Forced large amplitude periodic vibrations of non-linear Mathieu resonators for microgyroscope applications

This paper describes a comprehensive non-linear multiphysics model based on the Euler–Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parametric analyses are presented in order to investigate the effect of the proof mass frequency on the bifurcation topology. The model shows that the optimal sensitivity is reached for resonant microgyroscopes designed with sensing frequency four times faster than the actuation one.     

Nonlinear vibrations of a thin ring

Gorkii University. Translated from Prikladnaya Mekhanika, Vol. 25, No. 3, pp. 76–81, March, 1989.     

Control of cross-flow-induced vibrations of square cylinders using linear and nonlinear delayed feedbacks

We investigate the effectiveness of linear and nonlinear time-delay feedback controls to suppress high amplitude oscillations of an elastically mounted square cylinder undergoing galloping oscillations. A representative model that couples the transverse displacement and the aerodynamic force is used. The quasi-steady approximation is used to model the galloping force. A linear analysis is performed to investigate the effect of linear time-delay controls on the onset speed of galloping and natural frequencies. It is demonstrated that a linear time-delay control can be used to delay the onset speed of galloping. The normal form of the Hopf bifurcation is then derived to characterize the type of the instability (supercritical or subcritical) and to determine the effects of the linear and nonlinear time-delay parameters on their outputs near the bifurcation. The results show that the nonlinear time-delay control can be efficiently implemented to significantly reduce the galloping amplitude and suppress any dangerous behavior by converting any subcritical Hopf bifurcation into a supercritical one.     

A nonlinear subspace-prediction error method for identification of nonlinear vibrating structures

The industrial structural systems always contain various kinds of nonlinear factors. Recently, a number of new approaches have been proposed to identify those nonlinear structures. One of the promising methods is the nonlinear subspace identification method (NSIM). The NSIM is derived from the principals of the stochastic subspace identification method (SSIM) and the internal feedback formulation. First, the nonlinearities in the system are regarded as internal feedback forces to its underlying linear dynamic system. The linear and nonlinear components of the identified system can be decoupled. Second, the SSIM is employed to identify the nonlinear coefficients and the frequency response functions of the underlying linear system. A typical SSIM always consists of two steps. The first step makes a projection of certain subspaces generated from the data to identify the extended observability matrix. The second one is to estimate the system matrices from the identified observability matrix. Since the calculated process of the NSIM is non-iterative and this method poses no additional problems on the part of parameterization, the NSIM becomes a promising approach to identify nonlinear structural systems. However, the result generated by the NSIM has its deficiency. One of the drawbacks is that the identified results calculated by the NSIM are not the optimal solutions which reduce the identified accuracy. In this study, a new time-domain subspace method, namely the nonlinear subspace-prediction error method (NSPEM), is proposed to improve the identified accuracy of nonlinear systems. In the improved version of the NSIM, the prediction error method (PEM) is used to reestimate those estimated coefficient matrices of the state-space model after the application of NSIM. With the help of the PEM, the identified results obtained by the NSPEM can truly become the optimal solution in the least square sense. Two numerical examples with local nonlinearities are provided to illustrate the effectiveness and accuracy of the proposed algorithm, showing advantages with respect to the NSIM in a noise environment.     

Two nonlinear models of a transversely vibrating string

Modeling transverse vibration of nonlinear strings is investigated via numerical solutions of partial-differential equations and an integro-partial-differential equation. By averaging the tension along the deflected string, the classic nonlinear model of a transversely vibrating string, Kirchhoff’s equation, is derived from another nonlinear model, a partial-differential equation. The partial-differential equation is obtained via neglecting longitudinal terms in a governing equation for coupled planar vibration. The finite difference schemes are developed to solve numerically those equations. An index is proposed to compare the transverse responses calculated from the two models with the transverse component calculated from the coupled equation. A steel string and a rubber string are treated as examples to demonstrate the differences between the two models of transverse vibration and their deviation from the full model of coupled vibration. The numerical results indicate that the differences increase with the amplitude of vibration. Both models yield satisfactory results of almost the same precision for vibration of small amplitudes. For large amplitudes, the Kirchhoff equation gives better results.     

Flexural vibrations of a nonlinear elastic plate

International Applied Mechanics -     

Analysis of forced vibrations by nonlinear modes

The combination of Rausher method and nonlinear modes is suggested to analyze the forced vibrations of nonlinear discrete systems. The basis of the Rausher method is iterative procedure. In this case, the analysis of a nonautonomous dynamical system reduces to the multiple solutions of the autonomous ones. As an example, the forced vibrations of shallow arch close to equilibrium position are considered in this paper. The results of the analysis are shown on the frequency response.     

Derivation of nonlinear damping from viscoelasticity in case of nonlinear vibrations

Nonlinear Dynamics - Experiments show a strong increase in damping with the vibration amplitude during nonlinear vibrations of beams, plates and shells. This is observed for large size structures...     

电容式MEMS环形振动陀螺结构设计及加工

微机械环形振动陀螺仪采用四波腹工作原理,具有精度高、抗冲击性能好等优点。通常情况下为了提高电容值和信噪比,环形结构会采用高深宽比方案,因此带来的footing效应直接影响了结构加工的成品率。本文在设计环形结构的基础上提出了一种在硅下表面溅射一层Al金属层的方法,能够避免footing效应的发生。实验结果表明,该方法有效提高了结构加工精度。同时,为了验证所设计结构的正确性,对加工出的结构进行了扫频测试,结构驱动模态谐振频率与设计值相差仅0.13%,并在此基础上搭建了测控系统,进一步进行了静态实验,结果表明其零偏稳定性指标为101(°)/h证明了设计和加工的可行性。     

Vehicle vibrating on a soft compacting soil half-space: Ground vibrations, terrain damage, and vehicle vibrations

The point of departure of the present work may be either an interest in vehicle vibrations themselves, or in ground vibrations and terrain damage due to vehicles traveling off-road. The vibrations of a vehicle traversing dry, soft terrain, which is either rough or undulating, may be significantly modified by the dynamic interaction of the vehicle with the soil, particularly due to losses of energy by soil compaction and as elastic waves. The present work provides a prediction methodology for both vehicle and soil vibrations, accounting for the effects mentioned above. An expedient linear method is compared to a rheologically-based non-linear method. In the linear method, the soil compaction is incorporated as a loss factor in the dynamic stiffness of the otherwise elastic half-space; the imaginary part of that dynamic stiffness already includes the effects of wave damping. The non-linear model treats the compaction using a general rheological model for soils exhibiting both viscous and thixotropic effects, and requires iterative solution. A key feature of the latter model is the hypothesis that the stress distribution may be approximately regarded as quasi-static when calculating compaction losses; that approximation is expected to hold at low frequencies, since the P-wavelength in the soil is then much greater than the dimensions of the zone in which most compaction occurs. The methods predict that the soil compaction and excited ground vibrations have maxima at the vehicle bounce and hop resonances, and at high frequencies at which the Rayleigh wavelength approaches the order of the contact patch diameter. Moreover, sufficiently soft, compactable soils, but fully realizable in nature, control the vehicle response at the hop resonance, and possibly also at the bounce resonance.     

On interaction of vibrating beam with essentially nonlinear absorber

The paper deals with interaction of elastic beam with essentially nonlinear vibration absorber. Forced vibrations of the beam are described by 2DOF model. We treat the motions favorable for vibration absorption as nonlinear modes in a configuration space and compute them by a modification of Rausher method. Stability of these modes is analyzed numerically with the help of the Floquet theory.     

Pervasive nonlinear vibrations due to rod-obstacle contact

Nonlinear Dynamics - Simple mechanical contact between a deformable body and a rigid obstacle can lead to rich nonlinear behavior even when the equations governing the body’s motion are...     

Studying the nonlinear vibrations of statically compressed plates

The paper outlines a method for studying the vibrations of plates of complex geometry subjected to in-plane loading. The method is based on the R-function and variational methods. It is used to plot frequency response of plates with complex geometry and different boundary conditions __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 124–132, December, 2006.     

Stability of vibrations of finite amplitude in an electron-ion ring

The article discusses transverse vibrations of finite amplitude in an electron-ion ring. Far from the region of linear resonances, an equation is obtained for slowly varying amplitudes, and the conditions are found for the excitation of instability of the negative pressure type. Near the lower boundary of the region of linear instability, the conditions are found under which nonlinearity breaks down the stability of the vibrations with finite amplitudes.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 170–172, May–June, 1974.The author is indebted to B. V. Chirikov for his evaluation of the results of the work.     

Natural frequencies and modes of out-of-plane vibrations a fixed ring

The natural frequencies and modes of a flexible elastic ring fixed at one point are determined numerically by solving an eigenvalue problem for the differential operator of the equation of its motion. The ring models a large circular antenna that slowly expands under zero gravity. Bending and torsional vibrations in a plane perpendicular to the antenna plane are studied