External Harmonic Force Research Articles

Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium

In this study, the nonlinear vibration analysis of the new generation nanostructures is investigated. The composite nanoplate is fabricated from the functional-graded (FG) core and two lipid layers on top and bottom of the FG core as face sheets. The nonlinear vibration analysis is studied in the presence of the external harmonic excitation force. The porosity effect on the free and force vibration analysis of the composite nanoplate is investigated. The nonlocal elasticity theory is utilized to obtain the nonlinear differential governing equation. The Kelvin–Voigt model is used to model the viscoelastic effect of the lipid layers. The Hamilton's principle is utilized to obtain the differential governing equation. The Galerkin's method is used to discrete the nonlinear partial differential governing equation to a nonlinear ordinary differential equation. The multiple scale method is used to solve the ordinary differential equation. The numerical results are compared with the reported results in the literature. A comparison between the presented numerical results and the Runge–Kutta results is done and good agreement is obtained. In the presence and absence of the porosity, the system vibration behavior is studied in the primary and secondary resonance cases. The results show that the porosity distribution types play an important role in the mechanical behavior of the composite nanoplate. Also, the numerical results show that the nonlinear frequency of the system decreases by passing time. This study can be useful to product the sensors and devices at the nanoscale with high biocompatibility.

Theoretical Design and Analysis of a Nonlinear Electromagnetic Vibration Isolator with Tunable Negative Stiffness Characteristic

A novel tunable negative stiffness nonlinear electromagnetic isolator is presented, it includes a vertical positive stiffness spring, an electromagnetic spring constructed by two permanent magnets and an electromagnet. The electromagnetic isolator can act as a passive isolator or a semi-active control one. The mathematical expression of system stiffness-current-displacement is derived by analyzing the mechanical characteristics. The motion differential equation under an external harmonic excitation force is established. The amplitude-frequency relationship of the system is deduced by the averaging method. The global dynamic behavior on the Van der pol plane and the force transmissibility are researched. The results show that the nonlinear electromagnetic isolator with appropriate parameters can enlarge the low frequency isolation range and make the system have better isolation performance in the low frequency band. The nonlinear electromagnetic isolator with suitable system parameters has the advantages of widening the vibration isolation range and improving the low-frequency vibration isolation effect.

Dynamic instability of electroconductive cylindrical shell in a magnetic field

The dynamic instability of electroconductive cylindrical shells interacting with an external magnetic fields is considered in the paper. Two cases of applied external dynamic loads are discussed, namely (i) a harmonic mechanical force, and (ii) a harmonic magnetic field force. Analytical descriptions of the two-dimensional equations and associate conditions of dynamic instability are presented. On the basis of the formulated problems, specific issues related to the dynamic instability of electroconductive cylindrical shells in a magnetic field are offered. The study illustrated the effect of magnetoelastic interaction. Specifically it is shown that there exist a minimum value of the given magnetic field intensity, above which the parametric resonance due to external harmonic force is eliminated; furthermore, in a presence of time-harmonic magnetic field the parametric resonance with a resonance frequency can be generated not only near the first frequency of the external magnetic field (which is equal to the natural frequency of vibrations), but near the double frequency of natural vibrations. When the forced vibrations of conductive shells caused by external forces of non-electromagnetic origin is considered, including the effect of a time-harmonic magnetic field, resonance can occur in the presence of a non-stationary harmonic magnetic field. Results reveal that the rapid increase of the amplitude of vibrations occurs when the frequency of the external magnetic field is in close proximity to the first natural frequency of the magnetoelastic vibrations of the plate, as well as when the frequency of the magnetic field, is equal to half of the shell natural frequency.

Dynamics of Nonlinear Primary Oscillator with Nonlinear Energy Sink under Harmonic Excitation: Effects of Nonlinear Stiffness

The dynamics of a system consisting of a nonlinear primary oscillator, subjected to a harmonic external force, and a nonlinear energy sink (NES) are investigated. The analytical solutions for the steady-state responses are obtained by the complexification-averaging method and the analytical model is confirmed by numerical simulations. The results indicate that the introduction of the NES can effectively suppress the vibrations of the primary oscillator. However, as the excitation amplitude increased, the NES may lose its efficiency within certain frequency range due to the appearance of the high response branches. Following the results analysis, it is concluded that this failure can be eliminated by reducing the nonlinear stiffness of the NES properly. The effects of nonlinear stiffness of the primary oscillator on the corresponding responses are also studied. The increase in this nonlinear stiffness can reduce the response amplitude and alter the frequency band where the high branches exist.

Nonlinear vibration analysis of a circular micro-plate in two-sided NEMS/MEMS capacitive system by using harmonic balance method

In this study, forced nonlinear vibration of a circular micro-plate under two-sided electrostatic, two-sided Casimir and external harmonic forces is investigated analytically. For this purpose, at first, von Karman plate theory including geometrical nonlinearity is used to obtain the deflection of the micro-plate. Galerkin decomposition method is then employed, and nonlinear ordinary differential equations (ODEs) of motion are determined. A harmonic balance method (HBM) is applied to equations and analytical relation for nonlinear frequency response (F–R) curves are derived for two categories (including and neglecting Casimir force) separately. The analytical results for three cases: (1) semi-linear vibration; (2) weakly nonlinear vibration; (3) highly nonlinear vibration, are validated by comparing with the numerical solutions. After validation, the effects of the voltage and Casimir force on the natural frequency of two-sided capacitor system are investigated. It is shown that by assuming Casimir force in small gap distances, reduction of the natural frequency is considerable. The influences of the applied voltage, damping, micro-plate thickness and Casimir force on the frequency response curves have been presented too. The results of this study can be useful for modeling circular parallel-plates in nano/microelectromechanical transducers such as microphones and pressure sensors.

Stability and receptivity of boundary layers in a swirl flow channel

The analysis of the disturbances on a spiraling base flow is relevant for the design, operation, and control of technological devices such as parallel-disk turbines and swirl flow channel heat sinks. Spiraling inflow inside an annular cavity closed at the top and bottom is analyzed in the framework of modal and nonmodal stability theories. Local and parallel flow approximations are applied, and the inhomogeneous direction is discretized using the Chebyshev collocation method. The optimal growth of initial disturbances and the optimal response to external harmonic forcing are characterized by the exponential and the resolvent of the dynamics matrix. As opposed to plane Poiseuille flow, transient growth is small, and consequently, it does not play a role in the transition mechanism. The transition is attributed to a crossflow instability that occurs because of the change in the shape of the velocity profile due to rotational effects. Agreement is found between the critical Reynolds number predicted in this work and the deviation of laminar behavior observed in the experiments conducted by Ruiz and Carey (J Heat Transfer 137(7):071702, 2015). For the harmonically driven problem, an energy amplification of $$\textit{O}(100)$$ is observed for spiral crossflow waves. Transition to turbulence should be avoided to ensure the safe operation of a parallel-disk turbine, whereas large forcing amplification may be sought to promote mixing in a swirl flow channel heat sink. The analysis presented predicts and provides insight into the transition mechanisms. Due to its easy implementation and low computational cost, it is particularly useful for the early stages of engineering design.

Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

This paper focuses on a novel rotating mechanical model which provides a cylindrical example of transition from smooth to discontinuous dynamics. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting both smooth and discontinuous characteristics due to the geometry configuration. By using nonlinear dynamical technique, the unperturbed dynamics of the proposed system are studied including the irrational restoring force, stability of equilibria, Hamiltonian function and phase portraits. Note that a pair of double heteroclinic-like orbits connecting two non-standard saddle points are proposed in discontinuous case. For the perturbed system, we introduce a cylindrical approximate system for which the analytical solutions can be obtained successfully to reflect the nature of the original system without barrier of the irrationalities. Melnikov method is employed to detect the chaotic thresholds for the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing in smooth regime. Finally, numerical simulations show the efficiency of the proposed method and demonstrate the predicated periodic solution and chaotic attractors. It is found that a good degree of correlation is demonstrated in the bifurcation diagram, the phase portraits of periodic solution, the chaotic attractor’ structures and the Lyapunov characteristics between the original system and approximate system.

Active vibration suppression of a novel airfoil model with fractional order viscoelastic constitutive relationship

This paper aims to investigate the active vibration suppression of a fractional two-degree-of-freedom viscoelastic airfoil (TDOFVA) model with a harmonic external force by means of the sliding mode control (SMC) scheme. The viscoelastic behavior is described as a fractional-order derivative, leading to a new fractional TDOFVA model. Subsequently, an averaging technique is extended to derive the amplitude-frequency relations, and its correctness is verified by Monte Carlo simulations. In addition, effects of the system parameters on the dynamics are explored. To achieve a vibration suppression, we convert the TDOFVA system into a series of fractional-order differential equations. Then, a SMC strategy is employed, in which a fractional-order integral sliding surface is presented and asymptotical stability analysis of the SMC is performed. Several numerical results are presented to illustrate the performances of the proposed SMC scheme, which indicate that the given SMC methodology is effective to realize a vibration suppression of the TDOFVA model.

Inducing and destruction of chimeras and chimera-like states by an external harmonic force

We study the phenomena of chimera destruction and inducing of chimera-like states in an ensemble of nonlocally coupled chaotic Rössler oscillators under an external harmonic force. The localized harmonic influence can lead to both destruction and changing of the spatial topology of chimeras. At the same time this influence can cause the emergence of stable chimera-like states (induced chimeras) for the regime of partial coherent chaos. Induced chimeras are also observed for the global influence. We show the possibility of controlling the chimera-like state topology by varying the parameters of localized external harmonic influence.

Nonlinear forced vibration of FG-CNTs-reinforced curved microbeam based on strain gradient theory considering out-of-plane motion

The main goal of this research is to examine the in-plane and out-of-plane forced vibration of a curved nanocomposite microbeam. The in-plane and out-of-plane displacements of the structure are considered based on the first order shear deformation theory (FSDT). The curved microbeam is reinforced by functionally graded carbon nanotubes (FG-CNTs) and thus the extended rule of mixture is employed to estimate the effective material properties of the structure. Also, the small scale effect is captured using the strain gradient theory. The structure is rested on a nonlinear orthotropic viscoelastic foundation and is subjected to concentrated transverse harmonic external force, thermal and magnetic loads. The derivation of the governing equations is performed using energy method and Hamilton's principle. Differential quadrature (DQ) method along with integral quadrature (IQ) and Newmark methods are employed to solve the problem. The effect of various parameters such as volume fraction and distribution type of CNTs, boundary conditions, elastic foundation, temperature changes, material length scale parameters, magnetic field, central angle and width to thickness ratio are studied on the frequency and force responses of the structure. The results indicate that the highest frequency and lowest vibration amplitude belongs to FGX distribution type while the inverse condition is observed for FGO distribution type. In addition, the hardening-type response of the structure with FGX distribution type is more intense with respect to the other distribution types.

Optimum Design of a Nonlinear Vibration Absorber Coupled to a Resonant Oscillator: A Case Study

This paper presents the optimal design of a passive autoparametric cantilever beam vibration absorber for a linear mass‐spring‐damper system subject to harmonic external force. The design of the autoparametric vibration absorber is obtained by using an approximation of the nonlinear frequency response function, computed via the multiple scales method. Based on the solution given by the perturbation method mentioned above, a static optimization problem is formulated in order to determine the optimum parameters (mass and length) of the nonlinear absorber which minimizes the steady state amplitude of the primary mass under resonant conditions; then, a PZT actuator is cemented to the base of the beam, so the nonlinear absorber is made active, thus enabling the possibility of controlling the effective stiffness associated with the passive absorber and, as a consequence, the implementation of an active vibration control scheme able to preserve, as possible, the autoparametric interaction as well as to compensate varying excitation frequencies and parametric uncertainty. Finally, some simulations and experimental results are included to validate and illustrate the dynamic performance of the overall system.

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.

Blades Forced Vibration Under Aero-Elastic Excitation Modeled by Van der Pol

This paper employs a new analytical approach to model the influence of aerodynamic excitation on the dynamics of a bladed cascade at the flutter state. The flutter is an aero-elastic phenomenon that is linked to the interaction of the flow and the traveling deformation wave in the cascade when only the damping of the cascade changes. As a case study the dynamic properties of the five-blade-bunch excited by the running harmonic external forces and aerodynamic self-excited forces are investigated. This blade-bunch is linked in the shroud by means of the viscous-elastic damping elements. The external running excitation depends on the ratio of stator and rotor blade numbers and corresponds to the real type of excitation in the steam turbine. The aerodynamic self-excited forces are modeled by two types of Van der Pol nonlinear models. The influence of the interaction of both types of self-excitation with the external running excitation is investigated on the response curves.

Escape of a harmonically forced particle from an infinite-range potential well: a transient resonance

The paper considers a process of escape of classical particle from a one-dimensional potential well by virtue of an external harmonic forcing. We address a particular model of the infinite-range potential well that allows independent adjustment of the well depth and of the frequency of small oscillations. The problem can be conveniently reformulated in terms of action-angle variables. Further averaging provides a nontrivial conservation law for the slow flow. Thus, one can consider the problem in terms of averaged dynamics on primary 1:1 resonance manifold. This simplification allows efficient analytic exploration of the escape process, and yields a theoretical prediction for minimal forcing amplitude required for the escape, as a function of the excitation frequency. This function exhibits a single minimum for certain intermediate frequency value. Numeric simulations are in complete qualitative and reasonable quantitative agreement with the theoretical predictions.

Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance

The dynamical modeling, multi-pulse orbits and chaotic dynamics of cantilevered pipe conveying pulsating fluid with harmonic external force are studied using the energy-phase method for the first time. The nonlinear geometric deformation of the pipe and the Kelvin constitutive relation of the pipe material are considered. The nonlinear governing equation of motion for cantilevered pipe conveying pulsating fluid is determined using Hamilton principle. The four-dimensional averaged equation in the case of primary parametric resonance-1/2 subharmonic resonance and 1:2 internal resonance is obtained by directly using the method of multiple scales and the Galerkin approach to the partial differential governing equation of motion for cantilevered pipe conveying pulsating fluid. From the averaged equation, the normal form theory is used to find the explicit formulas of normal form. Based on normal form obtained here, the energy-phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics of cantilevered pipe conveying pulsating fluid. The analysis of global dynamics indicates that the multi-pulse jumping orbits exist in the perturbed phase space of the averaged equation. Based on the averaged equation, the chaotic motions and the multi-pulse orbits for the cantilevered pipe are found by using numerical simulations. The results described above indicate the existence of chaos in the Smale horseshoe sense for cantilevered pipe conveying pulsating fluid. It has been concluded that it is dangerous to induce the chaotic vibrations of the pipes conveying pulsating fluid because the amplitudes of the chaotic vibrations are larger than those of the periodic motions.

Dynamics of a small gap gas lubricated bearing with Navier slip boundary conditions

A gas lubricated bearing model is derived which is appropriate for a very small bearing face separation by including velocity slip boundary conditions and centrifugal inertia effects. The bearing dynamics is examined when an external harmonic force is imposed on the bearing due to the bearing being situated within a larger complex dynamical system. A compressible Reynolds equation is formulated for the gas film which is coupled to the bearing structure through an axial force balance where the rotor and stator correspond to spring–mass–damper systems. Surface slip boundary conditions are derived on the bearing faces, characterised by the slip length parameter. The coupled bearing system is analysed using a stroboscopic map solver with the modified Reynolds equation and structural equations solved simultaneously. For a sufficiently large forcing amplitude a flapping motion of the bearing faces is induced when the rotor and stator are in close proximity. The minimum bearing gap over the time period of the external forcing is examined for a range of bearing parameters.

Non-linear argumental oscillators: Stability criterion and approximate implicit analytic solution

The behaviour of a space-modulated, so-called “argumental” oscillator, is studied. The oscillator is submitted to an external harmonic force, which is amplitude-modulated by the oscillator's position in space. An analytic expression of a stability criterion is given. Using the averaging method, an integrating factor and a Van der Pol representation in the (amplitude, phase)-space, an exact implicit analytic solution is given when there is no damping, and an approximate implicit analytic solution is given when there is damping, allowing the plotting of the separatrix curve. An attractor is identified.

Vibration analysis of geometrically imperfect three-layered shear-deformable microbeams

The vibration behaviour of a geometrically imperfect three-layered shear-deformable microbeam is analysed via model development and numerical simulations. Taking into account all the translational and rotational motions, considering continuous variations through the thickness for the displacement field, employing the modified couple stress theory for including small-size effects, and using constitutive relations for both stress and the deviatoric part of the couple stress tensors, the size-dependent elastic energy stored in the three-layered shear-deformable microbeam is obtained. The kinetic energy, work of an internal damping mechanism, and the work due to an external harmonic excitation force of the three-layered microsystem are also obtained and dynamically balanced by the size-dependent elastic energy by means of Hamilton's principle. The continuous expressions obtained for the axial, transverse, and rotational motions of the three-layered microsystem are truncated to high-dimensional reduced-order models with the help of a weighted-residual method. Numerical simulations are conducted by means of the backward differentiation formula (BDF) in conjunction with a continuation method. The nonlinear vibration behaviour of the microsystem is then analysed by plotting the coupled frequency-responses. The effect of the length-scale parameters as well as the material fraction and thickness of each layer on the vibration behaviour of the microsystem is highlighted.

Fast and accurate determination of the relative binding affinities of small compounds to HIV-1 protease using non-equilibrium work.

The fast pulling ligand (FPL) out of binding cavity using non-equilibrium molecular dynamics (MD) simulations was demonstrated to be a rapid, accurate and low CPU demand method for the determination of the relative binding affinities of a large number of HIV-1 protease (PR) inhibitors. In this approach, the ligand is pulled out of the binding cavity of the protein using external harmonic forces, and the work of pulling force corresponds to the relative binding affinity of HIV-1 PR inhibitor. The correlation coefficient between the pulling work and the experimental binding free energy of R=-0.95 shows that FPL results are in good agreement with experiment. It is thus easier to rank the binding affinities of HIV-1 PR inhibitors, that have similar binding affinities because the mean error bar of pulling work amounts to δW=7%. The nature of binding is discovered using the FPL approach. © 2016 Wiley Periodicals, Inc.

The non-linear vibrations of rotating functionally graded cylindrical shells

This paper reports the result of an investigate on the non-linear vibrations of rotating functionally graded cylindrical shell in thermal environment, based on Hamilton’s principle, von Karman non-linear theory and the first-order shear deformation theory. The formulation includes the initial hoop tension, the centrifugal and Coriolis forces due to rotation of the shell. The effects of in-plane and rotary inertia are taken into account in the equations of motion. Galerkin’s method is utilised to convert the governing partial differential equations to non-linear ordinary differential equations. A reduction in the model is presented to investigate non-linear dynamics, including primary resonance responses, quasi-periodic and chaotic responses to harmonic transverse external forces. The modal coefficients of quadratic and cubic nonlinearities are calculated by Galerkin integration and superimposed on the linear part of equation to establish the non-linear reduction equation. To validate the approach proposed in this paper, a series of comparison are performed and the investigations demonstrate good reliability and low computational cost of the present approach.