Proportional Damping Research Articles

A finite element analysis of ship sections subjected to underwater explosion

This paper presents numerical simulations of dynamic responses of a ship section to non-contact underwater explosion using ABAQUS. The finite element model of the ship section including the size of fluid mesh, initial and boundary conditions etc. has been built up. Comparisons of the acceleration and velocity response between the experimental and numerical results have been investigated. The numerical results agree well with the measured results. Furthermore, the effect of the mass proportional damping factor on the velocity response have been investigated numerically. The dynamic response modes of the ship section subjected to a side-on non-contact underwater explosion are discussed.

Band structure of phononic crystals with general damping

In this paper, we present theoretical formalisms for the study of wave dispersion in damped elastic periodic materials. We adopt the well known structural dynamics techniques of modal analysis and state-space transformation and formulate them for the Bloch wave propagation problem. First, we consider a one-dimensional lumped parameter model of a phononic crystal consisting of two masses in the unit cell whereby the masses are connected by springs and dashpot viscous dampers. We then extend our analysis to the study of a two-dimensional phononic crystal, modeled as a dissipative elastic continuum, and consisting of a periodic arrangement of square inclusions distributed in a matrix base material. For our damping model, we consider both proportional damping and general damping. Our results demonstrate the effects of damping on the frequency band structure for various types and levels of damping. In particular, we reveal two intriguing phenomena: branch overtaking and branch cut-off. The former may result in an abrupt drop in the relative band gap size, and the latter implies an opening of full or partial wavenumber (wave vector) band gaps. Following our frequency band structure analysis, we illustrate the concept of a damping ratio band structure.

모형개선을 위한 감쇠행렬 추정법의 비교

Finite element models of dynamic systems can be updated in two stages. In the first stage, mass and stiffness matrices are updated neglecting damping, and in the second stage, damping matrices are estimated with the mass and stiffness matrices fixed. Three methods to estimate damping matrices for this purpose are proposed in this paper. The methods include one for proportional damping systems and two for non-proportional damping systems. Method 1 utilizes orthogonality of normal modes and estimates damping matrices using the modal parameters extracted from the measured responses. Method 2 estimates damping matrices from impedance matrices which are the inverse of FRF matrices. Method 3 estimates damping using the equation which relates a damping matrix to the difference between the analytical and measured FRFs. The characteristics of the three methods are investigated by applying them to simulated discrete system data and experimental cantilever beam data.

State space modeling of non-proportional passive damping in machine tools

In the context of virtual development processes of mechatronic systems like machine tools, the preventive assessment of the effects of lightweight structural components on the characteristics of controlled feed systems and therefore on the power and energy requirement of machine tools is of great importance. The modeling of the damping is often limited in practical computation of real machine tools, because they are mostly limited to proportional damping characteristics. Yet the presence of discrete dampers, as for instance, in damping units in guides, or known diverse material damping in construction (components made of glass or carbon fiber-reinforced plastic), demands that the possibility of observance of non-proportional damping, in regard to later model updating, has to be given. The article demonstrates the theoretical state-space modeling of non-proportional damped mechanics using the example of a biaxial vertical lathe.

Improved reduced order solution techniques for nonlinear systems with localized nonlinearities

This paper examines the modeling and solution of large-order nonlinear systems with continuous nonlinearities which are spatially localized. This localization is exploited by a combined component mode synthesis (CMS)—dynamic substructuring approach for efficient model reduction. A new ordering method for the Fourier coefficients used in the Harmonic Balance Method (HBM) is proposed. This allows the calculation of the slave dynamic flexibility matrix, using simple analytical expressions thus saving considerable computational effort by avoiding inverse calculation. This procedure is also capable of handling proportional damping. A hypersphere-based continuation technique is used to trace the solution, and hence track bifurcations since it has the advantage that the augmented Jacobian matrix remains square. The reduced system is also solved using a time-variational method (TVM) which generates sparse Jacobian matrices when compared with HBM. Several systems including those with parametric excitation and internal resonances are solved to demonstrate the capability of the proposed schemes. A comparison of these techniques and their effectiveness in solving extremely strong nonlinear systems with continuous nonlinearities is discussed.

The transient dynamic response of multiphase magneto-electroelastic sensors bonded to a shell structure

The dynamic response of a magneto-electro-elastic sensor bonded to a cylindrical shell is addressed in this paper. A semi-analytical finite-element method is used to model the structure. The cylinder is subjected to an internal pressure and the response is studied for clamped—free and clamped—clamped boundary conditions with different sensor locations. Proportional damping is assumed and the Newmark-beta method is used as the solution technique. The response is found to be the maximum when the sensor is placed at the clamped end. It is advisable to place the sensor at the free end than in the middle to replicate the nature of load history.

A Reduced Second-Order Approach for Linear Viscoelastic Oscillators

This paper proposes a new approach for the reduction in the model-order of linear multiple-degree-of-freedom viscoelastic systems via equivalent second-order systems. The assumed viscoelastic forces depend on the past history of motion via convolution integrals over kernel functions. Current methods to solve this type of problem normally use the state-space approach involving additional internal variables. Such approaches often increase the order of the eigenvalue problem to be solved and can become computationally expensive for large systems. Here, an approximate reduced second-order approach is proposed for this type of problems. The proposed approximation utilizes the idea of generalized proportional damping and expressions of approximate eigenvalues of the system. A closed-form expression of the equivalent second-order system has been derived. The new expression is obtained by elementary operations involving the mass, stiffness, and the kernel function matrix only. This enables one to approximately calculate the dynamical response of complex viscoelastic systems using the standard tools for conventional second-order systems. Representative numerical examples are given to verify the accuracy of the derived expressions.

Advanced state space modeling of non-proportional damped machine tool mechanics

In the context of virtual development processes of mechatronic systems like machine tools, the preventive assessment of the effects of light-weight structural components on the characteristics of controlled feed systems and therefore on the power- and energy requirement of machine tools is of great importance. The modelling of the damping is often limited in practical computation of real machine tools, because they are mostly limited to proportional damping characteristics. Yet the presence of discrete dampers, as for instance, in damping units in guides, or known diverse material damping in construction (components made of glass- or carbon fibre reinforced plastic), demands that the possibility of observance of non-proportional damping, in regard to later model-updating, has to be given. The article demonstrates the theoretical state-space modelling of non-proportional damped mechanics using the example of a biaxial vertical lathe.

Effects of damping on the linear stability of a free-free beam subjected to follower and transversal forces

In this paper a free-free uniform beam with damping effects subjected to follower and transversal forces at its end is considered as a model for a space structure. The effect of damping on the stability of the system is first investigated and the effects of the follower and transversal forces on the vibration of the beam are shown next. Proportional damping model is used in this work, hence, the effects of both internal (material) and external (viscous fluid) damping on the system are noted. In order to derive the frequency of the system, the Ritz method has been used. The mode shapes of the system must therefore be extracted. The Newmark method is utilized in the study of the system vibration. The results show that an increase in the follower and transversal forces leads to an increase of the vibrational motion of the beam which is not desirable.

Comparison on numerical solutions for mid-frequency response analysis of finite element linear systems

Mid-frequency response analysis often faces computational difficulties when a conventional modal approach is used for a finite element linear system. In this paper, the computational burden is relieved by frequency sweep algorithm or mode acceleration method for a reduced-order system constructed by algebraic substructuring, a variant of model order reduction. The two methods are compared with the help of numerical experiments and their computational complexity. As demonstrated by the finite element simulations, in which proportional damping is assumed, of a turbo-prop aircraft and a ring resonator, the frequency sweep algorithm for reduced-order systems shows the best performance among all considered numerical solutions, including the conventional approach.

동적 시스템의 감쇠행렬 추정

Finite element models of dynamic systems can be updated in two stages. In the first stage, mass and stiffness matrices are updated neglecting damping. In the second stage, a damping matrix is estimated with the mass and stiffness matrices fixed. Methods to estimate a damping matrix for this purpose are proposed in this paper. For a system with proportional damping, a damping matrix is estimated using the modal parameters extracted from the measured responses and the modal matrix calculated from the mass and stiffness matrices from the first stage. For a system with non-proportional damping, a damping matrix is estimated from the impedance matrix which is the inverse of the FRF matrix. Only one low or one column of the FRF matrix is measured, and the remaining FRFs are synthesized to obtain a full FRF matrix. This procedure to obtain a full FRF matrix saves time and effort to measure FRFs.

Numerical Investigation of the Dynamic Response of Symmetric Laminated Composite Beams to Harmonic Excitations

The aim of this study is to investigate the dynamic response of a laminated composite beam subjected to a harmonic excitation by a numerical time integration method known as Newmark method. The finite element method based on the classical laminated plate theory is used in order to obtain structural stiffness. The structural damping is modelled as proportional damping which is referred to as Rayleigh damping and two different damping ratios are used. The effect of damping on the frequency response of the beam is investigated for a broad range of excitation frequency. The effect of excitation point on the harmonic response is also considered. Four different lay-up configurations namely [0]2s, [0/90]s, [45/-45]s and [90]2s are considered in order to show the effect of the stacking sequence on the frequency response of the beam. The numerical results presented in this study show that, the magnitude of the harmonic response of the beam reduces considerably as the damping ratio increases and [90]2s lay-up produces largest dynamic response due to the reducing flexural rigidity. Numerical results also show that the location and frequency of the harmonic excitation has important role on the dynamic response of the beam.

A practical method for proper modeling of structural damping in inelastic plane structural systems

This study addresses some of the pitfalls of conventional numerical modeling of Rayleigh-type damping in inelastic structures. A practical modeling approach to solve these problems is proposed. Conventional modeling of Rayleigh-type damping for inelastic structures generates responses in which unrealistic damping forces are present that results in underestimation of peak displacement demands, overestimation of peak strength demands, and underestimation of buildings' collapse potential. The approach proposed in this paper avoids these problems by modeling each structural element with an equivalent combination of one elastic element with stiffness-proportional damping, and two springs at its two ends with no stiffness proportional damping.

Comparison of external damping models in a large deformation problem

In many applications of flexible multibody dynamics, the magnitudes of damping forces are very small in comparison with the elastic and inertial forces. However, these small forces may have a very significant influence on responses near resonant frequencies. The role of damping is to remove the energy of a system by dissipation, and dissipative forces in structures can be the result of either internal or external damping. External damping includes aerodynamic and hydrodynamic drag and dissipation in the supports of structures, and internal damping is usually related to energy dissipation in materials. In large deformation problems, because of the flexibility of very thin structures, external damping is more important. Two types of damping models, proportional damping and quadratic damping, have been widely applied to flexible multibody dynamics. The advantages and weaknesses of the two damping models are considered in this study. To make up for the common drawbacks in these two models, a frequency-dependent generic damping model based on experimental modal analysis is proposed. The proposed damping model leads to a accurate correlation with experimental results because it directly uses the modal parameters of each mode obtained by experiment, and can represent exact high frequency behaviors simultaneously. To define and formulate a large deformation problem, the absolute nodal coordinate formulation (ANCF) was used, and computer simulations with the ANCF were compared to experimental results. Using the proposed experimental method, modal parameters and damping behaviors are extracted until 5th mode, which has a frequency of 89 Hz. It is shown that the common drawbacks of proportional and quadratic damping are complemented by the proposed generic damping model.

On the modeling of vibration transmission over a spatially curved cable with casing

We propose an approach to the vibration modeling of spatially curved steel wires with a casing and a contact between the outer casing and the inner steel wire. For the mathematical model of the steel wire and the outer casing, the Euler–Bernoulli beam theory with no axial pre-load is used, and for the discretisation, finite elements are used. The excitation of the steel wire and the outer casing is in the form of random kinematic excitation. For the energy dissipation the proportional viscous damping model and the structural damping model are used. The damping parameters are identified from the Nyquist diagram and from the continuous wavelet transform. For the identification of the frequency dependence of the dynamic modulus of elasticity a method is proposed that uses the measured natural frequencies and the experimentally determined natural modes. The contact between the steel wire and the outer casing is modelled using the penalty method with the friction in a tangential direction. We show that higher values of the friction coefficient have a significant influence on lowering the level of vibration transmission. The model also predicts that the dynamic modulus of the elasticity of a steel wire does not have a major influence on the level of vibration transmission, which was also validated by experiment. On the basis of an experimental validation the model of a steel wire with an outer casing proved to be suitable for the prediction of the vibration transmission.

On the relationship between viscous and hysteretic damping models and the importance of correct interpretation for system identification

In engineering practice, both viscous and hysteretic damping models are usually employed to characterize damped dynamic properties of linear mechanical systems, the identification of which has long been an active area of research in experimental modal analysis. This paper is devoted to investigate the relationship between viscous and hysteretic damping models. In order to show whether the arbitrary choice of damping model is reasonable, the relationship has been derived based on two normalization procedures and the modal parameter identification procedure. In case of proportional damping, there exists an exact relationship between hysteretic and viscous damping models as η r = 2 ξ r . As for non-proportional damping, the equivalent mode shapes and their counterparts of the original system are almost the same except differing by a complex scaling factor. The numerical cases based on the simulated modal identification procedure have demonstrated that the error in the estimation of modal parameters caused by arbitrarily choosing damping model is quite small. Furthermore, the equivalent damping matrix is physically sensible in the case of the system with distributed damping whereas, no physically sensible equivalent damping matrix exists in the case where the damping is localized. The validity of the relationship between the two damping models has been further verified by a practical experimental example.

Experimental Identification of Generalized Proportional Viscous Damping Matrix

A simple and easy-to-implement algorithm to identify a generalized proportional viscous damping matrix is developed in this work. The chief advantage of the proposed technique is that only a single drive-point frequency response function (FRF) measurement is needed. Such FRFs are routinely measured using the standard techniques of an experimental modal analysis, such as impulse test. The practical utility of the proposed identification scheme is illustrated on three representative structures: (1) a free-free beam in flexural vibration, (2) a quasiperiodic three-cantilever structure made by inserting slots in a plate in out-of-plane flexural vibration, and (3) a point-coupled-beam system. The finite element method is used to obtain the mass and stiffness matrices for each system, and the damping matrix is fitted to a measured variation of the damping (modal damping factors) with the natural frequency of vibration. The fitted viscous damping matrix does accommodate for any smooth variation of damping with frequency, as opposed to the conventional proportional damping matrix. It is concluded that a more generalized viscous damping matrix, allowing for a smooth variation of damping as a function of frequency, can be accommodated within the framework of standard finite element modeling and vibration analysis of linear systems.

Optimum study on wind-induced vibration control of high-rise buildings with viscous dampers

In this paper, optimum methods of wind-induced vibration control of high-rise buildings are mainly studied. Two optimum methods, genetic algorithms (GA) method and Rayleigh damping method, are firstly employed and proposed to perform optimum study on wind-induced vibration control, six target functions are presented in GA method based on spectrum analysis. Structural optimum analysis programs are developed based on Matlab software to calculate wind-induced structural responses. A high-rise steel building with 20-storey is adopted and 22 kinds of control plans are employed to perform comparison analysis to validate the feasibility and validity of the optimum methods considered. The results show that the distributions of damping coefficients along structural height for mass proportional damping (MPD) systems and stiffness proportional damping (SPD) systems are entirely opposite. Damping systems of MPD and GAMPD (genetic algorithms and mass proportional damping) have the best performance of reducing structural wind-induced vibration response and are superior to other damping systems. Standard deviations of structural responses are influenced greatly by different target functions and the influence is increasing slightly when higher modes are considered, as shown fully in section 5. Therefore, the influence of higher modes should be considered when strict requirement of wind-induced vibration comfort is needed for some special structures.

Seismic Analysis of Non Proportionally Damped Two-Way Asymmetric Elastic Buildings Under Bi-Directional Seismic Ground Motions

This study investigates the effectiveness of the modal analysis using three-degree-of freedom (3DOF) modal equations of motion to deal with the seismic analysis of two-way asymmetric elastic systems with supplemental damping. The 3DOF modal equations of motion possessing the non proportional damping property enable the two modal translations and one modal rotation to be non proportional in an elastic state. The simple approximation method is to use the single degree-of-freedom (SDOF) modal equations of motion, which are obtained by neglecting the off-diagonal elements of the transformed damping matrix. One, one-story and one, three-story non proportionally damped two-way asymmetric buildings under the excitation of bi-directional seismic ground motions are analyzed. The analytical results are obtained by using the proposed method, noted simple approximation method, and direct integration of the equation of motion. It is seen that the proposed method can significantly improve the accuracy of the analytical results compared with those obtained by using the simple approximation method. Moreover, the proposed method does not substantially increase the computational efforts.

Causality and mathematical models in vibration and acoustics: A realistic perspective.

Acoustic and vibrations applications require stricter causality conceptions than primitive causality (effect never precedes cause). Relativistic causality (nothing travels faster than light) is irrelevant; there is no practical reason that a satisfactory continuum‐mechanical model, holding for low to moderate frequencies, be relativistically invariant. Modifying the requirement so that the speed of light is replaced by some acoustic speed is not satisfactory, as the insertion of dissipative mechanisms into any continuum‐mechanical model invariably results in small precursors which may precede sonic‐velocity wavefronts. These are small, but they are formally nonzero at arbitrarily large distances in advance of the front. The present paper follows Ginzberg (1955) and advocates acoustic causality: the requirements that (i) the precursors die out rapidly with distance, (ii) up to some frequency of interest acoustic disturbances genuinely propagate, with the attenuation per wavelength being substantially less than unity, and (iii) vibrations and propagation are governed by coupled partial differential equations. With these principles as a guide, approximate relations involving extrapolations into the complex plane are derived, and it is shown how error bounds can be placed on applications of various members of a derived family of Kramers–Kronig relations. Very low‐frequency relaxation processes account for proportional damping in vibrations.