Linear Structural Dynamics Research Articles

An analysis of high order FEM and IGA for explicit dynamics: Mass lumping and immersed boundaries

SummaryWe investigate the behavior of different shape functions for the discretization of hyperbolic problems. In particular, we consider classical Lagrange polynomials and B‐splines. The studies focus on the performance of the these functions as a spatial discretization approach combined with an explicit time marching scheme. In this regard, a major concern is the maximum eigenvalue that imposes restrictions on the critical time step size and suitable lumping techniques that yield a diagonal mass matrix. The accuracy of the discretization methods is assessed in an asymptotic manner in terms of the convergence of eigenvalues and eigenvectors. Further, the global accuracy is investigated in terms of the full spectrum. The results show that B‐spline discretization with a consistent mass matrix are more accurate than those based on Lagrange shape functions, which holds true in the boundary‐fitted as well as in the immersed setting. On the other hand, Lagrange shape functions are more robust with respect to standard lumping techniques, which cannot be directly applied for B‐splines without loss of accuracy. In general, we observe that none of the standard lumping schemes yields optimal results for B‐splines, even in the boundary‐fitted setting. For the immersed setting, also Lagrange shape functions show a drop in accuracy which depends on the position of the boundary that cuts the element. Several remedies are considered in order to overcome these issues, including interpolatory B‐spline bases as well as eigenvalue stabilization methods. While accuracy and stability can be improved using these remedies, we conclude from our study that it is still an open question, how to design a discretization method that achieves large critical time step sizes in combination with a diagonal mass matrix and high accuracy in the immersed setting. We note that these considerations primarily relate to linear structural dynamics applications, such as for example, structural acoustics. In nonlinear problems, such as automotive crash dynamics, other considerations predominate. An example of a one‐dimensional elastic‐plastic bar impacting a rigid wall is illustrative.

Analyzing the Impact of Wing Flexural Axis Position on the Dynamics of a Very Flexible Airplane Using Strain-based Formulation

New technologies and the needs of specific missions have allowed and stimulated the development of airplanes with great structural flexibility. In these vehicles, there is a significant coupling between aeroelastic phenomena and flight dynamics. This has been a topic of research in recent years. Different methodologies have been studied to model the complete dynamics of flexible aircraft. One of these is the NFNS_s methodology (Non linear flight dynamics, non linear structural dynamics, strain based formulation). This article presents the use of this methodology in the analysis of the aeroelastic response and flight dynamics of a transport category aircraft with high structural flexibility. The effects of the wing elastic axis and the flexural axis positions were analyzed. Trimming, calculus of eigenvalues and time-marching simulations were carried out and the results were analyzed in detail. Some results were compared with those published in the literature. Similar trends were observed. This served as a qualitative validation of the methodology used. Some limitations of the study are described. The contributions of this work lie in the use of the NFNS_s methodology to analyze the effects of the elastic and flexural axis positions. In addition, the results of time marching simulations of the airplane modeled and the analysis carried out in great detail are other important novelty.

Analytical Sensitivity Analysis of Dynamic Problems with Direct Differentiation of Generalized-α Time Integration

In this paper, the direct differentiation of generalized-α time integration is derived, equations are introduced and results are shown. Although generalized-α time integration has found usage, the derivation and the resulting equations for the analytical sensitivity analysis via direct differentiation are missing. Thus, here, the sensitivity equations of generalized-α time integration via direct differentiation are provided. Results with generalized-α are compared with Newmark-β time integration and their sensitivities with numerical sensitivities via forward finite differencing in terms of accuracy and performance. An example is shown for each linear structural dynamics and flexible multibody dynamics.

Improved two-stage implicit time integration methods with unconventionally determined parameters for analyses of linear and nonlinear structural dynamics

Improved two-stage implicit time integration methods with unconventionally determined parameters for analyses of linear and nonlinear structural dynamics

Nonlinear system identification of a UAV model with distributed aerodynamics and flexible structure

This paper presents the nonlinear system identification of a slightly flexible 25 kg fixed-wing UAV in the time-domain using a computationally efficient distributed aerodynamics model and a linear structural dynamics representation. The equations of motion are formed by making use of the free vibration modes of the structure and the mean axes formulation. The structural modes and mode shapes are determined from ground vibration tests. The distributed aerodynamics, accounting for elastic deformations, are modeled using a quasi-steady stability and control derivative approach and by applying strip theory. Initial distributions for the derivatives are obtained from vortex-lattice-method calculations. For matching the model response to the measured response, parameters for scaling the initial derivative distributions are introduced. The flexible model is subsequently identified based on flight test data using the output error method in the time-domain and maximum-likelihood estimation. A good overall identification result is achieved with a close match of the fast aircraft dynamics. Finally, an evaluation is given on the suitability of the identified model for real-time simulation, loads’ estimation, and active load control law design.

Dynamic response analysis of nonlinear structures with hybrid uncertainties

The physical systems of nonlinear structural dynamics are usually expressed mathematically as ordinary differential equations. When hybrid uncertainties are in the system, it is necessary to understand the effects of uncertainties in the system response to accurately predict its behavior using mathematical models. To address the hybrid uncertainty of random and interval parameters often encountered in engineering applications, we propose the nonlinear structural dynamics analysis with hybrid uncertainty (NSDAHU) method. The expressions for calculating the statistical characteristics (i.e., mean and variance) of nonlinear structural dynamic responses with hybrid uncertainties are derived using random interval moment and random interval perturbation methods. The applicability, accuracy, and efficiency of the NSDAHU method are verified using Monte Carlo simulation method (MCSM). The NSDAHU method can not only solve hybrid random interval problems but also solve single interval, single random, and linear structural dynamics problems.

Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics

We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method, particle methods, and other spatial methods on a single body sub-divided into multiple subdomains. This is in conjunction with implementing the well known Generalized Single Step Single Solve (GS4) family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework. In the current state of technology, the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as the Newmark methods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier. However, the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses, overcomes such barriers and allows a wide variety of arbitrary implicit-implicit, implicit-explicit, and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement, velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains. By selecting an appropriate spatial method and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body, the proposed work has the potential to better capture the physics of a given simulation. The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility. The accuracy and efficacy of the present work have not yet been seen in the current field, and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.

The first International Conference on Mechanical System Dynamics grandly convened in Nanjing, China

The first International Conference on Mechanical System Dynamics grandly convened in Nanjing, China

Normal form transformations for structural dynamics: An introduction for linear and nonlinear systems.

The aim of this paper is to provide an introduction to using normal form transformations for linear and nonlinear structural dynamics examples. Starting with linear single-degree-of-freedom systems, a series of examples are presented that eventually lead to the analysis of a system of two coupled nonlinear oscillators. A key part of normal form transformations are the associated coordinate transformations.This review includes topics such as Jordan normal form and modal transformations for linear systems, while for nonlinear systems, near-identity transformations are discussed in detail. For nonlinear oscillators, the classical methods of Poincaré and Birkhoff are covered, alongside more recent approaches to normal form transformations. Other important topics such as nonlinear resonance, bifurcations, frequency detuning and the inclusion of damping are demonstrated using examples. Furthermore, the connection between normal form transformations and Lie series is described for both first and second-order differential equations. The use of normal form transformations to compute backbone curves is described along with an explanation of the relationship to nonlinear normal modes. Lastly, conclusions and possible future directions for research are given.

Combining set propagation with finite element methods for time integration in transient solid mechanics problems

The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear structural dynamics problems modeled with partial differential equations (PDEs). While different algorithms for direct integration of the equations of motion exist, exploring all feasible behaviors for varying loads, initial states and fluxes in models with large numbers of degrees of freedom remains a challenging task. In this article we propose a novel approach, based in set propagation methods and motivated by recent advances in the field of Reachability Analysis. Assuming a set of initial states and inputs, the proposed method consists in the construction of a union of sets (flowpipe) that enclose the infinite number of solutions of the spatially discretized PDE. We present the numerical results obtained in five examples to illustrate the capabilities of our approach, and compare its performance against reference numerical integration methods. We conclude that, for problems with single known initial conditions, the proposed method is accurate. For problems with uncertain initial conditions included in sets, the proposed method can compute all the solutions of the system more efficiently than numerical integration methods.

Highly precise and efficient solution strategy for linear heat conduction and structural dynamics

AbstractTime‐dependent parabolic and hyperbolic equations are widely encountered in practical engineering. Accurate and fast solution methods for such equations have always attracted attention. To reach this aim, this article proposes a strategy to construct highly precise and efficient time integration methods (TIMs) for linear heat conduction and structural dynamic systems. In the proposed strategy, an integrated amplification matrix of a TIM is created to precisely transfer the free responses of the previous step, and the Gauss–Legendre quadrature is employed to precisely compute the forced responses of the current step. To reduce computational cost and rounding error, a 2m algorithm and a method of storing incremental matrix are applied in the construction of the integrated matrix. Moreover, the L‐stable backward difference formula for heat conduction and the generalized trapezoidal rule for structural dynamics, which is A‐stable and has controllable dissipation, are utilized to conduct this strategy. Numerical experiments validate that compared with some existing TIMs, the TIMs generated by the proposed strategy enjoy advantages both in precision and efficiency.

A Galerkin Time quadrature element formulation for linear structural dynamics

A well-posed time weak form for linear structural dynamics is used to construct a Galerkin time quadrature element formulation. Radau quadrature rule and the generalized differential quadrature analog are used to turn the well-posed weak form into a set of linear equations. The stability and accuracy properties of the formulation are discussed. Numerical examples are given to show the high computational efficiency of the well-posed weak form time quadrature element formulation, as compared with a time finite element solution based on the same weak form using third-order Hermite interpolations.

Orthogonal spline expansions for uncertainty quantification in linear dynamical systems

This paper leverages recent progress on orthonormal splines for solving uncertainty quantification (UQ) problems from linear structural dynamics. The resulting methods, premised on spline chaos expansion (SCE) and spline dimensional decomposition (SDD), both construe Fourier-like expansion of a dynamic system response of interest with respect to measure-consistent orthonormalized basis splines in input random variables and standard least-squares regression for estimating the expansion coefficients. The SCE and SDD methods are capable of capturing high nonlinearity and non-smoothness, if they exist, in a stochastic dynamic response markedly better than the polynomial chaos expansion (PCE) method. However, due to the tensor-product structure, SCE, like PCE, also suffers from the curse of dimensionality. In contrast, SDD, equipped with a desirable dimensional hierarchy of input variables, deflates the curse of dimensionality to a great extent. Numerical results from frequency response analysis of a two-degree-of-freedom dynamic system indicate that a low-order SCE with fewer basis functions removes or markedly reduces the spurious oscillations generated by high-order PCE in estimating the response statistics. Finally, a high-dimensional modal analysis of a fighter jet comprising 110 random variables was conducted, demonstrating the ability of SDD in solving large-scale UQ problems.

A weak form temporal quadrature element formulation for linear structural dynamics

PurposeVarious time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.Design/methodology/approachA variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.FindingsThe weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.Originality/valueThe weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.

Polynomial chaos based rational approximation in linear structural dynamics with parameter uncertainties

Surrogate models enable efficient propagation of uncertainties in computationally demanding models of physical systems. We employ surrogate models that draw upon polynomial bases to model the stochastic response of structural dynamics systems. In linear structural dynamics problems, the system response can be described by the frequency response function. It is well known that standard polynomial chaos expansions of the frequency response present slow convergence around system eigenfrequencies, due to the highly nonlinear nature of the frequency response for low damping. To overcome this issue, we develop a rational approximation that expresses the system response as a rational of two polynomials with complex coefficients. To estimate the latter, we propose a regression approach that is non-intrusive and can be easily coupled with existing deterministic solvers. We demonstrate the effectiveness of the proposed method with two examples, a two-degree-of-freedom system and a finite element model of a cross laminated timber plate.

Feasible Probabilistic Learning Method for Model-Form Uncertainty Quantification in Vibration Analysis

Recently, a novel probabilistic method for modeling and quantifying model-form uncertainties in a deterministic high-dimensional computational model was proposed and demonstrated for numerous applications in linear vibrations and nonlinear structural dynamics. The method relies on a stochastic projection-based reduced-order model (SPROM) grounded in a randomized reduced-order basis to accelerate the stochastic computations it incurs, and it accounts for all modeling uncertainties introduced by model reduction. It builds the probability model of the SPROM using a small number of hyperparameters, and it determines them so that the mean value and statistical fluctuations of quantities of interest predicted using the SPROM match target values obtained from available data. Hence, it can be interpreted as a probabilistic model learning or updating method. Unfortunately, even though it is performed using a reduced-order model, the hyperparameter identification process can be cost prohibitive. For this reason, the method is reformulated in this paper around the concept of hyper-reduction. Specifically, a reduced version of the mesh underlying the high-dimensional model is constructed and used to hyper-reduce the SPROM. The feasibility of the resulting approach is demonstrated for the model-form uncertainty quantification of generalized eigencomputations performed during what-if scenarios associated with shape changes of a jet engine nozzle.

A New Family of Explicit Model-Based Integration Algorithms for Structural Dynamic Analysis

A new family of explicit model-based integration algorithms for solving the equations of motion for linear and nonlinear systems is developed. These algorithms are also known as structure-dependent algorithms because the integration parameters are functions of the complete model of the structural system. A variety of numerical properties of the proposed algorithms, including consistency and local truncation error, stability, numerical dispersion and energy dissipation, overshooting, and frequency response under arbitrary excitation, are investigated using the discrete control theory and amplification matrix for linear elastic systems. In addition, the discrete control theory is applied for assessing the stability of the proposed algorithms for nonlinear structural systems. It is observed that the proposed algorithms exhibit the same numerical characteristics as the well-known Newmark family of integration algorithms. Compared with three existing model-based integration algorithms, i.e. the Chen–Ricles, modified Chen–Ricles, and Gui’s algorithms, the proposed algorithms possess more general and versatile numerical features. As a result, the new family of explicit model-based integration algorithms can be potentially used to solve complicated linear and nonlinear structural dynamics problems.

Iterative Component Mode Synthesis Using a Priori and a Posteriori Criteria

An iterative algorithm of component mode synthesis for a rational reduced-order modeling is proposed. The present algorithm consists of three steps: initial mode selection with a priori criterion, model reduction using the selected modes, and a posteriori error evaluation. The algorithm is based on the Craig–Bampton method, which is one of the most widely used techniques. Specifically, the present algorithm uses, in a judicious manner, both an a priori mode selection criterion employing the moment-matching strategy and an a posteriori error estimation technique in evaluating the relative eigenvalue errors. If desired, an optional step of boundary interface reduction offers an additional possibility of obtaining a more compact model. The proposed iterative algorithm is recommended to researchers as well as commercialized software users in developing linear reduced-order structural dynamics models. Numerical examples are presented for its practical usage and performance.

Development of Non-Dissipative Direct Time Integration Method for Structural Dynamics Application

A direct time integration scheme based on Gauss-Legendre quadrature is proposed to solve problems in linear structural dynamics. The proposed method is a one-parameter non-dissipative scheme. Improved stability, accuracy, and dispersion characteristics are achieved using appropriate values of the parameter. The proposed scheme has second-order accuracy with and without physical damping. Moreover, its stability, accuracy, and dispersion are analyzed. In addition, its performance is demonstrated by the two-dimensional scalar wave problem, the single-degree-of-freedom problem, two degrees-of-freedom spring system, and beam with boundary constraints. The wave propagation problem is solved in the high frequency wave regime to demonstrate the advantage of the proposed scheme. When the proposed scheme is applied to solve the wave problem, more accurate solutions than those of other methods are obtained by using the appropriate value of the parameter. For the single-degree-of-freedom system, two degrees-of-freedom system, and the time responses of beam, the proposed scheme can be used effectively owing to its high accuracy and lower computational cost.

Dominant vibration modes for broadband frequency analysis of multiscale structures with numerous local vibration modes

SummaryIn this paper, an extension of modal analysis in linear computational structural dynamics is presented to deal with complex structures characterized by the presence of numerous local vibration modes and for which the high modal density leads to high‐dimension reduced‐order models. These local modes consist of isolated vibrations that are dissociated from the global vibrations of the structure skeleton and that turn out to often be of negligible contribution to the global dynamics. Therefore, an automatic mode sorting procedure is proposed to extract the dominant modes that represent the global dynamics of the skeleton. Next, an alternative filtering methodology based on a modified eigenvalue problem is presented, which allows to build, a priori, a small‐dimension reduced‐order basis of dominant global modes. The two methods are compared to classic modal analysis using three applications, namely, a heterogeneous plate, a simplified nuclear fuel assembly, and a detailed boiling water reactor fuel assembly.