Damped Plate Research Articles

Lifespan of solutions to a damped plate equation with logarithmic nonlinearity

<p style='text-indent:20px;'>This paper is devoted to the lifespan of solutions to a damped plate equation with logarithmic nonlinearity</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}+\Delta^2u-\Delta u-\Delta u_t+u_t = |u|^{p-2}u\ln|u|. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Finite time blow-up criteria for solutions at both lower and high initial energy levels are established and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived.</p>

Adhesion enhancement between aluminum and butyl rubber by (3-mercaptopropyl) trimethoxy silane for vibration damping plate

A silane coupling agent for an aluminum-based vibrational damping plate with a butyl rubber damper, which shows excellent viscoelasticity and thermal stability, was studied to enhance the adhesion strength between aluminum and butyl rubber. Considering the chemical structure of the butyl rubber, we introduced (3-mercaptopropyl) trimethoxysilane (MPTMS) as the silane coupling agent because it has a thiol functional group that can bind to the carbon chain of butyl rubber via thiol-ene reaction under heating. Based on the results of various surface analyses and the T-peel strength test, the MPTMS treatment condition was optimized, and a 180% enhancement of peel strength was obtained. Finally, a central supporting vibration test was conducted to prove the damping performance of aluminum damping plated with MPTMS. As a result, a 7-fold enhancement in the damping loss factor was seen in the damping with MPTMS compared to without.

Nonlinear vibrations of a polar-orthotropic thin circular plate subjected to circularly moving point load

Here, we formulate and study nonlinear vibrations of a thin, polar-orthotropic circular plate subjected to a circularly moving point load at a fixed radius and constant angular velocity. The governing equations and boundary conditions are obtained following Kirchhoff’s plate theory, incorporating von Kármán nonlinearity, and employing extended Hamilton’s principle. The external damping is introduced via Rayleigh dissipation function. The governing equations are solved using mode summation procedure. The mode shapes and the natural frequencies of the polar-orthotropic circular plate are found using Frobenius series method. Mode summation procedure results in coupled nonlinear ordinary differential equations. These equations are solved using the Runge–Kutta method for time response and method of harmonic balance with the arc continuation method for frequency response. The spectrum of the undamped linear vibration response of isotropic and polar-orthotropic plates exhibits natural frequencies of plates and angular velocity of the rotating load. The damped response contains the frequency of the angular velocity of the rotating load only. The nonlinear transverse vibrations of the undamped and damped plate due to rotating point load reveal frequency rich spectrums. The frequency response function shows strong modal interactions.

Quantification of morphodynamic variability and sea state damping of plates at the nearshore area in the East Frisian North Sea

At the nearshore area of Norderney, the ebb-tidal delta (ETD) sandbanks exert a huge impact on the local wave climate, which is of particular interest for coastal protection. Besides an eastward migration of the sandbanks with a mean velocity from 435-491 m/y, depending on the plates position, the ETD sandbanks had a high variability in shape and size. Therefore, the influence of those morphodynamic changes on the sea state damping effect was quantified by dynamic bathymetries using validated SWAN simulations. The impacts of the bathymetries from the years 1995, 2005, 2006, 2008, and 2015 with highest morphological variability were considered for five different boundary conditions, representing storm surge events of the Norderney nearshore wave climate. A mean variation of 14 ± 2% of the relative sea state damping effect was found for the different bathymetries of the ETD sandbanks at the various boundary conditions. Swell waves showed a range of 16 ± 2% for the damping effect, which increased with increasing incoming wave height. Buoy data off the Norderney coast from 1992 to 2017 were investigated and a mean buoy-measured damping effect of 0.37 was determined over the entire period. General results of SWAN modeling were also found in the in-situ measurements.

Asymptotic profiles for damped plate equations with rotational inertia terms

We consider the Cauchy problem for plate equations with rotational inertia and frictional damping terms. We derive asymptotic profiles of the solution in [Formula: see text]-sense as [Formula: see text] in the case when the initial data have high and low regularity, respectively. Especially, in the low regularity case of the initial data one encounters the regularity-loss structure of the solutions, and the analysis is more delicate. We employ the so-called Fourier splitting method combined with the explicit formula of the solution (high-frequency estimates) and the method due to [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differential Equations 257 (2014) 2159–2177.] (low-frequency estimates). In this paper, we will introduce a new threshold [Formula: see text] on the regularity of the initial data that divides the property of the corresponding solution to our problem into two parts: one is wave-like, and the other is parabolic-like.

Dependency of critical damping on various parameters of tapered bidirectional graded circular plates rested on Hetenyi medium

In industrial machinery and automobiles the critical damping plates are used as shock absorbers to dampen system and return rest position, in the shortest period of time. The paper is focused on the effects of various parameters on critical damping of a complicated system including a tapered bidirectional graded circular plate subjected to radially variable in-plane pre-load and rested on visco-Hetenyi elastic medium with semi-rigid restraint. By employing a Chebyshev-Ritz method, dependency of critical damping is investigated for the first time on various parameters including bidirectional arbitrary material gradation, variable in-plane pre-load, Hetenyi elastic medium parameters, tapering, semi-rigid restraint and Poisson’s ratio. For the Chebyshev-Ritz method developed here, the proper scale factor and orthogonal shifted Chebyshev polynomials of the first kind without auxiliary function requirement are used. The conventional rule of mixture and Mori-Tanaka homogenization scheme are used to model transverse gradation. The characteristic equation is calculated by vanishing determinant of the total potential energy Hessian. An equivalent ABAQUS model is introduced to eliminate necessity of complicated modeling of original structure.

On the initial and terminal value problem for a class of semilinear strongly material damped plate equations

We study the semilinear strongly damped plate equation by considering its two different problems. For initial value problem, we prove the local well-posedness and blow-up results of solution for the problem with polynomial nonlinear source terms. For terminal value problem, given the ill-posedness in the sense of Hadamard we propose a regularization method for the problem with logarithmic nonlinearities. Under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method to stabilize the ill-posed problem, also by establishing some stability estimates of logarithmic type in Lq space.

Damping Determination by Half-Power Bandwidth Method for a Slightly Damped Rectangular Steel Plate in the Mid-Frequency Range

This paper presents a novel methodology for measuring the Damping Loss Factor (DLF) of a slightly damped plate in the mid-frequency range (400-1000 Hz) by the Half Power Bandwidth Method (HPBM). A steel flat plate of 650 x 550 x 2 mm was considered as the test case, which was excited by both a shaker and an impact hammer to quantify the effect of the excitation type for slightly damped plate. Since the HPBM is based on extracting the damping data from the modal resonance peaks, working with the correct Frequency Response Functions (FRF) was found to be a crucial factor. Therefore, the effects of coherence and resolution of the sampling frequency were examined in detail in the measurements. The obtained DLF results were statistically analysed and then applied in SEA simulations. Comparison of the simulation and experimental results showed that the method of extracting the DLF data from the measurements can have as much as 10 dB influence on the simulation results. The best results, with only 2 dB difference between measurement and simulation, were obtained when the statistical expected value of the data was used as the input in the SEA simulations.

Numerical study of acoustic transmission across heavily damped plate using hybrid Ray-Tracing-SEA method

Energy transmission across lightly damped structures has been well studied including the approved success of statistical energy analysis in mid and high frequency bands. For heavily damped elements, the diffuse field theory, which is used in computing coupling loss factors, tends to fail. Energy attenuation with distance becomes more significant for such elements and hence the energy is less likely to be evenly distributed within those elements. A ray tracing algorithm is developed taking account of this phenomenon by tracking the travel history of a great number of discrete rays. The predicted transmitted energy is used in a modified statistical energy analysis model to calculate energy level difference between different subsystems. Numerical validation and comparison on a concrete five-plate system are conducted in both lightly damped and heavily damped cases. Both the classic and the hybrid models show good agreement for lightly damped system and differ for heavily damped system. The difference tends to become larger with increasing frequency and internal damping level. The parameter “effective length ratio” is proposed to describe the phenomena of energy concentration along the edge and as in indicator of whether the application of diffuse field theory is appropriate.

COMPLEX EIGENFREQUENCIES AND DAMPING PROPERTIES OF AN ELONGATED PLATE WITH AN INTEGRAL DAMPING COATING

This paper considers the classical methods of surface damping of bending vibrations of thin-walled structures and a promising integral version of a damping coating consisting of two layers of a material with pronounced viscoelastic properties separated by a thin reinforcing layer of a high modulus material. A finite element with 14 degrees of freedom for modeling an elongated plate with the specified damping coating has been developed with consideration of the transverse compression of damping layers under high-frequency vibrations of the plate. The generalized problem of complex eigenvalues in the lower part of the spectrum of complex modes and frequencies of free vibrations of a damped plate is solved by the subspace iteration method taking into account the frequency dependence of the dynamic elastic moduli of the material. The damping properties of the plate are determined from the imaginary parts of the complex eigenfrequencies and the relative energy dissipation at resonance.

Broadband vibration damping of non-periodic plates by piezoelectric coupling to their electrical analogues

Several solutions for multimodal vibration damping of thin mechanical structures based on piezoelectric coupling have been developed over the years. Among them, piezoelectric network damping consists in using piezoelectric transducers to couple a structure to an electrical network, where the transferred electrical energy can be dissipated. In particular, the effectiveness of coupling rods, beams and plates to their analogous electrical networks has been proven. This work is the first step going towards more complex structures. After defining and experimentally validating a fully passive electrical analogous network of a simply-supported plate, the study is extended to the damping of a non-periodic plate. The non-periodicities here studied include the addition of a local mass and a variable thickness. Numerical simulations and experiments show that in these cases, a broadband damping is achieved once the piezoelectric transducers are coupled to an adequate analogous network. A finite element model of the structure coupled to a 2D non-periodic electrical network is concurrently developed and validated, which is another contribution of the present work.

Определение демпфирующих свойств удлиненной пластины с интегральным демпфирующим покрытием на основе исследования комплексных собственных частот

We describe the structure of a perspective integral damping coating consisting (with respect to the thickness) of two layers of a viscoelastic material with a thin reinforcing layer in-between. We propose a four-layer finite element model with fourteen degrees of freedom for a plate with a mentioned damping coating. This model allows us to take into account the effect of transversal compression of damping layers under high-frequency vibrations of the plate. For determining some lower complex modes and frequencies of free vibrations of the damped plate, we solve a generalized complex eigenvalue problem using the method of iterations in a subspace.

Asymptotic profiles and critical exponents for a semilinear damped plate equation with time-dependent coefficients

In this paper we study the asymptotic profile (as t → ∞) of the solution to the Cauchy problem for the linear plate equation u t t + Δ 2 u − λ ( t ) Δ u + u t = 0 when λ = λ ( t ) is a decreasing function, assuming initial data in the energy space and verifying a moment condition. For sufficiently small data, we find the critical exponent for global solutions to the corresponding problem with power nonlinearity u t t + Δ 2 u − λ ( t ) Δ u + u t = | u | p . In order to do that, we assume small data in the energy space and, possibly, in L 1 . In this latter case, we also determinate the asymptotic profile of the solution to the semilinear problem for supercritical power nonlinearities.

Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process

An ergodic control problem is studied for controlled linear stochastic equations driven by cylindrical Levy noise with unbounded control operator in a Hilbert space. A family of optimal controls is shown to consist of those asymptotically achieving the feedback form that employs the corresponding Riccati equation. The formula for optimal cost is given. The general results are applied to stochastic heat equation with boundary control and to stochastic structurally damped plate equations with point control.

Homogenised formulation for plates with thick constrained viscoelastic core

The present work analyses the dynamic behaviour of constrained layer damping (CLD) plates with thick viscoelastic layer. The bending modes and transverse response of such plates are obtained by adapting the Kirchhoff–Love thin plate formulation so that it considers the shear stiffness by using a frequency dependent equivalent flexural stiffness.The developed formulation is introduced in a finite element model and compared to the widely used RKU model and a reference 3D solid model in terms of eigenpairs and amplitude of the response for different boundary conditions.It is concluded that the developed formulation can substitute either a 3D or RKU model as it requires less computational resources than the former and provides more accurate results than the latter.

Spectral inequality and resolvent estimate for the bi-Laplace operator

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called “clamped” boundary conditions, that is, homogeneous Dirichlet and Neumann conditions simultaneously.We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp (C \mu^{1/4}) where \mu is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application, we deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator, in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.

Design and Implementation of a Novel Tilt Sensor Based on the Principle of Variable Reluctance.

Tilt angle measurement in dynamic systems is problematic because the rotation of the measured platform is coupled with translation. Therefore, when some sensors are applied in dynamic systems, their output signals are often submerged in the noise signals generated by translation. To enhance the ability of tilt sensors to resist translational noise, a dynamic tilt sensor is proposed based on the principle of variable reluctance from the perspective of sensor structure. The eccentric structure of the sensor constructed with a shell, liquid, and internal damping plate was designed according to the principles of mechanics. The characteristic of translational acceleration restraint determined by the sensor structure was established theoretically. In addition, the magnetic circuit of the sensor was analyzed to illustrate the sensor’s working principles. A Clapp oscillator circuit was designed to convert mechanical motion into a measureable electrical signal. A method to determine the sensor’s direction of rotation is proposed. A waveform conversion circuit was designed to convert the sine wave output of the Clapp oscillator to a square wave, and a square-wave frequency measurement circuit was designed based on the C8051 micro-control unit. A translation–rotation experimental hardware platform was constructed. The data acquisition program was designed on a PC platform, and the translation–rotation experiments were conducted with an MTi attitude measurement unit as a reference. The validity of the tilt angle measurements and the effect of the translational acceleration restraint of the sensor were verified by the experimental data. The theoretical results obtained were consistent with the experimental data, verifying the validity of the theoretical analysis and experimental devices employed. A measurement range of −180 to 180° was achieved.

Dynamic analysis of plates with thick unconstrained layer damping

In this work the dynamic behaviour of free layer damping plates with thick unconstrained viscoelastic layer is analysed. With this aim, the Kirchhoff–Love thin plate formulation is adapted to take shear stiffness into account by using a frequency dependent equivalent flexural stiffness.To check the validity of the proposed formulation, it is implemented in a finite element model and compared to both the Oberst model and a reference 3D solid model in terms of eigenvalues and dynamic response across a wide range of boundary conditions – clamped in all edges, free, simply supported at edges, clamped in a single edge and simply supported at corners. In all the cases the material of the viscoelastic layer presents fractional damping, its modulus thus being complex and frequency dependent.The results show that the proposed model is in good agreement with the 3D model and that it provides better accuracy than the Oberst model, specially as the thickness of the damping layer, and consequently the effect of the shear, increases. Hence, the need of developing a 3D solid model can be avoided as well as the storage and computation time problems it arises, that are specially critical in industrial practical applications.

Study on the influence of damping plates on structural vibration and radiation characteristics of the railway box girder

The influence of damping plates on structural vibration, noise and the contribution of the box girder bridge panels was analysed using the model test and numerical calculation methods. In this paper, a 1/10 scale box girder model test system was designed and constructed using a simply supported 32 m box girder from the Beijing–Shanghai high-speed railway as a prototype, and damping plates were laid on the box girder web side after the model was verified as being similar to the prototype by means of a series of model tests. The effects of the damping plates on the reduction of the structural vibration and noise were evaluated according to scaled model acoustic tests. In addition, the influence of the damping plates on the sound contribution of the panel was analysed based on the transient boundary element method. The study shows that the damping plates had an obvious vibration reducing effect on the concrete box girder, though their noise reduction properties were less than ideal. The vibration level reduction and time–history noise level reduction were kept within 10.2–29.5 dB and around 1.0 dB (A), respectively. The noise would decrease to its lowest value at 250 Hz – down to 4.0 dB. The damping plates were helpful in effectively reducing the sound power contribution and the sound pressure contribution of the concrete panel. At the peak frequencies of 17 Hz and 251 Hz, the sound power contributions of the panel on the damper side were reduced by 3.4 dB and 3.7 dB, respectively, compared with the normal side, while the sound pressure contributions on the panel decreased by 4.9 dB and 4.2 dB, respectively.

Active vibration control of thin constrained composite damping plates with double piezoelectric layers

Vibrations and the damping behaviour of thin constrained composite plates with double piezoelectric layers are analytically explored by using Fourier transformation and classical laminated plate theory. Electric potential equations in the double piezoelectric layers are solved with respect to closed and open circuit boundary conditions, an exterior dielectric slab and active control. The natural frequencies and loss factors of the constrained smart composite plates with passive control methods are not notably changed in comparison with those of the constrained composite plates without piezoelectric effects since vibrational energy does not efficiently convert to electrical energy. The loss factors of the composite plates with active constrained damping increase and the natural frequencies have significant variations as the proportional derivative gains increase. Transverse displacement power spectra of the piezoelectric composite plates with active control are compared with those of the piezoelectric composite plates with passive control showing that active control has the best suppression performance of vibrations for the constrained laminated plates with double piezoelectric layers. Radial power spectral density, phase angles and cylindrical-wave power spectral density are calculated. Interesting patterns of wave propagation are explained when plane wave expansion is used to obtain Bessel cylindrical waves.