The accuracy of the generalized-α method in the time integration of non-linear single- and two-DOF forced systems

This paper deals with the accuracy of a numerical time integration scheme, the implicit Generalized-α method, when applied near resonant conditions in periodic steady-state vibrations of elastic linear and non-linear systems. In order to evaluate errors, analytical solutions of frequency response functions (FRFs) are determined by using the Harmonic Balance method in single- and two-degrees-of-freedom viscously damped systems, where the non-linearity is introduced through hardening Duffing oscillators. Successively, the Generalized-α method is implemented in conjunction with the Harmonic Balance method to trace numerical solutions of FRFs. It is shown that the effective resulting algorithm, the Algorithmic Harmonic Balance-ρ_∞ method, can define non-linear FRFs and allows the errors exhibited by the integration scheme near resonance in terms of frequency location and amplitude of the resonant peak to be quantified. The accuracy estimates demonstrate the robustness of the Generalized-α method also in the forced case and confirm its capability to reproduce amplitudes at resonance in the low frequency range and damp out them in the high frequency range.     

Qualitative analysis of classes of motion for multiresonant systems II. A geometrical method

Summary. Classes of motion of general multiresonant systems are derived through a geometrical algorithm based on a set representation. First, the elementary classes existing under simple resonance conditions are evaluated; rules governing the interaction between elementary classes belonging to different resonance conditions are then drawn up as applications of a unique theorem. Illustrative examples are given. The method also permits a hierarchical ordering of the amplitudes of the resonant modes, according to their participation in the classes; it also makes it possible to ascertain in advance the existence of a standard form for the amplitude modulation equations. The stability analysis of incomplete steady solutions is then addressed. Three classes of perturbation are distinguished, namely: in-class perturbations, out-of-class resonant perturbations, and out-of-class nonresonant perturbations. The structure of the Jacobian variational matrix is studied. The Jacobian matrix is shown to comprise three diagonal blocks associated with the three perturbation classes, so that the stability equations are uncoupled. Further possible uncouplings of one of the blocks are analyzed in relation to some of the geometrical properties of the classes.     

直齿圆柱齿轮啮合耦合振动系统参数振动研究

齿轮副啮合耦合振动系统是一个多自由度参数振动系统。该文考虑啮合刚度时变性,传动轴、轴承和箱体等支撑刚度和阻尼,轮齿传动误差以及输入转矩非线性等因素的影响,建立了直齿圆柱齿轮副啮合耦合动力学模型。将动力学方程转换到正则模态下,利用多尺度法对其进行动力稳定性分析,推导出主共振和亚谐共振条件下系统的组合共振频率以及稳定性边界。数值模拟系统非参数和参数共振响应,与摄动法结果吻合较好。结果表明：当轮齿啮合频率接近和型共振频率时,系统发生参数共振,存在着不收敛的无界解。系统的非参数共振响应为概周期响应,包含着多种组合频率成分。     

Saturation in asymmetric structures under internal resonance

In this paper, a novel model for a single-storey irregular-plan building, subjected to earthquake and harmonic translational excitation, is presented to consider non-linear inertial coupling terms, which are ignored in conventional linear models. Furthermore, the effect of internal resonance is investigated. The governing equations of motion for a model of this type of structure are achieved in a non-inertial rotational system of reference, attached to the centre of mass of the rigid diaphragm of its floor. Non-linearities are caused by non-linear inertial coupled lateral and torsional responses. Using the method of multiple scales, the non-linear equations are reduced to approximate linear equations, and subsequently the dynamic stability of the steady-state responses near a two-to-one ratio of internal resonant conditions is studied. The difference between the responses of the proposed non-linear model and the conventional linear one is shown in time history and frequency domain analyses. By inspecting the effects of amplitude and frequency detuning parameters in the responses, some non-linear phenomena, such as saturation, hysteretic and jumping, are observed.     

Nonlinear free vibrations of thin-walled beams in torsion

Nonlinear torsional vibrations of thin-walled beams exhibiting primary and secondary warpings are investigated. The coupled nonlinear torsional–axial equations of motion are considered. Ignoring the axial inertia term leads to a differential equation of motion in terms of angle of twist. Two sets of torsional boundary conditions, that is, clamped–clamped and clamped-free boundary conditions are considered. The governing partial differential equation of motion is discretized and transformed into a set of ordinary differential equations of motion using Galerkin’s method. Then, the method of multiple scales is used to solve the time domain equations and derive the equations governing the modulation of the amplitudes and phases of the vibration modes. The obtained results are compared with the available results in the literature that are obtained from boundary element and finite element methods, which reveals an excellent agreement between different solution methodologies. Finally, the internal resonance and the stability of coupled and uncoupled nonlinear modes are investigated. This study can be a preliminary step in the understanding of complex dynamics of such systems in internal resonance excited by external resonant excitations.     

Transport Properties of Two-dimensional Snowballs Below the Surface of 4He-II Under Rotation

Kelvin-waves play an important role for the dissipation of quantum turbulence at low temperatures. Here the plasma resonance of two-dimensional (2D) snowballs trapped below the surface of rotating superfluid ⁴He are measured for the first time in order to examine whether 2D snowballs could be a new probe for study of vortex dynamics. Below 200 mK, a positive shift of the resonant frequency f ₁ and linewidth broadening are observed as small variations in the absorption spectra under rotation. Both f ₁ and the linewidth Δf increase linearly with the rotation speed, and the slopes of f ₁ and Δf against the rotation speed have no temperature dependence. The increase of Δf suggests that an additional dissipation is caused by the coupling between the snowballs and vortices. We provide a qualitative explanation for the linear increase of Δf in the context of Kelvin-waves excited by the motion of snowballs.     

Defect-Enhanced Polarization Switching in the Improper Ferroelectric LuFeO3

Results of switching behavior of the improper ferroelectric LuFeO₃ are presented. Using a model set of films prepared under controlled chemical and growth-rate conditions, it is shown that defects can reduce the quasi-static switching voltage by up to 40% in qualitative agreement with first-principles calculations. Switching studies show that the coercive field has a stronger frequency dispersion for the improper ferroelectrics compared to a proper ferroelectric such as PbTiO₃. It is concluded that the primary structural order parameter controls the switching dynamics of such improper ferroelectrics.     

Assessment of explicit and semi‐explicit classes of model‐based algorithms for direct integration in structural dynamics

The ‘model‐based’ algorithms available in the literature are primarily developed for the direct integration of the equations of motion for hybrid simulation in earthquake engineering, an experimental method where the system response is simulated by dividing it into a physical and an analytical domain. The term ‘model‐based’ indicates that the algorithmic parameters are functions of the complete model of the system to enable unconditional stability to be achieved within the framework of an explicit formulation. These two features make the model‐based algorithms also potential candidates for computations in structural dynamics. Based on the algorithmic difference equations, these algorithms can be classified as either explicit or semi‐explicit, where the former refers to the algorithms with explicit difference equations for both displacement and velocity, while the latter for displacement only. The algorithms pertaining to each class are reviewed, and a new family of second‐order unconditionally stable parametrically dissipative semi‐explicit algorithms is presented. Numerical characteristics of these two classes of algorithms are assessed under linear and nonlinear structural behavior. Representative numerical examples are presented to complement the analytical findings. The analysis and numerical examples demonstrate the advantages and limitations of these two classes of model‐based algorithms for applications in structural dynamics. Copyright © 2015 John Wiley & Sons, Ltd.     

Cyclic steady states of nonlinear electro‐mechanical devices excited at resonance

We present an efficient numerical method to solve for cyclic steady states of nonlinear electro‐mechanical devices excited at resonance. Many electro‐mechanical systems are designed to operate at resonance, where the ramp‐up simulation to steady state is computationally very expensive – especially when low damping is present. The proposed method relies on a Newton–Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naïve transient time‐stepping from zero initial conditions. We use a recently developed high‐order Eulerian–Lagrangian finite element method in combination with an energy‐preserving dynamic contact algorithm in order to solve the coupled electro‐mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro‐electro‐mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed‐ups of the form S  = 0.6·ξ  ^? 0.8, where ξ is the linear damping ratio of the corresponding resonance frequency. Copyright © 2016 John Wiley & Sons, Ltd.     

Electric analog of magnetic resonance

Resonant rf absorption was observed during an experiment with a negative point-to-plane corona discharge at a small-amplitude rf voltage combined with a high dc voltage. The absorption is maximum if the frequency of the rf voltage is equal to or a multiple of the repetition rate of Trichel pulses. A number of resonances are detected by recording the Q factor of a resonant circuit, which includes the interelectrode gap, against the dc voltage at a fixed frequency of the rf voltage. The rf voltage frequency was plotted against the electrostatic field strength, both measured under the resonance conditions at different interelectrode spacings. For the most part of the interelectrode gap, the initial portions of these graphs are linear with the proportionality factor being independent of the interelectrode spacing. The proportionality factor is evaluated as 1.0±0.2 Hz m/V.     

A hybrid variational method for multibody dynamics

This paper presents a hybrid variational method to minimize computational effort in forming and solving the equations of motion for broad classes of rigid multibody mechanical systems. The hybrid method combines the O(n) and O(n³) recursive variational methods for forming the equations of motion in terms of joint relative co-ordinates. While the O(n³) method is more efficient than the O(n) method for systems with short chains and decoupled loops, the converse is true when the number of bodies in chains is large. The computational complexity of the O(n³) and O(n) methods in forming and solving the equations of motion is analysed as a function of the numbers of bodies, decoupled loops, joints, cut joints, cut-joint constraint equations and force elements. Based on complexity estimates, the method presented in this paper uses either the O(n) or O(n³) variational method to formulate the equations of motion for each open chain and decoupled loop in the system, to minimize the computational effort.     

A vibrating-wire densimeter for measurements in fluids at high pressures

The paper describes a new type of densimeter especially designed for the accurate measurement of fluid densities at pressures up to 400 MPa. The densimeter makes use of the buoyancy force exerted on a mass immersed in the test fluid to alter the resonant frequency of a thin wire from which the mass is suspended. The resonant frequency of the wire carrying the mass is related to the fluid density by means of working equations which are based on a complete analysis of the fluid motion around the wire. Preliminary results are presented for n-octane at pressures up to about 100 MPa near ambient temperature. The results show that the instrument has a precision of ±0.1 % in density at elevated pressures when evaluated on a relative basis, while the accuracy is estimated to be one of ±0.2%.     

Squeezing from Raman Scattering in Optical Parametric Oscillators

Abstract

The squeezing properties of the Raman scattering process which takes place when the lower output frequency from a non-degenerate optical parametric interaction approaches a resonance of the nonlinear medium placed inside an optical resonant cavity are presented. The linear stability analysis is performed and the spectrum for the output fields is given in terms of the quadrature phase components in the Wigner representation. Perfect squeezing for the amplitude difference between the output Stokes and anti-Stokes modes at the Hopf bifurcation point is obtained.     

On the oscillations near and at resonance in open pipes

Summary This paper is concerned with resonance oscillations occurring when a piston executes small oscillations on one end of a pipe which is open to the atmosphere at the other end. According to linear theory very large amplitudes of pressure and velocity oscillations in the gas in the pipe result when the piston is oscillated with an angular frequency neara_o/2L, where a_o is the sound velocity of the gas and L the length of the pipe. In the theory of resonators, due to Helmholtz and Rayleigh and discussed in section 1, radiation from the open end is taken into account. Then resonance occurs at a frequency slightly below _o , and amplitudes are still very large, as is shown in section 1. Therefore a nonlinear theory is developed here, analogous to previous work on resonance oscillations in closed pipes. In section 2 the boundary conditions at the open end are formulated based on the fact that the reservoirconditions are constant at inflow but vary at outflow, since the gas issues as a jet. This difference results in a net efflux of energy to be balanced by the work done by the piston. In sections 3–7 a perturbation theory is developed in terms of the characteristics of motion. The pertinent perturbation parameter is suggested by the energy balance. An ordinary differential equation for the first order perturbation in the quasi-steady state is obtained in section 7. In section 8 experimental results are presented together with results obtained from numerical integration of the above mentioned equation. The results, showing a satisfactory agreement, indicate that further experimental investigation on the conditions at the open end are needed.     

Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.     

Vibration suppression of a CLD treated plate subject to a harmonic traveling load

Abstract

The vibrations of an annular plate with constrained layer damping (CLD) treatment subject to a traveling load are investigated. The equation of motion, after employing the assumed‐mode method, Donnell‐Mushtari‐Vlasov assumption and the Hamilton principle yielded terms of three plates’ displacements. The response is eventually, for each n, in terms of a single degree of freedom (SDOF) linear oscillator with hysteretic damping. The solution to a harmonic traveling load is then solved and discussed. Numerical results showed that the CLD treatment imposed significant damping onto the plate, especially as the plate reached its resonance. The interaction of harmonic driving frequency and traveling speed was also looked into. The results showed that to have the best damping effect, a relatively thin Visco Elastic Material (VEM) layer was enough and the damping was the most significant for n=0 and n=1 modes.     

On non-symmetric vibration of deep spherical sandwich shells

Summary This paper deals with the free non-symmetric vibration of deep spherical sandwich shells. The sandwich shell considered herein consists of three layers. A variational technique is utilized to obtain the equations of motion as well as the appropriate boundary conditions. The effects of transverse shear deformation and rotary inertia have been included in this analysis.New deformation functions have been introduced which considerably simplify the system of differential equations. The final solution is obtained in terms of Legendre functions.Numerical computations have been performed for the symmetric case and graphs are included to show the frequency variation with andh/R for various modes.     

Stressed Silicon Nitride Nanomechanical Resonators at Helium Temperatures

We have characterized the mechanical resonance properties (both linear and nonlinear) of various doubly-clamped silicon nitride nanomechanical resonators, each with a different intrinsic tensile stress. The measurements were carried out at 4 K and the magnetomotive technique was used to drive and detect the motion of the beams. The resonant frequencies of the beams are in the megahertz range, with quality factors of the order of 10⁴. We also measure the dynamic range of the beams and their nonlinear (Duffing) behaviour.     

A qualitative analysis of the behaviour of the Galerkin equations relevant to non-linear eddy current problems

The Galerkin equations relevant to the eddy currents induced in an iron body are considered. These equations are obtained by formulating the field problem in terms of the magnetic vector potential and by applying the Galerkin method. They are shown to have a unique steady-state solution if a certain condition on the magnetic constituitive relationship is satisfied. In particular a T-periodic source gives rise to a unique T-periodic solution to which all other solutions converge asymptotically independently from the initial conditions. Under the same condition the exponential decay of the ‘transients’ is shown, and an explicit lower and upper bound for its rate is given. These structural properties allow us to exclude a priori that qualitatively different asymptotic behaviours, including even chaotic solutions, may occur. Numerical simulation, when based on qualitative information of this type, enables us to obtain the quantitative properties in an efficient manner. In order to demonstrate the practical use of these results some numerical experiments are presented.     

面内弹性绳系卫星系统的内共振

研究了绳系卫星系统因系绳弹性和俯仰运动相耦合而引起的非线性内共振问题。首先应用Kane方程建立了面内椭圆轨道两体弹性绳系卫星系统在状态保持阶段的非线性动力学模型,分析了可能发生的内共振条件。然后,基于多尺度方法获得内共振下的调制方程及其Jacobi椭圆函数表示的解析解。结果表明,轨道摄动因素不会影响内共振,系统参数引起的内共振耦合振动会在两个模态之间相互传递,其调制周期取决于初始模态幅值。在一定的调谐参数上,模态振幅出现饱和现象。