Damping Matrices Research Articles

Closed-form derivation of aerodynamic damping matrix and pitch vector of an aero-servo-elastic wind turbine system

This paper develops closed-form expressions of aerodynamic damping matrix and pitch controller of an aero-servo-elastic wind turbine (WT) system. This facilitates establishing a linearized WT model completely analytically at an arbitrary operational point, without resorting to extensive numerical linearizations or fittings. The rotor blade flexibility and pitch controller are included in the model development, thus unfolding a full understanding of the aerodynamic coupling effects. The linearized WT model is verified by good agreement with results from the original nonlinear WT model. Based on the verified linearized model, it is seen that aerodynamic coupling between tower fore–aft and side–side vibrations is solely due to the tower top rotations, and the top rotations also significantly enlarge the diagonal aerodynamic damping coefficients. Both tower top rotations and flexible blade–tower interaction play important roles in tower aerodynamic damping ratios. The developed formulas of load coefficients are also readily applicable to a FE modeling framework of the tower–foundation system.

Evaluation of the effect of local rotation and vertical displacements modes in the nonlinear seismic response of steel frames

Seismic analysis of buildings is normally based on a damping matrix derived from the Rayleigh Model (CR). However, it is accepted that the damping matrix (CS) derived from the superposition of modal damping matrices (SMDM) gives more accurate results. Another issue related to seismic analysis of structures consists in the overlooking of the contribution of the modes associated to local rotations (LR) and vertical displacements (VD). The main purposes of this paper are to illustrate the inconvenience of using the CR matrix and to evaluate the contribution of the LR and VD modes. To this aim, the nonlinear seismic responses of three steel building models idealized as complex 2D MDOF systems under de action of several seismic records are calculated. The major finding of the paper are: (a) The underestimation of axial loads and bending moments can be larger than 40% and 30%, respectively, if the matrix CR is used; (b) if only lateral displacements (LD) modes are used to generate CS, some damping should be given in LR and VD modes to avoid amplification of the response; (c) the combined contribution of the LR and VD modes to axial loads, bending moment can be, on an average basis, larger than 50 % and 17 %, respectively, while for interstory shears and drifts it can be larger than 20 %. In light of the findings of this study, it is strongly recommended to use the SMDM procedure to form the damping matrix and to consider the contributions of the LR and VD modes when calculating the seismic response. The presented study was limited to plane frames that are regular in-plan and elevation. Other aspects like irregularity and 3D models need to be considered to get more general conclusions.

Soil–Structure Interaction Consideration for Base Isolated Structures under Earthquake Excitation

This study aims to analytically implement base isolation with soil–structure interaction (SSI) on a sample structure and to develop a very simple solution to add these combined effects into the mass, damping and stiffness matrices of the structure. A spectrum analysis is also carried out considering the base-isolated structures and SSI. Dynamic simulations are performed throughout the study. In these simulations, three shear frame structures with different properties are considered. The strong ground motions selected for these analyses are eighteen different events with far-fault and near-fault components. In addition, four different base and soil structure combination cases are taken into account. These four analytical cases are a conventional structure with a fixed base and with SSI and a seismically isolated structure with or without the SSI. The numerical results showed that when SSI is considered, the effectiveness of the base isolation system may decrease, and the effect is prominent in softer soil conditions.

Rapid Computation of Resonant Frequencies for Nonproportionally Damped Systems Using Dual Oscillators

AbstractMany oscillatory systems of engineering and scientific interest (e.g., mechanical metastructures) exhibit nonproportional damping, wherein the mass-normalized damping and stiffness matrices do not commute. A new modal analysis technique for nonproportionally damped systems, referred to as the “dual-oscillator approach to complex-stiffness damping,” was recently proposed as an alternative to the current standard method originally developed by Foss and Traill-Nash. This article presents a critical comparison of the two approaches, with particular emphasis on the time required to compute the resonant frequencies of nonproportionally damped linear systems. It is shown that, for degrees-of-freedom greater than or equal to nine, the dual-oscillator approach is significantly faster (on average) than the conventional approach, and that the relative computation speed actually improves with the system’s degree-of-freedom. With 145 degrees-of-freedom, for example, the dual-oscillator approach is about 25% faster than the traditional approach. The difference between the two approaches is statistically significant, with attained significance levels less than machine precision. This suggests that the dual-oscillator approach is the faster of the two algorithms for computing resonant frequencies of nonproportionally damped discrete linear systems with large degrees-of-freedom, at least within the limits of the present study. The approach is illustrated by application to a model system representative of a mechanical metastructure.

A weak-form spectral Chebyshev technique for nonlinear vibrations of rotating functionally graded beams

This study presents the spectral Chebyshev technique (SCT) for nonlinear vibrations of rotating beams based on a weak formulation. In addition to providing a fast-converging and precise solution for linear vibrations of structures with complex geometry, material, and physics, this method is further advanced to be able to analyze the nonlinear vibration behavior of continuous systems. Rotational motion and material gradation further complicate this nonlinear behavior. Accordingly, the beam is considered to be axially functionally graded (FG) and a model representing the forced nonlinear vibrations of the beam about steady-state equilibrium deformations (SSEDs) is developed. The model includes Coriolis, centrifugal softening, and nonlinear stiffening effects caused by coupling of the axial, chordwise, and flapwise motions, and large amplitude deformations. The integral boundary value problem for the rotating structure is discretized using the SCT and element-wise multiplication definition. As a result, mass, damping, and stiffness matrices, as well as internal nonlinear forcing functions and external forcing vectors, are obtained for a given rotating beam. This formulation provides a general representation of nonlinear strain relations in matrix form and circumvents the complexity rising from obtaining and solving the partial differential equations directly. In addition, nonlinear forcing functions are obtained in matrix form which facilitates the application of harmonic balance method easier to obtain the forced nonlinear response.

Performance evaluation for the domain decomposition method in nonlinear vibration of the composite hard-coating cylindrical shell.

The applications of the modified domain decomposition method in nonlinear vibration analysis of the composite hard-coating cylindrical shells are still at a relatively superficial level, owing to the fact that its performance under different decomposition parameters has not been thoroughly investigated for achieving sufficient precision. A parametric domain decomposition method is developed to facilitate self-performance evaluation in nonlinear vibration analysis of the shell. Correspondingly, in order to avoid a mass of redundant computation of the segment stiffness and material damping matrices during iterations, a specialized preprocessing scheme is designed by pre-establishing the parametric analytical expressions and matrix databases. The resonant response is sensitive to the circumferential segment number, but weakly affected by the axial segment number. The optimum circumferential segment number in the present study is suggested to be Nθ = 70, which can achieve good calculation accuracy and efficiency. Highly consistency is shown for the distributions of axial equivalent strain under different axial segment numbers. Smaller circumferential segment numbers would result in larger equivalent strain and bad solution accuracy. The sufficient solution accuracy of nonlinear vibration of the composite hard-coating cylindrical shell can't be achieved by increasing the axial segment number with constant segment width, but only by enough circumferential segment number, which is fundamentally determined by its equivalent strain distributions and gradients, and is with close relation to the axial and circumferential wave numbers of the shell.

Simplified complex-valued modal model for operating wind turbines through aerodynamic decoupling and multi-blade coordinate transformation

An efficient modelling methodology for steady-state operating wind turbines is proposed, combining aerodynamic decoupling, multi-blade coordinate transformation, and modal reduction. This leads to a complex-valued, reduced-order modal model for prediction of dynamic responses of the tower-rotor-blade wind turbine system, considering rotor rotation, blade flexibility, and vibration coupling between rotor and tower. A fully coupled finite element model was first developed, with aerodynamic forces linearised and expressed as a viscous aerodynamic damping matrix. A time-invariant state space model is formed using multi-blade coordinate transformation, allowing standard modal analysis. The complex-valued eigenvalues and mode shapes were obtained, and it is shown that the operating wind turbine modes exhibit a combination of tower and blade vibrations. Various degrees of modal reduction are applied to the state space model to obtain a modal model with fewer degrees of freedom, whose performance was evaluated in the time and frequency domain for operating wind turbines in normal condition. The displacement and stress responses are in close agreement with those of the fully coupled model with the first 21 modes included. The model already performs well with 8 modes considered to capture relevant fundamental frequency peaks. This allows significantly reduced computational effort and can be particularly beneficial for fatigue prediction, reliability analysis, and structural identification.

Nonlinear model reduction for a cantilevered pipe conveying fluid: A system with asymmetric damping and stiffness matrices

We construct reduced-order models (ROMs) for a geometrically nonlinear viscoelastic cantilevered pipe conveying fluid, a dynamical system that includes asymmetric damping and stiffness matrices and nonlinear internal forces. The asymmetries of the damping and stiffness matrices are resulted from flow-induced gyroscopic and follower forces respectively. The ROMs are constructed using spectral submanifolds (SSMs). Specifically, we use a simulation-free approach implemented in SSMTool, an open-source package that automates the computation of SSMs, to construct the SSM-based ROMs. This approach takes equations of motion of the pipe as inputs and no simulation of the full system is involved. The ROMs enable efficient and accurate predictions of nonlinear dynamics of the system, including free vibration, isolas in forced response curves, bifurcations of periodic and quasi-periodic orbits, and heteroclinic and homoclinic orbits.

Drift simulation of a floating offshore wind turbine with broken mooring lines in a dynamic sea condition

This study investigates the aero-hydro-mooring coupled dynamics of the National Renewable Energy Laboratory (NREL) offshore 5-MW baseline wind turbine supported by the Offshore Code Comparison Collaboration Continuation (OC4) DeepCwind semisubmersible floating platform by considering the effects of suddenly broken mooring behaviour. The wind and wave loads can be selected as the Cummins time-domain equation inputs based on the Norwegian Petroleum Directorate wind and Joint North Sea Wave Project wave spectrums. The nonlinear viscous drag was estimated using the quadratic damping matrix rather than Morison's element. Additionally, the quadratic transfer function matrix was used to calculate the slow-drift force. Compared to the multi-segmented quasi-static mooring model, the lump-mass method is a dynamic mooring model that can solve real-time motion on the platform. Once a single or a pair of mooring lines is intentionally disconnected from the floating platform at a certain time, the transient responses of mooring line tensions and turbine performance will be evaluated. The novelty of this study is that the drift trajectories and power performances of OC4 DeepCwind semisubmersible floating offshore wind turbine under different conditions of environmental loads and broken mooring lines can be approximately estimated in the offshore wind farm. The analysis can be used for the environmental impact assessment in harsh sea conditions.

Nonlinear transient response of elastoplastic sandwich beam in underwater blast and the fluid-structure interaction

This work focuses on a new theory on the transient response of elastoplastic sandwich beam in underwater blast, and two fully coupled fluid-structure interaction (FSI) models are proposed, to gain insight into the propagation and dissipation of underwater blast wave, and the optimal design for sandwich structure with excellent underwater blast resistance. Based on the nonlinear dynamic high-order sandwich panel theory (EHSAPT) according to the authors’ preceding work [1], the governing equation of motion is constructed and modified by integrating the rarefaction-wave-related term into the structural damping matrix. So, the underwater blast analysis is transformed into the dynamic response of sandwich structure with structural damping under the superposition of incident and reflected waves in vacuum. Furthermore, this governing equation is simplified to get the analytical solution of impulse transferred to the sandwich structure, by implementing displacement and core rigidity assumptions. Finite element simulation and two existing FSI models are used to validate the accuracy. Conclusions are drawn that, the key aspects dominating FSI mechanism and time-domain response include pressure signatures, fluid characteristics, front-face mass, core density, and core elastic modulus, and the quantitative relationships are provided. Moreover, the core elastic modulus dominates the impulse transmitted to the sandwich structure, and the core strength and strain hardening govern the pressure delivered to the supports.

Nonlinear Hamiltonian Systems Under Sampling

This article investigates the transformation of Hamiltonian structures under sampling. It is shown that the exact sampled equivalent model associated to a given port-Hamiltonian continuous-time dynamics exhibits a discrete-time representation in terms of the discrete gradient, with the same energy function but modified damping and interconnection matrices. By construction, the proposed sampled-data dynamics guarantees exact matching of both the state evolutions and the energy-balance at all sampling instants. Its generalization to port-controlled Hamiltonian dynamics leads to characterize a new power conjugate output so recovering the concept of average passivation. On these bases, energy-management control strategies can be proposed. An energetic interpretation of the approach is confirmed by its formulation in the Dirac formalism. Two classical examples are worked out to validate the proposed sampled-data modeling in a comparative way with the literature.

Analytical study on the aerodynamic and hydrodynamic damping of the platform in an operating spar-type floating offshore wind turbine

Accurate evaluation of aerodynamic damping and hydrodynamic damping is critical to response prediction and vibration control of spar-type floating offshore wind turbines. This paper presents a comprehensive analytical study of these two damping sources with respect to platform motions. Aerodynamic damping is evaluated by a newly derived aerodynamic damping matrix based on linearisation of aerodynamic resultant forces at tower top. Both radiation and viscous drag effects are considered for hydrodynamic damping. The former is analytically expressed in the form of a radiation damping matrix, while the latter is derived based on Morison's equation and strip theory. A simplified model is established based on the analytical damping expressions and successfully verified against FAST and Aqwa. Under various operational conditions, the damping ratios for different degrees of freedom of the platform are estimated using complex modal analysis. It is found that the modal damping ratios arising from different sources can be very different for different vibration modes of the platform, and the tower flexibility is proved to have negligible impact on the platform damping. For a typical operational state, surge, sway, pitch and yaw motions are highly damped, with the roll motion intermediately damped and the heave motion nearly undamped.

A direct method for the simultaneous updating of finite element mass, damping and stiffness matrices

This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices and with , for , , and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that . Problem OAP: Given real symmetric matrices and , find such that , where is the solution set of Problem IQEP and is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data.

Оптимальный гироскопический стабилизатор многомерной вибрационной системы

We study the stabilizing problem for a multidimensional vibration system using a gyroscopic stabilizer. A vibration system is given by a second order differential equation with symmetrical matrix coefficients: a positive definite stiffness matrix and an indefinite damping matrix; in general, such a system is unstable. We need find an optimal gyroscopic stabilizer for it represented by a skew-symmetric matrix coefficient in a speed term. The choice of this control type is dictated by the tendency to avoid additional vibrations caused by slip-slide friction. Its feature is a reduced order of the controller, regardless of the dimension of the system and the number of tunable parameter. Our goal is to elucidate the most important properties of the gyroscopic stabilizers variety. It is described by a polynomial equations system. The dimension of the general solution variety in the regular case is easily found and some of its points can be calculated numerically. We start with an example of dimension 3, which leads to a system of 6th order ordinary differential equations (ODE) and then a system of five polynomial equations with regard to six unknowns. Its general solution turns out to be a one-dimensional algebraic variety presented in a table form. The second example has dimension 5; it corresponds to a tenth-order system of differential equations and nine polynomial equations in fifteen unknowns. The dimension of the solution manifold is equal to six; we find a one-dimensional subvariety and some singular points. The main difficulty is the divergence of numerical calculations near the multiple poles of a closed system. One of the important properties, that manifested itself in both examples, was the presence of complex conjugate poles and occasionally multiples of real ones; thus, almost all solutions for an optimal gyroscopic stabilizer are made up of relatively rapidly decaying oscillations. In both examples, the variety of solutions consists mostly of simple poles and allows one to choose a stabilizer that does not create resonant effects.

Damping Optimization of Linear Vibrational Systems with a Singular Mass Matrix

We present two novel results for small damped oscillations described by the vector differential equation Mx¨+Cx˙+Kx=0, where the mass matrix M can be singular, but standard deflation techniques cannot be applied. The first result is a novel formula for the solution X of the Lyapunov equation ATX+XA=−I, where A=A(v) is obtained from M,C(v)∈Rn×n, and K∈Rn×n, which are the so-called mass, damping, and stiffness matrices, respectively, and rank(M)=n−1. Here, C(v) is positive semidefinite with rank(C(v))=1. Using the obtained formula, we propose a very efficient way to compute the optimal damping matrix. The second result was obtained for a different structure, where we assume that dim(N(M))≥1 and internal damping exists (usually a small percentage of the critical damping). For this structure, we introduce a novel linearization, i.e., a novel construction of the matrix A in the Lyapunov equation ATX+XA=−I, and a novel optimization process. The proposed optimization process computes the optimal damping C(v) that minimizes a function v↦trace(ZX) (where Z is a chosen symmetric positive semidefinite matrix) using the approximation function g(v)=cv+av+bv, for the trace function f(v)≐trace(ZX(v)). Both results are illustrated with several corresponding numerical examples.

Digital Current Controller With a Novel Active Damping Design for IPMSM

Internal model control gets wide attentions in electric drives for its outstanding robustness. Underdamped oscillations however, always can be seen in the synchronous <i>d</i>- and <i>q</i>- axis components of the stator currents if without proper extra damping elements included, especially under the low sampling to fundamental frequency ratios. The reasons for these oscillations are analyzed in the discrete-time domain other than as usually done in the continuous-time domain. Following that, a distinctive active damping matrix is proposed for the current closed-loop control of interior permanent magnet synchronous motors (IPMSMs). As a result, the anti-disturbance capability is enhanced, allowing the oscillations, often seen in the <i>d</i>- and <i>q</i>-axis current components, suppressed very well. Compared with other damping methods, the typical PI structure is still retained, which is desired on the point of engineering view. At last, the analysis as well as the damping effectiveness is validated through experiments.

Construction of a special damping matrix for the direct step-by-step integration method

Construction of a special damping matrix for the direct step-by-step integration method

Analysis of nonlinear vibrations of thin cylindrical shells subjected to supersonic flow

This paper presents a numerical model for analysis of the aeroelastic stability of a thin cylindrical shell subjected to external supersonic airflow including consideration of the effect of geometric nonlinearity of the shell on the dynamic behaviour of the structure. The nonlinear stiffness matrix is derived based on a new formulation developed for thin cylindrical shells, which accounts for the shell curvature effect in the circumferential direction of the displacement field, and the effect of coupling between the different nonlinear modes on the dynamic behaviour of the shell. Nonlinear strain–displacement relationships are inferred from Novozhilov’s theory. The linear and nonlinear mass and stiffness matrices are calculated by adopting the displacement functions derived from exact solutions of linear Sander’s theory equilibrium equations for thin cylindrical shells. The classic finite element method is employed to derive the global mass and stiffness matrices. The aerodynamic damping and stiffness matrices are evaluated using the first–order potential (piston) theory. The initial stiffening effect due to radial pressure and/or axial loading is also taken into account. The governing equations of motion are derived using the Lagrange method and solved numerically with the help of a direct iterative method. Numerical studies are conducted to illustrate the effects of different parameters including internal pressure, nonlinear coupling, circumferential wave number, radius-to-thickness ratio, length-to-radius ratio and boundary conditions on flutter boundaries and on the nonlinear dynamic behaviour of the cylindrical shell at the onset of flutter and during the flutter stage. Good agreement is found between the results obtained using the presented approach and those published in the literature.

The Impact of Damping in Second-Order Dynamical Systems with Applications to Power Grid Stability

We consider a broad class of second-order dynamical systems and study the impact of damping as a system parameter on the stability, hyperbolicity, and bifurcation in such systems. We prove a monotonic effect of damping on the hyperbolicity of the equilibrium points of the corresponding first-order system. This provides a rigorous formulation and theoretical justification for the intuitive notion that damping increases stability. To establish this result, we prove a matrix perturbation result for complex symmetric matrices with positive semidefinite perturbations to their imaginary parts, which may be of independent interest. Furthermore, we establish necessary and sufficient conditions for the breakdown of hyperbolicity of the first-order system under damping variations in terms of observability of a pair of matrices relating damping, inertia, and Jacobian matrices, and propose sufficient conditions for Hopf bifurcation resulting from such hyperbolicity breakdown. The developed theory has significant applications in the stability of electric power systems, which are one of the most complex and important engineering systems. In particular, we characterize the impact of damping on the hyperbolicity of the swing equation model which is the fundamental dynamical model of power systems, and demonstrate Hopf bifurcations resulting from damping variations.

High-order implicit time integration scheme based on Padé expansions

A single-step high-order implicit time integration scheme for the solution of transient as well as wave propagation problems is presented. It is constructed from the Padé expansion of the matrix exponential solution of a system of first-order ordinary differential equations formulated in the state–space. By exploiting the techniques of polynomial factorization and partial fractions of rational functions, and by decoupling the solution for the displacement and velocity vectors a computationally efficient scheme is developed. An important feature of the novel algorithm is that no direct inversion of the mass matrix is required. Based on the diagonal Padé expansion of order M a time-stepping scheme of order 2M is constructed. When M is odd, one system of equation with a real matrix and (M−1)/2 systems of equations with complex matrices are solved recursively. When M is even, there are M/2 systems of equations with complex matrices. These systems are sparse and comparable in complexity to the standard Newmark method, i.e., the effective system matrix is a linear combination of the static stiffness, damping, and mass matrices. The derived second-order scheme is analytically equivalent to Newmark constant average acceleration method. The proposed time integrator has been implemented in MATLAB and FORTRAN using direct linear equation solvers. In this article, numerical examples featuring nearly one million degrees of freedom are presented to highlight the exceptional accuracy and efficiency of the novel time-stepping scheme in comparison with established second-order methods. The MATLAB- and FORTRAN-implementations are available from the authors upon request.