Freedom Vibration System Research Articles

Input-output finite-time stabilization of periodic piecewise systems with multiple disturbances

In this work, the problems of input-output finite-time stability and disturbance rejection for continuous-time periodic piecewise systems with linear fractional uncertainty, matched and mismatched disturbances are investigated. In detail, the matched disturbances that are ensuing from the exogenous systems can be tackled by modeling a periodic piecewise disturbance observer (PPDO), which effectively estimates the disturbance with high precision. The mismatched part can be sorted out by implementing H∞ control protocol and meanwhile the state feedback is quantized in accordance with the logarithmic quantizer. On the whole, by combining the quantized state-feedback control law with the output of the PPDO, the anti-disturbance control protocol is developed. Moreover, by constructing a Lyapunov function with periodic piecewise positive definite matrices, a collection of adequate criteria affirming the system’s input-output finite-time stability are procured in the context of linear matrix inequalities (LMIs). Subsequently, the time-varying periodic piecewise gain values of the crafted disturbance observer and developed controller are acquired by working on the LMIs. Conclusively, the simulation results are provided, including the 2-degree of freedom vibration system, to verify the potential of the developed control strategy.

Impact response analysis of vibration system with inerter

This research investigates using an inerter system to control the impact-induced vibration response of a single degree of freedom(SDOF) and two degrees of freedom(TDOF) vibration system. An analytic model of the SDOF system with an inerter is first derived, and simulation is performed to compare the inerter performance in suppressing the maximum acceleration response under shock load. Next, a more complex TDOF vibration model with an inerter is derived, and the effectiveness of the inerter in reducing the large acceleration amplitude is evaluated. The energy analysis is conducted to investigate the energy transfer mechanism during impact. The simulation study shows that one can significantly decrease the maximum acceleration response with an appropriate selection of the inerter parameters. For the SDOF vibration system with an inerter, response reduction is greatly influenced by the system’s natural frequency. In the TDOF vibration system, the maximum acceleration response attenuation is caused by two factors. For a large inerter radius, the acceleration response reduction is mainly influenced by energy transfer between unsprung and inerter mass. However, for a big inerter mass, attenuation of the acceleration response is caused by decreasing the system’s natural frequency. It is observed from the simulation that the maximum acceleration response can be reduced up to 35% by adjusting the inerter radius of gyration. Furthermore, the optimum condition for reducing the acceleration response occurs when the inerter mass is half the unsprung mass ( m i = 0.5 m u). An experimental study is conducted to validate the simulation results.

Applications of the integral equation method on nonlinear multi degree of freedom vibration systems

This paper compares the accuracy of the integral equation method and the multiple scales method in solving the amplitude equation of multi degree of freedom nonlinear vibration systems. We consider three examples: a two-degree-of-freedom stainless-steel beam system controlled by a saturation controller, a three-degree-of-freedom rotating compressor blade model and a four-degree-of-freedom horizontally supported Jeffcott-rotor system controlled by a PPF controller. The amplitude equations are obtained by applying the integral equation method and the method of multiple scales. The stability analysis is achieved based on the Floquet theory together with Routh–Hurwitz criterion. Furthermore, we modified the iterative procedure of the integral equation method to make the analytic approximate solution more accurate. Finally, the analyses show that in most cases, the analytical solutions obtained by the integral equation method are more excellent agreement with the numerical solutions than the most commonly used method of multiple scales. Therefore, the integral equation method is worth popularizing to obtain the approximate analytical solutions of the multi-degree-of-freedom nonlinear vibration system.

Vibration Analysis Using Vibration Coupling Power Transmissibility of Two-degree-of-freedom System

In recent years, there has been a growing demand in the manufacturing field for more efficient product design and development, and for the development of more valuable products. To achieve this, it is necessary to have technologies that can achieve both diverse performance at an early stage of design, such as by eliminating "Temodori," as well as technologies that encourage the creation of new value. The models in “1D CAE” and “MBD” are useful as the technology. We have been studing in utilizing these technologies for designing highly efficient, Temodori-free designs through multi-performance optimization at an early stage of design, and creating new value in the process. In this paper, energy transfer characteristics（CLF）between two masses of two degree of freedom vibration system are newly derived. And the interpretation of vibration phenomena based on the CLF and its application to design are demonstrated. Finally, an example of vibration reduction of an automobile engine shake is discussed in comparison with the results of a reference.

Oscillations simulation of the vibration protection suspended load with a movable base

Reducing the vibration effects on a human operator of a construction, road or handling machine is an urgent task, since the vibration loads produced by the internal combustion engine, vehicle engine interacting with the support surface microrelief, as well as the machine working attachment, adversely affect the machine operator and increase the machine parts and assemblies wear. The operator may develop occupational diseases caused by vibrations. In this regard, the research objective is to develop the mathematical apparatus simulating the dynamic processes of the passive vibration protection systems oscillations at the design stage. Therefore, the problem developing a mathematical model that makes it possible to study the oscillations of a single degree of freedom passive vibration protection systems with any given static characteristic, was solved. On the basis of the dynamic equations of a single translational degree of freedom vibration system describing the vibration protection suspended operator’s seat, the mathematical model solving the differential equation of the system oscillations taking into account the spring loaded mass displacements damping and three-segment static characteristics of the vibration isolation mechanism with limiters was simulated using the package Simulink of MATLAB system. An example of using the developed mathematical model for modelling vibrations of a spring-loaded mass under the sinusoidal external influences is given. The preset vertical displacements of the machine base, i.e. its base chassis act as the external influences. The examples of the obtained simulation results, namely the vertical displacements and accelerations of the spring-loaded mass are presented. The maximum accelerations of the spring-loaded mass were found out to significantly increase, when the machine base forced displacements amplitude exceeds by half of the horizontal section of the quasi-zero static characteristic of the vibration protection suspension.

Vortex-induced vibration of elastically connected multiple circular cylinders in a side-by-side arrangement in steady flow

Vortex-induced vibration (VIV) of multiple circular cylinders elastically connected together in a side-by-side arrangement subject to steady flow is investigated numerically at a low Reynolds number of 150 and a mass ratio of 2. Simulations are conducted for two-, five- and ten-cylinder systems over a wide range of reduced velocities. The aim of the study is to identify the high-amplitude response range of the reduced velocity for the multiple degree of freedom vibration system and identify the difference between the responses of the single- and multiple-degree-of-freedom vibrations. Unlike the single cylinder case, distinct lock-in between the response frequency and any of the structural natural frequencies in a wide range of reduced velocity is not observed in the multiple-cylinder cases. Instead, the response frequency increases continuously with increasing reduced velocity. High response amplitudes are found when the response frequency is between the first and the highest modal frequencies. In a multiple-cylinder system, the single-mode response, where the vibration is dominated by one mode, can be only found in low reduced velocity range. In the single-mode branch, the dominance of a single mode shape in the response can be clearly identified except at the boundary reduced velocity between two modes. The maximum response amplitude occurs in the multiple-mode response and interaction between the vortices in the wake of the cylinders is strong when the response amplitudes are high.

Research on the dynamic characteristic of semi-active hydraulic damping strut

To reduce the vibration and shock of powertrain in the process of engine key on/off and vehicle in situ shift, a novel semi-active hydraulic damping strut is developed. The purpose of this paper is to study and discuss the dynamic stiffness model of the semi-active hydraulic damping strut. In this study, the dynamic characteristics of semi-active hydraulic damping strut were analyzed based on MTS 831 test rig first. Then, the dynamic stiffness model of semi-active hydraulic damping strut was established based on 2 degrees of freedom vibration system. In this research, a linear, fractional derivative and friction model was used to represent the nonlinear rubber bushing characteristic; the Maxwell model was used to describe the semi-active hydraulic damping strut body model; and the parameters of rubber bushing and semi-active hydraulic damping strut body were identified. The dynamic stiffness values were calculated with solenoid valve energized and not energized at amplitudes of 1 mm and 4 mm, which were consistent with experimental results in low-frequency range. Furthermore, the simplified dynamic stiffness model of the semi-active hydraulic damping strut was discussed, which showed that bushing can be ignored in low-frequency range. Then, the influence of equivalent spring stiffness, damping constant, and rubber bushing stiffness on the stiffness and damping capacity of the semi-active hydraulic damping strut were analyzed. Finally, the prototype of the semi-active hydraulic damping strut was developed and designed based on the vehicle in situ shift and engine key on/off situations, and experiments of the vehicle with and without semi-active hydraulic damping strut were carried out to verify its function.

Some significant improvements for interval process model and non-random vibration analysis method

Recently the authors proposed the interval process model for dynamic uncertainty quantification and based on this further developed a kind of non-probabilistic analysis method called ‘non-random vibration analysis method’ to deal with the important random vibration problems, in which the excitation and response are both given in the form of interval process rather than stochastic process. Since it has some attractive advantages such as easy to understand, convenient to use and small dependence on samples, the non-random vibration analysis method is expected to become an effective supplement to the traditional random vibration theory. In this paper, some significant improvements are made for the interval process model and the non-random vibration analysis method, making them not only more rigorous in theory but also more practical in engineering. Firstly, the definitions and relevant conceptions of interval process model are further standardized and improved, and in addition some new conceptions such as the interval process vector and the cross-covariance function matrix are complemented. Secondly, this paper proposes the important conceptions of limit and continuity of interval process, based on which the differential and integral of interval process are defined. Thirdly, the analytic formulation of dynamic response bounds is deduced for both of the linear single degree of freedom (SDOF) vibration system and the multiple degree of freedom (MDOF) vibration system, providing an important theoretical basis for non-random vibration analysis. Fourthly, this paper also gives the formulation and corresponding numerical methods of structural dynamic response bounds based on finite element method (FEM) for complex continuum problems, effectively enhancing the applicability of non-random vibration analysis in engineering. Finally, four numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Uncertain vibration analysis based on the conceptions of differential and integral of interval process

Recently, the authors proposed a new mathematical model called as the “interval process model” for quantifying uncertainty of time–varying parameters by making extension of the interval method into the time domain. In the interval process model, the imprecision of a time-varying parameter at arbitrary time point is described using an interval rather than the precise probability distribution, which makes the interval process model having some advantages over the traditional stochastic process in uncertainty quantification. Further, the authors proposed the important conceptions of limit, continuity, differential and integral of interval process, enriching the theory of interval process model. This paper applies the newly developed conceptions of differential and integral of interval process into the vibration analysis of mechanical structures or systems subjected to uncertain external excitations. By means of this application, the formulations of dynamic bounds of the velocity and acceleration responses are derived for the linear/multiple single degree of freedom (SDOF/MDOF) vibration systems subjected to dynamic uncertain excitations, which can provide some important reference information for reliability analysis and safety design of many practical mechanical structures or systems. The effectiveness of the proposed method are validated by investigating a spring-mass-damper system and a vehicle vibration problem.

Fatigue life due to non-Gaussian excitation – An analysis of the Fatigue Damage Spectrum using Higher Order Spectra

The fatigue damage spectrum (FDS) is an established tool for the assessment of the fatigue damage potential of any kind of vibration loading exerted on vibratory structures. It is especially useful for describing the fatigue damage potential of non-Gaussian random vibration signals, because for this type of signal no comprehensive, theoretically based statistical description is available. As the current definition of the FDS is based on the assumption of single degree of freedom vibration systems, the presented work analyses the resulting limitations for general vibration systems with multiple resonant frequencies. Improvements based on findings from higher-order spectra are proposed.

Brief Assessment for Natural Frequency of Two Degrees of Freedom Vibration Systems

Brief Assessment for Natural Frequency of Two Degrees of Freedom Vibration Systems

Resonance response interaction without internal resonance in vibratory energy harvesting

Abstract Resonance response interaction in nonlinear multiple degrees of freedom vibration systems has always involved internal resonance. In this article, bubble shaped response curve is uncovered without pre-designated resonances relationship that is a necessary condition for internal resonance. The objective of this article is twofold: first to explore the connection between the resonance response interaction and bubble shaped response curve that may appear in the forced response of a nonlinear magnetoelectric coupled system; and, second, to exploit resonance response interaction to enhance the vibratory energy harvesting bandwidth. A nonlinear magnetoelectric oscillator coupled linearly to an additional oscillator is used to demonstrate the phenomena and the enhanced performance. The lateral springs could produce the geometrical nonlinear stiffness and its stiffness to adjust the two resonance responses such that the bubble shaped response curve appears in the power-frequency response plot. The method of harmonic balance is well established method for the analysis of such the power-frequency response. The results are also validated by some numerical work. The power-frequency response curves are generated for distinct harvesting mass, manifesting the resonance response interaction could increase the bandwidth and output power. At last, the resonance response interaction designed energy harvesting from Gaussian white noise excitation is numerically addressed to illustrate positive effects of geometrical nonlinearity.

An approach to simplify the vehicle load in excavation-supporting structures design

In the design of underground excavation supporting structures, vehicle loads are generally considered as a constant distributed load of 20 kPa in the practical design method, and the equivalent soil layer method is also used. However, these methods do not take into account the dynamic properties of vehicle loads, and are relatively conservative in considering the influence of vehicle loads. Aiming at this problem, an improved method is presented to better solve this issue. Following the vehicle dynamics theory, vehicle loads are composed of static and dynamic loads; the quarter-truck model with two-degrees of freedom is introduced to simplify vehicle loads. According to motion differential equations of the two degrees of freedom vibration system, the dynamic vehicle load is deduced and expressed as a simple harmonic load. In an underground subway station, this simple harmonic load conversion method was introduced to convert the vehicle load, and was compared to the equivalent soil layer method and practical design method through numerical simulations. Some lateral displacement curves of supporting structures and settlements of road surfaces were obtained. The results revealed that the simple harmonic load conversion method can effectively reflect the influence of vehicle loads in the design of underground excavation-supporting structures..

Brief Assessment for Natural Frequency of Two Degrees of Freedom Vibration Systems

Brief Assessment for Natural Frequency of Two Degrees of Freedom Vibration Systems

Singular spectrum analysis in multi-degree of freedom vibration system

Singular spectrum analysis in multi-degree of freedom vibration system

Helical gear wear monitoring: Modelling and experimental validation

Gear tooth surface wear is a common failure mode. It occurs over relatively long periods of service nonetheless, it degrades operating efficiency and leads to other major failures such as excessive tooth removal and catastrophic breakage. To develop accurate wear detection and diagnosis approaches at the early phase of the wear, this paper examines the gear dynamic responses from both experimental and numerical studies with increasing extents of wear on tooth contact surfaces. An experimental test facility comprising of a back-to-back two-stage helical gearbox arrangement was used in a run-to-failure test, in which variable sinusoidal and step increment loads along with variable speeds were applied and gear wear was allowed to progress naturally. A comprehensive dynamic model was also developed to study the influence of surface wear on gear dynamic response, with the inclusion of time-varying stiffness and tooth friction based on elasto-hydrodynamic lubrication (EHL) principles. The model consists of an 18 degree of freedom (DOF) vibration system, which includes the effects of the supporting bearings, driving motor and loading system. It also couples the transverse and torsional motions resulting from time-varying friction forces, time varying mesh stiffness and the excitation of different wear severities. Vibration signatures due to tooth wear severity and frictional excitations were acquired for the parameter determination and the validation of the model with the experimental results. The experimental test and numerical model results show clearly correlated behaviour, over different gear sizes and geometries. The spectral peaks at the meshing frequency components along with their sidebands were used to examine the response patterns due to wear. The paper concludes that the mesh vibration amplitudes of the second and third harmonics as well as the sideband components increase considerably with the extent of wear and hence these can be used as effective features for fault detection and diagnosis.

Parametric study of pendulum type dynamic vibration absorber for controlling vibration of a two DOF structure

Passive dynamic vibration absorbers (DVAs) are often used to suppress the excessive vibration of a large structure due to their simple construction and low maintenance cost compared to other vibration control techniques. A new type of passive DVA consists of two pendulums connected with spring and dashpot element is investigated. This research evaluated the performance of the DVA in reducing the vibration response of a two degree of freedom shear structure. A model for the two DOF vibration system with the absorber is developed. The nominal absorber parameters are calculated using a Genetic Algorithm(GA) procedure. A parametric study is performed to evaluate the effect of each absorber parameter on performance. The simulation results show that the optimum condition for the absorber frequencies and damping ratios is mainly affected by pendulum length, mass, and the damping coefficient of the pendulum´s hinge joint. An experimental model validates the theoretical results. The simulation and experimental results show that the proposed technique is able be used as an effective alternative solution for reducing the vibration response of a multi degree of freedom vibration system.

Theoretical Investigation on the Dynamic Characteristics of One Degree of Freedom Vibration System Equipped with Inerter of Variable Inertance

Theoretical Investigation on the Dynamic Characteristics of One Degree of Freedom Vibration System Equipped with Inerter of Variable Inertance

モータと質点を用いた一自由度振動系による免震と制振

This paper describes a cart-type one degree of freedom vibration system using a DC motor and electrical circuit instead of a spring. The stiffness and the damping ratio of the proposed method can be tuned using an LR circuit and an LRC circuit. This proposed method solves the problem of occurring the deflection and buckling of a spring, and occupying the space with the spring. This paper describes the method that uses the proposed vibration systems as a seismic isolator and dynamic vibration absorber. The governing equations of the analytical models were formulated, and the optimum values of the proposed methods were derived. The effectiveness of the proposed methods and theoretical analysis were verified through simulations and experiments.

Proposal of a real-time data processing method for structural health monitoring (Examination of accuracy of proposed method due to differences in vibration environment)

A smart sensor is required if we are to popularize health monitoring using the vibration of a structure. It is used to evaluate the vibration characteristics of the structure. To detect the variance in the vibration characteristics from the vibration data of a structure, a time-frequency analysis technique that can detect the change of the frequency components of the vibration data over time is needed. In a previous report, we have proposed a real-time data processing method with arrayed narrowband band-pass filters based on a single degree of freedom vibration system for the evaluation of structural vibration characteristics. This method is applicable to processors with low computation loads and memory capacities. In this paper, the relation between the parameter setting and the accuracy of the evaluation result when the proposed method is applied to several vibrational environments is examined. Concretely, accuracy and the stability of the estimated result of the natural frequency of the structure were examined when a steady state noise is included to the response of the structure. Furthermore, a similar examination was conducted for the measurement data of the microtremor response of the pier model where the unsteady input is applied. Moreover, to obtain the criterion of the evaluation accuracy of the proposed method, the estimate was compared with AR model. As a result, it is clarified how each parameter of the proposed method was related to the estimation accuracy of the natural frequency. The proposal method showed that the influence on the noise was less than the AR model.