Design of piezo electric actuated cutting tools requires carefully controlling the response characteristics while minimizing the energy input for an efficient design. Many times, nonlinearities are seen as a burden in cutting tool design process as they can result in unpredictable and extremely sensitive response. As such, engineers make design changes to effectively minimize or completely eliminate the nonlinear behavior and that has been a practice in biomedical device design industry. In this paper, the periodic response of a buckled beams under piezo-excitation is considered. An optimization approach, as first demonstrated in \cite{shukla2016} and \cite{shukla2017}, is utilized that achieves maximum amplitude, periodic, and stable responses of the beam systems. Case studies are presented that demonstrate the utility of this optimization approach to exploit the nonlinear dynamics to achieve desired responses.This work presents a new, global optimization driven, computational design approach for medical devices while exploiting nonlinearity in response dynamics.

The problem of geometrically nonlinear, steady-state vibrations of beams made of viscoelastic (VE) materials is considered in this paper. The Euler–Bernoulli and the von Kármán theories are used to describe the dynamic behavior of beams. The VE material of the beams is modeled using the Zener model. Two harmonics are present in the assumed steady-state solution of the problem at hand, which enables an analysis of both the primary and secondary resonances. The virtual work equation and the harmonic balance method are used to derive the amplitude equations in the explicit form. The response curves are determined using the continuation method and treating the frequency of excitation as the main parameter. The results of several examples, which illustrate the dynamic behavior of the considered beams, are presented and discussed.

The superharmonic resonance of fractional-order Mathieu–Duffing oscillator subjected to external harmonic excitation is investigated. Based on the Krylov–Bogolubov–Mitropolsky (KBM) asymptotic method, the approximate analytical solution for the third superharmonic resonance under parametric-forced joint resonance is obtained, where the unified expressions of the fractional-order term with fractional order from 0 to 2 are gained. The amplitude–frequency equation for steady-state solution and corresponding stability condition are also presented. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional-order term, excitation amplitudes, and nonlinear stiffness coefficient on the superharmonic resonance response of the system are analyzed in detail. The results show that the KBM method is effective to analyze dynamic response in a fractional-order Mathieu–Duffing system.

We investigate modeling the dynamics of an electrostatically actuated resonator using the perturbation method of multiple time scales (MTS). First, we discuss two approaches to treat the nonlinear parallel-plate electrostatic force in the equation of motion and their impact on the application of MTS: expanding the force in Taylor series and multiplying both sides of the equation with the denominator of the forcing term. Considering a spring–mass–damper system excited electrostatically near primary resonance, it is concluded that, with consistent truncation of higher-order terms, both techniques yield same modulation equations. Then, we consider the problem of an electrostatically actuated resonator under simultaneous superharmonic and primary resonance excitation and derive a comprehensive analytical solution using MTS. The results of the analytical solution are compared against the numerical results obtained by long-time integration of the equation of motion. It is demonstrated that along with the direct excitation components at the excitation frequency and twice of that, higher-order parametric terms should also be included. Finally, the contributions of primary and superharmonic resonance toward the overall response of the resonator are examined.

This manuscript investigates the effect of nonconservative electromechanical energy transfer on the onset of interwell motions in multistable piezoelectric energy harvesters. Multistable piezoelectric energy harvesters have been proven to outperform their linear counterparts when they undergo interwell oscillations. The conditions for interwell oscillations in such harvesters are generally characterized in terms of their potential energy function. This is accurate for a stand-alone mechanical oscillator but when the piezoelectric patches and a load resistance are included, a part of the kinetic energy supplied to the system is converted into electrical energy. In this manuscript, the Melnikov necessary conditions for interwell oscillations are derived, considering the effect of this nonconservative piezoelectric energy transfer. Through Melnikov theoretic analysis, it is shown that in a tristable harvester with all the three potential wells having the same depth, a higher excitation level is required to enable exits from the middle well to the outer wells when compared to the exits from the outer wells to the middle well. This is in stark contrast to a stand-alone tristable mechanical oscillator wherein interwell motions are simultaneously enabled for all the wells having the same depth.

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

Tooth friction is unavoidable and changes periodically in gear engagement. Friction excitation is an important excitation source of a gear transmission system. They are different than the friction coefficients of any two points on the same contact line of a helical/herringbone gear. In order to obtain the influence of the friction excitation on the dynamic response of a helical/herringbone planetary gear system, a method that uses piecewise solution and then summing them to analyze the friction force and frictional torque of tooth surfaces is proposed. Then, the friction coefficient is obtained based on the mixed elastohydrodynamic lubrication (EHL) theory. A dynamic model of a herringbone planetary gear system is established considering the friction, mesh stiffness, and meshing error excitation by the node finite element method. The influence of friction excitation on the dynamic response of the herringbone planetary gear is studied under different working conditions. The results show that friction excitation has a great influence on the vibration acceleration of the sun and planetary gear. However, the effect on the radial and tangential vibration acceleration of a planetary gear is the opposite. In addition, the friction excitation has a slight effect on the meshing force.

In this paper, dynamic response of a rotating shaft with geometrical nonlinearity under parametric and external excitations is investigated. Resonances, bifurcations, and stability of the response are analyzed. External excitation is due to shaft unbalance and parametric excitation is due to periodic axial force. For this purpose, combination resonances of parametric excitation and primary resonance of external force are assumed. Indeed, simultaneous effect of nonlinearity, parametric, and external excitations are investigated using analytical method. By applying the method of multiple scales, four ordinary nonlinear differential equations are obtained, which govern the slow evolution of amplitude and phase of forward and backward modes. Eigenvalues of Jacobian matrix are checked to find the stability of solutions. Both periodic and quasi-periodic motion were observed in the range of study. The influence of various parameters on the response of the system is studied. A main contribution is that the parametric excitation in the presence of nonlinearity can be used to suppress the forward synchronous vibration. Indeed, in the presence of combination parametric excitation, the energy is transferred from forward whirling mode to backward one. This property can be applied in control of rotor unbalance vibrations.

We characterize the complex, heavy-tailed probability density functions (pdfs) describing the response and its local extrema for structural systems subject to random forcing that includes extreme events. Our approach is based on recent probabilistic decomposition-synthesis (PDS) technique (Mohamad, M. A., Cousins, W., and Sapsis, T. P., 2016, “A Probabilistic Decomposition-Synthesis Method for the Quantification of Rare Events Due to Internal Instabilities,” J. Comput. Phys., 322, pp. 288–308), where we decouple rare event regimes from background fluctuations. The result of the analysis has the form of a semi-analytical approximation formula for the pdf of the response (displacement, velocity, and acceleration) and the pdf of the local extrema. For special limiting cases (lightly damped or heavily damped systems), our analysis provides fully analytical approximations. We also demonstrate how the method can be applied to high dimensional structural systems through a two-degrees-of-freedom (TDOF) example system undergoing extreme events due to intermittent forcing. The derived formulas can be evaluated with very small computational cost and are shown to accurately capture the complicated heavy-tailed and asymmetrical features in the probability distribution many standard deviations away from the mean, through comparisons with expensive Monte Carlo simulations.

There have been a variety of attempts to model the quasi-static and high energy impact dynamics of vertically aligned carbon nanotube (VACNT) pads. However, very little work has focused on identifying the behavior at the midlevel frequencies that may occur in materials handling or vibration suppression applications. Moreover, the existing models are predominantly very complex, and yet provide only a very rough approximation of the bulk behavior. While several of the existing models make attempts at ascribing physical relevance, an adequate first principles approach has yet to be demonstrated. In this work, a close-fitting continuous model of these midfrequency dynamics is developed utilizing a combination of phenomenological- and identification-based methodologies. First, a set of specially fabricated carbon nanotube pads are preconditioned and subjected to various position-controlled compression experiments. The measured position and force responses are used to develop load–displacement curves, from which several characteristic features are identified. Based on these observations, a preliminary version of the proposed model is introduced. This simplified model is then systematically refined in order to demonstrate completely both the modeling approach and parameter identification scheme. The accuracy of the model is demonstrated through a comparison between the modeled and experimental responses including a normalized vector correlation of >0.998 across all sets of sinusoidal experimental data. A brief analysis utilizing a Lyapunov linearization approach follows, as well as a discussion of the advantages and limitations of the final model.

Electrostriction is a recent actuation mechanism which is being explored for a variety of new micro- and millimeter scale devices along with macroscale applications such as artificial muscles. The general characteristics of these materials and the nature of actuation lend itself to possible production of very rich nonlinear dynamic behavior. In this work, principal parametric resonance of the second mode in in-plane vibrations of appropriately designed electrostrictive plates is investigated. The plates are made of an electrostrictive polymer whose mechanical response can be approximated by Mooney Rivlin model, and the induced strain is assumed to have quadratic dependence on the applied electric field. A finite element model (FEM) formulation is used to develop mode shapes of the linearized structure whose lowest two natural frequencies are designed to be close to be in 1:2 ratio. Using these two structural modes and the complete Lagrangian, a nonlinear two-mode model of the electrostrictive plate structure is developed. Application of a harmonic electric field results in in-plane parametric oscillations. The nonlinear response of the structure is studied using averaging on the two-mode model. The structure exhibits 1:2 internal resonance and large amplitude vibrations through the route of parametric excitation. The principal parametric resonance of the second mode is investigated in detail, and the time response of the averaged system is also computed at few frequencies to demonstrate stability of branches. Some results for the case of principal parametric resonance of the first mode are also presented.

Nonlinear vibration characteristics of three-blade wind turbines are theoretically investigated. The wind turbine is modeled as a coupled system, consisting of a flexible tower with two degrees-of-freedom (2DOF), and three blades, each with a single degree of freedom (SDOF). The blades are subjected to steady winds. The wind velocity increases proportionally with height due to vertical wind shear. The natural frequency diagram is calculated with respect to the rotational speed of the wind turbine. The corresponding linear system with parametric excitation terms is analyzed to determine the rotational speeds where unstable vibrations appear and to predict at what rotational speeds the blades may vibrate at high amplitudes in a real wind turbine. The frequency response curves are then obtained by applying the swept-sine test to the equations of motion for the nonlinear system. They exhibit softening behavior due to the nonlinear restoring moments acting on the blades. Stationary time histories and their fast Fourier transform (FFT) results are also calculated. In the numerical simulations, localization phenomena are observed, where the three blades vibrate at different amplitudes. Basins of attraction (BOAs) are also calculated to examine the influence of a disturbance on the appearance of localization phenomena.

Vibrational resonance (VR) is a nonlinear phenomenon which occurs when a bistable system is subjected to a biharmonic excitation consisting of a small-amplitude resonant excitation and a large-amplitude high-frequency excitation. The result is that, under some conditions, the high-frequency excitation amplifies the resonant response associated with the slow dynamics. While VR was studied extensively in the open literature, most of the research studies used optical and electrical systems as platforms for experimental investigation. This paper provides experimental evidence that VR can also occur in a mechanical bistable twin-well oscillator and discusses the conditions under which VR is possible. The paper also demonstrates that the injection of the high frequency excitation can be used to change the effective stiffness of the slow response. This can be used for amplification/deamplification of the output signal which can be useful for sensitivity enhancement and/or vibration mitigation.

The resonant behavior of fractional-order Mathieu oscillator subjected to external harmonic excitation is investigated. Based on the harmonic balance (HB) method, the first-order approximate analytical solutions for primary resonance and parametric-forced joint resonance are obtained, and the higher-order approximate steady-state solution for parametric-forced joint resonance is also obtained, where the unified forms of the fractional-order term with fractional order between 0 and 2 are achieved. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional order and parametric excitation frequency on the resonance response of the system are analyzed in detail. The results show that the HB method is effective to analyze dynamic response in a fractional-order Mathieu system.

This paper presents the coupled axial-transverse-rotational nonlinear forced vibrations of Timoshenko tapered beams made of an axially functionally graded (AFG) material subjected to an external harmonic excitation. Two sources of nonlinearities are considered in modeling and numerical simulations: (i) the geometric nonlinearities arising from induced nonlinear tension due to the clamped–clamped boundary conditions and large deformations, and (ii) nonlinear expressions to address the nonuniform geometry and mechanical properties of the beam along the length. More specifically, a nonlinear model is developed based on the Timoshenko beam theory accounting for shear deformation and rotational inertia. Exponential distributions are presumed for the cross-sectional area, moduli of elasticity, mass density, and Poisson's ratio of the AFG tapered Timoshenko beam. The kinetic and potential energies, the virtual work of the external harmonic distributed load, and the one done by damping are implemented into Hamilton's energy principle. The resultant nonuniform nonlinearly coupled partial differential equations are discretized into a set of nonlinear ordinary differential equations utilizing Galerkin's technique. In the discretization scheme, a large number of modes, both symmetric and asymmetric, are employed due to the asymmetric characteristic of the nonuniform beam with respect to its length. The effect of different parameters, including the gradient index and different taper ratios, on the force-vibration-amplitude and frequency-vibration-amplitude diagrams is examined; the effect of these parameters on the natural frequencies, internal resonances, and asymmetric characteristics of the AFG system is investigated as well.

In our former work developed by Yang et al. (2017, “Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators,” ASME J. Comput. Nonlinear Dyn. , 12 (5), p. 051011), we put forward the rescaled vibrational resonance (VR) method in fractional duffing oscillators to amplify a weak signal with arbitrary high frequency. In the present work, we propose another method named as twice sampling VR to achieve the same goal. Although physical processes of two discussed methods are different, the results obtained by them are identical completely. Besides the two external signals excitation case, the validity of the new proposed method is also verified in the system that is excited by an amplitude modulated signal. Further, the dynamics of the system reveals that the resonance performance, i.e., the strength and the pattern, depends on the fractional order closely.

Nonlinear dynamics, control, and stability analysis of dry friction damped system under state feedback control with time delay are investigated. The dry friction damped system is harmonically excited, and the nonlinearities in the equation of motion arise due to nonlinear damping and spring force. In this paper, a frequency domain-based method, viz., incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary responses which may be present in such system. The IHBC is then reformulated in a manner to treat the dry friction damped system under state feedback control with time delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses is obtained by Floquet's theory using Hsu' scheme, and the stability of the controlled responses is obtained by applying a semidiscretization method for delay differential equation (DDE).

This paper investigates the possibility of energy generation via pendulum rotations when the source of vertical excitation is chaotic in nature. The investigations are conducted using an additional height-adjustable mechanism housing a secondary spring to optimize a configuration of experimental pendulum setup. Chaotic oscillations of the pendulum pivot are made possible at certain excitation conditions due to a piecewise-linear stiffness characteristic introduced by the modification. A velocity control method is applied to maintain the rotational motion of the pendulum as it interacts with the vertical oscillator. The control input is affected by a motor, and a generator is used to quantify the energy extraction. The experimental results imply the feasibility of employing a pendulum device in a chaotic vibratory environment for energy harvesting purpose.

The analysis of the shape memory prosthesis (SMP) of the middle ear is presented in this paper. The shape memory prosthesis permits the adjustment of its length to individual patient needs, but sometimes the prosthesis cannot be properly fixed to the stapes. In this case, the impact between the prosthesis and stapes is important. Therefore, the reconstructed middle ear is modeled as a two degree-of-freedom system with a nonlinear shape memory element and soft impact to represent its behavior when the prosthesis is not properly placed or fixed. The properties of the shape memory prosthesis, in the form of a helical spring, are represented by a polynomial function. The system exhibits advisable periodic and undesirable aperiodic and irregular behavior depending on the excitation amplitude, the frequency, and the prosthesis length. The prosthesis length can change, resulting in a modification of the distance between the prosthesis and the stapes. The results of this study provide an answer in terms of how the prosthesis length, which produces the ossicular chain tension, influences the system dynamics and its implication in medical practice.

A vibrating system with impacts, which can be applied to model the cantilever beam with a mass at its end and two-sided impacts against a harmonically moving frame, is investigated. The objective of this study is to determine in which regions of parameters characterizing system, the motion of the oscillator is periodic and stable. An analytical method to obtain stable periodic solutions to the equations of motion on the basis of Peterka's approach is presented. The results of analytical investigations have been compared to the results of numerical simulations. The ranges of stable periodic solutions determined analytically and numerically with bifurcation diagrams of spectra of Lyapunov exponents show a very good conformity. The locations of stable periodic solution regions of the system with a movable frame and two-sided impacts differ substantially from the locations of stable periodic solution regions for the system: (i) with a movable frame and one-sided impacts and (ii) with an immovable frame and two-sided impacts.