Method Of Time Scales Research Articles

Primary and combined multi-frequency parametric resonances of a rotating thin-walled composite beam under harmonic base excitation

A clamped rotating thin-walled composite elastic blade subjected to periodic base excitation is considered in this paper. The mathematical model of the structure captures the complex coupled bending–shear–torsional deformations of the beam resulting from the layout of the composite material. The partial differential equations of motion are formulated based on the Hamilton’s principle and next discretized according to the Galerkin’s method. The coefficients of the final state equations are found to be time dependent and have two distinct frequencies, namely the base excitation and the angular velocity of the system. The derived and initially coupled set of governing equations of motion is uncoupled and the method of multiple time scales is used to determine the transition lines separating stability regions of the system on Ince-Strut diagram. Numerical results are presented to illustrate the influence of the composite reinforcing fibres orientations as well as rotating speed and base excitation frequency on the dynamic stability of the structure.

Causation of Supersonic Limit Cycle Oscillations in Atmospheric Entry Vehicles

A successful atmospheric entry vehicle (AEV) design demonstrates manageable aerodynamic heating, bearable structural load, smooth deceleration, and intended trajectory during a descent into the atmosphere. Interestingly, the supersonic regime of the AEV manifests limit cycle oscillations (LCO) that restrict the maneuver potential and deployment of the drag chute. In this paper, efforts are made numerically and analytically to link the causation of LCO with external geometric variables of AEV. Computational fluid dynamics is used to calculate damping derivatives. The numerical results are validated with the Orion Crew Exploration Vehicle. The multiple time scales method, which belongs to the class of perturbation methods, is explored to develop an approximate closed-form solution of the nonlinear dynamic behavior of AEV. The analytical solution identifies that higher-order nonlinearities associated with pitch damping and static lift govern the onset of LCO. Finally, a parametric interaction study is carried out to determine the effect of two design variables, apex angle and length, on the vehicle’s dynamic stability. The mean data values from the main geometric effects plot show the condition of finite-amplitude oscillations. The results indicate that variation in these two parameters significantly impacts the magnitude of identified higher-order nonlinearities.

Amplitude resonance response and feedback control of cantilever beams with tip-mass under aerodynamic load

The dynamic of a cantilever beam with tip mass is studied under an aerodynamic loading. The effects of coupling is investigated by tacking into account the fluid flow. Using the multiple time scale method, the approximative solutions are found and the study of their stability is made by the Routh-Hurwitz stability criterion. The influence of parameters on the system is studied at the harmonic and subharmonic resonances. The results show that, the effects of tip mass can be neglected in harmonic resonance case, while they are more important in subharmonic resonance cases. The results equally demonstrate that an increase of the stable state fluid velocity reduces the amplitude of vibrations. In addition, the hysteresis phenomenon studies show that it is principally induced by nonlinearity coefficients. Finally, time-delay feedback control is applied and the effects of controlling are observed on amplitude response curve at the harmonic resonance, from where we note that optimized choice of control parameters can be useful in controlling vibrations.

Optimal Neuro Fuzzy Classification for Arrhythmia Data Driven System

This paper presents a data driven system used for cardiac arrhythmia classification. It applies the Neuro-Fuzzy Inference System (ANFIS) to classify MIT-BIH arrhythmia database electrocardiogram (ECG) recordings into five (5) heartbeat types. In fact, in order to obtain the input feature vector from recordings, a time scale method based on a Discrete Wavelet Transform (DWT) was investigated. Then, the time scale features are selected by applying the Principal Component Analysis (PCA). Therefore, the selected input feature vectors are classified by the Neuro-Fuzzy method. However, the ANFIS configuration needs mainly the choice of an initial Fuzzy Inference System (FIS) and the training algorithm. Indeed, two clustering algorithms which are the fuzzy c-means (FCM) and the subtractive ( SUBCLUST) algorithms, are applied to generate the initial FIS. Besides, for tuning the ANFIS membership function and rule base parameters, Gradient descent and evolutionary training algorithms are also evaluated. Gradient descent consists of the backpropagation (BP) method and its hybridization with the least square algorithm (Hybrid). However, the evolutionary training methods involve the Particle Swarm Optimization (PSO) and the Genetic Algorithm (GA). Therefore, eight (8) ANFIS are configured and assessed. Accordingly, a comparison study between their obtained Root Mean Square Error (RMSE) is analyzed. At the end, we have selected an optimal ANFIS which uses the SUBTRUCT algorithm to generate the initial FIS and the GA to tune its parameters. Moreover, to guarantee the effectiveness of this work, a comparison study with related works is done.

Nonlinear Vibration and Stability Analysis of Functionally Graded Nanobeam Subjected to External Parametric Excitation and Thermal Load

Analysis of vibration stability of simply supported Euler-Bernoulli functionally graded (FG) nanobeam embedded in viscous elastic medium with thermal effect under external parametric excitation is presented in this work. An attempt has been made for the first time is investigating the effect of thermal load on dynamic behavior, amplitude response, instability region and bifurcation points of functionally graded nanobeam. Thermal loads are supposed to be uniform, linear or nonlinear distribution along the thickness direction. Nonlocal continuum theory and the principle of the minimum total potential energy are applied to derive the governing equations. The partial differential equations (PDE) are transported to the ordinary differential equations (ODE) by using the Petrov-Galerkin method and the multiple time scales method are manipulated to solve the motion equation. To study the effect of external parametric excitation and thermal effect, different temperature distributions along the thickness such as uniform, linear, and nonlinear distribution are considered. Moreover, stable and unstable regions and bifurcation points are determined. It is obtained that the thermal load can affect the amplitude response of FG nanobeam. Also, it is observed that the instability of the system is affected by the detuning parameter and the parametric excitation amplitude plays great role in the instability of system. Nanobeams are used in many devices like nanoresonators, nanosensors and nanoswitches. This paper is helpful for designing and manufacturing nanoscale structures specially nanoresonators under different thermal loads.

Time-Scale Decomposition Techniques Used in the Ship Path-Following Problem with Rudder Roll Stabilization Control

The motion control of a surface ship based on a four degrees of freedom (4-DoF) (surge, sway, roll, and yaw) maneuvering motion model is studied in this paper. A time-scale decomposition method is introduced to solve the path-following problem, implementing Rudder Roll Stabilization (RRS) at the same time. The control objectives are to let the ship to track a predefined curve path under environmental disturbances, and to reduce the roll motion at the same time. A singular perturbation method is used to decouple the whole system into two subsystems of different time scales: the slow path-following subsystem and the fast roll reduction subsystem. The coupling effect of the two subsystems is also considered in this framework of analysis. RRS control is only possible when there is the so-called bandwidth separation characteristic in the ship motion system, which requires a large bandwidth separation gap between the two subsystems. To avoid the slow subsystem being affected by the wave disturbances of high frequency and large system uncertainties, the L1 adaptive control is introduced in the slow subsystem, while a Proportion-Differentiation (PD) control law is adopted in the fast roll reduction subsystem. Simulation results show the effectiveness and robustness of the proposed control strategy.

Analysis of a quarter car suspension based on a Kelvin–Voigt viscoelastic model with fractional-order derivative

The dynamical analysis of a quarter car suspension based on the Kelvin–Voigt​ viscoelastic model with fractional order derivative is analytically and numerically studied in this paper. Depending on the non-dimensional quadratic stiffness parameter, the potential configuration associated with the system under study is possible to obtain monostable and, asymmetric bistable configurations. Thanks to the multiple time scale method, the effects of viscoelastic parameters on the vibration amplitude of the quarter car suspension model are analytically investigated and the numerical simulations confirmed the analytical results. Moreover, the vibration amplitude of the quarter car suspension model can display periodic and dissipative chaotic behaviors by varying the non-dimensional quadratic stiffness, the fractional-order, and the road profile parameters. One scroll and double scroll chaotic attractors are generated by varying the non-dimensional quadratic stiffness parameter. Finally, the threshold conditions for the appearance of homoclinic bifurcation (left well and right well) in the case of asymmetric bistable potential are investigated using Melnikov’s criterion. Melnikov’s predictions are validated numerically by using the basin of attraction of initial conditions. It is found that the decrease of fractional order contributes to increasing the regular motion domain up to a critical value of the fractional-order and after it, decreases rather.

An analytical solution for nonlinear vibration of floating plate on the fluid by modified multiple scales method

The aim of the present paper is to analytically study the nonlinear forced vibration of a rectangular plate floating on the fluid by Modified Multiple Time Scales method for the first time. It is assumed that the fluid is stationary, incompressible, non-viscous, and non-rotational, and the plate is subjected to transversal excitation. The boundary condition is considered to be simply supported. Using von Karman nonlinear strain displacement relationships, the extended Hamilton principle, and FSTD plate theory, the partial differential equations of motion are derived. The fluid is mathematically modeled by Bernoulli equation and the velocity potential function. Galerkin method is then applied for converting the nonlinear partial differential equations into time-dependent nonlinear ordinary differential equations. The resulted equations are solved analytically by the Modified Multiple Scales Method, thereafter. Despite the large number of derivatives and calculations of the conventional multiple scale method, this approach is very simple and straightforward. The results reveal an excellent agreement with the traditional Multiple Scales method results and existing studies, and are more accurate than other available results. The effect of the presence of fluid near the plate on natural frequency and amplitude of vibration of plate are studied. The effects of some key parameters of the system are also examined.

Bi-Fo time scaling method in the numerical simulation of transient conjugate heat transfer

Reliable transient thermal analysis plays a very important role in the engine safety analysis. Transient conjugate heat transfer simulation is an important way of temperature analysis. But there exists a great disparity in the time scales between solid conduction and fluid convection. The calculation cost of transient conjugate heat transfer analysis is very huge because of the tiny time step of computational fluid dynamics. The Bi-Fo time scaling method is proposed to improve the computational efficiency of transient conjugate heat transfer. On the one hand, this method carries out a similar transformation on solid heat conduction, scaling the calculation time with the product of density and specific heat capacity to maintain the consistency of Fourier number. On the other hand, it takes very short time for the fluid domain to recover stability after a boundary disturbance. Based on the above characteristic, the flow time is directly compressed to the same as that of the solid domain. It is verified by Mark Ⅱ vane that increasing the solid thermal diffusivity can reduce the time scale of heat conduction. In the situation of rapidly stable flow field, scaling flow time does not affect the solid thermal boundary under corresponding dimensionless time. Within the application scope, the Bi-Fo time scaling method can greatly reduce the time cost of transient conjugate heat transfer simulation while maintaining the accuracy of transient temperature analysis.

Effect of the second invariant of the Cauchy–Green deformation tensor on the local dynamics of dielectric elastomers

This paper aims to investigate the local oscillations of a dielectric elastomer with emphasis on the effect of the second invariant of the deformation tensor in finite strain theory on such oscillations. Four hyperelastic constitutive models are utilized in this study, namely the classic Gent model, a modified Gent model, a modified version of the Pucci–Saccomandi model, and the Mooney–Rivlin model. We derive the governing equations for all the models through the use of the Euler–Lagrange formulation. We explore the local responses of the system in the framework of the nonlinear frequency responses by solving the equations of motion using the multiple time scales method. In addition, to provide an in-depth analysis, we analyze static and dynamic electromechanical instabilities by plotting the voltage-stretch diagram and the time signature. The numerical results for damped and undamped dielectric elastomers are simulated and compared. We analyze the system’s response for the aforementioned hyperelastic constitutive functions, and compare the differences and similarities of models. Based on the results, two saddle–node bifurcations occur in the system. Generally, increasing the second invariant parameters of the hyperelastic models, decreases the response amplitude. As contribution of the second invariant in the Mooney–Rivlin model is increased, for both damped and undamped systems, softening turns into hardening nonlinearity. We also show that the second invariant tunes the bifurcation points and instability of the system. Moreover, it can control the static and dynamic pull-in and snap-through voltages.

Sound Radiation Mitigation of Geometrically Nonlinear Plates Subjected to Subsonic Airflow

This paper gives a theoretical analysis to obtain and reduce the acoustic pressure generated from plates with geometric nonlinearities subjected to subsonic airflow and external excitation. von-Kármán assumptions are applied considering the nonlinear terms in strain-displacement relations. Airflow passing through the plate is considered as an incompressible, irrotational, and inviscid flow. Galerkin’s method is employed to acquire the governing equations of time-dependent coefficients. Multiple Time Scale Method (MTSM) is then used to obtain the response of the plate. Suppressing undesirable vibration is carried out using an optimal tuned mass damper (TMD) system and an analytical solution is proposed based on Laplace transform and Adomian Decomposition Method (ADM). The acoustic pressure received from the plate is calculated by solving the Rayleigh integral using Boundary Element Method (BEM). A parametric study is carried out, and the effects of the flow speed, the aspect ratio, the thickness of the plate, the material of the plate, the forcing frequency, and the effectiveness of the designed TMD system on the sound pressure are examined. According to the results, using the TMD system reduces the amplitudes of the plate vibrations and, consequently, reduces the acoustic pressure around the vibrating plate. In this study, the passive control strategy leads to a significant decrease in the sound pressure level (about 35%) in some airflow speeds. Results show good efficiency in the control strategy. It is also found that the acoustic pressure generated from steel plates is significantly larger than that generated from aluminum plates. Moreover, increasing the aspect ratio and the plate thickness reduces the acoustic pressure. On the other hand, the external excitations with lower frequencies and the airflows with higher densities can generate lower sound pressures around the plate.

Vibration Control for a Coupled Pitch- roll Ship Model Via a Negative Cubic Velocity Feedback Control

One of the most essential ship reactions to waves is roll motion. Due to the intricacy of ship wave interactions and their sensitivity, predicting such a reaction is extremely challenging. Because vibration motion is an undesirable occurrence, it must be removed, decreased, or controlled. A coupled Pitch- roll ship model with negative cubic velocity feedback control subjected to parametric excitations is premeditated and solved in this paper. The method of multiple time scales is applied to scrutinize the response of the two modes of the system neighbouring the simultaneous sub-harmonic, and internal resonance situation. Besides, the steady-state solution is determined through the Rung-Kutta Method (RKM) of fourth order. Stability of the steady state solution near this resonance case is discussed and studied applying Lyapunov’s first indirect method and Routh- Hurwitz criterion. The influences of the different parameters on the steady state solution are reconnoitred and discussed. The controller effects on the stability are clarified. Simulation results are accomplished with the help of MATLAB and Maple software programs.

Study on gamma spectrum wavelet noise reduction based on scintillation detector

Wavelet transform is a time-scale signal analysis method, which holds the characteristics of the same window size, arbitrary shape change and multi-resolution analysis. Most applications of wavelet transform need to be based on signal decomposition and reconstruction, that is, through the combination of low-pass and high-pass filters to achieve frequency division, the signal is decomposed into signal detail components and large-scale approximate components. In the measurement process of the original energy spectrum data, the denoting process plays a role in reducing the incidence of statistical fluctuations to a certain extent, which is of profound significance to improve the data accuracy. In this paper, wavelet transform is used as the theoretical basis to denoise the actual energy spectrum, at the same time based on the MATLAB platform, simulation of signal with noise, using the different parameters on the signal processing. Firstly, the signal-to-noise ratio (SNR) of the hard and soft thresholds to the signal processing under the same conditions was compared, and then the change of the SNR was observed when the number of decomposition layers was changed under the wavelet basis of db3 (soft threshold). The experimental results show that: 1) under the same conditions, the soft threshold method has higher SNR than hard threshold; 2) The number of decomposition layers and the elimination moment affect the signal-to-noise ratio and smoothness of the processed signal respectively, and when the number of decomposition layers is 7-8 layers, the signal-to-noise ratio is about 29, the signal processing effect is the best. This study has shown that Wavelet method is an effective way to process energy spectrum.

Model predictive current control with adaptive-adjusting timescales for PMSMs

A model predictive current control (MPCC) with adaptive-adjusting method of timescales for permanent magnet synchronous motors (PMSMs) is proposed in this paper to improve the dynamic response and prediction accuracy in transient-state, while lessening the computational burden and improving the control performance in steady-state. The timescale characteristics of different parts of MPCC, such as signal sampling, prediction calculation, control output, model error correction, are analyzed, and the algorithm architecture of MPCC with multi-timescale is proposed. The difference between reference and actual speed, and the change rate of actual speed are utilized to discriminate the transient state of speed change and load change, respectively. Adaptive-adjusting method of control period and prediction stepsize are illustrated in detail after operation condition discrimination. Experimental results of a PMSM are presented to validate the effectiveness of proposed MPCC. In addition, comparative evaluation of single-step MPCC with fixed timescale and proposed MPCC is conducted, which demonstrates the superiority of proposed control strategy.

Nonlinear interaction of longitudinal and transverse vibrations of a rod at an internal combinational resonance in view of opto-thermal excitation of N/MEMS

In this paper, the character of coupled longitudinal-flexural vibrations of a rod is investigated in the case of an internal combinational resonance between two flexural and lower longitudinal vibration modes. An approximate analytical solution is obtained by a multiple time-scales method applied directly to a system of nonlinear partial differential equations. It is shown that the initial perturbation in the longitudinal direction can effectively excite bending vibration modes, which, on the whole, leads to a longitudinal-transverse beating regime with noticeable amplitudes. It is noted that the period and amplitude of these beats substantially depend on the frequency detuning parameter between the sum of the bending vibration frequencies and the longitudinal vibration frequency. This effect can be used as a new principle of operation of opto-thermally excited N/MEMS resonant sensors.

Modelling and energy transfer in the coupled nonlinear response of a 1:1 internally resonant cable system with a tuned mass damper

Considering the vibration of a Tuned Mass Damper (TMD), a dynamic model composed of the cable and TMD is investigated. Different from the other studies, this paper is mainly devoted to nonlinear behaviours of the model by considering the participation of the damper in energy transfer and coupling interaction between the cable and damper. According to the extended Hamilton’s principle, the classical equations of motion of the cable and TMD are derived. Based on the equations of motion of the cable and TMD, the one-to-one internal resonance of the system is studied when external primary resonance of the cable occurs. By applying the Galerkin’s method, a set of ordinary differential equations (ODEs) are obtained. To solve the ODEs, the multiple time scale method is used and the modulation equations are derived. The stable solutions of the modulation equations are acquired by Newton-Raphson method and continued by pseudo arclength algorithm. Meanwhile, the parametric analyses of some key parameters, such as the excitation amplitude, the spring stiffness, the damping ratio and position of the TMD and the sag of the cable, are carried out through frequency-/force-response curves to explore the nonlinear behaviours of the system. The results show that the TMD plays an important role in both energy consumption and energy transfer.

On resonances in a weakly nonlinear microbeam due to an electric actuation

In this paper, the oscillations of an actuated, simply supported microbeam are studied for which it is assumed that the electric load is composed of a small DC polarization voltage and a small, harmonic AC voltage. Bending stiffness and mid-plane stretching are taken into account as well as small viscous or structural damping. No tensile axial force is assumed to be present. By using a multiple time-scales perturbation method, approximations of the solutions of the initial-boundary value problem for the microbeam equation are constructed. This analysis is performed without truncating the infinite series representation in advance as is usually done in the existing literature. It is shown in which cases truncation is allowed for this problem. Moreover, accurate and explicit approximations of the natural frequencies up to order \(\varepsilon ^3\) of the actuated microbeam are also obtained. Intriguing and new modal vibrations are found when the frequency of the harmonic AC voltage is (near) half or twice a natural frequency of the microbeam, i.e., near a superharmonic or a subharmonic resonance.

Nonlinear feedback anti-control of limit cycle and chaos in a mechanical oscillator: theory and experiment

Though engineers believe that self-excited oscillation and chaos is detrimental, recent research has revealed that self-excited periodic and chaotic oscillation can be effectively utilized in many engineering processes and devices for significant benefits. Then a simple nonlinear acceleration feedback controller is proposed to generate self-excited periodic and chaotic oscillation in a mechanical oscillator. Analytical expressions relating the amplitude of limit cycles and control parameters are obtained by the method of multiple time scales. Analytical results are verified by numerical simulations performed in MATLAB–SIMULINK. Existence of chaotic oscillations is confirmed by numerical simulations. An extensive parametric study is carried out to reveal the nature of the chaotic oscillations and its dependence on the controller parameters. Theoretical results are finally verified by experiments. It is generally observed that the same controller can be used to induce both periodic and chaotic oscillations just by flipping the phase of the controller. This is possibly the first example where the same controller can be utilized to generate periodic and chaotic oscillation in a mechanical system. The findings of the paper are believed to be used in macro- and micro-mechanical systems.

Fabrication and resonance simulation of 3D-printed biocomposite mesoporous implants with different periodic cellular topologies

In order to achieve more efficient additive materials having precise desired porosities with excellent performance, advanced manufacturing techniques can be put to use. In the present study, 3D printing technology is employed to fabricate mesoporous bioceramic-based bony scaffolds made of various periodic cellular topologies including cubic, cylindrical and hexagonal porosity configurations. The samples are composed of polylactic acid-hydroxyapatite (PLA-HA) mesoporous bioceramic-based nanocomposites. Via the respective experiments, the mechanical properties of the fabricated bony scaffolds are extracted corresponding to each porosity configuration with different pore sizes. Moreover, by using the scanning electron microscopy (SEM), X-ray diffraction (XRD) and the permeability characteristics of the fabricated bio-composite samples after soaking in the simulated body fluid (SBF) are captured. Thereafter, based on the experimentally obtained mechanical properties, the multiple time-scales method is utilized to develop an analytical solution for the nonlinear secondary resonance of PLA-HA mesoporous beam-type implants under subharmonic and superharmonic excitations. The frequency-response and amplitude response associated with the both types of hard excitations are depicted corresponding to different periodic cellular topologies and pore sizes. It is observed that among various periodic cellular topologies, the gap between two branches of the subharmonic frequency-response associated with the cubic and hexagonal ones is maximum and minimum, respectively. In the case of superharmonic excitation, it is found that through reduction of the pore size, the peak of frequency-response and its associated value of the detuning parameter decrease.

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

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