Receptances of non-proportionally and continuously damped plates—Reduced dampers method

This paper presents a method for the dynamic analysis of continuously and non-proportionally damped plates in bending modes. The damping can be in the form of constrained or unconstrained layers. The method is an extension of the equivalent dampers method discussed in a previous paper, in which the damping matrix of a discretized plate is replaced by a diagonal equivalent damping matrix. Each diagonal element represents an equivalent damper inserted between the structure and ground. In this method the number of equivalent dampers is reduced so that the receptance matrix of the damped structure can be obtained economically by a direct matrix method. The receptances of two different partially coated plates in transverse directions are computed by the method suggested. The verification of the results is demonstrated by comparison with the experimental values and also with the theoretical values obtained by the equivalent dampers method. The method presented can also be applied to the transverse vibration analysis of plates with discrete damping inserts.     

A new method for harmonic response of non-proportionally damped structures using undamped modal data

A method of calculating the receptances of a non-proportionally damped structure from the undamped modal data and the damping matrix of the system is presented. The method developed is an exact method. It gives exact results when exact undamped receptances are employed in the computation. Inaccuracies are due to the truncations made in the calculation of undamped receptances. Numerical examples, demonstrating the accuracy and speed of the method when truncated receptance series are used are also presented. Advantages of the method over classical methods are discussed, and it is concluded that the method is most advantageous when used for a structure with frequency and/or temperature dependent damping properties, or when the non-proportional part of the damping is local. The technique suggested can easily be applied to structural modification problems if there is no additional degree-of-freedom due to the modifying structure.     

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.     

A method of treatment the damped quantum oscillator

A functional-integration method allowing to treat the damped quantum oscillator in equilibrium without making special choice of the heat-bath Hamiltonian is introduced. Exact results for the damped quantum oscillator calculated recently are shown to represent a special case of the present treatment.     

A novel method to quantize systems of damped motion

A novel method to quantize systems of damped motion is proposed in the frameworks of canonical quantization and path integral quantization. It can be afforded by considering a Lagrangian multiplied by a time-dependent function, which may represent an effective interaction with “environment.”     

Discrete element method model and damping performance of bean bag dampers

Bean bag dampers (BBDs) have been widely applied in engineering to attenuate the vibration of a structural system, but the theoretical analysis on BBDs has been scarcely reported because of their nonlinear damping performance and complex mechanism. In this work, a three-dimensional model of a BBD was established based on the discrete element method (DEM); its flexible boundary was discretized. The model was verified by comparing simulation with test data. Based on the model, the selection of proper particle diameter on the flexible boundary of the BBD was discussed first, and then the effects of internal particle size of the BBD, the BBD?s tightness and the gap between BBD and the inner wall of its enclosure on the energy dissipation capacity were studied. Moreover, the filling ratio of BBD (total internal particles? volume/the flexible boundary?s capacity) was defined to quantitatively describe the tightness of BBD, and the effects of the internal particle size, the natural frequency of primary system and the enclosure size on the optimum tightness of the BBD were also considered. The results can be used as a guide in the design of BBDs.     

A method for modal identification of lightly damped structures

In many cases modal tests are conducted on individual components of complex engineering structures where interest is confined to deriving an undamped model of the structure. A method is proposed for this task which demands a minimum of input data and which, in particular, does not require accurate measurements around resonance. The method is simple to program and its application to various practical structures is described.     

Numerical solution of fractionally damped beam by homotopy perturbation method

This paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.     

Conformal structure-preserving method for damped nonlinear Schrdinger equation

In this paper,we propose a conformal momentum-preserving method to solve a damped nonlinear Schrodinger(DNLS) equation.Based on its damped multi-symplectic formulation,the DNLS system can be split into a Hamiltonian part and a dissipative part.For the Hamiltonian part,the average vector field(AVF) method and implicit midpoint method are employed in spatial and temporal discretizations,respectively.For the dissipative part,we can solve it exactly.The proposed method conserves the conformal momentum conservation law in any local time-space region.With periodic boundary conditions,this method also preserves the total conformal momentum and the dissipation rate of momentum exactly.Numerical experiments are presented to demonstrate the conservative properties of the proposed method.     

Particle path tracking method in two- and three-dimensional continuously rotating detonation engines

The particle path tracking method is proposed and used in two-dimensional（2D） and three-dimensional（3D） numerical simulations of continuously rotating detonation engines（CRDEs）. This method is used to analyze the combustion and expansion processes of the fresh particles, and the thermodynamic cycle process of CRDE. In a 3D CRDE flow field, as the radius of the annulus increases, the no-injection area proportion increases, the non-detonation proportion decreases, and the detonation height decreases. The flow field parameters on the 3D mid annulus are different from in the 2D flow field under the same chamber size. The non-detonation proportion in the 3D flow field is less than in the 2D flow field. In the 2D and 3D CRDE, the paths of the flow particles have only a small fluctuation in the circumferential direction. The numerical thermodynamic cycle processes are qualitatively consistent with the three ideal cycle models, and they are right in between the ideal F–J cycle and ideal ZND cycle. The net mechanical work and thermal efficiency are slightly smaller in the 2D simulation than in the 3D simulation. In the 3D CRDE, as the radius of the annulus increases, the net mechanical work is almost constant, and the thermal efficiency increases. The numerical thermal efficiencies are larger than F–J cycle, and much smaller than ZND cycle.     

A new damped least-squares method for the calculation of molecular force fields

A modified Gauss method for the calculation of molecular force fields from spectroscopic data is presented. Comparison with other methods shows that it is capable of overcoming many of the frequently occurring convergence problems in least-squares force field optimization. The method has been tested by calculating valence and Urey-Bradley force fields for formaldehyde. The results are in good agreement with other force field calculations on the same molecule.     

An efficient method for determining the effects of mass modifications in damped systems

An analytical procedure is presented for determining the effects of mass modifications in viscously damped vibratory systems. A very general but straightforward formulation is given which simplifies considerably for positive definite systems with distinct eigenvalues, the most common case. The procedure permits successive modifications, is reversible, provides accurate results, and significantly reduces computation time. The complete solution of the unmodified system is not a prerequisite for applying the method, although if it is available an exact solution for the modified system can be obtained. A stable numerical scheme for locating the complex roots of the characteristic equation is given, and numerical results are provided to demonstrate the theory.     

Lie transformation method on quantum state evolution of a general time-dependent driven and damped parametric oscillator

A variety of dynamics in nature and society can be approximately treated as a driven and damped parametric oscillator. An intensive investigation of this time-dependent model from an algebraic point of view provides a consistent method to resolve the classical dynamics and the quantum evolution in order to understand the time-dependent phenomena that occur not only in the macroscopic classical scale for the synchronized behaviors but also in the microscopic quantum scale for a coherent state evolution. By using a Floquet U-transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamic behaviors of a driven and damped parametric oscillator to obtain the optimal solutions by means of invariant parameters of K$K$s to combine with Lewis–Riesenfeld invariant method. This approach can discriminate the external dynamics from the internal evolution of a wave packet by producing independent parametric equations that dramatically facilitate the parametric control on the quantum state evolution in a dissipative system. In order to show the advantages of this method, several time-dependent models proposed in the quantum control field are analyzed in detail.     

A real decoupled method and free interface component mode synthesis methods for generally damped systems

This paper reports on the development of a new transformation method. In contrast to most existing mode transformation methods in which the first-order state-space equation of the damped vibration system is transformed into a decoupled form with complex coefficient matrices, using the decoupled method presented in this paper, the equation of the damped system can be decomposed into a decoupled equation with real coefficient matrices. Two new free interface component mode synthesis methods are also presented. The equivalent full-mode matrix of the damped structure is used to capture the effects of the higher-order modes. Additionally, this work modifies the compatibility conditions at the junctions that are employed in most of the previous component mode synthesis methods for generally damped systems. The first component mode synthesis method is performed in complex space, whereas the second method can be applied in real space. Because the coefficient matrices of the coupled equation constructed by the second component mode synthesis method are all real-valued, the solution of the eigenproblem for this coupled equation can be performed in real space as well. Additionally, numerical examples demonstrate the accuracy and validity of these two component mode synthesis methods.