Equivalent Stiffness Coefficient Research Articles

Primary resonance of Duffing oscillator with fractional-order derivative

In this paper the primary resonance of Duffing oscillator with fractional-order derivative is researched by the averaging method. At first the approximately analytical solution and the amplitude–frequency equation are obtained. Additionally, the effect of the fractional-order derivative on the system dynamics is analyzed, and it is found that the fractional-order derivative could affect not only the viscous damping, but also the linear stiffness, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. This conclusion is remarkably different from the existing research results about nonlinear system with fractional-order derivative. Moreover, the comparisons of the amplitude–frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two parameters of the fractional-order derivative, i.e. the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated, which are different from the traditional integer-order Duffing oscillator.

Optimal Parameters Analysis of Dynamical Vibration Absorption Lathe Tool with Large Length to Diameter Ratio

The multi flexible body dynamical simulation model of dynamical vibration absorption lathe tool with large length to diameter ratio is built up according to the actual experiment lathe tool by using the software ADAMS and ANSYS. The experiment data which are consistent with the simulation result verify the correctness of the multi flexible body dynamical model. Aiming at reducing the peak value of system frequency response, the dynamical vibration absorption system is optimized. The optimized frequency curve shows that the system frequency response is improved obviously. On this basis the influence of the mass of heavy turning body on the vibration system is discussed. Also, the relationship of the mass of heavy turning body and the optimal system parameters including equivalent stiffness coefficient and equivalent viscous damping coefficient is studied. This offers the design considerations about the structure of the dynamical vibration absorption lathe tool with large length to diameter ratio.

Torsion/cantilever-based MEMS bistable mechanisms with different support configurations: structure design and comparison

Three types of torsion/cantilever-based MEMS bistable mechanisms (BMs) with different support configurations have been constructed, modeled and experimented. For the support configuration, there is a crisscross-shaped, a ring-shaped and a diamond-shaped support beam, respectively. The proposed MEMS BMs consist of a free–free torsion-based cantilever which forms a symmetrical rocker lever. The free–free cantilever is suspended by a support skeleton which in turn is attached to a torsion cantilever. A permanent magnet is attached beside for holding the closed state with a permalloy soft magnetic circuit. The different special support configurations account for a low torsional compliance with the overhanging beams. In order to deduce the equivalent stiffness coefficient of BM systems, mechanical modeling of three types of torsion/cantilever-based MEMS BMs was performed by the classical beam theorem. Meanwhile, the magnetostatic latching force was also deduced by the Maxwell electromagnetism theory. The performances of these MEMS BMs have been compared by the evaluation of static deformation variations, equivalent stiffness coefficients and dynamical switching characterizations. Finally, mechanical performance was characterized by atomic force microscopy, combined with a Nanoindentation Tester. In addition, bistabilities of the MEMS BMs were proved by theoretical analysis as well as experimental results. Among these BMs, the ring-shaped MEMS BM is extremely prone to deflect due to relatively low stiffness compared with other types. The torsion/cantilever-based MEMS BMs have potential application in the field of latching relays with low power consumption.

Noise and Vibration Analysis of Compressor for Air-Conditioning Machine Equipped with a Concentrated Winding Motor with a Press-Fitted Segment Core

The compressor which has a concentrated winding motor with the press-fitted segment core was developed for the first time domestically. It was necessary to grasp the dynamic characteristics of the press-fitted segment core to establish the low noise design technique of the developed compressor. This paper proposes a method that matches a calculated natural frequency of (2, 2) mode to the measured natural frequency based on the theoretical formula of the natural frequency of the ring, and presumes the equivalent stiffness coefficient needed for the vibration analysis. Using the method, it was clarified that the equivalent stiffness coefficient of press-fitted segment core changed from 206GPa to 165GPa by shrink-fit. Afterwards, the noise analysis of the remodeled compressor was carried out by using the equivalent stiffness coefficient. The calculated results agree well with the measured results. Therefore, it was clarified that the noise characteristics of a compressor which has a concentrated winding motor with the press-fitted segment core could be predicted.

108 空調用圧縮機の振動騒音解析(連成系・振動騒音,OS-9 モード解析とその応用関連技術,総合テーマ「伝統を,未来へ!」)

Mitsubishi Electric Corporation developed a compressor equipped with a concentrated winding motor that adopted a press-fitted segment core for the first time domestically. It was necessary to grasp the dynamic characteristics of the press-fitted segment core to establish the low noise design technique of the developed compressor. This paper proposes a method that a calculated natural frequency of (2, 2) mode was matched to the measured natural frequency based on the theoretical formula of the natural frequency of the ring to presume the equivalent stiffness coefficient needed for the vibration analysis. Afterwards, the noise analysis of the structure-changing model was carried out by using the equivalent stiffness coefficient. The calculated results agree well with the measured results. Therefore, it was clarified that the noise characteristics of a compressor equipped with a concentrated winding motor that adopted the press-fitted segment core could be predicted.

Identification of time-varying hysteretic structures using wavelet multiresolution analysis

In this paper, a wavelet multiresolution technique is proposed to identify time-varying properties of hysteretic structures. It is well known that arbitrary transient functions can be effectively and accurately approximated using wavelet multiresolution expansions due to wavelet's good time–frequency localization property. By decomposing the time-varying parameters with wavelet multiresolution expansion, a time-varying parametric identification problem can be transformed into a time-invariant non-parametric one. The identification in the time-invariant wavelet multiresolution domain can be achieved by choosing a wavelet basis function and performing a suitable parameter estimation technique. Since wavelet representation of arbitrary signal uses only a small number of terms, the orthogonal forward regression algorithm can be adopted for significant term selection and parameter estimation. Single and multiple degrees of freedom Bouc–Wen hysteretic structures with gradual and abrupt varying properties are used to illustrate the proposed approach. Results show that the wavelet multiresolution technique can identify and track the time-varying hysteretic parameters quite accurately. The effect of measurement noise is also studied. It is found that the presence of noise would affect more on the damping ratios and the Bouc–Wen parameters but less on the equivalent stiffness coefficients.

English

Hexagonal lattice pattern formed by helical and circumferential ribs is the most common among different possible lattice patterns. An energy-based smeared stiffener model (SSM) is developed to obtain equivalent stiffness coefficients of a composite lattice cylindrical shell with such hexagonal lattice patterns. Using the equivalent stiffness coefficients, Ritz buckling analysis was carried out. Extensive finite element modelling covering different representative sizes have been carried out. SSM is validated by comparing the estimated buckling loads. Variation of material properties of rib unidirectional composites from those of normal unidirectional composites is accounted for in the energy formulations. Defence Science Journal, 2009, 59(3), pp.230-238 , DOI:http://dx.doi.org/10.14429/dsj.59.1516

Stochastic averaging of energy envelope of Preisach hysteretic systems

A new stochastic averaging technique for analyzing the response of a single-degree-of-freedom Preisach hysteretic system with nonlocal memory under stationary Gaussian stochastic excitation is proposed. An equivalent nonhysteretic nonlinear system with amplitude-envelope-dependent damping and stiffness is firstly obtained from the given system by using the generalized harmonic balance technique. The relationship between the amplitude envelope and the energy envelope is then established, and the equivalent damping and stiffness coefficients are expressed as functions of the energy envelope. The available range of the yielding force of the system is extended and also the strong nonlinear stiffness of the system is incorporated so as to improve the response prediction. Finally, an averaged Itô stochastic differential equation for the energy envelope of the system as one-dimensional diffusion process is derived by using the stochastic averaging method of energy envelope, and the Fokker–Planck–Kolmogorov equation associated with the averaged Itô equation is solved to obtain stationary probability densities of the energy envelope and amplitude envelope. The approximate solutions are validated by using the Monte Carlo simulation.

Analysis on Dynamic Performance of Hydrodynamic Tilting-Pad Gas Bearings Using Partial Derivative Method

Tilting-pad gas bearings are widely used in high-speed rotating machines due to their inherent stability characteristics. This paper advances the analytical method for prediction of the dynamic performances of tilting-pad gas bearings. The main advantage of the analytical method is that the complete set of dynamic coefficients of tilting-pad gas bearings can be obtained. The predictions show that the perturbation frequency has the strong effects on the dynamic coefficients of gas bearings. In general, at lower perturbation frequency, the equivalent direct stiffness coefficients increase with frequency, whereas equivalent direct damping coefficients dramatically reduce. For higher perturbation frequency, the dynamic coefficients are nearly independent of the frequency. Moreover, the equivalent dynamic coefficients of four-pad tilting-pad gas bearing obtained by the method in this paper are in good agreement with those obtained by Zhu and San Andres [(2007), “Rotordynamic Performance of Flexure Pivot Hydrostatic Gas Bearings for Oil-Free Turbomachinery,” ASME J. Eng. Gas Turbines Power, 129(4), pp. 1020–1027] in the published paper. The results validate the feasibility of the method presented in this paper in calculating the dynamic coefficients of gas-lubricated tilting-pad bearings.

Nonlinear stochastic optimal control of Preisach hysteretic systems

The nonlinear stochastic optimal control of Preisach hysteretic systems is studied, and the control procedure is illustrated with an example of the single-degree-of-freedom Preisach system. The Preisach hysteretic system subjected to a stochastic excitation is first replaced by an equivalent non-hysteretic nonlinear stochastic system with displacement-amplitude-dependent damping and stiffness, by using the generalized harmonic balance technique. Then, the relationship between the displacement amplitude and total system energy is established, and the equivalent damping and stiffness coefficients are expressed as functions of the system energy. The averaged Itô stochastic differential equation for the system energy as one-dimensional controlled diffusion process, is derived by using the stochastic averaging method of energy envelope. For the semi-infinite time-interval ergodic control, the dynamical programming equation is obtained based on the stochastic dynamical programming principle, and is solved to yield the optimal control force. Finally, the Fokker–Planck–Kolmogorov equation associated with the averaged Itô equation is established, and the stationary probability density of the system energy is obtained, from which the variances of the controlled system response and the optimal control force are predicted and the control efficacy is evaluated. Numerical results show that the proposed control strategy for Preisach hysteretic systems is very effective and efficient.

Rheological–dynamical analogy: Prediction of damping parameters of hysteresis damper

This research aims to predict the damping parameters of hysteresis damper based on an analytical rheological–dynamical (RDA) visco-elasto-plastic solution of one-dimensional longitudinal continuous vibrations of a bar. A visco-elasto-plastic bar or damper is an energy dissipation device. An attempt is made to estimate quantitatively the influence of material physical parameters of materials on the damping ratio in both the linear visco-elastic analysis and the nonlinear visco-elasto-plastic analysis of damper subjected to external vibration forces. Two types of damping are considered: viscous damping in the case of linear analysis, defined as stiffness and/or mass proportional and, in the case of nonlinear analysis, hysteresis damping caused by inelastic deformations of damper. Owing to the visco-elastic nature of the materials of the damper and the frequency dependence of the viscous damping ratio ξ, it is useful to consider separately the situations arising when ξ is positive (the system is stable) and when it is negative. A negative damping ratio means that the complementary solution of the response would not die away (the system is unstable because of factor e ξ · ω · t ). In the case of nonlinear analysis, the force–displacement relation is nonlinear, so it is very difficult to predict the actual damping and stiffness coefficients, even if the force–displacement characteristic is simply perfect elasto-plastic. Using the RDA method, which takes into account the rate of release of visco-elasto-plastic energy of the dissipation devices; nonlinear behaviors are linearized, enabling to obtain the equivalent damping and stiffness coefficients and the effective period for the damper.

Response Spectrum Method for Nonlinear Surface Ground Analysis

An equivalent linear response spectrum approach is proposed for nonlinear surface ground analysis. In contrast to the conventional method which deals with only a single ground motion for equivalent linearization of soil properties, it is argued in this paper that a design response spectrum defined at the upper level (bottom of the surface ground) of an engineering bedrock can be handled as the target design earthquake. The effective shear strain in each soil layer is evaluated by means of a response spectrum method in which the mean peak shear strain is computed by a response spectrum mode-superposition procedure. The soil hysteretic damping is substituted by the corresponding viscous damping. The stiffness coefficient and damping ratio of each soil layer are obtained iteratively from the nonlinear relation of stiffness reduction factors and damping ratios with respect to the strain level. After the evaluation of the equivalent stiffness coefficient and damping ratio of every soil layer, the surface ground response can be described via the one-dimensional wave propagation theory. The reliability and accuracy of the proposed analysis method is examined through comparison with the results obtained by the conventional method (represented by the SHAKE program) for many simulated spectrum-compatible ground motions.

Describing Functions For Effective Stiffness and Effective Damping of Hystersis Structures

For a hysteresis structure with energy dissipation devices, the force-displacement relation is nonlinear such that it is very difficult to evaluate the actual damping and stiffness coefficients, even if the forcedisplacement characteristic is simply perfect elasto-plastic. With the describing function method, we can linearize the nonlinear behavior of the energy dissipation devices and then obtain the equivalent damping and stiffness coefficients; In turn, the effective period and equivalent damping ratio can be attained. It is stressed that with this approach, the imaginary part of the describing function is just the energy dissipation term, which corresponds to the conventional hystersis damping derived by the energy method. Simulation results confirm the effectiveness of this proposed method.

Dynamic Stiffness Formulation and Its Application for a Combined Beam and a Two Degree-of-Freedom System

This paper is concerned with the dynamic stiffness formulation and its application for a Bernoulli-Euler beam carrying a two degree-of-freedom spring-mass system. The effect of a two degree-of-freedom system kinematically connected to the beam is represented exactly by replacing it with equivalent stiffness coefficients, which are added to the appropriate stiffness coefficients of the bare beam. Numerical examples whose results are obtained by applying the Wittrick-Williams algorithm to the total dynamic stiffness matrix are given and compared with published results. Applications of the theory include the free vibration analysis of frameworks carrying two degree-of-freedom spring-mass systems.

Acoustical validation of the rheological models for a structural bond

The purpose of this study is to characterize by ultrasound a bonded structure metal/adhesive/metal. At first, we investigate the guided modes of the structure for the adhesive layer in its entirety; this is the exact model. Secondly, we assume that the adhesive layer is described by one geometrical interface, with a superficial distribution of longitudinal and transversal springs with or without mass; this is the rheological model. A comparison of the guided modes for the two models allows to determinate the validity limits for the rheological modelling and define the equivalent stiffness coefficients and the mass as a function of the frequency. Some cutoff-frequencies are better evaluated when the springs mass is taken into account.

Comfort and Stability of Vehicles with Stochastic Excitation

AbstractAn efficient method to determine comfort and road holding stability of vehicles is discussed. By using the frequency spectrum of the vertical excitation, modeling of a stochastic road becomes superfluous. In the case of a linearized model, the frequency response yields the acceleration of the chassis and the dynamic wheel forces. They are used as measures for comfort and stability, if applied to the spectrum of the road excitation. In order to include buffer effects with nonlinear characteristics, an approximation method is discussed to determine the equivalent stiffness coefficient of a, linear spring.

Sparse Plate-Trusses

Strategies of reducing the number of bars of a sparse plate-truss, while increasing structural height and improving structural performance are considered. Four strategies are presented by considering seven plate-trusses. The morphological and structural properties of each of the plate trusses with reduced number of bars are discussed. An approximate and quick method to find the stiffness coefficients of a periodic unit of the plate-truss is presented and an equivalent single stiffness coefficient is introduced. The stiffness coefficients enable the designer to understand the existing relationship between the morphology of a truss and its structural ability. They are very useful for a comparison of structural ability of different sparse-trusses and permit a determination of the relative structural efficiency of each strategy of reducing the bar density.

THE FORCED VIBRATION OF A BEAM WITH VISCOELASTIC BOUNDARY SUPPORTS

A method of analysis for the forced vibration of a beam with viscoelastic boundary supports is proposed based on complex normal mode analysis. The viscoelastic support regions are first described in terms of equivalent complex stiffness coefficients, and then using the complex modes of the beam system with complex stiffness at the boundary points, the equations of motion are completely uncoupled. The modal equation has precisely the same form as the equation of a structurally damped single-degree-of-freedom system. Effects of the viscoelastic supports on the forced vibration response are discussed as an example.

Stochastic analysis of nonlinear vehicle systems using a generalized harmonic linearization technique

The discrete harmonic linearization method which is applicable to stochastic systems could not provide an equivalent stiffness coefficient for a nonlinear restoring element. In this paper, this technique is generalized to obtain linear representations of both nonlinear restoring and damping elements, based on a principle of energy similarity of dynamic elements. A numerical iterative procedure is presented to compute the local linear coefficients of nonlinear dynamic elements. A nonlinear system is then represented by a set of its complex frequency response function matrices, as functions of the excitation frequency. Stochastic analysis of general multi-DOF nonlinear vehicle systems is established in terms of response PSD characteristics.

鋼構造骨組の弾塑性域における変形と安定性に関する考察 (第 I 報)

A study is made herein for the evaluation of the criical horizontal load that will cause side way buckling of an inelastic portal frame hinged at its base, subjected to symmetrical vertical loads concentrated on the column only. The equivalent stiffness coefficient (β), which is based on the curvature of unit length, is used in pursuing the load deflection relation. Several numerical examples are shown herein, that obtained by using digital computer.