Non-proportional Viscous Damping Research Articles

Finite element model updating of non-proportional and exponential non-viscous damping systems using complex eigenvalues and eigenvectors

This paper proposes a finite element model updating method for non-proportional viscous and exponential non-viscous damping systems using complex eigenvalues and eigenvectors. The exponential non-viscous damping provides complex eigenvalues/vectors for elastic modes and real-valued eigenvalues/vectors for non-viscous modes. This study first highlights the challenges in accurately identifying real-valued eigenvalues/vectors for non-viscous modes with a commonly used system identification method. In contrast, complex eigenvalues/vectors for elastic modes are reliably identified and, therefore, utilized in the proposed model updating approach. Consequently, an optimization formulation is proposed to minimize the difference between the simulated and experimental complex eigenvalues/vectors. The method applies large structures that can contain multiple substructures with different damping properties. For brevity, damping properties are assumed to be uniform over each substructure. In addition, mass-proportional viscous damping and stiffness-proportional non-viscous damping are adopted for each substructure, which results in overall damping being non-proportional. To improve computational efficiency, the analytical gradient of the proposed optimization formulation is derived and implemented. For validation, the model updating of a full-scale steel pedestrian bridge is performed. The method is first validated in simulation and further validated by experimental data considering the statistical properties of system identification results. The proposed method successfully identifies stiffness and damping parameters using only a limited number of measured DOFs and modes.

Model Updating of a Euler-Bernoulli Beam Using the Response Method

In this study, the computational model is updated using an analytical model instead of an experimental one. Continuous and discrete parameter models of a Euler–Bernoulli beam are constructed analytically and computationally. To construct the computational models, Ansys™ software is employed, and 1-D beam elements are chosen to get the finite element model of a cantilever beam. To get analytical solutions for the continuous and discrete parameter models, a state-space representation is employed. In the first step, only mass and stiffness matrices are considered to model the beam. Eigenfrequencies and eigenvectors of the beam are calculated. The analytical and computational eigenfrequencies of continuous and discrete parameter models are compared. In the seconds step, non-proportional viscous damping and non-proportional structural damping matrices are introduced into the analytical discrete parameter model. Then, the frequency response functions of the model are generated. The damping matrices are identified using the generated frequency response functions. The damping matrices used in the analytical model, and the damping matrices identified using the frequency response functions are compared. It is observed that the assigned damping matrices and the identified damping matrices are identical, which shows that the computational model can be accurately updated provided the FRFs are available.

Particle swarm optimization–based characterization technique of nonproportional viscous damping parameter of a cantilever beam

A complex eigenvector is a result of nonproportional damping present in a structural system. However, it is difficult to identify the accurate damping matrix considering the modal sparsity and coordinate sparsity. A nonproportional viscous damping parameter identification is formulated as an unconstrained optimization problem in the present study. The damping coefficient of each element is considered as the design variable for the optimization problem. The objective function is defined using the incomplete complex eigenvectors, which are generated because of the presence of external damping devices in the structure. This objective function is then minimized using standard particle swarm optimization to identify the damping coefficient of the damping matrix. The accuracy and efficiency of the particle swarm optimization are investigated by solving a few numerical problems with simulated measured data. The proposed method works well with the incomplete measured modal data. The current methodology performs satisfactorily with and without noisy data. A comparison study is performed with the existing gradient-based method, and the results show that the proposed method performs better than the existing gradient-based method for the present problem with and without noisy measurement data.

Repeated eigenvalues and their derivatives of structural vibration systems with general nonproportional viscous damping

Repeated vibration modes often occur in practice in structural vibration systems with general nonproportional viscous damping. However, the topic remains perhaps the least understood in vibration analysis. Some researchers have suggested that the inclusion of nonproportional viscous damping renders a system defective in the case of repeated eigenvalues, while others have assumed that a complete set of linearly independent eigenvectors can always be found regardless of the forms and magnitudes of viscous damping. This paper seeks to first establish that for light non-proportional viscous damping, a system with repeated eigenvalues generally does not become defective based on perturbation theory and realistic numerical examples since rigorous theoretical proof is believed to be difficult. Once non-defectiveness is confirmed, a method for computing the eigen derivatives with repeated eigenvalues in the case of general viscous damping is developed. Effect of mode truncation on numerical accuracy has been discussed. When the magnitude of non-proportional viscous damping becomes considerably high however, it is possible for a damped system to become defective when repeated modes occur. This can have profound practical implications where very high damping is desired as in the cases of vibration suspension and absorber designs, since system defectiveness can lead to major difficulties in the applications of conventional vibration analyses.

Gibbs Sampling for Damage Detection Using Complex Modal Data from Multiple Setups

This paper presents a novel Gibbs sampling approach for structural health monitoring (SHM) with detection of structural changes/damages using incomplete complex modal data measured with a limited number of sensors. The usual difficulty with the availability of sensors in SHM practices and enforcing data acquisition in multiple setups is thoroughly addressed. Structural modeling incorporated with damping is considered in this proposed inverse problem exercise to calibrate damping parameters along with the stiffness and mass parameters facilitating SHM. Both proportional and nonproportional viscous damping are adopted in structural modeling. Detailed formulations on the probabilistic detection of changes/damages are presented in detail. Moreover, a Gibbs sampling technique is introduced to quantify uncertainties of the various sets of uncertain parameters, where samples of the conditional probability density function of a parameter set are obtained iteratively. The proposed approach retains the typical advantage of the nonrequirement of mode-matching. A validation exercise is performed using a three-dimensional building structure (attached with supplementary viscous dampers) and a laboratory steel structure considering multiple damage cases and different sensor placements. The proposed methodology is observed to be efficient for SHM using incomplete complex modal data measured with a limited number of sensors.

Análise dinâmica no domínio da frequência por meio do método do balanço harmônico

The analysis of structures in the frequency domain has been gaining ground in the last decades, first, because of the computational development and strategies to evaluate the Discrete Fourier Transform more efficiently. The advantages of evaluating the frequency response are that there is no need to know the initial conditions, and the response to systems with frequency-dependent properties can be evaluated more accurately, eg, soil-structure interaction. This work aims to evaluate the dynamic response of a discrete structural system in the frequency domain through the resonance curves using the Harmonic Balance Method HBM. The response is evaluated in the permanent phase for actions and / or harmonic loads for systems with proportional and non-proportional viscous damping. The properties of the system, such as mass and rigidity, are taken as constants. For the models developed and analyzed in this study are presented the respective spectra of frequency response and history of displacement in time through the HBM. The system responses obtained through the HBM are compared with two other methods also based on the frequency domain (ImFT and PseudoForces), allowing to evaluate the amplitudes and critical frequencies of the system.

Experimental direct spatial damping identification by the Stabilised Layers Method

A new method is developed for direct damping matrix identification using experimental receptance-matrix data together with physical connectivity constraints based on what are described as layers. A number of spectral lines are considered in symmetric frequency bands around the damped peaks of the receptances and the identification procedure is shown to be ill-conditioned (singular) when one of the spectral lines coincides with an undamped natural frequency. A test is developed that makes use of a stabilisation diagram to ensure not only that a solution exists, but also that it is stable when the number and frequency range of the spectral lines is changed. An experimental parametric three degrees of freedom lumped mass system connected by springs and air dampers is considered and the matrix of non-proportional viscous damping terms is identified under different damper configurations and levels.

Analytical investigation of railway overhead contact wire dynamics and comparison with experimental results

Abstract In this paper an analytical method for studying the free response of continuous vibrating systems with distributed and possibly non-proportional viscous damping is proposed. The most general case the method refers to is a piece-wise homogeneous Euler-Bernoulli beam, with lumped elastic and inertial elements and subjected to tensile load. The practical application of the method to a contact wire is also presented, aiming at analysing its dynamic response. Contact wires are typically used in the overhead contact line of the railway electrification system but, despite their wide diffusion, their damping properties have not been exhaustively studied. This study aims at experimentally validate the analytical method to define a reliable dynamic model of overhead contact lines. The wire is modelled as an axially loaded homogeneous beam, with lumped elastic and inertial elements (i.e. droppers and clamps). 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. Experimental measurements have been carried out in the Dynamics & Identification Research Group (DIRG) laboratory of Politecnico di Torino considering two different damping scenarios, and the modal properties of the test bench have been extracted by using a linear subspace identification technique. The damping distribution is finally investigated starting from the experimental data, in order to seek for the most appropriate damping model.

Measurement of FRFs of coupled geared rotor system and the development of an accurate finite element model

An experimental study has been attempted to find out the effect of gear pair contact on the modal behaviour of an actual geared rotor system mounted on rolling element bearings. This is important as gear pair contact considerably affects the dynamic characteristics of the system. Frequency response functions of the uncoupled and coupled geared rotor systems have been measured to find out the effect of gear pair contact on the natural frequencies of the system. Moreover, a finite element model of the geared rotor system has been developed by modelling the shaft continuum, bearing coefficients and gear pair contact (using mesh stiffness and damping). Non-proportional viscous damping has been considered in modelling bearing and mesh damping coefficients. The results obtained after numerical simulation of finite element model are validated with the experimental results to check the accuracy of the developed finite element model.

Identification of non-proportional viscous damping matrix of beams by finite element model updating

A methodology has been proposed to estimate non-proportional viscous damping matrix of beams from measured complex eigendata using finite element model updating technique. Representation of damping through a proportional damping matrix ignoring the complexity of eigenvectors may not be appropriate when external damping devices are employed. The current literature of determination of non-proportional damping matrix demands measurement of a large number of complex modes which is extremely difficult in practice. A gradient based finite element model updating algorithm implementing inverse eigensensitivity method has been presented through a series of numerically simulated cantilever beams. The method can accurately predict the non-proportional damping matrix even if the measured eigenvectors are polluted with random noise. The novelty of the current method is that it can sustain a high level of modal and coordinate sparsity in measurement. The method assumes prior determination or updating of the mass and stiffness matrices.

Effect of mesh stiffness of healthy and cracked gear tooth on modal and frequency response characteristics of geared rotor system

Geared rotor systems are used in industrial machinery, automotive applications etc. to provide rotational speed changes and torque variations by transmitting rotational motion from driving shaft to the driven shaft. Gear tooth contact, as modelled using gear mesh stiffness and damping, considerably affects the dynamic characteristics of the system. This work primarily studies the effect of mesh stiffness and damping due to gear pair on the modal characteristics of the geared flexible rotor-shaft system supported on compliant bearing. The rotor-shaft is modelled using Timoshenko beam elements; whereas non-proportional viscous damping model is used to represent bearing and gear mesh damping. Fatigue loading causes development of gear tooth crack, which results in reduction of mesh stiffness. The effect of various cases of cracked gear tooth has also been investigated on the modal characteristics and frequency response functions of the system. The changes in the modal and frequency response characteristics due to cracked gear tooth have been compared with that of healthy geared rotor system. The study may prove helpful in detection of faults developing in the gear tooth by observing the changes in the dynamic characteristics of the system.

Updating Bearing Stiffness and Damping Coefficients of a Rotor System

Finite element (FE) models of structures have been quite useful in both static and dynamic analyses of structures. However, quite often, these models are not reliable enough since predictions based on them may not be found to have acceptable correlation with experimentally obtained data. This paper attempts updating of bearing radial and tilt stiffness as well as damping parameters of a rotor system by using inverse eigen sensitivity method (IESM). Non-proportional viscous damping model has been used in modelling damping coefficients of bearings. The state space form of equations of motion of the system is used in applying the IESM for model updating. The results show that both stiffness and damping coefficients of bearings can be effectively found out by using the IESM. The method is found to update the eigenvalues quite well even under the presence of measurement noise.

Model updating of rotors supported on ball bearings and its application in response prediction and balancing

This work attempts experimental studies in finite element model updating of an actual rotor system mounted on ball bearings by using Inverse Eigen Sensitivity Method (IESM). The IESM is applied on state space representation of equations of motion and is used to identify bearing stiffness, damping and shaft material damping parameters. Non-proportional viscous damping model is used to model the bearing and shaft material damping. The experimental identification of viscous coefficient of shaft material damping was not found in the available literature and this work attempts the same as well. The updated model is validated for its accuracy by comparing the predicted frequency response with that obtained from the experiments. Finally, it is shown that the updated finite element model of the rotor system can be efficiently used to predict the unbalance in the rotor.

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.

Modal analysis by free vibration response only for discrete and continuous systems

This work aims to develop the algorithm for modal analysis by free vibration response only (MAFVRO), in particular for the general or non-proportional viscous damping system model. If the structural displacement or acceleration response due to free vibration can be measured, the system response matrices, including the displacement, velocity and acceleration, can be obtained through numerical differential or integration methods. These response matrices can then be applied to the developed MAFVRO method to determine the structural modal parameters. The numerical differential and integration methods are introduced and adopted to establish the modal parameter prediction program for the non-proportional damping model of MAFVRO. This work also shows the applications of MAFVRO to the multiple degree-of-freedom (mdof) systems and the cantilever beam, respectively. Both the discrete and continuous systems are demonstrated for the feasibility of the MAFVRO algorithm. The developed method uses the free vibration output response only and can obtain the structural modal parameters successfully.

Comparative study of damped FE model updating methods

Most of finite element (FE) model updating techniques do not employ damping matrices and hence, cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, a detailed comparison of two approaches of obtaining damped FE model updating methods are evaluated with the objective that the FRFs obtained from damped updated FE models is able to predict the measured FRFs accurately. In the first method, damped updating FE model is obtained by complex parameter-based updating procedure, which is a single-step procedure. In the second method, damped updated model is obtained by the FE model updating with damping identification, which is a two-step procedure. In the first step, mass and stiffness matrices are updated and in the second step, damping matrix is identified using updated mass and stiffness matrices, which are obtained in the previous step. The effectiveness of both methods is evaluated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by a case involving actual measured data for the case of F-shaped test structure. The updated results have shown that the complex parameter-based FE model updating procedure gives better matching of complex FRFs with the experimental data.

Finite element model updating with damping identification

Most of finite element model updating techniques do not employ damping matrices and hence, cannot be used for accurate prediction of complex frequency response functions (FRFs). In this paper, damped finite element model updating procedure is proposed and tested with the objective that the damped finite element updated model is able to predict the measured FRFs accurately. The proposed damped updating procedure is a two-step procedure. In the first step, mass and stiffness matrices are updated using FRF data and in the second step, damping matrix is identified using updated mass and stiffness matrices that are obtained in the previous step. The effectiveness of the proposed procedure is demonstrated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by case involving actual measured data for the case of F-shaped test structure. The results have shown that the proposed procedure can be used for accurate prediction of the complex FRFs.

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.

Fast frequency response analysis of partially damped structures with non-proportional viscous damping

This paper presents a new method for the modal frequency response analysis of partially damped structural system with non-proportional viscous damping. The basic idea is to handle the modal viscous damping matrix by noting that the rank of the viscous damping matrix is typically very low for problems of interest. Then, the Sherman–Morrison–Woodbury formula provides a convenient expression for the inverse of equation which includes the low-rank matrix. The new method, fast frequency response analysis (FFRA) algorithm, dramatically improves the performance of the modal frequency response analysis compared to conventional methods in industry with the same accuracy.

Modal identification of linear non-proportionally damped systems by wavelet transform

A time-frequency identification technique based on wavelet transform is formulated and applied to free-decay responses of linear systems with non-proportional viscous damping. The Cauchy mother wavelet is used. Frequencies, modal damping ratios and complex mode shapes are identified from output-only free vibration signals. This identification technique has also shown to be effective when the (non-proportional) damping is significant.