Structural Dynamic Systems Research Articles

On the Numerical Discretization in Space and Time: Part 1 - Hamilton's Law of Varying Action Involving Lagrangian/Hamiltonian/Total Energy Framework

The focus in Part 1 of this exposition is strictly restricted to holonomic-sceleronomic systems, and the applications of interest are elastodynamics, which are routinely encountered in a wide class of structural dynamics problems in engineering. Restricting attention to these considerations, new and different perspectives and equivalences are described which deal with finite element space discretization aspects employing three distinctly different frameworks with scalar formalism. It encompasses the Lagrangian mechanics, the Hamiltonian mechanics, and as an alternative, the framework with a built-in measurable quantity based on the Total Energy. Historically, traditional practices routinely employ the weighted residual method or equivalently the principle of virtual work in dynamics for developing finite element formulations. In contrast, the present developments stem from Hamilton's law of varying action (HLVA) as a starting point and involve distinctly different scalar descriptive functions (the Lagrangian [ℒ( q , [qdot] ) : TQ →ℝ], the Hamiltonian [ℋ( p , q ) : T* Q →ℝ], or the Total Energy [ℰ( q , [qdot] ):TQ →ℝ]). These developments naturally embody the weak form in space and the statement of the weighted residual in time. Complicated structural dynamical systems such as a rotating bar and the Timoshenko beam are particularly shown here simply for illustration. In Part 2, we describe the satisfaction of conservation properties of the fully discretized equations of motion in space and time with particular attention to the Total Energy framework (in contrast to the Lagrangian and the Hamiltonian), primarily because it is very natural and is computationally attractive and meaningful to conducting the time discretization [1] process. Instead of starting from the continuous form of representations, the particular focus is upon directly employing the discrete formulations for enabling algorithmic designs for the class of LMS methods for linear dynamic problems for illustration purposes. This is a necessary first step in the design of algorithms for integrating the equations of motion, and they serve as the parent algorithms for subsequent extensions to nonlinear dynamic problems.

Multiagent bayesian forecasting of structural time-invariant dynamic systems with graphical models

Time series are found widely in engineering and science. We study forecasting of stochastic, dynamic systems based on observations from multivariate time series. We model the domain as a dynamic multiply sectioned Bayesian network (DMSBN) and populate the domain by a set of proprietary, cooperative agents. We propose an algorithm suite that allows the agents to perform one-step forecasts with distributed probabilistic inference. We show that as long as the DMSBN is structural time-invariant (possibly parametric time-variant), the forecast is exact and its time complexity is exponentially more efficient than using dynamic Bayesian networks (DBNs). In comparison with independent DBN-based agents, multiagent DMSBNs produce more accurate forecasts. The effectiveness of the framework is demonstrated through experiments on a supply chain testbed.

Order reduction of forced nonlinear systems using updated LELSM modes with new Ritz vectors

Enhanced modal-based order reduction of forced structural dynamic systems with isolated nonlinearities has been performed using the updated LELSM (local equivalent linear stiffness method) modes and new Ritz vectors. The updated LELSM modes have been found via iteration of the modes of the mass normalized local equivalent linear stiffness matrix of the nonlinear systems. The optimal basis vector of principal orthogonal modes (POMs) is found via simulation and used for POD-based order reduction for comparison. Two new Ritz vectors are defined as static load vectors. One of them gives a static displacement to the mass connected to the periodic forcing load and the other gives a static displacement to the mass connected to the nonlinear element. It is found that the use of these vectors, which are augmented to the updated LELSM modes in the order reduction modal matrix, reduces the number of modes used in order reduction and considerably enhances the accuracy of the order reduction. The combination of the new Ritz vectors with the updated LELSM modes in the order reduction matrix yields more accurate reduced models than POD-based order reduction of the forced nonlinear systems. Hence, the LELSM modal-based order reduction is enhanced via new Ritz vectors when compared with POD-based and linear-based order reductions. In addition, the main advantage of using the updated LELSM modes for order reduction is that, unlike POMs, they do not require a priori simulation and thus they can be combined with new Ritz vectors and applied directly to the system.

Nonparametric Stochastic Modeling of Uncertainty in Rotordynamics—Part I: Formulation

A systematic and rational approach is presented for the consideration of uncertainty in rotordynamics systems, i.e., in rotor mass and gyroscopic matrices, stiffness matrix, and bearing coefficients. The approach is based on the nonparametric stochastic modeling technique, which permits the consideration of both data and modeling uncertainty. The former is induced by a lack of exact knowledge of properties such as density, Young’s modulus, etc. The latter occurs in the generation of the computational model from the physical structure as some of its features are invariably ignored, e.g., small anisotropies, or approximately represented, e.g., detailed meshing of gears. The nonparametric stochastic modeling approach, which is briefly reviewed first, introduces uncertainty in reduced order models through the randomization of their system matrices (e.g., stiffness, mass, and damping matrices of nonrotating structural dynamic systems). Here, this methodology is extended to permit the consideration of uncertainty in symmetric and asymmetric rotor dynamic systems. More specifically, uncertainties on the rotor stiffness (stiffness matrix) and/or mass properties (mass and gyroscopic matrices) are first introduced that maintain the symmetry of the rotor. The generalization of these concepts to uncertainty in the bearing coefficients is achieved next. Finally, the consideration of uncertainty in asymmetric rotors is described in detail.

On improving weakly coupled cavity models for vibro-acoustic predictions and design

The vibro-acoustic characteristics of a cavity based on its vibro-acoustic FE model at times do not correlate well with the corresponding measured vibro-acoustic characteristics. Structural dynamic modeling errors that are often associated with the structural domain of the cavity are the main reasons for this discrepancy. This paper addresses this issue and deals with the improvement of the vibro-acoustic FE models of weakly coupled cavities. This particular application, hitherto, has not been much addressed, though the improvement of purely structural dynamic systems has been researched a lot. An experimental example of a 3D rectangular-box cavity with a flexible plate is considered. The study demonstrates that the improvement of the vibro-acoustic FE models through FE model updating can be an effective approach to obtain more accurate vibro-acoustic predictions. The study further addresses the question whether these improved vibro-acoustic models are suitable for vibro-acoustic design. This is answered by analyzing the accuracy with which the improved vibro-acoustic FE model predicts the effects of the structural modifications. It is finally concluded that the predictions based on the unimproved vibro-acoustic models may not be reliable and can have significant error, while the improved vibro-acoustic models give an improvement in the predictive capability of the model and are also found suitable for vibro-acoustic design.

Granger Causality and Dynamic Structural Systems

Using a generally applicable dynamic structural system of equations, we give natural definitions of direct and total structural causality applicable to both structural vector autoregressions (VARs) and recursive structures representing time-series natural experiments. These concepts enable us to forge a previously missing link between Granger (G-)causality and structural causality by showing that, given a corresponding conditional form of exogeneity, G-causality holds if and only if a corresponding form of structural causality holds. We introduce a variety of structurally informative extensions of G-causality and provide their structural characterizations. Of importance for applications is the structural characterization of finite-order G-causality, which forms the basis for most empirical work. We show that conditional exogeneity is necessary for valid structural inference and prove that, in the absence of structural causality, conditional exogeneity is equivalent to G noncausality. These characterizations hold for both structural VARs and natural experiments. We propose practical new G-causality and conditional exogeneity tests and describe their use in testing for structural causality. We illustrate with studies of oil and gasoline prices, monetary policy and industrial production, and stock returns and macroeconomic announcements.

Frequency Reliability Sensitivity for Dynamic Structural Systems#

Parameter uncertainty of structural systems is inherent in most engineering problems. Therefore, reliability analysis of uncertain structural systems is of great significance, and frequency reliability analysis is important to uncertain vibration structure systems. Based on the frequency reliability theory, this article introduces an approach to calculate the reliability of a mechanical structure under dynamic excitation. With the criterion that the absolute value of the difference between the natural frequency and forcing frequency, the reliability mode and the safety probability of random systems are defined, and then the frequency reliability sensitivity analysis method is presented. The structure is supposed to fail if the frequency of the external force is within a predefined distance of structural resonant frequencies. The frequency reliability sensitivity analysis method for resonant problem of dynamic structural systems reliability is proposed by using the random perturbation technique, the reliability theory, and sensitivity approach. The formulas of quasi-failure probability sensitivity are derived when random variables are normally distributed. The reliability is evaluated, and its sensitivity to mean values, variances, and covariances is mathematically expressed, then the efficiency and accuracy of the proposed method are demonstrated by a numerical example.

Self-equilibration of inverse system in damage detection for dynamic structures

In this study, a modal method coupled with mathematical programming is presented for the assessment of error propagation in damage detection for linear dynamic structural systems. Finite element method is applied to inverse problems in which the structural modifications to the baseline system are sought for the characteristic changes of the perturbed structure. Unlike structural optimization, system identification and damage detection require a unique inverse system. When the exact modal responses are available, accurate structural changes can be obtained through iterative procedures with nonlinear perturbation equations. In practice, however, the modal data may contain numerical errors and consequently yield incorrect inverse solutions that are physically infeasible or quite different from the real perturbed system. Due to the intrinsic instability of eigenproblem, minor errors tend to cause significant element deviations. Hence, mathematical programming techniques are applied to search for a self-equilibrating inverse solution in the close vicinity of the measured data. The proposed approach may provide a breakthrough in the effort to alleviate the numerical difficulty of error propagation in the modal method for the inverse problem of damage detection. The linear perturbation equation is used to obtain an initial approximation, which can be improved through nonlinear iterations.

Novel generalized- [formula omitted] methods for interfield parallel integration of heterogeneous structural dynamic systems

A novel partitioned algorithm able to solve ODEs arising from transient structural dynamics is presented. The spatial domain is partitioned into a set of disconnected subdomains owing to computational or physical considerations, and continuity conditions of velocity at the interface are modelled using a dual Schur formulation, where Lagrange multipliers represent reaction forces. Interface equations along with subdomain equations lead to a system of DAEs for which an interfield parallel procedure is developed. The algorithm solves interface Lagrange multipliers, which are subsequently used to advance the solution in subdomains. The proposed coupling algorithm that enables arbitrary Generalized- α schemes to be coupled with different time steps in each subdomain is an extension of a method originally proposed by Pegon and Magonette. Thus, subcycling permitting to deal also with stiff and nonstiff subsystems is allowed. In detail, the paper presents the convergence analysis of the novel interfield parallel scheme for linear single- and two-degrees-of-freedom systems because a multi-degrees-of-freedom system is too difficult for a mathematical treatment. However, the insight gained from the analysis of these coupled problems and the consequent conclusions are confirmed by means of numerical experiments on a four-degrees-of-freedom system.

Spectra and pseudospectra of neutral delay differential equations with application to real-time substructuring

This paper deals with the computation of pseudospectra of neutral delay differential equations (NDDEs) with fixed finite delays. This method provides information on the sensitivity of eigenvalues of the system under perturbations of a given size, allowing one to analyse uncertainties in, for example, structural dynamical systems. Furthermore, pseudospectra computations are a fast and efficient method by which to identify the spectra of NDDEs in any chosen part of the complex plane. This is of particular advantage as the spectra of NDDEs consists of infinitely many eigenvalues. We illustrate these methods by considering a scalar second-order NDDE with one or more fixed delays. Such systems are used to model real-time substructuring experiments, where the delay arises from the response time of actuator(s) forming a coupling between a numerical model and the substructure being tested. We identify changes in the stability and sensitivity to perturbation of this system both as the delay time and as the mass of the substructure are varied. In particular, the latter reveals an essential instability in which infinitely many eigenvalues move to the right-half plane. Finally, we investigate the relationship between pseudospectra and the effect of periodic forcing on the substructured system.

Volterra methods for constructing structural dynamic observables for nonlinear systems: An extended calculation

Although a great deal of work has been carried out on structural dynamic systems under random excitation, there has been a comparatively small amount of this work concentrating on the calculation of the quantities commonly measured in structural dynamic tests. Among the existing work, the Volterra series, a means of predicting nonlinear system response for weakly nonlinear systems, has allowed the computation of various measurable quantities of interest for structural dynamics, including: auto- and cross-spectra, FRFs, coherences and higher-order spectra. These calculations are quite intensive and are typically only possible using computer algebra. A previous calculation by the authors for the coherence for a Duffing oscillator yielded results which showed some qualitatitive disagreement with numerical simulation; the object of the current paper is simply to extend the calculation in order to see if better agreement can be achieved.

Approximation of the Duffing oscillator frequency response function using the FPK equation

Although a great deal of work has been carried out on structural dynamic systems under random excitation, there has been a comparatively small amount of this work concentrating on the calculation of the quantities commonly measured in structural dynamic tests. Perhaps the most fundamental of these quantities is the Frequency Response Function (FRF). A number of years ago, Yar and Hammond took an interesting approach to estimating the FRF of a Duffing oscillator system which was based on an approximate solution of the Fokker-Planck-Kolmogorow equation. Despite reproducing the general features of the statistical linearization estimate, the approximation failed to show the presence of the poles at odd multiples of the primary resonance which are known to occur experimentally. The current paper simply extends the work of Yar and Hammond to a higher order of approximation and is thus able to show the existence of a third `harmonic' in the FRF.

Finite element formulations using the theorem of power expended: total energy framework with a differential formulation

Traditional practices involving variational calculus have historically dominated most finite element formulations to-date, and have no doubt served as indispensable tools. Besides these practices, our recent contributions in Acta Mechanica (Har and Tamma, 2009, in press) described new alternatives and developments emanating from Hamilton’s Law of Varying Action (HLVA) as a starting point with a measurable built-in scalar function, namely, the Total Energy. The associated framework (in contrast to Lagrangian or Hamiltonian mechanics framework) demonstrated certain new advances, and also provided some fundamental insight into explaining traditional practices of finite element discretization. Here we additionally provide other advances, new directions, and viable alternatives in contrast to all these past practices which routinely employ variational concepts. In particular, focusing on elastodynamics applications, in this paper we provide for the first time finite element formulations stemming instead from a differential formulation and the theorem of power expended with a measurable built-in scalar function, namely, the Total Energy \({[\mathcal{E}({\varvec q},\dot{{\varvec q}}):TQ\to \mathbb{R}]}\) , as a starting point to capitalize on certain added advantages. The autonomous total energy has time/translational/rotational symmetries for the continuum/N-body dynamical systems. The proposed concepts: (i) can be shown to yield the same governing mathematical model equations of motion that are continuous in space and time together with the natural boundary conditions just as balance laws such as linear momentum or Hamilton’s principle are routinely used to derive such equations, but without resorting to any variational concepts, or approaches such as variational principles, (ii) explain naturally how the classical Bubnov–Galerkin weighted-residual form that is customarily employed for discretization can be readily constructed for both space and time, and alternately, (iii) circumvents relying on traditional practices of conducting numerical discretizations starting either from the balance of linear momentum (Newton’s law) involving Cauchy’s equations of motion (governing equations) arising from continuum mechanics or via (1) and (2) above, and instead provides new avenues of discretization for continuum-dynamical systems. For illustration, numerical discretizations are presented for the modeling of complicated structural dynamical systems.

Finite element formulations via the theorem of expended power in the Lagrangian, Hamiltonian and total energy frameworks

Traditionally, variational principles and variational methods have been employed in describing finite element formulations for elastodynamics applications. Here we present alternative avenues emanating from the theorem of expended power, using the differential calculus directly. We focus on scalar representations under three distinct frameworks: Lagrangian mechanics, Hamiltonian mechanics, and a new framework involving a built-in measurable quantity, called the total energy in the configuration space. All three frameworks are derivable from each other, since they represent the same physics as Newton’s second law; however, the total energy framework which we advocate inherits features that are comparable and competitive to the usual Newtonian based finite element formulations, with several added advantages ideally suited for conducting numerical discretization. The present approach to numerical space-time discretization in continuum elastodynamics provides physical insight via the theorem of expended power and the differential calculus involving the distinct scalar functions: the Lagrangian L.q;P q/V T Q! R, the Hamiltonian H.p;q/V T Q! R, and the total energy E.q;P q/V T Q! R. We show that in itself the theorem of expended power naturally embodies the weak form in space, and after integrating over a given time interval yields the weighted residual form in time. Hence, directly emanating from the theorem of expended power, this approach yields three differential operators: a discrete Lagrangian differential operator, a Hamiltonian differential operator, and a total energy differential operator. The semidiscrete ordinary differential equations in time derived with our approach can be readily shown to preserve the same physical attributes as the corresponding continuous systems. This contrasts with traditional approaches, where such proofs are nontrivial or are not readily tractable. The modeling of complicated structural dynamical systems such as a rotating bar and the Timoshenko beam are shown for illustration. Variational concepts have long played a significant role in the development of numerical discretization techniques. First, variational principles in physics and mechanics have been used to derive the governing differential equations from which numerical discretizations are routinely conducted. The principles — including Hamilton’s principle in dynamics, in the electromagnetic and gravitational fields, and even in quantum mechanics; the principle of stationary potential energy and the Hu‐Washizu’s variational principle in continuum statics; Fermat’s principle of least time leading to Snell’s law in optics; Maupertuis’ principle of least action, Jacobi and Lagrange’s principle of least action in dynamics again; and Gurtin’s

Wavelet analysis for identification of damping ratios and natural frequencies

This study addresses the identification of damping ratios and natural frequencies in linear structural dynamic systems. A previously proposed method based on the Morlet wavelet transform of the system's response is investigated analytically. The method can be applied to single- or multi-degree of freedom, lightly or heavily damped systems. It utilizes a relationship between natural frequency, damping ratio and the continuous Morlet wavelet transform of the system response. It is found that the validity of the relationship depends on the scaling factor a, translation factor b, and the frequency parameter ω 0 of the Morlet wavelet transform. A general guideline of choosing a, b, and ω 0 is provided in this paper. Both numerical and experimental results verify the theoretical analysis.

An Agent Infrastructure over HLA for Distributed Simulation of Reconfigurable Systems and its Application to UAV Coordination*

This paper proposes an agent infrastructure (Theatre) centered on Java for distributed simulations over High-Level Architecture/RunTime Infrastructure (HLA/RTI). The architecture rests on actors (agents) as the basic building blocks. Actors have a public message interface and encapsulate a state of local variables and a behavior patterned as a finite state machine. Actors interact with one another by asynchronous message passing. At the system level, theatres are used as the execution platforms for actors. Theatres naturally map on to HLA federates. Actors can migrate between theatres for load-balancing concerns, or in response to (re)configuration operations of dynamic structure systems. The paper introduces Theatre and demonstrates its application to a complex simulation model based on unmanned aerial vehicles (UAVs). The case study is an open agent-based distributed model where mobile agents follow communication patterns established at runtime.

Modeling Loose Joints in Elastic Structures— Simulation Algorithm and Results

This is the third paper in a four companion paper sequence. In these papers, a hybrid parameter multiple body system (HPMBS) methodology is utilized to model the frictional contact/impact of a loose bolted joint between two sections of a cantilever beam undergoing planar slewing motion. In this preliminary model, a “rigid” joint is utilized, allowing the members of the joint to be rigid, each attached to an elastic beam section. Frictional contact/impact is modeled at four contact points. The contact constraints and momentum transfer are modeled with the idea of instantly applied nonholonomic constraints. In this paper, the contact and switching algorithms, the numerical solution scheme, and simulation results are presented. The theoretical model of the variable structure dynamic system, the momentum transfer equations and the experimental comparison are presented elsewhere. The motivation for this work is the need for accurate low order models of complicated nonlinear phenomena in structural systems.

Optimal boundary control of dynamics responses of piezo actuating micro-beams

Optimal control theory is formulated and applied to damp out the vibrations of micro-beams where the control action is implemented using piezoceramic actuators. The use of piezoceramic actuators such as PZT in vibration control is preferable because of their large bandwidth, their mechanical simplicity and their mechanical power to produce controlling forces. The objective function is specified as a weighted quadratic functional of the dynamic responses of the micro-beam which is to be minimized at a specified terminal time using continuous piezoelectric actuators. The expenditure of the control forces is included in the objective function as a penalty term. The optimal control law for the micro-beam is derived using a maximum principle developed by Sloss et al. [J.M. Sloss, J.C. Bruch Jr., I.S. Sadek, S. Adali, Maximum principle for optimal boundary control of vibrating structures with applications to beams, Dynamics and Control: An International Journal 8 (1998) 355–375; J.M. Sloss, I.S. Sadek, J.C. Bruch Jr., S. Adali, Optimal control of structural dynamic systems in one space dimension using a maximum principle, Journal of Vibration and Control 11 (2005) 245–261] for one-dimensional structures where the control functions appear in the boundary conditions in the form of moments. The derived maximum principle involves a Hamiltonian expressed in terms of an adjoint variable as well as admissible control functions. The state and adjoint variables are linked by terminal conditions leading to a boundary-initial-terminal value problem. The explicit solution of the problem is developed for the micro-beam using eigenfunction expansions of the state and adjoint variables. The numerical results are given to assess the effectiveness and the capabilities of piezo actuation by means of moments to damp out the vibration of the micro-beam with a minimum level of voltage applied on the piezo actuators.

Modeling Loose Joints in Elastic Structures-Variable Structure Motion Model Development

In this article, a hybrid parameter multiple body system (HPMBS) methodology is utilized to model the frictional contact/impact of a loose bolted joint between two sections of a cantilever beam undergoing planar slewing motion. In this preliminary model, a "rigid" joint is utilized which allows the members of the joint to be rigid, each attached to the elastic beam sections. Frictional contact/impact is modeled at four contact points. The contact constraints and momentum transfer are modeled on a basis of instantly applied nonholonomic constraints. The theoretical model of the variable structure dynamic system is developed in this article, while the momentum transfer equations (utilizing the concept of instantly applied nonholonomic constraints), numerical solution scheme, and experimental comparisons are presented in companion articles. The underlying theme of this work is to produce low order models of complicated nonlinear structural dynamics, specifically, frictional contact and impact in joints with clearance, based on first principles.

Total energy framework, finite elements, and discretization via Hamilton’s law of varying action

Within the total energy framework which we introduce here for the first time (in contrast to Lagrangian or Hamiltonian mechanics framework), we provide an alternative and have developed in this paper a general numerical discretization for continuum-elastodynamics directly stemming from Hamilton’s law of varying action (HLVA) involving a measurable built-in scalar function, namely, Total Energy \({[{\mathcal{E}}\left({{\boldsymbol{q}},\dot{{\boldsymbol{q}}}}\right): TQ\rightarrow {\mathbb{R}}]}\). The Total Energy we use herein for enabling the space discretization is defined as the kinetic energy plus the potential energy for N-body systems, and the kinetic energy plus the total potential energy for continuum-body systems. It thereby provides a direct measure and sound physical interpretation naturally, while enabling this framework to permit general numerical discretizations such as with finite elements. In the variational formulation proposed here, we place particular emphasis upon the notion that the scalar function which represents the autonomous total energy of the continuum/N-body dynamical systems can be a crucial mathematical function and physical quantity which is a constant of motion in conservative systems. In addition, we prove that the autonomous total energy possesses the three invariant properties and can be viewed as the so-called total energy version of Noether’s theorem; therefore, the autonomous total energy has time/translational/rotational symmetries for the continuum/N-body dynamical systems. The proposed concepts directly emanating from HLVA inherently involving the scalar function, namely, total energy: (i) can be shown to yield the same governing mathematical model equations of motion that are continuous in space and time together with the natural boundary conditions just as Hamilton’s principle (HP) is routinely used to derive such equations, but without obvious inconsistency via such a principle as explained in the paper; (ii) explain naturally the Bubnov–Galerkin weighted-residual form that is customarily employed for discretization for both space and time, and alternately, (iii) circumvent relying on traditional practices of conducting numerical discretizations starting either from the balance of linear momentum (Newton’s second law) involving Cauchy’s equations of motion (governing equations) arising from continuum mechanics or via (i) and (ii) above if one chooses this option, and instead provides new avenues of discretization for continuum-dynamical systems. The present developments naturally embody the weak form in space and time that can be described by a discrete Total Energy Differential Operator (TEDO). Thereby, a novel yet simple, space-discrete Total Energy formulation proposed here only needs to employ the discrete TEDO which provides new avenues and directly yields the semi-discrete ordinary differential equations in time which can be readily shown to preserve the same physical attributes as the continuous systems for continuum-dynamical applications unlike traditional practices. The modeling of complicated structural dynamical systems such as Euler-Bernoulli beams and Reissner–Mindlin plates is particularly shown here for illustration.