518 On Vibration Characteristics of Shallow Shells Including Damping
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An analysis for vibration of cylindrical shallow shells with a constrained viscoelastic layer and metal layers is presented. In this paper, vibration damping properties of three layer type shallow shells with viscoelastic material are analyzed from both of SEM (Strain energy method) and CEM (Complex eigenvalue method). Furthermore, behavior and accuracy of each solution are examined. Eigenvalue and eigenvector (displacement functions) are analyzed based on the Ritz method, when natural frequency, modal loss factor and strain energy are obtained. In method (CEM), the elastic modulus of viscoelastic material is dealt with complex quantity considering material loss factor. The eigenvalue problem that is derived by means of minimizing the energy functional, is solved to determine the natural frequencies and modal loss factors. In method (SEM), modal loss factor is defined as a ratio of damping energy (strain energy consumed in viscoelastic layer) dissipated during a vibration cycle and total strain energy of the shell. The accuracy and validity of the present results from two methods are illustrated through investigation of convergence and comparison with the established results from the literature. From thebehavior of the solution, consideration and interpretation are tried with respect to accuracy and characteristic of each solutionmethod.Keywords:
Loss factor
Ritz method
Constrained-layer damping
Strain energy
Natural frequency
Optimal damping layout of the constrained viscoelastic damping layer on beam is identified with temperatures by using a gradient-based numerical search algorithm. An optimal design problem is defined in order to determine the constrained damping layer configuration. A finite element formulation is introduced to model the constrained layer damping beam. The four-parameter fractional derivative model and the Arrhenius shift factor are used to describe dynamic characteristics of viscoelastic material with respect to frequency and temperature. Frequency-dependent complex-valued eigenvalue problems are solved by using a simple re-substitution algorithm in order to obtain the loss factor of each mode and responses of the structure. The results of the numerical example show that the proposed method can reduce frequency responses of beam at peaks only by reconfiguring the layout of constrained damping layer within a limited weight constraint.
Constrained-layer damping
Loss factor
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Constrained-layer damping
Loss factor
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Optimum design of a viscoelastic damping layer which is unconstrainedly cohered on a steel plate is discussed from the viewpoint of the modal loss factor. Themodal loss factor is analyzed by using the energy method to the base steel plate and cohered damping layer. Optimum distributions of the viscoelastic damping layer for modes are obtained by sequentially changing the position of a piece of damping layer to another position which contributes to maximizing the modal loss factors. Analytical procedure performed by using this method simulated for 3 fundamental modes of an edge-fixed plate. Simulated results indicate that the modal loss factor ratios can be increase by as much as 210%, or more, by optimizing the thickness distribution of the damping layer to two times of the initial condition which is entirely covered. Optimum configurations for the modes are revealed by positions where added damping treatments become most effective. The calculated results by this method are validated by comparison with the experimental results and the calculated results obtained by the Ross-Ungar-Kerwin's model in the case of the layer is uniformly treated over the steel plate.
Loss factor
Constrained-layer damping
Position (finance)
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The optimum configuration of the composite plate covered with an unconstrained viscoelastic damping layer is discussed from the viewpoint of the modal loss factor. When the configuration of a damping layer is optimized to an arbitrary mode, it's modal loss factor becomes maximum. In general, however, loss factors of the other modes are much lower than those obtained by optimization to each mode. Therefore, in this study, the optimum configuration of a damping layer is investigated under consideration of modal loss factor to some modes. The optimum distribution of the damping layer is obtained by sequentially changing the position of a section of the layer to another position to optimize the composite modal loss factor. Results are compared with the modal loss factors of (1, 1), (1, 2) and (1, 3) modes, which are optimized to their individual modes. These results indicate that although the maximum loss factor is decreased by 2∼8% compared to those optimized only to each mode, those of the other modes could be increased by 5∼23%. The results are validated by suitable experimental results.
Loss factor
Constrained-layer damping
Mode (computer interface)
Position (finance)
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In the present study, optimum distributions of unconstrained viscoelastic damping layer elements for plates are discussed from the viewpoint of the modal loss factor. The modal loss factor is expressed in terms of both the mechanical properties of the plate and damping layer and their thickness ratio. Optimum distributions of the thickness ratios of elements are obtained by sequentially changing the position of a piece of layer to another position which contributes to maximizing the modal loss factors. Results are presented for (1, 1) and (1, 3) modes of edge-fixed plates. These results indicate that the modal loss factor can be increased by as much as 200%, or more, by optimizing the thickness distribution of the damping treatment. The optimum configurations of the (1, 1) and (1, 3) mode of plates are revealed by positions where added damping treatments become most effective. The results are validated by both suitable experimental results and the calculated results obtained by the Ross-Kerwin-Ungar's model in the case of a uniformly treated viscoelastic damping layer over the entire area of the plate.
Loss factor
Constrained-layer damping
Position (finance)
Aspect ratio (aeronautics)
Mode (computer interface)
Damping ratio
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Uncontrolled vibration develops into a serious of problems in machinery and structures like the collapse of Tacoma Narrow bridge. Damping is the simplest method of control the vibrations by limiting the amplitudes and accelerations of the structure. Out of the two types of structural damping passive damping and active damping active damping involves complicated electronic gadgets and is suitable for highly sensitive structures. Structural damping in metals can be greatly enhanced by two passive damping techniques namely Free Layer Damping (FLD) and Constrained Layer Damping (CLD). In case of CLD the constraining layer thickness was optimized by Ross, Kerwin and Ungar (RKU) analysis. The constraining layer made of conventional metals increases the weight of the structure. Hence Fiber Reinforced Plastic (FRP) composite made of 70% epoxy and 30% E glass fibre were used as constraining layer as it is having higher complex Youngs modulus and shear modulus. Different beams were made using mild steel (MS), FLD by adding hard rubber as viscoelastic material and CLD using FRP composites as constraining layer. These beams were modelled using ANSYS and analyzed as per RKU method. Loss factor was calculated using OBERST beam technique and RKU equations. It was found that numerical analysis validates the experimental results. A constraining layer thickness of 2 mm of FRP composite increases the loss factor by 18.69% and reduces the frequency responses by 21%. In conclusion both FLD as well as CLD increases the loss factor and decreases the responses significantly CLD gives better performance.
Constrained-layer damping
Loss factor
Damping capacity
Damping ratio
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The composite with interleaved acrylonitrile butadiene rubber(NBR) layer was fabricated by the co-curing process.The temperature spectrum of loss factor for the composite were tested using dynamic mechanical analysis(DMA) equipment.The damping properties and damping mechanism of the composite were investigated at different temperatures.The result indicates that the loss factor of the composite is small and almost invariant with temperature when the temperature is within an elastomeric state and glassy state of the damping layer.The loss factor increases to the peak value and then decreases rapidly when the temperature is within a viscous state of the damping layer.The max loss factor of composites with interleaved damping layer is 19.2% and is about 13 times of the composite without interleaved damping layer.The damping properties of the cocured composite decrease with the decrease of the loss factor of the damping layer during the cocurig process,and the loss factor by DMA is less than that by prediction in the viscous state of the damping layer remarkably.The damping properties of the cocured composite increase owing to the damping of the interface,and the loss factor by DMA is higher than that by prediction when the damping layer is in the glassy state.
Loss factor
Constrained-layer damping
Damping capacity
Acrylonitrile butadiene styrene
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This paper analyzes the natural frequency and modal loss factor of stiffened plates with viscoelastic damping treatment by finite element method. Suitable arrangements of elastic and viscoelastic materials can control effectively structural vibration over a wide frequency range. In this paper two cases of stiffened plates with damping treatment are considered and compared: only with free damping layer and only with viscoelastic beam. In the finite element analysis, the degrees of freedom considered at each node of plate elements with four nodes are the displacement w , the slopes θ x and θ y , each node of beam elements with two nodes also has three degrees of freedom as the node of plate elements by omitting the displacements u and v , and the slope θ z of each node of the space beam elements considering influence of eccentricity. In the analysis, viscoelastic material has complex modulus and viscoelastic elements are the same as elastic elements. In each case, natural frequencies of the stiffened plate change and model loss factors increase. Results indicate that the influences of damping treatment are different. For different cases the numerical values of modal loss factors are different and the order of maximal modal loss factor is different. Suitable arrangements of viscoelastic damping treatments of the free damping layer and the viscoelastic beam can lead to good effect of controlling vibration and sound radiation from the stiffened plate. The influences of viscoelastic material modulus and loss factor, thickness of free damping layer and section of viscoelastic beam on the matural frequency and model loss factor of the stiffened plate also are analyzed.
Loss factor
Constrained-layer damping
Natural frequency
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An analysis for damping of shallow shells with a constrained viscoelastic layer and metal layers is presented. In this paper, vibration damping properties of three layered shallow shells are analyzed by SEM (Strain energy method). Eigenvalue and eigenvector (displacement functions) are analyzed based on the Ritz method, when natural frequency, modal loss factor and strain energy are obtained. The modal loss factor is defined as a ratio of damping energy (strain energy consumed in viscoelastic layer) dissipated during a vibration cycle and total strain energy of the shell. For numerical examples, damping energy distributions in the core are presented, and the concept of effective damping patch design are considered.
Loss factor
Constrained-layer damping
Strain energy
Damping ratio
Natural frequency
Ritz method
Damping capacity
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