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The principal parameter resonance of a ferromagnetic functionally graded (FG) cylindrical shell under the action of axial time-varying tension in magnetic and temperature fields is investigated. The temperature dependence of physical parameters for functionally graded materials (FGMs) is considered. Meanwhile, the tension bending coupling effect is eliminated by introducing the physical neutral surface. The kinetic and strain energies are gained with the Kirchhoff–Love shell theory. Based on the nonlinear magnetization characteristics of ferromagnetic materials, the electromagnetic force acting on the shell is calculated. The nonlinear vibration equations are obtained through Hamilton’s principle. Afterward, the vibration equations are discretized and solved by Galerkin and multiscale methods, respectively. The stability criterion for steady-state motion is established utilizing Lyapunov stability theory. After example analysis, the effects of magnetic field intensity, temperature and power law index on the static deflection are elucidated. Subsequently, the impacts of these parameters, as well as axial tension, on the amplitude-frequency characteristics, resonance amplitude, and multiple solution regions are discussed explicitly. Results indicate that the stiffness can be enhanced due to the generation of static deflection. The amplitude decreases with increasing magnetic field intensity, temperature, and power law index. When the magnetic field intensity surpasses a threshold, the resonance phenomenon disappears. [ABSTRACT FROM AUTHOR]

Featured Application: The first-order asymptotic method (FOAM) is used to analyze the dynamic instability in the area of principal parametric resonance (APPR) for the automotive driveshaft's (AD) forced torsional vibration (ADFTV). This paper aims to investigate using FOAM to analyze the dynamic instability in the APPR for ADFTV based on a dynamic model (DMADFTV). The DMADFTV considers the following aspects: AD kinematic nonuniformity (ADKN), AD geometric nonuniformity (ADGN) of inertial characteristics for the spinning movements (ICSM) of the AD elements (ADE), and the excitations induced by the gearbox–internal combustion engine modulations. The DMADFTV is considered the already-designed dynamic model developed by the first author of the ADFTV in a previous publication. This DMADFTV was used to compute the stationary frontiers of instability and the nonstationary spectral velocity amplitude (NSVA) versus nonstationary spectral amplitude (NSA) in the configuration space in transition through APPR, using the FOAM. The use of FOAM is much more versatile, from the analytical point of view, than the method of multiple scales and allows the computation of the NSA and the NSVA in the APPR. In contrast, these computations cannot be performed using the harmonic balance method. MATLAB Software R 2017 was developed based on DMADFTV and used the FOAM to compute the stationary frontiers of instability and the NSVA versus the NSA in transition through APPR for the ADFTV. The numerical results were compared with the experimental and numerical data published in the literature, finding agreements. The computation of the NSVA versus NSA in the configuration space using FOAM represents a method of detection of the chaotic manifestation of ADFTV. [ABSTRACT FROM AUTHOR]

In this paper, principal parametric resonance of an axially moving piezoelectric rectangular thin plate under thermal and electric field is investigated. Based on Kirchhoff–Love plate theory and Von Karman theory, the transverse vibration differential equation of a piezoelectric rectangular thin plate under thermal and electric field is derived by using Hamilton's principle. The dimensionless vibration equation of piezoelectric rectangular thin plate with parametric excitations is discretized by Galerkin's method. Then, the multiple scales method is applied to derive amplitude-frequency response equation and the stability conditions of the steady-state solution are obtained by Lyapunov stability theory. Numerical method is used to find the influences of specific parameters on the vibration performance and stability of the system. Based on the global bifurcation diagram and corresponding response diagram, the influences of bifurcation control parameters on the nonlinear dynamic characteristics of the system are discussed. Numerical results illustrate that the system amplitude frequency characteristic curve presents soft spring characteristics. There are periodic and chaotic motions with the increase of velocity and the central temperature difference, and the decrease of plate thickness and velocity will result in the decrease of chaotic threshold. The results also show that increase the velocity perturbation amplitude can prolong the chaotic motion. [ABSTRACT FROM AUTHOR]

This paper aims to investigate using FOAM to analyze the dynamic instability in the APPR for ADFTV based on a dynamic model (DMADFTV). The DMADFTV considers the following aspects: AD kinematic nonuniformity (ADKN), AD geometric nonuniformity (ADGN) of inertial characteristics for the spinning movements (ICSM) of the AD elements (ADE), and the excitations induced by the gearbox–internal combustion engine modulations. The DMADFTV is considered the already-designed dynamic model developed by the first author of the ADFTV in a previous publication. This DMADFTV was used to compute the stationary frontiers of instability and the nonstationary spectral velocity amplitude (NSVA) versus nonstationary spectral amplitude (NSA) in the configuration space in transition through APPR, using the FOAM. The use of FOAM is much more versatile, from the analytical point of view, than the method of multiple scales and allows the computation of the NSA and the NSVA in the APPR. In contrast, these computations cannot be performed using the harmonic balance method. MATLAB Software R 2017 was developed based on DMADFTV and used the FOAM to compute the stationary frontiers of instability and the NSVA versus the NSA in transition through APPR for the ADFTV. The numerical results were compared with the experimental and numerical data published in the literature, finding agreements. The computation of the NSVA versus NSA in the configuration space using FOAM represents a method of detection of the chaotic manifestation of ADFTV.

In this work, we explore internal resonances in a levitation force microelectromechanical system-based actuator assuming flexible cantilever and clamped–clamped microbeam configurations. The levitation force is generated through a special arrangement of two-side stationary charged substrates and a central grounded stationary strip. The design assumes as well an upper flexible microbeam strip. The DC part of the excitation voltage pushes up the moving microbeam away from its lower stationary electrode. A superimposed harmonic AC voltage lets the flexible strip vibrate around its equilibrium position. Energy exchange among the computed lower vibration frequencies and their respective frequency and force response curves are explored for possibilities of principal parametric and internal resonances interactions. The generated responses are computed using the bifurcation toolbox MatCont. The effect of Von Karman nonlinearity to power the energy exchange within vibration modes is explored and dominant factors in the MEMS design for resonances are found from a two-parameter bifurcation analysis. Our results allow optimizing the micro-actuator device performance for future applications. [ABSTRACT FROM AUTHOR]

• A parametrically excited energy harvester with a resonant attachment is proposed. • Interactions with the attachment expanded the harvesting frequency range. • Intentionally introduced perturbations facilitated entry into the unstable region. • Resonant energy transfer rapidly induced a high-amplitude steady-state response. Drawing inspiration from the enhancement of sensing capabilities in Micro-electro-Mechanical Systems (MEMS), the utilization of parametric resonance with nonlinear amplification features shows promise in vibration energy harvesting. However, harnessing parametric resonance for vibration energy harvesting presents several challenges. Initiating the resonant regime under low-energy conditions can be challenging, and the periodic modulation of system parameters requires additional time to accumulate the vibration amplitude. The range of modulation frequencies needs to trigger parametric resonance is notably narrow, especially when employing piezoelectric transducers for energy harvesting. To address these concerns, a parametrically excited piezoelectric energy harvester incorporating with a movable resonant attachment is proposed. The governing equations of this system were derived using the extended Hamilton principle and then numerically solved. Experimental testing was performed based on the design, and it displayed good agreement with numerical predictions. System parametric studies were carried out, revealing the mutual influence among critical system parameters. The findings demonstrate that the proposed device exhibited hardening dynamic responses and achieved more than a fivefold increase in overall bandwidth compared to a conventional parametrically excited counterpart. [ABSTRACT FROM AUTHOR]

Featured Application: Time–history analysis (THA), the Lyapunov Exponents Approach (LEA), and the Poincaré Map (PM) are enhanced to investigate the dynamic instability of forced torsional vibrations (FTV) for a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR). Based on a modified designed model of FTV for an automotive driveshaft (AD), a robust method comprising THA, LEA, and PM (THA–LEA–PM) was created that certifies the existence of deterministic chaos or an ergodic process for the dynamic behavior of DTID's FTV in transition through PPRR. These methods represent a design tool for the DTID's FTV control of the homokinetic driveline transmission in the industry. The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR) which is considered by the researchers in the field as one of the most important resonance regions for the systems having parametric excitations. The DTID's model for FTV considers the following effects: nonuniformities of inertial characteristics of the DTID's elements, the harmonic torque excitation induced by the asynchronous electrical motor used for a heavy-duty grain mill, and the harmonic reaction torque generated by different granulation of the substance needed to be milled. Based on these aspects, a model of the FTV for the DTID was designed which was a modified, physically consistent model already used by the authors to investigate the FTV of automotive driveshafts (homokinetic transmission). For the DTID elements, the dynamic instability for nonstationary FTV in the PPRR using time–history analysis (THA) was analyzed—THA represents the phase portraits. Time–history analysis is a detection method for possible chaotic dynamic behavior for the nonstationary FTV (NFTV) in transition through PPRR. If this dynamic behavior was seen, a new robust method LEA–PM was created to certify and confirm the deterministic chaos for the NFTV of DTID. The new method, LEA–PM, is composed of the Lyapunov exponent's approach (LEA) coupled with the Poincaré Map (PM) applied to the global system of differential equations that describe the FTV of DTID in the PPRR. This new robust method, which embeds LEA and PM, LEA–PM, establishes if the mechanical system has a deterministic chaotic dynamic behavior (strange attractor) or an ergodic dynamic process in this resonant region. LEA represents a new method that includes not only the maximal Lyapunov exponent method (MLEM) but also new mathematical criteria that is "the sum of all Lyapunov exponents has to be negative" which, coupled with MLEM, indicates the presence of deterministic chaos (strange attractors). THA–LEA–PM had been used for the NFTV of DTID computing the phase portraits, the Lyapunov exponents, and representing the Poincaré Maps of the NFTV for the DTID's elements in transition through PPRR, founding deterministic chaos or ergodic dynamic behavior. Based on the obtained results, numerical simulations revealed the pitting manifestations of the DTID's elements, typical for the geared systems transmission, mentioned recently in experimental data research for the homokinetic transmissions. Using the new robust method, THA–LEA–PM (time–history analysis coupled with LEA–PM) can be used in future research for chaotic dynamic analysis of DTID's NFTV transition through superharmonic resonances, subharmonic resonances, combination resonances, and internal resonances. Time–history analysis as a detection method for chaos and LEA–PM as a certifying method for deterministic chaos can be integrated as a design tool for DTID's FTV control of the homokinetic transmission. [ABSTRACT FROM AUTHOR]

Featured Application: The asymptotic method approach (AMA) is used to compute the amplitudes of forced bending vibrating (FBV) movements versus the excitation frequency in the region of principal parametric resonance (PPR) for multibody mechanical systems. An FBV movements model was designed for an automotive driveshaft to realize a powerful design tool for FBV movement control of the automotive transmission. This research's goal is to model and analyze the forced bending vibrating (FBV) movements for the elements of an automotive driveshaft using a perturbation technique, the asymptotic method approach (AMA), in the region of principal parametric resonance (PPR). The PPR region was chosen because the principal parametric resonance region is one of the essential resonance regions. The model of FBV movements for the automotive driveshaft (AD) considers the aspects of the following phenomena: geometric nonuniformity of the AD elements and shock excitation due to the road. To overcome the equations for the FBV movements of the AD elements, all inertia characteristics were reduced to the longitudinal ax of the midshaft using the variation of the geometric moments of inertia with the concurrent axis and Stener's theorem. The midshaft of the AD was considered a Timoshenko simply supported beam with a concentrated mass at both ends and springs and dampers for linear and rotational movements at both ends. To determine the equations describing the FBV movements of the AD elements, Hamilton's principle was used. After establishing the equations of motion for each AD element coupled with the specific boundary conditions, the amplitude and the phase angle were computed for stationary and nonstationary motion in the PPR region using the first order of the AMA, and the dynamic instability frontiers were determined based on the same equations. The dynamic behavior of the AD was investigated concerning the variation of the damping ratio and the variation of the parametric excitation coefficient. The AMA coupled with the model of FBV movements for the AD exhibits the future research directions for analyzing FBV movements for the AD in the regions of superharmonic resonances, subharmonic resonances, combination resonances, internal resonances, and simultaneous resonances. Additionally, the AMA can predict the endurance of the AD and design control of car damping systems. [ABSTRACT FROM AUTHOR]

Described is a closed-loop control scheme capable of stabilizing a parametrically excited nonlinear structure in several vibration modes. By setting the relative phase between the spatially filtered response and the excitation, the open-loop unstable solution branches are stabilized under a 2:1 parametric excitation of a chosen mode of vibration. For a given phase, the closed-loop automatically locks on a limit cycle, through an Autoresonance scheme, at any desired point on the solution branches. Axially driven slender beams and nanowires develop large transverse vibration under suitable amplitudes and frequency base-excitation that are sensitive to small potential coupled field. To utilize such a structure as a sensor, stable and robust operation are made possible by the control scheme. In addition, an optimal operating point with large sensitivity to the sensed potential field can be set using phase as a tunable parameter. Detailed analysis of the dynamical behavior, experimental verifications, and demonstrations sheds light on some features of the system dynamics. [ABSTRACT FROM AUTHOR]

Six different time-delayed controllers are introduced within this article to explore their efficiencies in suppressing the nonlinear oscillations of a parametrically excited system. The applied control techniques are the linear and nonlinear versions of the position, velocity, and acceleration of the considered system. The time-delay of the closed-loop control system is included in the proposed model. As the model under consideration is a nonlinear time-delayed dynamical system, the multiple scales homotopy method is utilized to derive two nonlinear algebraic equations that govern the vibration amplitude and the corresponding phase angle of the controlled system. Based on the obtained algebraic equations, the stability charts of the loop-delays are plotted. The influence of both the control gains and loop-delays on the steady-state vibration amplitude is examined. The obtained results illustrated that the loop-delays can play a dominant role in either improving the control efficiency or destabilizing the controlled system. Accordingly, two simple objective functions are introduced in order to design the optimum values of the control gains and loop-delays in such a way that improves the controllers' efficiency and increases the system robustness against instability. The efficiency of the proposed six controllers in mitigating the system vibrations is compared. It is found that the cubic-acceleration feedback controller is the most efficient in suppressing the system vibrations, while the cubic-velocity feedback controller is the best in bifurcation control when the loop-delay is neglected. However, the analytical and numerical investigations confirmed that the cubic-acceleration controller is the best either in vibration suppression or bifurcation control when the optimal time-delay is considered. It is worth mentioning that this may be the first article that has been dedicated to introducing an objective function to optimize the control gains and loop-delays of nonlinear time-delayed feedback controllers.

The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR) which is considered by the researchers in the field as one of the most important resonance regions for the systems having parametric excitations. The DTID’s model for FTV considers the following effects: nonuniformities of inertial characteristics of the DTID’s elements, the harmonic torque excitation induced by the asynchronous electrical motor used for a heavy-duty grain mill, and the harmonic reaction torque generated by different granulation of the substance needed to be milled. Based on these aspects, a model of the FTV for the DTID was designed which was a modified, physically consistent model already used by the authors to investigate the FTV of automotive driveshafts (homokinetic transmission). For the DTID elements, the dynamic instability for nonstationary FTV in the PPRR using time–history analysis (THA) was analyzed—THA represents the phase portraits. Time–history analysis is a detection method for possible chaotic dynamic behavior for the nonstationary FTV (NFTV) in transition through PPRR. If this dynamic behavior was seen, a new robust method LEA–PM was created to certify and confirm the deterministic chaos for the NFTV of DTID. The new method, LEA–PM, is composed of the Lyapunov exponent’s approach (LEA) coupled with the Poincaré Map (PM) applied to the global system of differential equations that describe the FTV of DTID in the PPRR. This new robust method, which embeds LEA and PM, LEA–PM, establishes if the mechanical system has a deterministic chaotic dynamic behavior (strange attractor) or an ergodic dynamic process in this resonant region. LEA represents a new method that includes not only the maximal Lyapunov exponent method (MLEM) but also new mathematical criteria that is “the sum of all Lyapunov exponents has to be negative” which, coupled with MLEM, indicates the presence of deterministic chaos (strange attractors). THA–LEA–PM had been used for the NFTV of DTID computing the phase portraits, the Lyapunov exponents, and representing the Poincaré Maps of the NFTV for the DTID’s elements in transition through PPRR, founding deterministic chaos or ergodic dynamic behavior. Based on the obtained results, numerical simulations revealed the pitting manifestations of the DTID’s elements, typical for the geared systems transmission, mentioned recently in experimental data research for the homokinetic transmissions. Using the new robust method, THA–LEA–PM (time–history analysis coupled with LEA–PM) can be used in future research for chaotic dynamic analysis of DTID’s NFTV transition through superharmonic resonances, subharmonic resonances, combination resonances, and internal resonances. Time–history analysis as a detection method for chaos and LEA–PM as a certifying method for deterministic chaos can be integrated as a design tool for DTID’s FTV control of the homokinetic transmission.

This research’s goal is to model and analyze the forced bending vibrating (FBV) movements for the elements of an automotive driveshaft using a perturbation technique, the asymptotic method approach (AMA), in the region of principal parametric resonance (PPR). The PPR region was chosen because the principal parametric resonance region is one of the essential resonance regions. The model of FBV movements for the automotive driveshaft (AD) considers the aspects of the following phenomena: geometric nonuniformity of the AD elements and shock excitation due to the road. To overcome the equations for the FBV movements of the AD elements, all inertia characteristics were reduced to the longitudinal ax of the midshaft using the variation of the geometric moments of inertia with the concurrent axis and Stener’s theorem. The midshaft of the AD was considered a Timoshenko simply supported beam with a concentrated mass at both ends and springs and dampers for linear and rotational movements at both ends. To determine the equations describing the FBV movements of the AD elements, Hamilton’s principle was used. After establishing the equations of motion for each AD element coupled with the specific boundary conditions, the amplitude and the phase angle were computed for stationary and nonstationary motion in the PPR region using the first order of the AMA, and the dynamic instability frontiers were determined based on the same equations. The dynamic behavior of the AD was investigated concerning the variation of the damping ratio and the variation of the parametric excitation coefficient. The AMA coupled with the model of FBV movements for the AD exhibits the future research directions for analyzing FBV movements for the AD in the regions of superharmonic resonances, subharmonic resonances, combination resonances, internal resonances, and simultaneous resonances. Additionally, the AMA can predict the endurance of the AD and design control of car damping systems.

• This study explores the nonlinear forced vibration of a nano-beam induced by a periodic thermal load. • The study is based on nonlocal strain gradient theory. • The dynamics, in the vicinity of primary and principal parametric resonance is studied. • The periodic motions are captured by Averaging method and verified by Shooting technique. • The proposed model can be used as either a sensitive remote temperature sensor or a high efficiency thermal energy harvester. This study explores heat-induced nonlinear vibration of a functionally graded (FG) capacitive nanobeam within the framework of nonlocal strain gradient theory (NLSGT). The elastic FG beam, which is firstly deflected by a DC voltage, is driven to vibrate about its deflected position by a periodic heat load. The nano-structure, which consists of a clamped-clamped nanobeam, is modeled assuming Euler–Bernoulli beam assumption which accounts for the nonlinear von-Karman strain and the electrostatic and intermolecular forcing. To simulate the static and dynamic responses, a model reduction procedure is carried out by employing the Galerkin method. The method of Averaging as a regular semi-analytic perturbation method is applied to obtain governing equations of the steady-state responses. With the purpose of establishing the validity of the solution, a Shooting technique in conjunction with the Floquet theory is used to capture the periodic motions and then examine their stability. The nonlinear resonance frequency of the FG nanobeam near its fundamental natural frequency (primary resonance) and near principal parametric resonance is investigated while the emphasis is placed on studying the effect of various parameters including DC voltage, amplitude of the periodic heat source, material index, damping ratio, and small scale parameters. The main objective of this study is to model a miniature structure which can be used as either a sensitive remote temperature sensor or a high-efficiency thermal energy harvester. [ABSTRACT FROM AUTHOR]

In the present work, both theoretical and experimental investigation of a vertical cantilever beam–based piezoelectric energy harvester are carried out under principal parametric resonance condition. A piezoelectric patch is attached near the fixed end of the cantilever beam along with an attached mass positioned at an arbitrary location. The extended Hamilton's principle is used to derive the spatio-temporal equation of motion, and generalized Galerkin's approximation is used to obtain the temporal nonlinear electromechanical governing equation of motion. The method of multiple scales is used to find the reduced modulation equations. Due to large transverse deflection and effect of rotary inertia of the attached mass, the system exhibits cubic and inertial nonlinearities. An experimental setup with slider crank mechanism–based shaker and a harvester consisting of a cantilever beam with piezoelectric patch and attached mass is designed and developed. The challenges posed by parametric resonance in crack development in the PZT and in the beam are reported. The theoretical and experimental output voltage and the power obtained are found to be in good agreement. Furthermore, a qualitative and quantitative comparative study of 17 energy harvesters has been carried out, and the normalized power densities have been compared. [ABSTRACT FROM AUTHOR]

In this present work, the dynamic behavior of a pneumatic artificial muscle (PAM) has been studied by varying different muscle parameters and air pressure into it. The governing equation of motion has been derived using Newton's law of motion to study the various responses in the system at principal parametric resonance condition. The temporal equation of motion contains various nonlinear parameters with forced and nonlinear parametric excitation. Then, the second-order method of multiple scales is used to find the approximate solutions and to study the dynamic stability and bifurcations of the system. The results are found to be in good agreement with the solutions obtained by solving the temporal equation of motion numerically. The instability regions by varying different system parameters have been plotted. The time responses and phase portraits have been plotted to study the system behavior with the nonlinearity. The influences of the different system parameters in the amplitude for the muscle have also been studied with the help frequency responses. In order to verify the solution, the basin of attraction has also been plotted. The obtained results will be very useful for designing the desired PAM use in different rehabilitation robotics and exoskeleton system. [ABSTRACT FROM AUTHOR]

Abstract In this work, the nonlinear dynamic analysis of a parametrically base excited cantilever beam based piezoelectric energy harvester is carried. The system consists of a cantilever beam with piezoelectric patches in bimorph configuration and attached mass at an arbitrary position. The attached mass is placed in such a way that the system exhibits 3:1 internal resonance. The governing spatio-temporal equation of motion is discretized to its temporal form by using generalized Galerkin's method. To obtain the steady state voltage response and stability of the system, Method of multiple scales is used to reduce the resulting equation of motion into a set of first-order differential equations. The response and stability of the system under principal parametric resonance conditions has been studied. The parametric instability regions are shown for variation in different system parameters such as excitation amplitude and frequency, damping and load resistance. Bifurcations such as turning point, pitch-fork and Hopf are observed in the multi-branched non-trivial response. By tuning the attached mass an attempt has been made to harvest the electrical energy for a wider range of frequency. Such kind of smart self-sufficient systems may find application in powering low power wireless sensor nodes or micro electromechanical systems. Highlights • Nonlinear dynamics of a cantilever-based piezoelectric energy harvester is studied. • The system is base excited with an attached mass leading to 1:3 internal resonance. • Parametric instability regions, time and frequency responses have been determined. • Bifurcations such as turning point, pitch-fork and Hopf are observed in this system. • The present energy harvester generates more voltage than conventional harvesters. [ABSTRACT FROM AUTHOR]

The transverse vibrations of axially moving Timoshenko beam, as suitable mathematical models for woodworking bandsaws, are investigated. Special attention is paid to the influence of longitudinal displacement effect, as opposed to most models which can be usually encountered in the literature. This influence is introduced through the integro-partial differential equations. The expressions for the mode shapes in the case of hybrid supports with different torsion spring stiffness on the support points are also derived. The influence of mean beam velocity and axial tension on its natural frequencies and mode shapes is also investigated. Based on the nonlinear model, the amplitudes of the steady-state response are calculated for the case of principal parametric resonance. Developed program solution was tested on a number of earlier known examples. Present theoretical considerations, with the help of the program solution, is also used to analyse an example from industrial practice.

The paper describes a novel parametric excitation scheme that acts as a tunable amplifier by controlling two pumping signals and two nonlinear feedback terms. By modulating the stiffness of a mechanical oscillator with a digital signal processor, low-frequency inputs are projected onto a higher resonance frequency, thus exploiting the natural selective filtering of such structures. Described is an optimized dual-term nonlinear stiffness resonator that enhances the input signal level and the sensitivity to changes in both amplitude and phase, while limiting the obtained response to desired levels. This amplifier is geared to cases when the frequency of the input is known or measurable, like in rotating structures, while the amplitude and phase are too weak to be detected without amplification. It is shown that by tuning the cubic and quadratic feedback terms, the amplifier benefits from a nearly linear response behavior, while exploiting the benefits of nonlinear and pumping signal enhancements. [ABSTRACT FROM AUTHOR]

This paper investigates the effect of low frequency parametric excitations on the coupled dynamics of a spinning disc bounded in a compressible fluid filled enclosure. Specifically, the effect of principal parametric resonance due to disc accelerations is studied. The angular acceleration of the disc results in the governing equations being represented as nonlinearly coupled non-autonomous gyroscopic system. Investigations into the bifurcation behaviour is first studied using Floquet theory on the linearised equations; the stable and unstable regions of the trivial solution are obtained in the amplitude–frequency parameter space corresponding to the acceleration, for various damping configurations. A codimension-two bifurcation is observed when two control parameters are considered in the bifurcation analysis. The bifurcation behaviour subsequently investigated through direct numerical integration of the governing equations, reveals that the coupled system exhibits a two-period quasi periodic oscillation in the post bifurcation regime. [ABSTRACT FROM AUTHOR]

Parametric resonance is one of the key topics in studying the dynamics of structures. In this paper, dynamic analysis of rotating beams with varying rotational speed in the presence of principal parametric resonance is investigated. The equations of motion are based on the von Karman strain–displacement relationship. The beam is made of isotropic material with rectangular cross section. The flapping and axial motions are considered along the thickness and length of the beam, respectively. The Galerkin discretization approach is implemented to determine the natural frequencies. The method of multiple scales is applied directly to the ordered equations of motion for the dynamic stability analysis. The method of multiple scales delivers a closed form relation for the stability region boundary in terms of the adimensional rotational speed, axial mode frequency and damping ratio coefficient. The differential quadrature method is employed to validate the multiple scales results. A comprehensive study is accomplished to find the effects of damping ratio coefficient and mode number on the critical parametric excitation amplitude and the parametric excitation frequency. Damping ratio coefficient, mode number and parametric excitation amplitude influences on the stability region boundaries are also examined. [ABSTRACT FROM AUTHOR]

The nonlinear dynamics of a cable-stayed beam driven by the subharmonic resonance of the beam and the principal parametric resonance of the cable are investigated. Considering the combined effects of the nonlinear terms caused by the geometry of the cable and the coupled behavior between the modes of the beam and cable, a spatial discrete model of the cable-stayed beam is developed. The frequency response curves are determined by applying the method of multiple time scales to the model. The effects of some key parameters of the cable-stayed beam, namely the mass, stiffness, and sag-to-span ratios and the initial tension force, are discussed. The bifurcation and chaos of cable-stayed beams with different parameters are also studied to understand the effects of external excitation. The results show that these parameters have a considerable effect on the dynamic behavior of the cable-stayed beam, particularly that of the cable. Changes in the stiffness ratio and initial tension force also lead to more complex dynamics. [ABSTRACT FROM AUTHOR]

A study on the principal parametric stimulation of rotating microbeams strengthened by means of graphene platelet is presented. The microbeam is exposed to a temperature gradient. On the basis of the assumptions of the hypothesis of the Timoshenko for beams alongside the modified couple stress hypothesis the nonlinear motion equations are achieved. It is supposed that the rotating speed of the microbeam varies harmonically about a mean quantity. The frequency of the harmonic term of the rotating velocity is presumed to be almost two times an axial or a transversal natural frequency. In such situation, the principal parametric agitation is stimulated. Assuming a proportional damping, a least square scheme is employed to define the Rayleigh's coefficients. The impacts of the rotating speed, the weight fraction as well as the scattering pattern of the graphene platelets, and the damping coefficient on the presented results are examined. The results illuminate that as a consequence of the addition of the damping coefficient, the critical incitement amplitude constant increases more for reinforced microbeams rather than not-reinforced microbeams. Moreover, by means of enlarging the weight fraction of the graphene platelets, the critical incitement amplitude constant develops more for a rotating X microbeam rather than an O microbeam. Additionally, at moderate to high magnitudes of the damping coefficient, the instability area border aligned with the fundamental axial mode is impressed by the graphene platelet scattering pattern although it is invariant for small values of the damping coefficient. Moreover, by the development of the temperature the instability area border aligned with the fundamental transversal mode gets broader, while the critical incitement amplitude coefficient decreases; the both more for on O scattering pattern for the graphene platelet. Furthermore, the instability region boundary is more influenced by the graphene platelet weight fraction, while the critical incitement amplitude coefficient is more impressed by the damping coefficient design value. • Parametric resonance of rotating GPL microbeams is presented considering damping. • Increasing damping enlarges critical amplitude more for reinforced microbeams. • Increasing GPL weight fraction enlarges critical amplitude more for X-GPL microbeams. • Instability region boundary is more influenced by graphene platelet weight fraction. • Critical incitement amplitude coefficient is more impressed by damping design value. [ABSTRACT FROM AUTHOR]

Nonlinear parametric vibration and stability is investigated for an axially accelerating rectangular thin plate subjected to parametric excitations resulting from the axial time-varying tension and axial time-varying speed in the magnetic field. Considering geometric nonlinearity, based on the expressions of total kinetic energy, potential energy, and electromagnetic force, the nonlinear magneto-elastic vibration equations of axially moving rectangular thin plate are derived by using the Hamilton principle. Based on displacement mode hypothesis, by using the Galerkin method, the nonlinear parametric oscillation equation of the axially moving rectangular thin plate with four simply supported edges in the transverse magnetic field is obtained. The nonlinear principal parametric resonance amplitude-frequency equation is further derived by means of the multiple-scale method. The stability of the steady-state solution is also discussed, and the critical condition of stability is determined. As numerical examples for an axially moving rectangular thin plate, the influences of the detuning parameter, axial speed, axial tension, and magnetic induction intensity on the principal parametric resonance behavior are investigated. [ABSTRACT FROM AUTHOR]

Abstract: A generalized vibrational model of cubic nonlinear continuous system with arbitrary parametric excitation is considered. The method of multiple scales (a perturbation method) is used to find an approximate analytical solution. The primary parametric resonance of the parametric excitation is considered. The amplitude and phase modulation equations are derived. Steady state solutions and their stability are discussed. The solution algorithm is applied to the problem of nonlinear vibrations of viscoelastic pipes conveying fluids. Natural frequencies of viscoelastic pipes are found. Frequency response curves and bifurcation points are drawn. Stable and unstable regions of trivial and nontrivial solutions are obtained. [Copyright &y& Elsevier]

Abstract: This paper treats the classical problem of radial motions of cylindrical and sphericalshells under pulsating pressures. The novelty in this work is that the shells are taken to be non-linearly viscoelastic(of strain-rate type). It is remarkable that this classical problem, which does not treat the loss of stability to non-radial motions (but which is essential for such treatments), has such a rich dynamics due to the often neglected effects of non-linear material response, to the role of prestress under the action of the mean pressure, and to the different effects of pressure on cylindrical and spherical shells. The study of radial motions near primary resonance (when the frequency of the pulsating pressure is near the natural frequency about an equilibrium state under a constant pressure) gives formulas ensuring that the motions are of hardening or softening type depending on the constitutive functions and whether the constant mean pressure is compressive or inflational. The method of multiple scales gives asymptoticformulas for the principal parametric instability regions (Mathieu tongues) and for the stable and unstable motions at twice the forcing frequency, which closely agree with those obtained by numerical continuation methods. The dependence of frequency on amplitude and the form of instability regions are critically influenced by deviations (even very slight deviations) of material response from that of linearly viscoelasticshells, by the constant mean pressure, and by the type of shell. This paper exhibits the rich diversity of postcritical periodic motions. [Copyright &y& Elsevier]

The principal parametric resonance of two van der Pol oscillators under coupled position and velocity feedback control with time delay is investigated analytically and numerically on the assumption that only one of the two oscillators is parametrically excited and the feedback control is linear. The slow-flow equations are obtained by the averaging method and simplified by truncating the first term of Taylor expansions for those terms with time delay. It is found that nontrivial solutions corresponding to periodic motions exist only for one oscillator if no feedback control is applied although the two oscillators are nonlinearly coupled. Based on Levenberg-Marquardt method, the effects of excitation and control parameters on the amplitude of periodic solutions of the system are graphically given. It can be seen that both of the two oscillators can be excited in periodic vibration with proper feedback. However, the amplitudes of the periodic vibrations are independent of the sign of feedback gains. In addition, the influence of time delay on the response of the system is periodic. In terms of numerical simulations, it is shown that both of the two oscillators can also have quasi-periodic motions, periodic motions about a new equilibrium position and other complex motions such as relaxation oscillation when feedback control is considered. [ABSTRACT FROM AUTHOR]

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. • Non-Linear feedback using digital signal processor to enhance sensitivity. • Combine parametric excitation of principle and combination types. • Detect imbalance's projection on specific modes. • Operates at low rotation speeds, safe operation. • Analytical, numerical and experimental verifications. [ABSTRACT FROM AUTHOR]

Abstract: The non-linear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated. The method of multiple scales is used to determine a uniform first-order expansion of the solution of equations. According to solvability conditions, the non-linear modulation equations for the principal parametric resonance are obtained. Firstly, The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is quantificationally resolved, in which, the modified Bessel function of the first kind is introduced. Results show that the increase of the bandwidth facilitates the almost sure stability of the trivial response and stabilizes the system for a lower acceleration oscillating amplitude but intensifies the instability of the trivial response for a higher one. Secondly, the first and second order non-trivial steady state response of the system is obtained by perturbation method and the corresponding amplitude–frequency curves are calculated when the bandwidth is very small. Results show that the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the hardening type for the first mode, whereas for the second mode the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the softening type. Finally, the stochastic jump and bifurcation is investigated for the first and second modal parametric principal resonance. The basic jump phenomena indicate that, under the conditions of system parameters with a smaller bandwidth, the most probable motion is around the non-trivial branch of the amplitude response curve, whereas with a higher bandwidth, the most probable motion is around the trivial one of the amplitude response curve. However, the stochastic jump is sometimes more sensitive to the change of the bandwidth, in other words, a small change of bandwidth may induce a series of stochastic jump and bifurcation. [Copyright &y& Elsevier]

According to the nonlinear dynamic equations of motion for slender cantilever beams axially excited by narrow-band random processes of the base, a set of nonlinear modulation equations for the principal parametric resonances is developed based on the method of multiple scales. The first and second order non-trivial steady state responses of the system are obtained by perturbation method and the corresponding amplitude-frequency curves are calculated by both approximate theory and direct numerical integration. Results show that for the first mode, the amplitude expectations of both the first and the second order steady state responses are of the hardening type, whereas for the second mode, the amplitude expectations for both the first and the second order steady state responses are of the softening type. Based on the results of both theory and numerical integration, the small changes of the narrow-band widths have almost no influences on the amplitude expectations of both the first and second order steady state responses. Furthermore, via a direct numerical integration method, the probability density of the displacement and the amplitude and the joint probability density function of the displacement and velocity of the stationary response of the system are obtained, and through which stochastic jumps between the non-trivial stationary response and the trivial stationary response are examined for the first time. The digital simulation shows that with the increase of the narrow-band width, the probability density near the non-trivial stationary solution gradually decreases whereas the probability density near the trivial one gradually increases. [ABSTRACT FROM AUTHOR]

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results. [ABSTRACT FROM AUTHOR]

Abstract: Flow-induced vibrations of a fluid-conveying cantilever pipe are examined theoretically under the condition that the fluid velocity has a small harmonic pulsatile component. More specifically, the case of principal parametric resonance is considered for the pipe free to undergo three-dimensional motions. The mean flow is considered to be near the critical flow rate at which the tube undergoes a Hopf bifurcation into self-excited oscillations. When the governing equations of motion for the tube with steady flow are reduced to those on the center manifold in the neighborhood of Hopf bifurcation, the normal form equations are O(2)-equivariant. The weak harmonic fluctuations due to pulsatile flow result in symmetry-breaking terms in the normal form. The eigenvalues of an O(2)-equivariant system undergoing a symmetry-breaking Hopf bifurcation have multiplicity two. When an additive linear term, arising from time-periodic modulations of the original dynamic system, is introduced into the normal form, the symmetry-breaking bifurcation structure for the trivial solution splits into three categories: a steady-state bifurcation giving rise to standing wave fixed-point solutions, a Hopf bifurcation giving rise to two-frequency solutions, and an O(2)-Takens–Bogdanov bifurcation. The resulting dynamics in each case are studied along with secondary and tertiary bifurcations. The dynamics of the tube system are studied as a function of the mean flow rate and the frequency of flow fluctuations. Amplitude response diagrams constructed for a specific example tube system using the continuation and bifurcation analysis software package AUTO illustrate the variety of possible behaviors. [Copyright &y& Elsevier]

Principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed. A nonlinear partial-differential equation governing the transverse vibration is derived from the dynamical, constitutive, and geometrical relations. Under certain assumption, the partial-differential reduces to an integro-partial-differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams. The method of multiple scales is applied to two equations to calculate the steady-state response. Closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelascity, and nonlinearity and to compare results obtained from two equations. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The nonlinear dynamics of a clamped-clamped/sliding inextensional elastic beam subject to a harmonic axial load is investigated. The Galerkin method is used on the coupled bending-bending-torsional nonlinear equations with inertial and geometric nonlinearities and the resulting two second order ordinary differential equations are studied by the method of multiple time seales and by direct numerical integration. The amplitude equations are analyzed for steady and Hopf bifurcations. Depending on the amplitude of excitation, the damping and the ratio of principal flexural rigidities, various qualitatively distinct frequency response diagrams are uncovered and limit cycles and chaotic motions are found. In the truncated two-degree-of-freedom system the transition from periodic to chaotic amplitude-modulated motions is via the process of torus doubling and subsequent destruction of the torus. [ABSTRACT FROM AUTHOR]

• A measurement technique is presented based on the motion-induced current generated in an encapsulated MEMS transducer. • The principal parametric resonance characteristics are examined with the aim of bifurcation-based sensing, providing ultra-sensitivity. • The out-of-plane motion of the MEMS is detected using the harmonic detection of resonance (HDR) method. • Different types of bifurcation points are addressed in the system dynamics, for two levels of the AC voltage amplitude. • A combination of shooting and arc-length continuation scheme is utilized to capture the frequency-displacement behaviors of the MEMS system. 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. Image, graphical abstract A gyroscopic MEMS-based transducer composing of a microcantilever, one side-wall, and one bottom electrode. The microbeam is assumed to be encapsulated, and is actuated by two electrostatic forces simultaneously through in-plane and out-of-plane directions. The total electric current induced by the variable cpacitors are collected through the bottom electrode and is considered as the output signal. [ABSTRACT FROM AUTHOR]