Proportional Damping Research Articles

Rayleigh Damping in Abaqus/Explicit Dynamic Analysis

In the nonlinear time-history analysis for high-rise building, different damping models may lead to results with great difference. Limited by the stable time increment, the stiffness proportional damping is usually neglected in Abaqus/Explicit, and only a simplified mass proportional damping is used, which can be derived from the fundamental frequency and the critical damping factor. In this paper, the influence on the results of this method has been researched by numerical examples, the numerical results also show that the stiffness proportional damping cannot be neglected. Based on these comparisons, a more reasonable damping model is presented for improving the results of high frequencies.

A Model for Analyzing the Startup Dynamics of a Belt Transmission Driven by a DC Motor

In this paper the vibration of a synchronous belt drive during start-up is analyzed and discussed. Besides considering the belt elasticity, the model here proposed also takes into consideration the electromagnetic response of the DC motor. The solution of the motion equations is obtained by means of the modal analysis in state space, which allows to obtain the decoupling of all equations, without introducing the hypothesis of proportional damping. The mathematical model of the transmission and the solution algorithms have been implemented within a computing software that allows the user to simulate the dynamics of the system and to evaluate the effects due to the elasticity of the belt branches and to the electromagnetic behavior of the DC motor. In order to show the details of the calculation procedure, the paper presents a case study developed with the aid of the above-mentioned software. Keywords—Belt drive, Vibrations, Startup, DC motor.

The Damping Forced Vibration of Rectangular Stiffened Plates with Elastic Boundary Edges Including Boundary Damping

The paper proposes an analytical-numerical method for damping forced vibration problem of the rectangular stiffened plates with elastic boundary restraints. By enforcing applicable continuity conditions between plate and beams, coupled equations of beams-plate system are established. The displacement function is expressed as a modified two-dimensional Fourier series, which converts the differential equations of plate-beam systems into linear algebraic equations. Furthermore, Rayleigh’s proportional damping is introduced into beams-plate system so that damping parameters of stiffened plate can be inserted into vibration equations. It is derived that the present method demonstrates good agreement with standard finite element analysis. In subsequent analysis, it can be found that the boundary damping can reduce vibration energy of stiffened plate to same extent.

Controls on Material and Mesh for Structural Nonlinear Time-History Analysis

<strong> </strong>In order to improve the solution of nonlinear time-history analysis for high-rise buildings, the concrete constitutive relation of Chinese code has been modified by using Najar damage theory, and based on the limitation of stable time increment in explicit dynamic analysis, a simplified Rayleigh damping form with tiny stiffness proportional damping is proposed to decrease the damage induced by those high frequencies, at last, the influence of mesh size on stress and damage is also discussed. The numerical results show that these principles are workable for more accurate results.

Nonlinear finite element analysis of three‐dimensional free and harmonically forced vibrations of stranded conductor cables

SUMMARYThe 3D finite deformation beam model originally developed by Simo is appropriately modified to derive a finite element formulation for the static and dynamic analysis of flexible electrical conductors. In contrast to what is currently carried out in the literature, a linear viscoelastic constitutive equation and an additional mass proportional damping mechanism are introduced to account for energy dissipation in a physically consistent way. The model is used for simulation of free and forced vibration tests performed on an actual electrical conductor. Energy balance calculations illustrate the reliability of the computations. The experiments reveal amplitude dependence of both stiffness and damping, pointing out the presence of material nonlinearity in the cable. The development of numerical models that can account for the amplitude dependence of bending stiffness and energy dissipation capacity is subject of current computational and experimental work. Copyright © 2014 John Wiley & Sons, Ltd.

Damping Matrices Revisited

AbstractAfter a brief review of the problems entailed by the so-called proportional damping matrices, this paper elaborates on a simple technique to construct damping matrices that guarantee the fraction of damping in each mode to be positive and adequate and accomplishes this without solving for either the modes or the natural frequencies. Although the resulting matrices do not generally satisfy the proportionality condition, this fact is irrelevant for the purposes to which such matrices are put.

Investigating the Performance of Hyperelastic Constitutive Models in Predicting Dynamic Characteristics of Elastomeric Components

In this paper, the performance of different hyperelastic constitutive models namely Money-Rivlin, Gent, Ogden and Arruda-Boyce to predict the dynamic characteristic of elastomeric components is investigated. An elastomeric engine mount is chosen as case study. Material Parameters of the different models are extracted by curve fitting on uni-axial stress-strain test data. Both static and dynamic response of the finite element model considering the hyperelastic constitutive models are compared with their measured counterparts and each other. To obtain frequency response function of the component for each model, at first considering proportional damping for elastomeric media, the transient analysis due to impulse excitation is performed on the component and then using FFT transformation, the frequency response function is achieved. The results show that hyperelastic constitutive models can be implemented in predicting both static and dynamic behavior of elastomeric components.

Modelling of a biologically inspired robotic fish driven by compliant parts

Inspired by biological swimmers such as fish, a robot composed of a rigid head, a compliant body and a rigid caudal fin was built. It has the geometrical properties of a subcarangiform swimmer of the same size. The head houses a servo-motor which actuates the compliant body and the caudal fin. It achieves this by applying a concentrated moment on a point near the compliant body base. In this paper, the dynamics of the compliant body driving the robotic fish is modelled and experimentally validated. Lighthill’s elongated body theory is used to define the hydrodynamic forces on the compliant part and Rayleigh proportional damping is used to model damping. Based on the assumed modes method, an energetic approach is used to write the equations of motion of the compliant body and to compute the relationship between the applied moment and the resulting lateral deflections. Experiments on the compliant body were carried out to validate the model predictions. The results showed that a good match was achieved between the measured and predicted deformations. A discussion of the swimming motions between the real fish and the robot is presented.

Characterization and synthesis of Rayleigh damped elastodynamic networks

We consider damped elastodynamic networks where the damping matrix is assumed to be a non-negative linear combination of the stiffness and mass matrices (also known as Rayleigh or proportional damping). We give here a characterization of the frequency response of such networks. We also answer the synthesis question for such networks, i.e., how to construct a Rayleigh damped elastodynamic network with a given frequency response. Our analysis shows that not all damped elastodynamic networks can be realized when the proportionality constants between the damping matrix and the mass and stiffness matrices are fixed.

Harmonic response analysis of symmetric laminated composite beams with different boundary conditions

Abstract This study deals with the determination of the harmonic response of symmetric laminated composite beams by the finite element method. The structural stiffness of the composite beam is determined by the classical laminated plate theory. Four different ply orientations, namely, [0]2s, [0/90]s, [45/-45]s, and [90]2s are used to examine the effect of the stacking sequence on the harmonic response of the beam. Proportional damping is used to model the structural damping, and the damped harmonic responses of the composite beams are obtained to show the effect of the damping on the harmonic response. The effect of the boundary conditions on the harmonic response is also investigated. The displacement maps calculated for varying excitation points are obtained for different boundary conditions and damping ratios at different vibrational modes. The numerical results presented in this study show that the magnitudes of the harmonic response of the composite beam increase as the flexural rigidity decreases, and the vibration magnitudes reduce considerably with damping. The vibration patterns created for varying excitation and observation locations change as the damping ratio and excitation frequency change.

Identification of modal parameters of non-stationary systems with the use of wavelet based adaptive filtering

The Operational Modal Analysis (OMA) is a common tool for identification of parameters of mechanical structures during operation. Modal analysis can be applied for linear, stationary and undamped systems or systems with small and proportional damping. To apply this technique to other systems, mainly to non-stationary systems, new procedures are required.The paper focuses on the application of time–frequency signal filtration to the recursive method of the modal parameters' identification based on operational measurements, dedicated for non-stationary systems. The presented technique uses an adaptive wavelet signal filtering method to separate signal components and reduce the model order. This approach considerably facilitates selection of the wavelet function parameters and significantly improves the quality of the separated modal components. Thanks to the reduction of model order, estimation of modal parameters can be performed using a relatively simple mathematical formula. This approach significantly reduces the demand for computing power which has a direct impact on system's costs and modal parameter's estimation time. This is particularly an important problem when the system parameters are changing rapidly and the information about this changes is required in real-time. The algorithm allows assessing the quality of the estimated parameters by simultaneous estimation of confidence bounds. The method has been tested on numerical models, experimental laboratory test rig and applied to real data.

Complex eigensolutions of coupled flexural and longitudinal modes in a beam with inclined elastic supports with non-proportional damping

Abstract Structure borne vibration and noise in an automobile are often explained by representing the full vehicle as a system of elastically coupled beam structures representing the body, engine cradle and body subframe where the engine is often connected to the chassis via inclined viscoelastic supports. To understand more clearly the interactions between a beam structure and isolators, this article examines the flexural and longitudinal motions in an elastic beam with intentionally inclined mounts (viscoelastic end supports). A new analytical solution is derived for the boundary coupled Euler beam and wave equations resulting in complex eigensolutions. This system is demonstrated to be self-adjoint when the support stiffness matrices are symmetric; thus, the modal analysis is used to decouple the equations of motion and solve for the steady state, damped harmonic response. Experimental validation and computational verifications confirm the validity of the proposed formulation. New and interesting phenomena are presented including coupled rigid motions, modal properties for ideal angled roller boundaries, and relationships between coupling and system modal loss factors. The ideal roller boundary conditions when inclined are seen as a limiting case of coupled longitudinal and flexural motions. In particular, the coupled rigid body motions illustrate the influence of support stiffness coupling on the eigenvalues and eigenfunctions. The relative modal strain energy concept is used to distinguish the contribution of longitudinal and flexural deformation modes. Since the beam is assumed to be undamped, the system damping is derived from the viscoelastic supports. The support damping (for a given loss factor) is shown to be redistributed between the system modes due to the inclined coupling mechanisms. Finally, this article provides valuable insight by highlighting some technical issues a real-life designer faces when balancing modeling assumptions such as rigid or elastic formulations, proportional or non-proportional damping, and coupling terms in multidimensional joint properties.

Modal analysis reduction of multi-body systems with generic damping

Modal analysis of multi-body systems is broadly used to study the behavior and controller design of dynamic systems. In both cases, model reduction that does not degrade accuracy is necessary for the efficient use of these models. Previous work by the author addressed the reduction of modal representations by eliminating entire modes or individual modal elements (inertial, compliant, resistive). In that work, the bond graph formulation was used to model the system and the modal decomposition was limited to systems with proportional damping. The objective of the current work is to develop a new methodology such that model reduction can be implemented to modal analysis of multi-body systems with non-proportional damping that were not modeled using bond graphs. This extension also makes the methodology applicable to realistic systems where the importance of modal coupling terms is quantified and potentially eliminated. The new methodology is demonstrated through an illustrative example.

A dynamic model of a double-helical planetary gear set

A linear, time-invariant model of a double-helical planetary gear set is proposed in this paper. The model formulation allows the analysis of a gear set with any number of planets, any planet phasing and spacing configurations and any support conditions. Torsional, transverse, axial and rotational (rocking) motions of gears and the planet carrier are included in this three-dimensional model. Planets are allowed to be positioned equally or unequally spaced around the sun gear. The natural modes are computed by solving the corresponding Eigen value problem. The modal summation technique is used to find the forced response due to periodic gear mesh excitations under constant or proportional modal damping conditions. An example gear set consisting of four equally-spaced planets is considered at the end to demonstrate the influence of key design parameters including right-to-left stagger angle, and support conditions on the dynamic response of the double-helical planetary gear system. Numerous modes with dominant tilting motions are predicted to underline the essence of using a 3D formulation for double helical gears. Staggering of gear teeth is shown to impact the dynamic response of the gear system substantially. Different values of stagger are shown excite different types of modes, resulting in different dynamic response curves. It is also found that left and right sides of double helical gears can carry equal or different dynamic load amplitudes based on the right-to-left stagger values. Likewise the influence of the sun support conditions is shown to be more pronounced for the case of a non-zero stagger.

Damping Models for Shear Beams with Applications to Spacecraft Wiring Harnesses

Spacecraft wiring harnesses can fundamentally alter a spacecraft’s structural dynamics, necessitating a model to predict the coupled dynamic response of the structure and attached cabling. Although a beam model including first-order transverse shear can accurately predict vibration resonance frequencies, current time domain damping models are inadequate. For example, common proportional damping models result modal damping that depends unrealistically on the frequency. Inspired by a geometric rotation-based viscous damping model that provides frequency independent modal damping in an Euler–Bernoulli formulation, a viscous damping model with terms associated with the shear and bending angles is presented. The model provides modal damping that is approximately constant in the bending-dominated regime (low mode numbers), increasing by at most 6% for a particular selection of bending and shear angle-based damping coefficients. In the shear-dominated regime (high mode numbers), damping values increase linearly with mode number and in proportion to the shear angle-based damping coefficient. A key feature of this shear beam damping model is its ready finite element implementation using only matrices commonly developed for an Euler–Bernoulli beam. Such an analysis using empirically determined damping coefficients generates damping values that agree well with existing spacecraft wiring harness cable data.

General shell model for a rotating pretwisted blade

A novel dynamic model for a pretwisted rotating compressor blade mounted at an arbitrary stagger angle using general shell theory and including the rotational velocity is developed to study the eigenfrequencies and damping properties of the pretwisted rotating blade. The strain–displacement relation and constitutive model based on the general (thick) shell theory are applied to bring out the strain energy of the rotating blade. Using Hamilton's principle, the variational form of the total energy is derived in order to obtain the corresponding weak form for the numerical simulation. The model is validated by comparing to the literature results and Ansys results, showing good agreement. Parametric analyses are carried out to study the influence of the rotation velocity, the stagger angle and the radius of the disk on the eigenfrequencies of the pretwisted blade. Proportional damping is included into the proposed model to investigate the influence of rotational velocity on the damping characteristics of the pretwisted rotating blade system. It is shown that, due to inertial and Coriolis effects, damping decreases as the rotation velocity increases for the lower part of the velocity range considered and either decreases or increases depending on the mode order for higher velocities. Furthermore, frequency loci veering as a result of the rotation velocity is observed. The proposed model is an efficient and accurate tool for predicting the dynamic behavior of compressor blades of arbitrary thickness, stagger angle and pretwist, potentially during the early designing stage of turbomachinery.

A viscous approach based on oscillatory eigensolutions for viscoelastically damped vibrating systems

Dissipative mechanisms in linear multiple-degree-of-freedom viscoelastic systems are characterized by the dependence on the history of the velocity degrees-of-freedom via convolution integrals over kernel functions. As a result, damping matrices are frequency-dependent and, in general, state-space approach based methods are widely used for their resolution. This paper proposes a new approach suitable for modelling such systems via a viscous model with proportional damping, which approximates the response of the original viscoelastic system using the undamped eigenvectors together with the complex eigenvalues with oscillatory nature, neglecting those modes with non-viscous nature (negative real eigenvalues). It is rigorously demonstrated that the transfer function of any viscoelastic system can be expanded as the sum of the transfer function of certain viscous model and a residual term. The obtained viscous model, named Equivalent Viscous Model, is characteristic of each viscoelastic system. In order to study the bound of the residual term two indexes are introduced to describe, on one hand, the non-proportionality, or the modal decoupling capability of the damping matrix and, on the other hand, the damping level of the system. The proposed viscous model is validated through numerical examples simulating different conditions of proportionality and damping.

In situ measurement of structural mass, stiffness, and damping using a reaction force actuator and a laser Doppler vibrometer

The problem of structural parameter measurements needs a total solution bridging theory and experiments. In this paper, a practical methodology for in situ measurements of structural mass, stiffness, and damping is presented for built-up structures. As for the experimental uniqueness of the methodology, a reaction force actuator and non-contact optical device are utilized respectively as an input force generator and output displacement measurer, providing a fundamental data set of the proposed numerical algorithm for data-driven structural parameter estimation. The algorithm autonomously estimates the diagonalized mass, symmetric stiffness, optimal non-proportional damping, and suboptimal proportional damping matrices for multi-degrees-of-freedom structures. Structural parameter measurements of two built-up structures followed by a comparison with conventional measurements are used as examples for verification of the accuracy of the proposed methodology.

Analysis of Damped Bloch Waves by the Rayleigh Perturbation Method

Bloch waves in viscously damped periodic material and structural systems are analyzed using a perturbation method originally developed by Rayleigh for vibration analysis of finite structures. The extended method, called the Bloch–Rayleigh perturbation method here, utilizes the Bloch waves of an undamped unit cell as basis functions to provide approximate closed-form expressions for the complex eigenvalues and eigenvectors of the damped unit cell. In doing so, we circumvent the solution of a quadratic Bloch eigenvalue problem and subsequent computationally intensive transformation to first order/state-space form. Dispersion curves of a one-dimensional damped spring-mass chain and a two-dimensional phononic crystal with square inclusions are calculated using the state-space method and the proposed method. They are compared and found to be in excellent quantitative agreement for both proportional and nonproportional viscous damping models. The perturbation method is able to capture anomalous dispersion phenomena—branch overtaking, branch cut-on/cut-off, and frequency contour transformation—in parametric ranges where state-space formulations encounter numerical issues. Generalization to other linear nonviscous damping models is permissible.

'Geometric' Viscous Damping Model for Nearly Constant Beam Modal Damping

This paper considers the damped transverse vibration of flexural structures. Viscous damping models available to date, such as proportional damping, suffer from the deficiency that the resulting modal damping is strongly frequency dependent, which is a situation not representative of experiments with built-up structures. The focus model addresses a viscous geometric damping term in which an internal resisting shear force is proportional to the time rate of change of the slope. Separation of variables does not lead directly to a solution of the governing partial-differential equation, although a boundary-value eigenvalue problem for free vibration can nevertheless be posed and solved. For small damping the method of weighted residuals provides an alternate approach to the development of approximate modal equations of motion and estimation of modal damping. In a discretized finite-element context the resulting damping matrix resembles the geometric stiffness matrix used to account for the effects of membrane loads on lateral stiffness. For beams having any combination of hinged and guided boundary conditions this model yields constant modal damping that is independent of frequency as well as real mode shapes. For more general boundary conditions modal damping varies somewhat, approaching the expected constant value with increasing mode number; furthermore, the mode shapes are complex. This viscous damping model should prove useful to researchers and engineers who need a simple time-domain damping model that exhibits more realistic variation of damping with frequency than the alternatives.