with an attached micropaddle, the T-beam has a larger surface for absorption of targeted analytes for mass sensing. We conclude that an electrostatic MEMS resonator with a “T”-shaped beam is potentially appropriate for the new generations of sensors and actuators.

Modal Interaction-Induced Parametric Resonance of Stayed Cable: A Combined Theoretical and Experimental Investigation

The cable-stayed bridge is widely used due to its strong spanning capacity and navigability. However, flexible cables parametrically resonated by external excitation may result in instability or even damage to the bridge. To prevent such undesirable resonance, this paper discusses an in-plane modal interaction-induced parametric resonance of the stayed cable excited by the bridge deck vibration via nonlinear dynamic analysis. Based on the nonlinear distributed model, two modal governing equations of the cable are established via the Galerkin method. A certain working condition, when the external excitation frequency is close to the second-order natural frequency of the stay cable while nearly twice the first-order natural frequency, is theoretically and experimentally investigated. Specifically, the frequency response equations are obtained by the multiscale method, and the stability of solutions is examined through the Routh Hurwitz criterion. Theoretical and experimental results show that bridge deck vibration can induce not only the primary and superharmonic resonance of the cable but also the principal parametric resonance. Parametric resonance-induced bifurcations are also observed in the system. Particularly, the energy exchange from second-order primary resonance to first-order principal parametric resonance is found, which can induce the parametric resonance with the response amplitude one to three times higher than that of the primary resonance. This paper also validates the superiority of the present modal interaction model over the traditional single-mode model in practical engineering applications.

Linear and nonlinear analysis of traditional and non-traditional piezoelectric vibration absorber with time delay feedback for simultaneous resonance conditions

In the present work, vibration suppression of a single degree of freedom nonlinear spring-mass primary (main) system subjected to simultaneous external harmonic and parametric excitation by using traditional and non-traditional active nonlinear vibration absorber (ANVA) is carried out. The traditional and non-traditional ANVA consists of a PZT (lead zirconate titanate) stack actuator in series with a nonlinear spring and a damper. In the traditional ANVA, the damper is connected to the main system whereas, in non-traditional ANVA, the damper is connected to the base of the main system. The governing nonlinear equations of motion for the system have been derived using weighted modal matrix approach considering novel actuation force by time delay in displacement, velocity and acceleration feedback of the main system by the PZT stack actuator and the nonlinear spring in series combination. The method of multiple scales (MMS) is used to obtain the reduced equations which are solved using Newton’s method. The responses of the system for the primary, principal parametric and simultaneous primary and principal parametric resonance conditions have been studied. The results are compared with those obtained by using the harmonic balance method (HBM), numerical method and with previous literature. From these results, two resonating peaks in the frequency response of the main system are shown by using the reduced equation of the MMS which was not reported in earlier literature. It is shown that for the active case the vibration of the undamped main system is completely reduced at the specified operating frequency for both the linear and nonlinear systems. Also, for a wider range of operating frequencies, the vibration of the main system is reduced by actively tuning the absorber for various mass ratios. Further, the efficacy of the active force is observed in reducing the instability regions and improving the critical value of excitations in the trivial state.

High-frequency vibrational control of principal parametric resonance of a nonlinear cantilever beam: Theory and experiment

The effects of high-frequency parametric excitation on the principal parametric resonance of a nonlinear beam are studied both theoretically and experimentally. Only the first mode of vibration under simultaneous slow-fast parametric excitation is considered for investigation. The method of direct partition of motion yields slow dynamics of the system which is further analyzed by multiple time scale method to obtain the amplitude and phase response of the system. It is observed that the effective damping and stiffness of the system is significantly modified by the high frequency parametric excitation. The analysis reveals that the parametric resonance curve shifts towards the higher range of slow parametric excitation frequency due to high-frequency excitation. The peak resonant response decreases with the increasing value of the strength of the high frequency excitation and also beyond a critical value of the strength of the high-frequency excitation, the jump resonance is eliminated. Overall, it is found that the principal Parametric resonance can be effectively controlled and suppressed by high frequency excitation. Analytical results are validated by direct numerical simulations and experiments. For some parameter values, vibrational resonance is also observed in theory and experiment. The parametric vibrational resonance of a cantilever beam is not previously observed experimentally and thus, it is a new finding of the present study. The vibrational resonance is believed to be utilized in amplifying the response of parametrically driven microcantilever beams found in many microsystems applications.

Piezoelectrically induced nonlinear resonances for dynamic morphing of lightweight panels

A rich variety of resonance scenarios induced in thin elastic plates by periodic axial strains is investigated to describe the morphing capability of flexible, ultra-lightweight panels. The strain-induced excitation is provided by embedded piezoelectric strips actuated by time-varying voltages. The study deals with plates characterized by geometric and mechanical symmetry entailing cylindrical motions so that an ad hoc approximate nonlinear model of elastic beams is adopted to formulate the incremental equation of motion about the nonlinear equilibrium under dead loads. An asymptotic approach based on the method of multiple scales is employed to study the plate dynamics and, primarily, to investigate the effect of the plate slenderness on the nonlinearity of the lowest normal modes. A bifurcation analysis is carried out to investigate the wide array of resonances induced by the harmonic axial excitation, acting in nontrivial equilibrium states, which include primary, super-harmonic, sub-harmonic, and principal parametric resonances, respectively. For selected slendernesses of the plate, corresponding to nonlinear softening, hardening and quasi-linear behaviors, respectively, comprehensive parametric studies on the effect of the PZT cross-sectional size, the plate damping, and the piezoelectric coupling coefficient are carried out to identify the threshold voltages triggering the different resonances.

Dynamical behavior of fractionalized simply supported beam: An application of fractional operators to Bernoulli-Euler theory

Abstract The complex structures usually depend upon unconstrained and constrained simply supported beams because the passive damping is applied to control vibrations or dissipate acoustic energies involved in aerospace and automotive industries. This manuscript aims to present an analytic study of a simply supported beam based on the modern fractional approaches namely Caputo-Fabrizio and Atanagna-Baleanu fractional differential operators. The governing equation of motion is fractionalized for knowing the vivid effects of principal parametric resonances. The powerful techniques of Laplace and Fourier sine transforms are invoked for investigating the exact solutions with fractional and non-fractional approaches. The analytic solutions are presented in terms of elementary as well as special functions and depicted for graphical illustration based on embedded parameters. Finally, effects of the amplitude of vibrations and the natural frequency are discussed based on the sensitivities of dynamic characteristics of simply supported beam.

Effect of spring mid-support condition on the vibrations of the axially moving string

In this study, the axially moving string with spring-loaded middle support is discussed. The supports assumed as simple support on the string both ends. The intermediate support shows the characteristics of the spring. The string velocity is accepted as harmonically varying around a mean value. The Hamiltonian principle is used to find the equations of motion. The equations of motion become nonlinear, considering the nonlinear effects caused by string extensions. The equations of motion and boundary conditions are become dimensionless by nondimensionalization. Approximate solutions were found by using multiple time scales which is one of the perturbation methods. By solving the linear problem that is obtained by the first terms of the perturbation series, the exact natural frequencies were calculated for the different locations of the mid-support, various spring coefficients, and various axial velocity values. The second-order nonlinear terms reveal the correction terms for the linear problem. Stability analysis is carried out for cases where the velocity change frequency is away from zero and two times the natural frequency. Stability boundaries are determined for the principal parametric resonance case.

Two-frequency parametric excitation and combination resonance of a nanocomposite laminated piezoelectric trapezoidal plate in subsonic airflow

In this study, an analytical approach is presented to analyze the bifurcations and nonlinear dynamics of a cantilevered piezoelectric nanocomposite trapezoidal actuator subjected to two-frequency parametric excitations in the presence of subsonic airflow. The assumption of uniformly distributed single-walled carbon nanotubes along the thickness is taken into the consideration. The governing equations are built by the von-Karman nonlinear strain-displacement relations to consider the geometrical nonlinearity and the linear potential flow theory. The present study focuses on a specific resonance case deals with the occurrence of simultaneous resonances in the principal parametric resonance of the first mode and combination of the parametric resonance of the difference type involving two modes. The multiple scales method is employed to obtain the four nonlinear averaged equations which are solved by using the Runge-Kutta method. Moreover, the frequency-response curves, bifurcation diagrams, time history responses, and phase portrait are obtained to find the nonlinear dynamic responses of the plate. The effects of the amplitude of piezoelectric excitation, piezoelectric detuning parameter, and aerodynamic pressure are also studied. The results indicate that, the chaotic, quasi-periodic and periodic motions of the plate exist under certain conditions and the variation of controlling parameters can change the form of motions of the nanocomposite piezoelectric trapezoidal thin plate.

Nonlinear Oscillations of Nonlinear Damping Gyros: Resonances, Hysteresis and Multistability

This paper addresses the issues on the dynamics of nonlinear damping gyros subjected to a quintic nonlinear parametric excitation. The fixed points and their stability are analyzed for the autonomous gyros equation. The number of fixed points of the system varies from one to six. The approximate equation of gyros is considered by expanding the nonlinear restoring force and parametric excitation for the study of the dynamics of gyros. Amplitude and frequency of possible resonances are found by using the multiple scales method. Also obtained are the principal parametric resonance and orders 4 and 6 subharmonic resonances. The stability conditions for each of these resonances are also obtained. Chaotic oscillations, multistability, hysteresis, and coexisting attractors are found using the bifurcation diagrams, the Lyapunov exponents, the phase portraits, the Poincaré section and the time histories. The effects of the damping parameter, the angular spin velocity and the parametric nonlinear excitation are analyzed. Results obtained by using the approximate gyros equation are compared to the dynamics obtained with the exact equation of gyros. The analytical investigations are complemented by numerical simulations.

Dynamic stability of an axially transporting beam with two-frequency parametric excitation and internal resonance

Abstracts In this paper, the dynamic stability of axially transporting viscoelastic beams with two-frequency parametric excitation and 1:3 internal resonance is investigated for the first time. The governing equation and corresponding inhomogeneous boundary conditions are developed by the Newton's second law. The viscoelastic characteristic of the transporting Euler-Bernoulli beam obeys the Kelvin-Voigt model. The axial tension is considered to vary longitudinally. Direct method of multiple scales is employed to obtain the solvability conditions in principal parametric resonances. The stability boundary conditions are obtained by the Routh-Hurwitz criterion. Numerical examples are shown to illustrate the effects of relevant parameters on the stability boundaries. Unusual and interesting phenomena of stability boundaries occur in two-frequency parametric excitation and 1:3 internal resonance. The accuracies of the approximate analytical results are verified by comparing with the numerical results, which obtained by a differential quadrature method.

Nonlinear Resonant Responses, Mode Interactions, and Multitime Periodic and Chaotic Oscillations of a Cantilevered Pipe Conveying Pulsating Fluid under External Harmonic Force

The nonlinear resonant responses, mode interactions, and multitime periodic and chaotic oscillations of the cantilevered pipe conveying pulsating fluid are studied under the harmonic external force in this research. According to the nonlinear dynamic model of the cantilevered beam derived using Hamilton’s principle under the uniformly distributed external harmonic excitation, we combine Galerkin technique and the method of multiple scales together to obtain the average equation of the cantilevered pipe conveying pulsating fluid under 1 : 3 internal resonance and principal parametric resonance. Based on the average equation in the polar form, several amplitude-frequency response curves are obtained corresponding to the certain parameters. It is found that there exist the hardening-spring type behaviors and jumping phenomena in the cantilevered pipe conveying pulsating fluid. The nonlinear oscillations of the cantilevered pipe conveying pulsating fluid can be excited more easily with the increase of the flow velocity, external excitation, and coupling degree of two order modes. Numerical simulations are performed to study the chaos of the cantilevered pipe conveying pulsating fluid with the external harmonic excitation. The simulation results exhibit the existence of the period, multiperiod, and chaotic responses with the variations of the fluid velocity or excitation. It is found that, in the cantilevered pipe conveying pulsating fluid, there are the multitime nonlinear vibrations around the left-mode and the right-mode positions, respectively. We also observe that there exist alternately the periodic and chaotic vibrations of the cantilevered pipe conveying pulsating fluid in the certain range.

Out-of-plane motion detection in encapsulated electrostatic MEMS gyroscopes: Principal parametric resonance

The impetus of this research is to numerically investigate a measurement method for out-of-plane motion detection in an encapsulated electrostatic MEMS gyroscope. This method is based on a variant of the harmonic detection of resonance (HDR) of the motion-induced current proposed by S. Park et al. [1]. The MEMS device consists of a cantilever microbeam having a paddle at its free end, and is under two simultaneous electrostatic actuations through side-wall and bottom electrodes along in-plane and out-of-plane directions, respectively. In common MEMS gyroscopes, the sense direction lays on the out-of-plane direction, and therefore, it is significant to detect the motion characteristics along this direction. To this aim, the current passing through the bottom electrode is collected. Because the resonance bandwidth for parametric resonance case is much narrower comparing to that of primary and secondary resonances, this motion characteristic is effectively applicable while measurement with ultra-sensitivity is purposed. In this research, it is shown that the out-of-plane resonance characteristics are retrieved through locking on a specific harmonic in the frequency content of the current signal, considering highest sensitivity to harmonic motions. To capture the frequency-amplitude response of the system, a combination of shooting and arc-length continuation scheme is applied to the governing dynamic motion's equations. Some intersections between the stable and unstable solutions branches as such, saddle-node, Hopf, and pitchfork (subcritical and supercritical) bifurcation points, occurring in the system behavior are addressed. Furthermore, for different levels of AC voltage amplitude, it is shown that the variations in the bifurcation points’ position can be figured out from the output current signal, so that this issue is defined as the fundamental basis for current sensing application.

Complex stability boundaries of axially moving beams with interdependent speed and tension

In this article, dynamic stabilities of axially accelerating viscoelastic beams with interdependent speed and tension is investigated. The effect of the interdependent speed and tension is highlighted. However, time dependent speeds and time dependent tensions are independent of each other in previous studies. The dynamic equilibrium approach is used to obtain the governing equation of the axially accelerating viscoelastic beam with internal and principal parametric resonance. Another highlight is that the simply supported boundary conditions are given in precise forms, that are, inhomogeneous forms. The nonhomogeneous terms are closely related to Kelvin–Voigt viscoelastic constitutive relation. The method of directly multiple scales with a first-order uniform expansion is employed. By using the technique of the modified solvability conditions, the complex variable modulation equations are deduced in detail. By some numerical examples, the influences of viscosity, internal resonance, axial tension perturbation amplitude, axial speed perturbation amplitude, and old and current models on the stability boundaries are given. In addition, the approximate analytical results are compared with the numerical integration results by applying the differential quadrature method.