Frequency-amplitude Relationship Research Articles

Dynamic analysis of a class of fractional‐order dry friction oscillators

This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin.

An Innovative Approach in Inspecting a Damped Mathieu Cubic–Quintic Duffing Oscillator

Abstract Purpose The objective of the present study is to analyze a damped Mathieu–cubic quintic Duffing oscillator as a parametric nonlinear oscillatory dynamical system. This equation has multiple applications in diverse fields, including optics, quantum physics, and general relativity. There are multiple concerns related to periodic motion and the analysis of boundary-value problems with elliptic symmetries. The current effort aims to determine the frequency amplitude of parametric nonlinear issues. Method The non-perturbative approach (NPA) is employed to transform the nonlinear ordinary differential equation (ODE) into a linear equation. The derivation of the approximate solutions is achieved without relying on typical perturbation approaches, separate from the series expansion. Hence, the objective of this study is to depart from traditional perturbation methods and acquire approximated solutions for minor amplitude parametric components without imposing any limitations. Furthermore, the technique is extended to ascertain optimal solutions for the nonlinear large amplitude of fluctuation. Results The current approach allows for rapid estimation of the frequency-amplitude relationship in order to attain successive approximations of the solutions for parametric nonlinear fluctuations. A validation is obtained for the derived parametric equation, demonstrating a high level of agreement with the original equation. An analysis of stability behavior is conducted in multiple scenarios. In addition, the Floquet theory is used to examine the transition curves. Conclusion The current technique is characterized by its clear principles, making it practical, user-friendly, and capable of achieving exceptionally high numerical precision. The current approach is highly beneficial for addressing nonlinear parametric problems due to its ability to minimize algebraic complexity during implementation.

Coupled nonlinear vibration characteristics of quasi-zero-stiffness Gough-Stewart isolation platform

The quasi-zero-stiffness (QZS) Gough-Stewart vibration isolation platform is designed for six degrees-of-freedom (DOFs) low-frequency vibration isolation. Lots of studies regarding the amplitude-frequency relationship of the platform are based on the assumption that interference with the platform in one direction only provokes a response in that direction. This assumption artificially decouples a six DOFs coupling model into six single DOF equations, ignoring the coupling relationship between each DOFs, making the analysis results inaccurate. Therefore, without simplifying the nonlinear terms and taking the coupling effect into consideration, the improved Incremental Harmonic Balance Method (IHBM) is employed to discuss the coupling relationship between different DOFs of the QZS Gough-Stewart vibration isolation platform. The results show that the platform has multiple equilibrium positions. When the platform is excited from different directions, it will vibrate at different equilibrium positions, and the closer these positions are to the zero-stiffness position, the more likely the system is to exhibit jump phenomenon and sudden increases in response. The platform exhibits a single-directional response only when excited from the z-direction. Through comparative study, it is found that the cubic configuration can weaken the coupling effect between each DOFs compared with the general configuration.

Nonlinear forced vibration of functionally graded hybrid three-phase nanocomposite toroidal shell segments reinforced by carbon nanotubes (CNTs) and graphene nanoplatelets (GPLs)

This study proposes a unique three-phase functionally graded (FG) hybrid nanocomposite material reinforcing the toroidal shell segments to investigate the nonlinear forced vibration using Reddy's higher-order shear deformation theory, von Karman geometrical nonlinearity and Stein-McElman's assumption along with the Hamilton principle. The examined toroidal shell's material consists of polymeric resin, carbon nanotubes (CNTs), and graphene platelets (GPLs) as nanocomposites. The material properties have been derived based on a modified Halpin-Tsai micromechanical model. Four distribution patterns have been studied: uniform (UD), FG-X, FG-O, and FG-V. The continuous distribution of GPLs and CNTs leads to inhomogeneous position-dependent properties throughout the shell thickness. The obtained differential equations of motion have been reduced to ordinary equations using Galerkin's technique. A multi-scales method (MSM) is used to estimate a closed-form solution representing the frequency-amplitude relation, and the state space representation is used along with numerical fourth-order Runge Kutta method (RK4) to obtain the nonlinear dynamic response of the toroidal shell. The accuracy of the current results obtained has been verified by comparing them with the relevant literature and numerical results using (RK4). In addition, the influence of both GPLs and CNTs weight fractions, nanofillers’ distribution types, longitudinal and circumferential wave numbers, elastic foundation parameters, static axial compression load, transverse excitation load, damping ratio, geometrical characteristics of the toroidal shell on the dimensionless natural frequency, nonlinear primary resonance (frequency-amplitude curve), and nonlinear dynamic response are carefully studied. The results found that combining (GPLs & CNTs) into the shell's matrix improves performance, especially in concave shells compared to convex ones. The FG-X pattern reduces peak resonance amplitude and improves natural frequency. A reduction of 12.3 % is observed for convex shells, and 21.5 % for concave shells compared to FG-O.

A review of the frequency-amplitude formula for nonlinear oscillators and its advancements

The frequency-amplitude relationship is pivotal in understanding oscillatory systems, dictating their dynamic behaviors and responses. This paper delves into the intricacies of the frequency amplitude formula, elucidating its foundational role in both linear and nonlinear oscillations. Through comprehensive analysis, we highlight the formula’s significance in predicting system behaviors, especially in environments with varying amplitudes. Our findings underscore the formula’s potential as a robust tool for enhanced system characterization, offering profound implications for diverse applications across scientific and engineering domains. This study delves deep into the formulation of the frequency-amplitude relationship, extending its application beyond the conventional cubic powers of the restoring force. We introduce three novel and equivalent formulations of the generalized frequency amplitude, breaking traditional boundaries by not confining the restoring force to just odd functions. The formula to determine the frequency of the singular oscillator has been set forth. Our approach, characterized by its simplicity, offers a robust tool for analyzing high nonlinearity vibrations, ushering in a richer, multidimensional perspective to vibration analysis.

Linear and Nonlinear Dynamics Responses of an Axially Moving Laminated Composite Plate-Reinforced with Graphene Nanoplatelets

The subharmonic resonances of an axially moving graphene-reinforced laminated composite plate are studied based on the Galerkin and multiscale methods. Graphene nanoplatelets (GPLs) are added into matrix material which acts as the basic layer of the plate, and a graphene-reinforced nanocomposite plate is thus obtained. Different GPL distribution patterns in the laminated plate are considered. The Halpin–Tsai model is selected to predict the physical properties of the nanocomposite. Hamilton’s principle is utilized to conduct the dynamic modeling of the plate and the von Kármán deformation theory is used. The velocity is assumed to be a combination of constant and harmonically varied velocities. The natural frequencies of the linear system with constant velocity can be obtained using the eigenvalues of the coefficient matrix of the ordinary differential equations after the governing partial differential equations of motion are discretized through the Galerkin method. The instability regions of the linear system and the amplitude–frequency relations of the nonlinear system considering the harmonically varied velocity are obtained based on the multiscale analysis. The effect of GPL reinforcement on the subharmonic resonances of the linear and nonlinear systems is analyzed in detail.

Mechanical and vibration characteristics investigation of the elastic ring squeeze film damper-rotor considering oil film sealing and leakage

Elastic ring squeeze film damper (ERSFD) is extensively used in industrial rotating systems due to its excellent noise control and vibration suppression characteristics. However, the oil film sealing and leakage has a profound effect on the mechanical and vibration characteristics of the ERSFD and the ERSFD-rotor during operation. The sealing and leakage reduce the axial flow of the oil film, leading to changes in the initial boundary conditions of the flow field, which in turn affects the oil film pressure and damping coefficient of ERSFD. Meanwhile, the sealing causes an increase in the oil film force of ERSFD, which in turn affects the response of the rotor system. But there is relatively little research on the impact of sealing and leakage on the dynamic performances of ERSFD-rotor. In this paper, a mathematical model of ERSFD under different sealing conditions (no end seal, complete end seal, piston ring seal, O-ring seal) is established by changing the initial boundary equations of the flow field of ERSFD and the coupled solution of the Reynolds equation of ERSFD with different sealing and leakage boundary equations is realized. Then, the characteristics of oil film pressure and force of ERSFD under different sealing conditions is studied. After that, an ERSFD-rotor model considering oil film sealing and leakage is established based on displacement compatibility conditions and force balance principles. Meanwhile, the impact of sealing and leakage on the amplitude-frequency relation and sudden imbalance of an ERSFD combined support rotor is researched. The research results reveal that good sealing can significantly increase the damping of the system, thereby reducing the vibration amplitude of the ERSFD-rotor system. In addition, good sealing conditions can also shorten the transient process of sudden unbalanced excitation. Finally, the effectiveness of some simulation results was verified through experiments on the ERSFD rotor test bench.

Nonlinear Dynamics and Combination Resonance of a Flexible Turbine Blade with Contact and Friction of Shrouds

Flexible shrouded blades are commonly adopted in the last stages of steam turbines where complicated dynamical behavior can be induced by dry friction force generated on contacting interfaces between adjacent shrouds and the geometric nonlinearity due to the structural flexibility of the blades. In this paper, combination resonance caused by contact and friction forces generated on shroud interfaces is investigated, which is a concurrence of 1:3 internal resonance involving the first and second modes in the flapwise direction and the primary resonance of the first flapwise mode. The stiffness and damping at the contact interface are obtained by linearizing the contact and friction forces between shrouds through the harmonic balance method. The vibrating blade is modeled as a beam with a concentrated mass of which the responses under the combination resonance are solved through the multiple-scale method. Sensitivities of response with respect to the angle of shrouds, contact stiffness and rotation speed are illustrated, and the influences of these parameters on the periodicity and amplitudes of steady responses are demonstrated. The parametric regions where the combination resonance occurs are pointed out. Finally, parametric analyses are presented to show how the amplitude–frequency relation of the multiple-scale solutions under the combination resonance vary with detuning and design parameters. The present research provides a designing basis for improving the dynamic performance of flexible shrouded blades and suppressing vibrations of blades by adjusting structural parameters in practical engineering.

Non-linear oscillation of a mass attached to a stretched elastic wire in a fractal space

The challenge for a non-linear vibration system in a fractal space is more fractal dimensions than frequency-amplitude relationship, the system energy consumption depends upon its fractal property, so its best-case scenario is to establish a relationship among the fractal dimensions, frequency and amplitude. For this purpose, this paper studies a fractal-fractional vibration system of a mass attached to a stretched elastic wire in a fractal space, and its asymptotic periodic property is elucidated, the effect of the fractal dimensions on the vibration system is discussed. This paper offers a new road to fast and reliable analysis of fractal oscillators with high accuracy.

Local fractional Duffing equation: Its periodic property and its application to energy harvesting

A local fractional modification of the Duffing equation is considered, and the homotopy perturbation method is employed to reveal its frequency-amplitude relationship, which is of paramount importance in the optimal design of the energy harvesting devices and chatter detection. Effects of the initial conditions on the periodic property is also discussed.

Applying numerical control to analyze the pull-in stability of MEMS systems

The micro-electro-mechanical system is widely used for energy harvesting and thermal wind sensor, its efficiency and reliability depend upon the pull-in instability. This paper studies a micro-electro-mechanical system using He-Liu [34] formulation for finding its frequency-amplitude relationship. The system periodic motion, pull-in instability and pseudo-periodic motion are discussed. This paper offers a new window for security monitoring of the system reliable operation.

Nonlinear vibration analysis of a double curved shallow sandwich shell in which the core made of three-phase nanocomposite and the two-outer layer of electromagnetic materials

This paper investigates the nonlinear vibrations of a double curved shallow sandwich shell with a three-phase nanocomposite core and two outer electromagnetic layers considering the thermal environment's influence. Relying on the Halpin–Tsai model, the core is assumed to be made of some composite plies incorporating three constituents: carbon nanotubes (CNTs) or graphene nanoplatelets (GPLs) as nanocomposites, fibers and polymeric matrix (resin). Governing equations calculating the nonlinear vibration of the double curved shallow shell were obtained by combining Reddy's first-order shear deformation theory (FSDT) and the Von Kármán geometrical nonlinearity. Later, this system of equations is transformed into the nonlinear ordinary differential expressions by utilizing the Galerkin procedure. Thus, the effects of material properties, temperature, geometrical properties, elastic foundations to the frequency of linear and nonlinear free vibration, and amplitude–frequency relations of the double curved shallow sandwich shell were discussed. The results demonstrate that the free vibration frequency of the shell increases if the magnetic field enhances, whereas that decreases when the voltage expands in both models as CNTs and GPLs. Simultaneously, the shallow shell has a better ability to force load compared to a plate with the same dimensions. These outcomes can make a profit in the design and manufacturing of new smart structures applied in energy harvesting, and medicine.

An improved one-step amplitude–frequency relation for nonlinear oscillators

This paper proposes an amplitude–frequency relation which improves He’s simple, one-step, approach for approximating the frequency in the case of the cubic–quintic Duffing oscillator; the relation we apply is a weighted average amplitude–frequency relation.

Tumbler-inspired electromagnetic generator for low-frequency ocean wave energy harvesting

Ocean wave is considered as one of most promising renewable energy sources due to the merits of wide distribution and high energy density. However, the low-frequency and large randomness characteristics make it hard to harvest. Here we report a tumbler-inspired electromagnetic generator (T-EMG) with a Halbach array to improve the low-frequency ocean wave energy harvesting performance. The theoretical model of the T-EMG is established. The amplitude-frequency relationship is derived. The simulations and experimental investigation are carried out. The maximum harvested voltage of 3 V with a lower frequency of 2.6 Hz is obtained, and the maximum average output power can reach 65 mW through adjusting the load and tumbler parameters. More significantly, the offshore tests in the East China sea were carried out and the experimental results showed that T-EMG can work effectively under ultra-low frequency (<1Hz) real ocean wave environments and achieve a maximum peak power of 120 mW, which could meet the requirement of low-power wireless sensors. In addition, the offshore tests also demonstrate the strong frequency adaptability of T-TMG that enables it to harvest random and low-frequency ocean waves efficiently. This work provides an effective solution to harvest low-frequency and random ocean wave energy and can be easily adapted to mechanical or aerospace engineering, etc.

Incorporating Geometric Nonlinearity in Theoretical Modeling of Muscle-Powered Soft Robotic Bio-Actuators

Abstract Biohybrid actuators aim to leverage the various advantages of biological cells over artificial components to build novel compliant machines with high performance and autonomy. Significant advances have been made in bio-fabrication technologies, enabling the realization of muscle-powered bio-actuators. However, the mechanics of muscle-scaffold coupling has been relatively understudied, limiting the development of bio-actuators to intuitive or biomimetic designs. Here, we consider the case of implementing muscle-based actuation for soft robotic swimmers operating at low Reynolds numbers. We develop an analytical model to describe the elasto-hydrodynamic problem and identify key design parameters. Muscle contraction dynamics is characterized experimentally and the implications of nonlinear amplitude-frequency relationship of muscle-based actuation are discussed. We show that a novel bio-actuator with high performance can be developed by introducing compliant flexural mechanisms undergoing large deflection. Geometric nonlinearities are accounted for in the analysis of the force-deflection relationship for the flexural mechanism. Our results show that for expected muscle contraction forces, this novel bio-actuator can outperform previous muscle-powered swimmers by up to two orders of magnitude in swimming speed.

Amplitude and inter-event time statistics for the island volcanoes Stromboli, Mount Etna, Yasur, and Whakaari

Detailed analyses of past major and minor seismo-volcanic events can help to understand the eruptive behavior of volcanoes and the underlying physical and chemical processes. Catalogs of these eruptions and, specifically, seismo-volcanic events may be generated using continuous seismic recordings at stations in the proximity of volcanoes. Here, we apply a recently-developed automated approach Adaptive-Window Volcanic Event Selection Analysis Module (AWESAM) to seismic data from Stromboli (Italy), Mount Etna (Italy), Yasur (Vanuatu) and Whakaari (New Zealand). We perform an inter-event time analysis to identify characteristic patterns in the events’ recurrence time and the volcanic activity. Using this identical approach for all volcanoes, we were able to discover that despite their differing types and activity, they exhibit similar statistical behaviors. For Whakaari, we noticed a bimodal inter-event time distribution for large events. Since this observation is based on single station data, further in-depth investigations are needed once more data is available in future. We also derive a new amplitude-frequency relationship from seismo-volcanic events. With this relation, we can confirm a change in slope for large events at Stromboli, which is based on 10 years of data. Additionally, we apply a classification model to events from Stromboli to differentiate between low-period (LP) events and high-frequency (HF) events and found an alternating behavior in the frequency of these events before and after the two paroxysms in 2019.

Dynamic modeling and characterization of compliant cable-driven parallel robots containing flexible cables

AbstractFlexible cables in cable-driven parallel robots (CDPRs) are easy to be excited and vibrate. Cable vibration will react on the end-effector, causing attitude deviation of the end-effector. The main objective of this study is to accurately model axially moving flexible cables and characterize the dynamic behaviors of associated compliant CDPRs. Firstly, a model for transverse vibration of the axially moving length-variable cable is developed. On this basis, an original nonlinear dynamic model of the CDPRs able to capture the vibration of the cables and the dynamics of the end-effector is proposed. Secondly, the frequency–amplitude relationship of the CDPR is obtained. Moreover, the significance of the excitation effect caused by the axially moving length-variable cables is demonstrated, by comparing the results with and without excitation effect at different frequencies. It turns out that, as the oscillation frequency of the end-effector increases, the end-effector and cables exhibit the dynamics process from steady state to unstable large-amplitude vibration and finally to stable small-amplitude vibration. This indicates that the dynamics of the CDPR exhibit non-linear characteristics, due to the influence of flexible cables. Finally, the proposed dynamic model of compliant CDPRs is validated by experiments performed in the laboratory.

Multi-field coupled dynamics for a movable tooth drive system integrated with shape memory alloys

The harmonic movable tooth drive system integrated with shape memory alloys has a small size and a large output torque. Its dynamics performance is the key factor for evaluating the drive system. Here, for the drive system, based on its structure and working principle, the coupled dynamics equations are deduced. Using the equations, changes of the natural frequencies of the drive system during the operation are investigated. Effects of the system parameters and SMA wires phase change process on the natural frequencies are analyzed. The nonlinear resonant frequencies of the drive system and its amplitude-frequency relationship are studied. Results show that natural frequencies of the drive system change periodically which is caused by SMA phase transformation during operation. The eccentricity, movable tooth radius, the wave generator radius and SMA wire length have also important effects on the natural frequencies of the drive system. The nonlinear resonant frequencies are smaller than linear resonant frequencies. In the design of the drive system, the coupled nonlinear effects of the temperature, phase change, stress and strain of the SMA wires, and the system parameters of the movable tooth drive system should be considered. In this paper, the coupled nonlinear dynamics model of the harmonic movable tooth drive system integrated with shape memory alloys is proposed in which the coupled effects of the temperature, phase change, stress and strain of the SMA, and the system parameters of the movable tooth drive system are considered.

Analysis of the fractional oscillator for a mass attached to a stretched elastic wire

In this paper, we consider a fractional nonlinear oscillator for a mass attached to a stretched elastic wire. The coupling technique based on the fractional complex transform and the global residue harmonic balance method is proposed for solving this fractional oscillator. The approximated periodic solutions and frequencies are provided without complicated computation and discretization. The frequency–amplitude relationship is investigated to show its monotonic property. Numerical approximations for the nonlinear oscillator with integer or fractional orders are compared with the approximations by some existing methods. Numerical results confirm the effectiveness and robustness of the proposed method.

A Nonlinear Free Vibration Analysis of Functionally Graded Beams Using a Mixed Finite Element Method and a Comparative Artificial Neural Network

Based on the Hamilton principle combined with the Timoshenko beam theory, the authors developed a mixed finite element (FE) method for the nonlinear free vibration analysis of functionally graded (FG) beams under combinations of simply supported, free, and clamped edge conditions. The material properties of the FG beam gradually and smoothly varied through the thickness direction according to the power-law distributions of the volume fractions in the constituents, and the effective material properties of the FG beam were estimated using the rule of mixtures. The von Kármán geometrical nonlinearity was considered. The FE solutions of the amplitude-frequency relations of the FG beam were obtained using an iterative process. Implementing the mixed FE method showed that its solutions converged rapidly and that the convergent solutions closely agreed with the accurate solutions reported in the literature. A multilayer perceptron (MP) back propagation neural network (BPNN) was also developed to predict the nonlinear free vibration behavior of the FG beam. After appropriate training, the prediction of the MP BPNN’s amplitude-frequency relations was entirely accurate compared to those obtained using the mixed FE method, and its central processing unit time was less time-consuming than that of the mixed FE method.