Vertical accelerations of simple beams due to successive loads traveling at resonant speeds

In this paper, the vertical acceleration response of a simple beam traveled by a series of equally spaced moving loads at constant speeds is studied by the superposition method. From the closed-form solution derived, the key parameters dominating the resonant response of the beam are identified, along with the effect of higher modes of vibration on the acceleration response investigated. For the loads moving at resonant speeds, the higher modes can have significant influence on the acceleration amplitude. This is true especially for beams with light damping, for which the maximum acceleration on the beam depends on which vibration mode is excited. As such, the maximum acceleration of the beam need not occur at the mid-point. By considering the resonant speeds associated with the first and second modes, a simplified formula is proposed for checking whether the maximum acceleration may occur at the mid-point of the beam. For the case when the structural damping is taken into account, the contribution of higher modes to the acceleration response tends to be damped out. It is concluded that for a beam properly damped, the maximum acceleration response of the beam is dominated by the fundamental vibration mode.     

Stationary response of combined linear dynamic systems to stationary excitation

The stationary response of a broad class of combined linear systems to stationary random excitation is determined by the normal mode method. The systems are characterized by a viscously damped simple beam (or string, membrane, thin plate or shell, etc.) connected at discrete points to a multiplicity of viscously damped linear oscillators and/or masses. The solution of the free vibration problem by way of Green functions and the deterministic forced vibration problem by modal analysis for both proportional and non-proportional damping is reviewed. The orthogonality relation for the natural modes of vibration is used to derive a unique relationship between the cross-spectral density functions of the applied forces and the cross-spectral density functions of the generalized forces. Finally, the response spectral density functions and the mean square responses of the beam and oscillators are derived in closed form, exact for the proportionally damped system and approximate for the non-proportionally damped system.     

Suppression of random vibration in flexible structures using a hybrid vibration absorber

A design method is proposed to suppress stationary random vibration in flexible structures using a hybrid vibration absorber (HVA). While the traditional vibration absorber can damp down the vibration mainly at the pre-tuned mode of the primary structure, active damping is generated by the proposed HVA to damp down all resonant modes of interest of the vibrating structure and the spatial average mean square motion of the vibrating structure can be minimized. Only one absorber and one feedback signal are required to achieve global vibration suppression of a flexible structure under stationary random excitation. A special pole-placement controller is designed such that all vibration modes of the flexible structures become critically damped. It is proved analytically that the proposed HVA damps the vibration of the entire structure instead of just the attachment point of the absorber. The proposed optimized HVA is tested on a beam structure and it shows a superior performance on global suppression of broadband vibration in comparison to other published designs of passive and hybrid vibration absorbers.     

Geometrical structures,bending potentials,and vibrational frequencies ν1 for 1A″ states of HSiBr,HSiCl, and DSiCl

The geometrical structures and the approximate bending potentials are determined for excited electronic states ¹A″ in HSiBr, and in HSiCl and DSiCl. A Hamiltonian for large amplitude bending vibration and K type rotation contains parameters which were adjusted to yield energy levels consistent with data observed spectroscopically. The effect of centrifugal stretching of the HSi bond was treated by a simple semiclassical method. In the process of applying this method it was found that the HSi stretching frequencies ν₁ do not have the values recommended by Herzberg and Verma, but have the alternate values given by these authors. The barriers to the linear conformation were found to be approximately 8700 cm^?1 for HSiBr, and 12 400 cm^?1 for HSiCl and DSiCl.     

Operator method for calculating the spectrum of states in the framework of the extended dicke model

A system of N two-level atoms interacting with a resonant single-mode quantum field (the Dicke model) is described using the operator method for solving the Schrödinger equation. The spectrum of states is calculated without recourse to the rotating-wave approximation and the assumption regarding smallness of the linear sizes of the system as compared to the radiation wavelength. Analytical approximate expressions are obtained for the energy spectrum. These expressions approximate the energy spectrum over the entire range of Hamiltonian parameters in the normal and collective states of the system and make it possible to calculate the thermodynamic characteristics.     

Inverse iteration for damped natural vibration

Inverse iteration is extended to internally and/or externally damped natural vibration. Each iteration involves one matrix multiplication and one linear equation solution of order n. The symmetric band form of the original undamped eigenvalue problem is preserved. If the undamped mode is taken as the first approximation, the inverse iteration will converse to the corresponding damped mode in about four iterations. However, the one step method is divergent for heavy damping. Therefore, it is advisible to subdivide the damping into successive steps if inverse iteration does not converge in say five iterations. The method is successful for both discrete systems and distributed systems. The implementation is very simple by means of complex arithmetic which is readily available in many FORTRAN compilers.     

Non-linear vibrations of three-layer beams with viscoelastic cores,II: Experiment

Experimental results are presented for large amplitude, forced motion of damped, three-layer beams. The beams are constructed with a viscoelastic material constrained between stiff, elastic, outer layers. The sandwich beam is axially restrained; therefore large amplitude displacements cause non-linear response. When the beam is forced at one-half of the lateral vibration resonant frequency, superharmonic response occurs. The experiment is briefly described and frequency response characteristics, spatial shapes and a measure of superharmonic response are presented. The results are compared with predictions from a previously developed theory.     

Parameter estimation and multivariable model building for the non-destructive, on-line determination of eggshell strength

In the non-destructive quality assessment of agro-products using vibration analysis, the resonant frequency and the damping of the vibration are the main interest. Those parameters are usually calculated starting from the frequency spectrum, obtained after a fast Fourier transformation (FFT) of the time signal. However, this method faces several drawbacks when applied to short-time signals, as in the case of impact testing of highly damped specimen. An alternative to the FFT method is used for the high-resolution estimation of both resonant frequency and damping. Furthermore, the mass-spring model that is used in the literature for non-destructive quality assessment of various agro-products is extended with the incorporation of the damping and a shape characteristic. As a practical example, eggshell stiffness was estimated using vibration measurements. A data set consisting of 229 eggs was measured. It is shown that both the damping and the shape characteristics are of major importance to explain eggshell strength. This paper makes clear that a univariable model, as is mostly used in the literature, is not always satisfactory to describe the vibration behaviour of biological products.     

On linear modeling of interface damping in vibrating structures

Dissipation of mechanical vibration energy at contact interfaces in a structure, commonly referred to as interface damping, is an important source of vibration damping in built-up structures and its modeling is the focus of the present study. The approach taken uses interface forces which are linearly dependent on the relative vibration displacements at the contact interfaces.The main objective is to demonstrate a straightforward technique for simulation of interface damping in built-up structures using FE modeling and simple, distributed, damping forces localized to interfaces where the damping occurs.As an illustration of the resulting damping the dissipated power is used for evaluation purposes. This is calculated from surface integrals over the contact interfaces and allows for explicit assessment of the effect of simulated interface forces for different cases and frequencies. The resulting loss factor at resonance is explicitly evaluated and, using linear simulations, it is demonstrated that high damping levels may arise even though the displacement differences between contacting surfaces at damped interfaces may be very small.     

Reduction of sound transmission into a circular cylindrical shell using distributed vibration absorbers and Helmholtz resonators

A modal expansion method is used to model a cylindrical enclosure excited by an external plane wave. A set of distributed vibration absorbers (DVAs) and Helmholtz resonators (HRs) are applied to the structure to control the interior acoustic levels. Using an impedance matching method, the structure, the acoustic cavity, and the noise reduction devices are fully coupled to yield an analytical formulation of the structural kinetic energy and acoustic potential energy of a treated cylindrical cavity. Lightweight DVAs and small HRs tuned to the natural frequencies of the targeted structural and acoustic modes, respectively, result in significant acoustic and structural attenuation when the devices are optimally damped. Simulations show that significant interior noise reduction can only be achieved by adding damping to both structural and acoustic modes, which are resonant in the frequency bandwidth of interest. In order to be independent of the azimuth angle of the excitation and to avoid unwanted modal interactions, the devices are distributed evenly around the cylinder in rings. This treatment can only achieve good performance if the structure and the acoustic cavity are lightly damped.     

H-infinity optimization of a variant design of the dynamic vibration absorber—Revisited and new results

The H_∞ optimum parameters of a dynamic vibration absorber (DVA) with ground-support are derived to minimize the resonant vibration amplitude of a single degree-of-freedom (sdof) system under harmonic force excitation. The optimum parameters which are derived based on the classical fixed-points theory and reported in literature for this non-traditional DVA are shown to be not leading to the minimum resonant vibration amplitude of the controlled mass. A new procedure is proposed for the H_∞ optimization of such a dynamic vibration absorber. A new set of optimum tuning frequency and damping of the absorber is derived, thereby resulting in lower maximum amplitude responses than those reported in the literature. The proposed optimized variant DVA is also compared to a ground-hooked damper of the same damping capacity of the damper in the DVA. It is proved that the proposed optimized DVA has better suppression of the resonant vibration amplitude of the controlled system than both the traditional DVA and also the ground-hooked damper if the proposed design procedure of the variant DVA is followed.     

非线性电容RLC串联电路的主共振研究

研究非线性电容RLC串联电路,应用多尺度法,得到非线性振动系统主共振的一次近似解并进行数值计算,分析电阻、电感、电容和电动势对主共振幅频响应的影响.结果表明:RLC串联电路的主共振响应有跳跃和滞后现象;随着电动势的增加,主共振的振幅和共振区增大;随着电阻的增大,主共振的振幅和共振区减小.     

Vibration of continuous systems with compliant boundaries

A theoretical study of two types of continuous systems with a general form of compliant boundary conditions is presented. The systems considered are elastic beams and circular plates with elastic damped edge constraints. Beam studies are restricted to those with identical boundary conditions at each end. The method of solution consists of formulating the edge condition of the system in terms of the impedance of the compliant boundary material and of using classical solution techniques to solve the equations of motion. The result of matching the boundary conditions of the system with constraining conditions is the system frequency equation in terms of the constraint impedances.A discussion is presented giving the influence of the compliant material on the vibration of the structure. The models give numerically the effect of elasticity and damping of the supports on the resonant frequencies of the systems. Parameters are obtained which indicate when one may assume simply supported or clamped boundaries for the actual case of elastic damped constraints without introducing large errors in the natural frequencies.     

Some properties of horn equation model of ultrasonic system vibration and of transfer matrix and equivalent circuit methods of its solution

Traditional technique of horn equation solved by transfer matrices as a model of vibration of ultrasonic systems consisting of sectional transducer, horn and load is discussed. Expression of vibration modes as a ratio of solutions of two Schrödinger equations gives better insight to the structure of a transfer matrix and properties of amplitudes of displacement and strain, and enables more systematic search for analytic solutions. Incorrectness of impedance matrix method and of equivalent circuit method on one hand and correctness and advantages of transfer matrix method in avoiding numerical artifacts and revealing the real features of the model on the other hand are demonstrated on examples. Discontinuous dependence of the nth resonant value on parameters of ultrasonic system, recently described in Sturm–Liouville theory, and consequently, a jump from half-wave to full-wave mode, is observed in a transducer model.     

Adiabatic Raman passage via two-photon resonant transitions in a five-level Λ system

We investigate a five-level Λ system for achieving the efficient population transfer between two ground-state levels without essentially populating other three intermediate excited-state levels. This five-level Λ system is found to have an approximate dark state characterized by very small (but not exactly zero) absolute eigen-values when we work in the parameter regime where all single-photon pump and Stokes transitions are well suppressed. Dynamically manipulating the approximate dark state in a proper way, we can either attain the complete population transfer from one ground-state level to another ground-state level or create the maximal coherence between the two ground-state levels. The dynamic process of a stimulated Raman adiabatic passage is implemented in fact by manipulating a two-photon resonant pump transition and a two-photon resonant Stokes transition, which is especially important when the two ground-state levels differ by Δm = ± 4 in the magnetic quantum number.     

Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

The parametric dynamic stability of resonant beams with various parameters under periodic axial force is studied.It is assumed that the theoretical formulations are based on Euler-Bernoulli beam theory.The governing equations of motion are derived by using the Rayleigh-Ritz method and transformed into Mathieu equations,which are formed to determine the stability criterion and stability regions for parametricallyexcited linear resonant beams.An improved stability criterion is obtained using periodic Lyapunov functions.The boundary points on the stable regions are determined by using a small parameter perturbation method.Numerical results and discussion are presented to highlight the effects of beam length,axial force and damped coefficient on the stability criterion and stability regions.While some stability rules are easy to anticipate,we draw some conclusions:with the increase of damped coefficient,stable regions arise;with the decrease of beam length,the conditions of the damped coefficient arise instead.These conclusions can provide a reference for the robust design of parametricallyexcited linear resonant sensors.     

Variation with distance of aircraft noise impact parameters

Recent developments in the use of statistical energy analysis for the prediction of vibration levels on spacecraft components have led to the need to measure the frequency averaged modal density. In the case of structural components having low damping it has been extremely difficult to make a successful measurement by using existing techniques. Success has usually been achieved by the addition of damping tape to the structure under test. The transient testing technique also requires extremely long data lengths on lightly damped structures, which can lead to a requirement for Fourier transforms considerably longer than normally available. In the technique described here random excitation and extremely short data lengths are used; the technique is shown to give good results on structures with no additional damping material, with a considerable saving in time and expense not only during the experimentation but also during data processing.     

A note on the vibrations of rectangular plates of variable thickness with two opposite simply supported edges and very general boundary conditions on the other two

An approximate solution for the title problem is obtained by making use of the Galerkin method. The plate displacement function is approximated by means of a sinusoid multiplied by a polynomial. Translational and rotational flexibilities are taken into account at x = ±a/2. It is shown that the free edge situation (Kirchoff's boundary condition) can be treated as a special case by means of the approach developed herein. A simple algorithm which allows evaluation of the fundamental frequency of vibration is derived and rough estimates of amplitudes and stress resultants are also given when the plate is subjected to a p₀cosωt-type excitation.     

Transient response optimization of vibrating structures by Liapunov's second method

Liapunov's second method is applied to minimize an integrated square performance measure for damped vibrating structures subjected to initial excitation. The method reduces the calculation of the performance measure and its derivatives with respect to design parameters to the solution of a set of linear algebraic equations. The computational effectiveness of the method is illustrated by applying it to the classical vibration absorber and to a cantilever beam carrying an absorber at its midpoint.     

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin-Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin-Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.