Investigation of multi-layer sandwich beams through single degree-of-freedom transformation

The dynamic behaviors of multi-layer sandwich beams are investigated through single degree-of-freedom (SDOF) transformation. The frequency response of the multi-layer sandwich beam is obtained using finite element code COMSOL and is transformed to a SDOF system with the same frequency response. Hence, the mass, spring constant and damping coefficient of the sandwich beams with different lengths and number of visco-elastic layers can be investigated. Further, viscous damping and structural damping models are individually employed to simulate the damping effect of the sandwich beam. The frequency responses from both models are compared with that from COMSOL and experiment. The resonant peak and resonant frequency of the SDOF system using structural damping model is more consistent with that from COMSOL. The experimental result demonstrates that the response of the sandwich beam can be predicted through COMSOL and SDOF transformation.     

Zur Grenzschichtdämpfung von Verbundwerkstoffen mit kurzfasrigen Einlagerungen

Composite materials with discontinuous fibres are theoretically examined under alternating tension. By the aid of simple models for a composite with elastic matrix, elastic fibres, and plastificable interface the behaviour of the composite material can be described. The mechanical damping, caused by slipping of the interface, is computed for the cases of a perfect-plastic, a viscous and a visco-plastic boundary layer. Damping peaks occur in dependence of the stress amplitude and the frequency, respectively. From the position of these peaks we can conclude to the parameters of the interface, in particular, to its shear strength. Thus damping measurements render it possible to determine the properties of the interface within the composite.     

A new analytical technique for vibration analysis of non-proportionally damped beams

Vibrating linear mechanical systems, in particular continuous systems, are often modelled considering proportional damping distributions only, although in many real situations this simplified approach does not describe the dynamics of the system with sufficient accuracy. In this paper an analytical method is given to take into account the effects of a more general viscous damping model, referred to as non-proportional damping, on a class of vibrating continuous systems. A state-form expansion applied in conjunction with a transfer matrix technique is adopted to extract the eigenvalues and to express the eigenfunctions in analytical form, i.e., complex modes corresponding to non-synchronous motions. Numerical examples are included in order to show the efficiency of the proposed method; non-proportional damping distributions of different type, such as internal and external lumped or distributed viscous damping, are tested on non-homogeneous Euler-Bernoulli beams in bending vibration with different boundary conditions. Finally, a discussion on root locus diagrams behaviour and on modal damping ratio significance for non-proportionally damped systems is presented.     

THE DYNAMIC STIFFNESS MATRIX METHOD IN FORCED VIBRATION ANALYSIS OF MULTIPLE-CRACKED BEAM

The dynamic behaviour of a beam with numerous transverse cracks is studied. Based on the equivalent rotational spring model of crack and the transfer matrix for beam, the dynamic stiffness matrix method has been developed for spectral analysis of forced vibration of a multiple cracked beam. As a particular case, when the excitation frequency is close to zero, the solution for static response of beam with an arbitrary number of cracks has been obtained exactly in an analytical form. In general case, the effect of crack number and depth on the dynamic response of beam was analyzed numerically.     

Sound transmission through finite lightweight multilayered structures with thin air layers

The sound transmission loss (STL) of finite lightweight multilayered structures with thin air layers is studied in this paper. Two types of models are used to describe the vibro-acoustic behavior of these structures. Standard transfer matrix method assumes infinite layers and represents the plane wave propagation in the layers. A wave based model describes the direct sound transmission through a rectangular structure placed between two reverberant rooms. Full vibro-acoustic coupling between rooms, plates, and air cavities is taken into account. Comparison with double glazing measurements shows that this effect of vibro-acoustic coupling is important in lightweight double walls. For infinite structures, structural damping has no significant influence on STL below the coincidence frequency. In this frequency region, the non-resonant transmission or so-called mass-law behavior dominates sound transmission. Modal simulations suggest a large influence of structural damping on STL. This is confirmed by experiments with double fiberboard partitions and sandwich structures. The results show that for thin air layers, the damping induced by friction and viscous effects at the air gap surfaces can largely influence and improve the sound transmission characteristics.     

Damping characteristics of a spray-deposited shape memory alloy beam

The damping characteristics of an Ni–Ti shape memory alloy (SMA) beam are theoretically and experimentally studied with interest in identifying an appropriate damping model for the material. The SMA beam is manufactured by a spray deposition method followed by heat treatment and found to have nanocrystalline structure in which damping capacity is high. The beam is then tested to obtain an impulse response and the frequency response function (FRF). By using the Hilbert transform technique it is shown that damping of the beam is almost amplitude independent in the tested range of displacement. It is also shown from the FRF that the damping of the spray-deposited shape memory alloy beam is well represented by a model including both linear viscous and hysteretic dampings.     

THE INFLUENCE OF CYLINDER LUBRICATION ON PISTON SLAP

A model has been developed for determining the time history of piston slap impact force. This model takes into account the influence of the oil film on the impact behaviour, which was found to be an important factor. However, it was also found that entrapped gas bubbles in the oil are equally significant. Three test rigs were designed and built to study these effects on the impact phenomenon and extensive tests were carried out. The impact force time history has been determined using Reynolds' theory. Results have shown that Reynolds' theory for fluid film squeezing can be applied for oil film damping determination. However, the experimental results have also shown that when gas is entrapped during the impact, this theory considerably overpredicts the magnitude of the impact. An eight-degree-of-freedom lumped parameter model was developed through the dynamic analysis of each component of an internal combustion engine's reciprocating system. The effective damping factor derived from this model was found to be inversely proportional to the oil film thickness cubed, as expected from Reynolds' theory. A dynamic model has been proposed, where the oil film mixed with bubbles is considered to be analogous to a serial spring and damping system. By incorporating a spring in series with this damper, the effect of the bubbles can also be predicted.     

Design and optimization of periodic structure mechanical filter in suppression of foundation resonances

The paper describes the development of periodic structure mechanical filter (PSMF) that has the potential to reduce vibration transmission and sound radiation at resonances of the foundation in a two-degree-of-freedom (2dof) vibration isolation system by using the band gaps of the periodic structure. The transmission matrix method is used to model vibration transmission of the 2dof system and an analytical expression of sound radiation from the foundation plate is derived. The multi-layer PSMF composed of rigid plates and curved beams is represented by an equivalent m-k-c (viscous damping) model. The propagation/attenuation zones and attenuation ability of PSMF are expressed in the propagation scenario and the iso-attenuation curves by exploiting the unit cell transfer matrix invariant. Influence of the number of unit cells, viscous damping on the mobility of PSMF and vibro-acoustic behavior of the 2dof system is extensively studied. And under the constraints of installation space and stability of the whole system, the more the number of the unit cells, the better attenuation ability in the band gap can be obtained. The interaction between PSMF and the 2dof system is analyzed by the substructure method and contribution of frequency component from different substructures is identified by setting different level of damping for each substructure. Factors influencing the first mounting frequency of the 2dof system with PSMF are discussed and three styles of installing PSMF are studied. The performance of piecewise periodic PSMF and quasi-periodic PSMF is also studied in an attempt to eliminate new-born resonances by PSMF. An optimization scheme involving sensitivity analysis is applied to obtain the optimal values of m and k. And the optimization is effective. The experiment of detecting the band gap of PSMF and the comparative trial of a 2dof system with a flexible plate as the foundation with/without PSMF are carried out. Both numerical and experimental simulation results have demonstrated that by use of PSMF, the vibration transmission at resonances is reduced and the radiation of the foundation at resonances is suppressed.     

Frequency modulation of waveguides in acoustic band-gap materials consisting of solid cylinders in viscous liquid

We investigated waveguides in acoustic band-gap materials consisting of steel cylinders in a liquid with viscous damping. Numerical results show that when the viscous penetration depth is comparable to the structural length scale, linear defect states fall in complete acoustic band gaps forming waveguides. It is also found that the magnitude of the viscosity in the liquid has an influence on the frequency of waveguides, that large viscous damping can make the defect modes ascend. An expected frequency of waveguides can be obtained by modulating the viscous damping parameter θ.     

Simplified models of the vibration of mannequins in car seats

A simplified two-dimensional modelling approach to predict the vibration response of mannequin occupied car seats about a static settling point is demonstrated to be feasible. The goal of the research is to develop tools for car seat designers. The two-dimensional model, consisting of interconnected masses, springs and dampers is non-linear due to geometric effects but, under the excitations considered, the model behaviour is linear. In this approach to modelling, the full system is initially broken down into subsystems, and experiments are conducted with subsystems to determine approximate values for the stiffness and damping parameters. This approach is necessary because of the highly non-linear behaviour of foam where stiffness changes with compression level, and because the simplified model contains more structure than is necessary to model the relatively simple measured frequency response behaviour, thus requiring a good initial starting point from which to vary parameters. A detailed study of the effects of changing model parameters on the natural frequencies, the mode shapes and resonance locations in frequency response functions is given, highlighting the influence of particular model parameters on features in the seat-mannequin system's vibration response. Reasonable qualitative as well as good quantitative agreement between experimental and simulation frequency response estimates is obtained. In particular, the two-dimensional motions at the peaks in the frequency response, a combination of up and down and rotational behaviour is predicted well by the model. Currently research is underway to develop a similar model with non-linear springs, surface friction effects and viscoelastic elements, that predicts the static settling point, a necessary step to aid in the subsystem modelling stage in this dynamic modelling approach.     

IDENTIFICATION OF DAMPING: PART 4, ERROR ANALYSIS

In previous papers (S. ADHIKARI and J. WOODHOUSE 2001 Journal of Sound and Vibration243, 43-61; 63-88; S. ADHIKARI and J. WOODHOUSE 2002 Journal of Sound and Vibration251, 477-490) methods were proposed to obtain the coefficient matrix for a viscous damping model or a non-viscous damping model with an exponential relaxation function, from measured complex natural frequencies and modes. In all these works, it has been assumed that exact complex natural frequencies and complex modes are known. In reality, this will not be the case. The purpose of this paper is to analyze the sensitivity of the identified damping matrices to measurement errors. By using numerical and analytical studies it is shown that the proposed methods can indeed be expected to give useful results from moderately noisy data provided a correct damping model is selected for fitting. Indications are also given of what level of noise in the measured modal properties is needed to mask the true physical behaviour.     

Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures

The vibrational response of finite periodic lattice structures subjected to periodic loading is investigated. Special attention is devoted to the response in frequency ranges with gaps in the band structure for the corresponding infinite periodic lattice. The effects of boundaries, viscous damping, and imperfections are studied by analyzing two examples; a 1-D filter and a 2-D wave guide. In 1-D the structural response in the band gap is shown to be insensitive to damping and small imperfections. In 2-D the similar effect of damping is noted for one type of periodic structure, whereas for another type the band gap effect is nearly eliminated by damping. In both 1-D and 2-D it is demonstrated how the free structural boundaries affect the response in the band gap due to local resonances. Finally, 2-D wave guides are considered by replacing the periodic structure with a homogeneous structure in a straight and a 90° bent path, and it is shown how the vibrational response is confined to the paths in the band gap frequency ranges.     

IDENTIFICATION OF DAMPING: PART 3, SYMMETRY-PRESERVING METHODS

In two recent papers (Adhikari and Woodhouse 2001 Journal of Sound and Vibration243, 43-61; 63-88), methods were proposed to identify viscous and non-viscous damping models for vibrating systems using measured complex frequencies and mode shapes. In many cases, the identified damping matrix becomes asymmetric, a non-physical result. Methods are presented here to identify damping models which preserve symmetry of the system. Both viscous and non-viscous models are considered. The procedure is based on a constrained error minimization approach and uses only experimentally identified complex modes and complex natural frequencies together with, for the non-viscous model, the mass matrix of the system. The methods are illustrated by numerical examples.     

Development of a 3D finite element acoustic model to predict the sound reduction index of stud based double-leaf walls

Building standards incorporating quantitative acoustical criteria to ensure adequate sound insulation are now being implemented. Engineers are making great efforts to design acoustically efficient double-wall structures. Accordingly, efficient simulation models to predict the acoustic insulation of double-leaf wall structures are needed. This paper presents the development of a numerical tool that can predict the frequency dependent sound reduction index R of stud based double-leaf walls at one-third-octave band frequency range. A fully vibro-acoustic 3D model consisting of two rooms partitioned using a double-leaf wall, considering the structure and acoustic fluid coupling incorporating the existing fluid and structural solvers are presented. The validity of the finite element (FE) model is assessed by comparison with experimental test results carried out in a certified laboratory. Accurate representation of the structural damping matrix to effectively predict the R values are studied. The possibilities of minimising the simulation time using a frequency dependent mesh model was also investigated. The FEA model presented in this work is capable of predicting the weighted sound reduction index R_w along with A-weighted pink noise C and A-weighted urban noise C_tr within an error of 1 dB. The model developed can also be used to analyse the acoustically induced frequency dependent geometrical behaviour of the double-leaf wall components to optimise them for best acoustic performance. The FE modelling procedure reported in this paper can be extended to other building components undergoing fluid–structure interaction (FSI) to evaluate their acoustic insulation.     

阻尼型高斯-牛顿法及其在高频电磁波测井反演中的应用

提出一种改进的阻尼型高斯牛顿优化算法,通过引入阻尼矩阵,对反演参数依其相对修改量不同而给以不同的阻尼作用,并将它用于高频电磁波测井资料的反演中.     

Interface stress effect tuning and enhancing the energy dissipation of staggered nanocomposites

ABSTRACT

Natural biological composites and artificial biomimetic staggered composites with nanoscale internal structures can exhibit extraordinary energy dissipation, compared with conventional composites. It is believed that the interface stress effects of the interfaces between hard platelets and a viscous matrix play an important role in the extraordinary damping properties of such nanocomposites. In this study, a viscoelastic model is established to investigate the mechanism of the influence that the interface stress effect has on the damping properties, based on the Gurtin-Murdoch interface model and the tension-shear chain model. An explicit analytical solution of the effective dynamic moduli characterising the damping properties is obtained by using the correspondence principle, which is also validated by comparison with a finite element analysis. From the obtained analytical solution, an interface factor is abstracted to characterise the synergistic effect of the feature size and material parameters on the damping properties. Based on the model established, the optimal size of the platelets and the optimal loading frequency can be designed to achieve superior energy dissipation, when the staggered nanocomposites bear the dynamic load. Therefore, the findings of the present study not only reveal the damping mechanism of biological structures at nanoscale but also provide useful guidelines for the design of biomimetic nanocomposites from the nanoscale to the macroscopic scale.     

Five-parameter fractional derivative model for polymeric damping materials

Fractional derivative models offer a powerful tool to describe the dynamic behaviour of real viscoelastic materials. A version of the fractional derivative models characterized by five parameters is presented and investigated in this paper in order to describe asymmetrical loss factor peak and the high-frequency behaviour of polymeric damping materials. The speculative derivation of the model constitutive equation containing time derivatives of stress and strain of different orders is given. The model behaviour is investigated in the frequency domain, the physical meaning of the model parameters is defined and constraints on the parameter values are made. It is shown that the asymmetry of loss peak and the high-frequency behaviour of the model are governed by the difference between the order of time derivatives of stress and strain. Moreover, it is shown that this difference is related to the high-frequency limit value of the loss factor. The model is fitted to experimental data on some polymeric damping materials to verify its behaviour.     

Passive-active vibration isolation systems to produce zero or infinite dynamic modulus: theoretical and conceptual design strategies

The application of mechanical springs connected in parallel and/or in series with active springs can produce dynamical systems characterised by infinite or zero value stiffness. This mathematical model is extended to more general cases by examining the dynamic modulus associated with damping, stiffness and mass effects. This produces a theoretical basis on which to design an isolation system with infinite or zero dynamic modulus, such that stiffness and damping may have infinite or zero values. Several theoretical designs using a mixture of passive and active systems connected in parallel and/or in series are proposed to overcome limitations of feedback gain experienced in practice to achieve an infinite or zero dynamic modulus. It is shown that such systems can be developed to reduce the weight supported by active actuators as demonstrated, for example, by examining suspension systems of very low natural frequency or with a very large supporting stiffness or with a viscous damper or a self-excited vibration oscillator. A more general system is created by combining these individual systems allowing adjustment of the supporting stiffness and damping using both displacement and velocity feedback controls. Frequency response curves show the effects of active feedback control on the dynamical behaviour of these systems. The theoretical design strategies presented can be applied to design feasible hybrid vibration control systems displaying increased control performance.     

Analytical model for viscous damping and the spring force for perforated planar microstructures acting at both audible and ultrasonic frequencies

The paper presents a model for the squeezed film damping, the resistance of the holes, and the corresponding spring forces for a periodic perforated microstructure including the effects of compressibility, inertia, and rarefied gas. The viscous damping and spring forces are obtained by using the continuity equation. The analytical formula for the squeezed film damping is applied to analyze the response of an ultrasonic transducer. The inclusion of these effects in a model significantly improves the agreement with measured results. Finally, it is shown that the frequency dependence of the total damping and total spring force for a cell are very similar to those corresponding to a rectangular open microstructure without holes. A separate analysis reveals the importance of each particular correction. The most important is the compressibility correction; the inertia has to be considered only for determining the spring force and the damping force for sufficiently high frequencies.