Damped Cylindrical Shell Research Articles

Solution of nonlinear eigenvalues for the viscoelastic damped cylindrical shell considering the frequency dependence of viscoelastic materials

The dynamic model of the cylindrical shell partially attached with a viscoelastic free damping layer is established by a semi-analytical method, and the complex boundary conditions of the cylindrical shell are simulated by using the non-uniform distribution of spring groups. Two new methods for solving nonlinear eigenvalue problems of the viscoelastic damped cylindrical shell are proposed. One method is named as the Vector Iteration method based on Approximate Eigenvalues (VIAE), and the solution process can be described as: the iteration equation is obtained by the characteristic equation; the approximate eigenvalue and the initial eigenvector are used for the iteration calculation; in each iteration, the eigenvectors are normalized and the nonlinear eigenvalues are obtained by using the Rayleigh quotient. Another method is named as the Secant Method of Characteristic Polynomials (SMCP). In this method, the iteration equation of the eigenvalue is derived by the secant method. Furthermore, the approximate eigenvalues of the system are used as the initial iteration eigenvalues and the nonlinear eigenvalues of the system can be approximated through several iteration calculations. Finally, the proposed modeling method and algorithm are verified by the experiment of the actual cylindrical shell attached the ZN-1 viscoelastic material at the free end. The two new methods were also compared with the classical Eigenvector Increment Method (EIM). The comparison results show that the VIAE algorithm has the highest efficiency, and both the VIAE and SMCP algorithms have wider applicability because they do not need to perform eigenvector calculations.

A unified method for the vibration and damping analysis of constrained layer damping cylindrical shells with arbitrary boundary conditions

In this paper, an accurate solution is developed for the vibration and damping characteristics of a three-layered passive constrained layer damping (PCLD) cylindrical shell with general elastically restrained boundaries. In this formulation, characteristic equations of the system are derived by using the modified Fourier–Ritz method in conjunction with Donnell shell assumptions and linear viscoelastic theory. Regardless of boundary conditions, the displacements of each layer are expanded as the linear combination of a standard Fourier series and closed-form functions introduced to eliminate all the relevant discontinuities with the displacements and derivatives at the edges. This method can be universally applicable to all classical boundaries, elastic boundaries and their combinations without any special change in the solution procedure. It provides an effective way to investigate the influence of restraints from different directions on the vibration and damping performance of PCLD shells. New results for elastic restraints and intermediate ring supports are presented, which may serve as benchmark solutions. Furthermore, the detailed effects of thickness of layers and shear parameter on natural frequencies and loss factors are also illustrated.

Modal analysis of thin cylindrical shells with cardboard liners and estimation of loss factors

Cardboard liners are often installed within automotive drive shafts to reduce radiated noise over a certain frequency range. However, the precise mechanisms that yield noise attenuation are not well understood. To overcome this void, a thin shell (under free boundaries) with different cardboard liner thicknesses is examined using analytical, computational and experimental methods. First, an experimental procedure is introduced to determine the modal behavior of a cylindrical shell with a cardboard liner. Then, acoustic and vibration frequency response functions are measured in acoustic free field, and natural frequencies and the loss factors of structures are determined. The adverse effects caused by closely spaced modes during the identification of modal loss factors are minimized, and variations in measured natural frequencies and loss factors are explored. Material properties of a cardboard liner are also determined using an elastic plate treated with a thin liner. Finally, the natural frequencies and modal loss factors of a cylindrical shell with cardboard liners are estimated using analytical and computational methods, and the sources of damping mechanisms are identified. The proposed procedure can be effectively used to model a damped cylindrical shell (with a cardboard liner) to predict its vibro-acoustic response.

Vibration Power Flow Analysis of a Submerged Constrained Layer Damping Cylindrical Shell

The vibration power flow in a submerged infinite constrained layer damping (CLD) cylindrical shell is studied in the present paper using the wave propagation approach. Dynamic equations of the shell are derived with the Hamilton principle in conjunction with the Donnell shell assumptions. Besides, the pressure field in the fluid is described by the Helmholtz equation and the damping characteristics are considered with the complex modulus method. Then, the shell-fluid coupling dynamic equations are obtained by using the coupling between the shell and the fluid. Vibration power flows inputted to the coupled system and transmitted along the shell axial direction are both studied. Results show that input power flow varies with driving frequency and circumferential mode order, and the constrained damping layer will restrict the exciting force inputting power flow into the shell, especially for a thicker viscoelastic layer, a thicker or stiffer constraining layer (CL), and a higher circumferential mode order. Cut-off frequencies do not exist in the CLD cylindrical shell, so that the exciting force can input power flow into the shell at any frequency and for any circumferential mode order. The power flow transmitted in the CLD cylindrical shell exhibits an exponential decay form along its axial direction, which indicates that the constrained damping layer has a good damping effect, especially at middle or high frequencies.

Analysis of Acoustic Radiation of Submerged Damped Cylindrical Shell Combined with Thin Shell Theory and Elasticity Theory

The acoustic radiation of submerged damped cylindrical shell is investigated by combining with thin shell theory and elasticity theory. The classical integral transform technique is used to derive the solutions of coupled equations. Numerical computations show the correctness of the proposed method, and also conclude the influence of thickness of inner shell and coated layer to acoustic radiation. The conclusions may be valuable to the application of damping material on noise and vibration control of underwater pipes.

Dynamic Performance Optimization of Cylindrical Shell with Constrained Damping Layer Using Topology

Topology optimization of the constrained damping cylindrical shell for the maximized modal damping coefficient is studied in this paper. According to the first 3 orders modes, a novel topological optimization model where the volume is taken as the constraint function and the function of the modal damping coefficient as the target function is proposed and analyzed. The reasonable topology configuration was obtained for the damping structure. The results show that this method can be applied to solve the problem of optimization of the constrained damping cylindrical shell with the damping materials volume constrained. Numerical simulations are performed to very the effectiveness of the presented method.

Acoustic radiation of damped cylindrical shell with arbitrary thickness in the fluid field

The insertion loss of acoustic radiation of damped cylindrical shell described by 3-D elasticity Navier equations under radial harmonic applied load in fluid is presented. The classical integral transform technique, potential theory and Lame resolution are used to derive the solutions of Navier equations. The higher precision inversion computation is introduced to solve the linear equations. Comparing with acoustic radiation of one-layer cylindrical shell, the influence of thickness, mass density, dilatational wave loss factor and Young’s modulus of damping material and circumferential mode number of the cylindrical shell on the insertion loss is concluded. The theoretical model in the paper can be used to deal with the arbitrary thickness and any frequency of the coated layer in dynamic problem. The conclusions may be of theoretical reference to the application of damping material to noise and vibration control of submarines and underwater pipes.

Analysis of the response of damped cylindrical shells carrying discontinuously constrained beam elements

A simple analysis is presented of the forced vibratory response of a cylindrical shell having a number of axial beams adhered to it by a viscoelastic material layer. The attached beams are identical, closely spaced and distributed around the full circumference of the shell. The excitation is a concentrated vibratory force acting radially at the mid-section on the surface of the shell. The end conditions of the shell and the attached beams are all assumed to be simply supported. The effects of the operational temperature and frequency on the viscoelastic material properties are considered. An experiment was conducted, for comparison, on a damped cylindrical shell suspended in air by lightweight elastic shock cords and driven at the mid-section by an electromechanical vibration shaker. Good correlations between the test data and analytical solutions were obtained over a wide range of frequencies.