Non-proportional Damping Research Articles

Analytical predictions for vibration phase shifts along fluid-conveying pipes due to Coriolis forces and imperfections

Resonant vibrations of a fluid-conveying pipe are investigated, with special consideration to axial shifts in vibration phase accompanying fluid flow and various imperfections. This is relevant for understanding elastic wave propagation in general, and for the design and trouble-shooting of phase-shift measuring devices such as Coriolis mass flowmeters in particular. Small imperfections related to elastic and dissipative support conditions are specifically addressed, but the suggested approach is readily applicable to other kinds of imperfection, e.g. non-uniform stiffness or mass, non-proportional damping, weak nonlinearity, and flow pulsation. A multiple time scaling perturbation analysis is employed for a simple model of an imperfect fluid-conveying pipe. This leads to simple analytical expressions for the approximate prediction of phase shift, providing direct insight into which imperfections affect phase shift, and in which manner. The analytical predictions are tested against results obtained by pure numerical analysis using a Galerkin expansion, showing very good agreement. For small imperfections the analytical predictions are thus comparable in accuracy to numerical simulation, but provide much more insight. This may aid in creating practically useful hypotheses that hold more generally for real systems of complex geometry, e.g. that asymmetry or non-proportionality in axial distribution of damping will induce phase shifts in a manner similar to that of fluid flow, while the symmetric part of damping as well as non-uniformity in mass or stiffness do not affect phase shift. The validity of such hypotheses can be tested using detailed fluid-structure interaction computer models or laboratory experiments.

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.

Model Reduction Methods for Rotor Dynamic Analysis: A Survey and Review

The focus of this literature survey and review is model reduction methods and their application to rotor dynamic systems. Rotor dynamic systems require careful consideration in their dynamic models as they include unsymmetric stiffness, localized nonproportional damping, and frequency-dependent gyroscopic effects. The literature reviewed originates from both controls and mechanical systems analysis and has been previously applied to rotor systems. This survey discusses the previous literature reviews on model reduction, reduction methods applied to rotor systems, the current state of these reduction methods in rotor dynamics, and the ability of the literature to reduce the complexities of large order rotor dynamic systems but allow accurate solutions.

Orthogonal polynomials or wavelet analysis for mechanical system direct identification

A unified formulation of a direct identification method for linear mechanical systems is proposed. Linear operators are applied to the set of motion differential equations, transforming it into an algebraic system. The cases of expansion on Chebyshev polynomials and of Cauchy continuous wavelet transform are studied with a focus on their similarities and differences in writing and performances. Both methods are illustrated and compared by applying them on numerical simulations of two different 3 degrees of freedom systems with non-proportional damping. The effect of additive white noise on signals is also investigated.

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.

Stochastic direct integration schemes for dynamic systems subjected to random excitations

In this work, a stochastic version of direct integration schemes is constructed, based on a general recursive state space formulation. The technique is applicable to evaluating the second order response statistics of systems subjected to non-stationary random excitations, and is potentially able to handle non-proportional damping. A specific group of these algorithms obtained using one of the most frequently used families of time stepping schemes is introduced. It is shown that statistics obtained for two multi-degree of freedom (MDOF) dynamic systems using the proposed method are in close agreement with those obtained from Monte Carlo simulation when a proper time step is used. Stability of the proposed scheme is systematically investigated and it is concluded that, for the response statistics to be bounded, the time step used in the algorithm has to lie within certain limits.

A Note on Nonproportional Damping

This note deals with three aspects of nonproportional damping in linear damped vibrating systems in which the stiffness and damping matrices are not restricted to being symmetric and positive definite. First, we give results on approximating a general damping matrix by one that commutes with the stiffness matrix when the stiffness matrix is a general diagonalizable matrix, and the damping and stiffness matrices do not commute. The criterion we use for carrying out this approximation is closeness in Euclidean norm between the actual damping matrix and its approximant. When the eigenvalues of the stiffness matrix are all distinct, the best approximant provides justification for the usual practice in structural analysis of disregarding the off-diagonal terms in the transformed damping matrix. However, when the eigenvalues of the stiffness matrix are not distinct, the best approximant to a general damping matrix turns out to be related to a block diagonal matrix, and the aforementioned approximation cannot be justified on the basis of the criterion used here. In this case, even when the damping and stiffness matrices commute, decoupling of the modes is not guaranteed. We show that for general matrices, even for symmetric ones, the response of the approximate system and the actual system can be widely different, in fact qualitatively so. Examples illustrating our results are provided. Second, we present some results related to the difficulty in handling general, nonproportionally damped systems, in which the damping matrix may be indefinite, by considering a simple example of a two degrees-of-freedom system. Last, we use this example to point out the nonintuitive response behavior of general nonproportionally damped systems when the damping matrix is indefinite. Our results point to the need for great caution in approximating nonproportionally damped systems by damping matrices that commute with the stiffness matrix, especially when considering general damping matrices. Such approximations could lead to qualitatively differing responses between the actual system and its proportionally damped approximation.

Effects of full-height nonstructural partitions on modal properties of two nominally identical building floors

This paper presents a combined experimental and numerical investigation of the modal properties of two full-scale and nominally identical steel–concrete composite floors. The floors were one above the other in the same fully operational multi-storey building. Both floors accommodated open-plan as well as partitioned offices. Multi-input-multi-output (MIMO) modal testing was employed to measure as-built modal properties of both floors. It was found that the two nominally identical floors had different modal characteristics, likely due to the different arrangement of partitions in the floor. It was also found that the measured modes on both floor levels experienced a considerable level of complexity, likely to be caused by nonproportional damping. Finite element (FE) models were developed in ANSYS for both floors using best engineering judgement and their features and properties were then tuned to match the measured counterparts. The tuning was done manually by trial-and-error and then automatically using sensitivity-based FE model updating procedure implemented in the FEMtools software. It was found that the initial and geometrically very detailed FE models, which did not feature any nonstructural components, underestimated the measured natural frequencies by up to a considerable 20%–25%, depending on the floor level. When full-height plasterboard and glass partitions were explicitly modelled as vertical springs connected to the floor and grounded at the other end, the correlation between the experimental and FE results improved considerably.

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.

Damping estimation using free decays and ambient vibration tests

This paper studies the quality of modal damping ratios estimates based on ambient and free vibration tests using both numerical simulations and data collected on large Civil Engineering structures where both tests were performed. The simulated data allowed to study the influence of factors like non-proportional damping or proximity of natural frequencies on the quality of the estimates and also to illustrate the influence of the identification algorithms parameters on the accuracy of the results. The analysis of data collected on a cable-stayed bridge, on the suspended roof of a stadium and on a footbridge permitted to compare the results of both testing approaches in real applications where some factors that cannot be realistically included in the numerical simulations can play an important role. The processing of ambient vibration responses is performed with two output-only identification approaches: frequency domain decomposition and stochastic subspace identification methods. The numerically simulated and the measured free decays were analyzed with a simple method based on filters and fitting of exponential decays and also with the use of subspace models.

Seismic History Analysis of Asymmetric Buildings with Soil–Structure Interaction

An efficient approximate method that uses the multi-degrees-of-freedom (MDOF) modal equations of motion for the seismic response history analysis of asymmetric elastic buildings with soil–structure interaction (SSI) is presented in this paper. The systems considered are two-way asymmetric shear buildings resting on the surface of an elastic half-space, which are excited by two-directional seismic ground motions. The SSI forces were simulated using frequency-independent soil springs and dashpots. First, the MDOF modal equations of motion for the SSI systems were derived. The modal response histories were obtained by solving the MDOF modal equations of motion using the step-by-step integration method. Subsequently, the seismic response histories of the whole SSI system were determined from the arithmetic summation of the modal response histories. The MDOF modal equations of motion retain the property of nonproportional damping of the original SSI system. The proposed method has the advantages of the convent...

SPECTRUM-BASED RESPONSE PREDICTION OF ASYMMETRIC STRUCTURE WITH SUPPLEMENTAL VISCOUS DAMPING

Seismic response of one-story structure with the significant stiffness eccentricity, developing a torsional mode, is studied in this paper. Viscous damping is added to reduce both the translational and torsional vibration of the structure. The structure in such a case possesses the so-called non-proportional damping, which makes the conventional modal analysis using undamped vibration mode more inaccurate. On the other hand, more accurate method using the complex-valued mode is not easy for practical analysis, and is not very effective to understand dynamic response characteristies of the structure. Hence, this paper proposes a new modal analysis which uses the real-valued mode and yet treats the non-proportional damping effect with a unique correction process. Many cases of non-proportional damping are examined, and the reasonable accuracy of the proposed method is demonstrated through comparison with the other existing methods.

Rayleigh Quotient and Dissipative Systems

Rayleigh quotients in the context of linear, nonconservative vibrating systems with viscous and nonviscous dissipative forces are studied in this paper. Of particular interest is the stationarity property of Rayleigh-like quotients for dissipative systems. Stationarity properties are examined based on the perturbation theory. It is shown that Rayleigh quotients with stationary properties exist for systems with proportional viscous and nonviscous damping forces. It is also shown that the stationarity property of Rayleigh quotients in the case of nonproportional damping (viscous and nonviscous) is conditional upon the diagonal dominance of the modal damping matrix.

Modal properties of beam-and-block pre-cast floors

This article describes a combined experimental and analytical approach to investigating modal properties of beam and block (B&B) building floors. A rarely performed dynamic testing of a representative sample of four nominally identical B&B open plan suspended ground floors in a real-life building has been carried out using state-of-the-art modal testing featuring a pair of simultaneously operating electrodynamic shakers and multi-channel data acquisition. The testing demonstrated that the floors have considerable transverse stiffness able to engage sizeable parts of the floor area in a manner similar to orthotropic plates. Although nominally identical, the four floors have somewhat different modal properties. In general, although open plan, the B&B floors tested developed considerable levels of non-proportional damping. This is possibly a consequence of friction between non-monolithic floor elements (in the beam-to-block and/or block-to-block contacts). By back analysis, using experimentally measured stat...

Sensitivity of friction-induced vibration in idealised systems

Friction-induced vibration occurs in many contexts: vehicle brake squeal in particular remains surprisingly unpredictable and poorly understood. Testing theory against measurements has been hindered by the difficulty in obtaining repeatable results suggesting that the phenomenon is sensitive to small changes in parameters. This paper explores highly idealised cases as a starting point to understanding sensitivity. Using a stability criterion based on the roots of the characteristic equations of the system, the sensitivity of predictions to parameter changes is studied, focussing on a single-mode model. The effects of contact stiffness, non-proportional damping and a velocity-dependent coefficient of friction are considered. It is found that each physical effect can significantly alter predictions; each physical effect can lead to extreme sensitivity; and high sensitivity can sometimes occur when modal amplitudes are small such that they might normally be considered insignificant. With a large body of literature focussing on reduced-order models this study provides an important warning when interpreting their results.

Efficient Dynamic Response Analysis Using Substructuring Reduction Method for Discrete Linear System with Proportional and Nonproportional Damping

The dynamic response analysis for large structures using finite element method requires a large amount of computational resources. This paper presents an efficient vibration analysis procedure by combining node-based substructuring reduction method with a response analysis scheme for structures with undamped, proportional or nonproportional damping. The iterative form of substructuring reduction scheme is derived to reduce the full eigenproblem and to calculate the dynamic responses. In calculating the time response, direct integration scheme is used because it can be applied directly to the reduced model. Especially for the non proportional damping matrix, the transformation matrices defined in the displacement space are used to reduce the system. The efficiency and the effectiveness of the present method are demonstrated through the numerical examples.

Effect of non-proportional damping on the torque roll axis decoupling of an engine mounting system

Several mounting system design concepts are conceptually used to decouple the engine roll mode though limited success has been observed in practice. One shortcoming of the existing theories or design methods is that they ignore non-proportional viscous damping in their formulations. It seems that the rigid-body vibrations are coupled whenever non-proportional damping is introduced to the mounting system even though the torque roll axis decoupling is still theoretically possible with proportional damping assumption. To overcome this deficiency, we re-formulate the problem for a non-proportionally damped linear system while recognizing that significant damping may be possible with passive (such as hydraulic) or adaptive mounts. The complex mode method is employed in our work and the torque roll axis decoupling paradigm is re-examined given mount rate ratios, mount locations and orientation angles as key design parameters. We derive a necessary axiom for a mode in the torque roll axis direction provided two eigenvalue problems, in terms of stiffness and damping matrices, are concurrently satisfied. Two numerical examples are chosen to examine both steady-state and transient responses and the extent of coupling or decoupling is quantified. Results show that the torque roll axis for a mounting system with non-proportional damping (under oscillating torque excitation) is indeed decoupled when one of the damped modes lies in the torque roll axis direction. Finally, eigensolutions are validated by using experimental data.

Reducing sparse nonlinear eigenproblems by automated multi-level substructuring

The automated multi-level substructuring method (AMLS) was recently suggested as an alternative to iterative projection methods for computing eigenpairs of huge matrix eigenproblems in the context of structural engineering. Taking advantage of a substructuring on several levels the method constructs a projected problem of much smaller dimension which still yields satisfactory accuracy over a wide frequency range of interest. In this paper we generalise the AMLS method to certain classes of nonlinear eigenvalue problems which can be partitioned into an essential linear and positive definite pencil and a small residual. The efficiency of the method is demonstrated by numerical examples modeling damped vibrations of a structure with nonproportional damping, a gyroscopic eigenproblem, and a rational eigenproblem governing free vibrations of a fluid–solid structure.

An Efficient Algorithm for Aeolian Vibration of Single Conductor With Multiple Dampers

A novel algorithm is developed in this paper for analyzing Aeolian vibration of a span of a single conductor with multiple dampers. The algorithm is characterized by a combination of an impedance transfer technique (ITT) and a forced vibration method (FVM). The ITT avoids singularities that may occur due to a conductor's very low bending stiffness with respect to its tension. On the other hand, the FVM can easily handle non-proportional damping (which is typical for the problem in concern) without having to solve a complex eigenvalue problem. In addition, the algorithm can easily accommodate a conductor's terminal flexibility as well as a damper's rotational effect that may exist due to its possible asymmetry. The algorithm is very efficient so that solution is almost instantaneous for most analysis problems. The algorithm has recently been implemented into a commercial Aeolian vibration analysis and design software-EPRI Vibration version 3.0.

Dynamic response of large structural system including non-proportional damping with recursive component mode synthesis method

The modal frequency response problem including non-proportional damping is solved with the recursive component mode synthesis (RCMS) method. For large-scale structures, the RCMS method is used not only for computing the eigensolution, but also for forming the modal frequency response problem. The efficiency and accuracy of this approach are verified with examples of structural Finite-element model compared with the conventional industry approach that uses the block Lanczos method.