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Coupling of substructures for dynamic analyses
Abstract
A method for treating a complex structure as an assemblage of distinct regions, or substructures, is presented. Using basic mass and stiffness matrices for the substructures, together with conditions of geometrical compatibility along substructure boundaries, the method employs two forms of generalized coordinates. Boundary generalized coordinates give displacements and rotations of points along substructure boundaries and are related to the displacement modes of the substructures known as "constraint modes." All constraint modes are generated by matrix operations from substructure input data. Substructure normal-mode generalized coordinates are related to free vibration modes of the substructures relative to completely restrained boundaries. The definition of substructure modes and the requirement of compatibility along substructure boundaries lead to coordinate transformation matrices that are employed in obtaining system mass and stiffness matrices from the mass and stiffness matrices of the substructures. Provision is made, through a RayleighRitz procedure, for reducing the total number of degrees of freedom of a structure while retaining accurate description of its dynamic behavior. Substructure boundaries may have any degree of redundancy. An example is presented giving a free vibration analysis of a structure having a highly indeterminate substructure boundary.
... The procedure of the proposed formulation can be summarized in three steps. 1) First, the Craig-Bampton CMS method [18,20] is used to reduce the number of generalized coordinates of substructures. ...
... In Section 2.1, problems associated with two conventional Hamiltonian formulations are discussed. In Section 2.2, the Craig-Bampton method [18,20] is reviewed. In Section 2.3, the proposed formulation is presented. ...
... The potential energy U is defined as U ≡ (q T Kq)/2. When M and K are constant matrices, the Craig-Bampton method [18,20] can be applied to reduce the degrees of freedom of q i . The coordinate transformation matrix T is defined as ...
... To address this, the Component Mode Synthesis (CMS) method has emerged as an efficient alternative. CMS reduces computational demands by dividing structures into substructures and combining their static and dynamic characteristics to create reduced order models [5][6][7][8][9][10][11][12]. This approach is particularly advantageous for transient analysis of large and complex systems. ...
... Among the various CMS methods, the Craig-Bampton (CB) method is particularly well known for its efficiency and accuracy [6,9]. However, as finite element models become larger and more complex, the accuracy of the reduced order models generated by the CB method can diminish, especially for systems with a large number of DOFs. ...
... Furthermore, to mitigate the increase in interface boundary size caused by algebraic substructuring, an interface boundary reduction technique [22] has been applied. While these advancements have been effective, most studies on the fixed-interface CMS method have primarily focused on eigenvalue problems to evaluate dynamic characteristics [6,11,13,14]. ...
The increasing demand for dynamic analysis of large-scale structural systems has highlighted the need for efficient model reduction methods. Reduced order modeling allows large finite element models to be represented with significantly fewer degrees of freedom while retaining essential dynamic characteristics. This paper investigates the Enhanced Craig–Bampton (ECB) method and further explores its application in dynamic analysis. The effectiveness of the ECB method is evaluated by comparing it with the conventional Craig–Bampton (CB) method and the full finite element model using benchmark examples. The numerical results demonstrate that the ECB method provides superior accuracy and computational efficiency, making it a valuable tool for dynamic analysis in complex engineering problems.
... To make this strategy computationally feasible, model order reduction techniques are applied to the FE model of each flexible substructure. In this context, the most common choice is CMS [15][16][17][18][19]. In practice, interface nodes, also called 'master' nodes, which are points of connection of the flexible body with other entities, are distinguished from all other nodes, called 'slave' nodes. ...
... In practice, interface nodes, also called 'master' nodes, which are points of connection of the flexible body with other entities, are distinguished from all other nodes, called 'slave' nodes. These CMS methods can be divided into those which employ free-interface normal modes and attachment modes [17][18][19] and those using fixedinterface normal modes and constraint modes [15,16]. In both cases, the normal modes are used to build a modal transformation matrix which filters out insignificant high-frequency vibration modes, thus reducing the problem's dimensionality. ...
Despite the spread of digital payment systems, banknotes still play a vital role in the lives of people, financial institutions, and businesses. Automation is crucial in all cases where a large number of deposits/withdrawals need to be handled. In spite of the large number of cash recyclers worldwide and their continuous evolution, one of the most significant parts of their design, i.e., the interaction with banknotes, is still predominantly based on a lengthy iterative process that includes testing. This has detrimental effects not only on time-to-market but also on the costs and on the willingness to explore multiple design solutions, thus potentially reducing the quality of the final product. The testing phase is made necessary by the lack of an effective and predictive model for the banknote and of its contact with the components of the cash recycler. The purpose of this work is to bridge this gap in the literature by introducing and validating an original multibody model of the banknote-cash recycler system. The proposed approach includes the possibility of including the nonlinear orthotropic paper material curves and of customizing quantities such as paper thickness and friction coefficients at specific locations, with potential applications in optimizing banknote validation, reducing wear in cash handling systems, and improving the design of next-generation cash recyclers.
... Once a suitable basis is chosen, the reducedorder model (ROM) is then obtained using Galerkin projection. Similar linear projection techniques have been devised for component-mode synthesis (CMS), such as the Craig-Bampton method [13]. An implicit assumption to all linear projection techniques is that the full system dynamics evolves in a lower-dimensional linear invariant subspace of the phase space of the system. ...
... Relative reduction error E calculated according from formula(13) ...
We apply two recently formulated mathematical techniques, Slow-Fast Decomposition (SFD) and Spectral Submanifold (SSM) reduction, to a von Karman beam with geometric nonlinearities and viscoelastic damping. SFD identifies a global slow manifold in the full system which attracts solutions at rates faster than typical rates within the manifold. An SSM, the smoothest nonlinear continuation of a linear modal subspace, is then used to further reduce the beam equations within the slow manifold. This two-stage, mathematically exact procedure results in a drastic reduction of the finite-element beam model to a one-degree-of freedom nonlinear oscillator. We also introduce the technique of spectral quotient analysis, which gives the number of modes relevant for reduction as output rather than input to the reduction process.
... However, these time-domain analyses are computationally expensive compared to time-invariant analyses. Therefore, to increase computational efficiency, NVH system simulation models are often implemented in an elastic multi body simulation environment based on modal reduction [11]. By reducing the degrees of freedom of flexible bodies through modal reduction, significant improvements in computational efficiency are achieved. ...
The acoustic behavior of technical products—referred to as Noise, Vibration, and Harshness (NVH)—forms an important criterion in customers’ purchasing decisions, e.g. in the automotive sector. Thus, optimizing the NVH behavior is essential during product development. NVH is influenced by dynamic excitations and the transfer of sound from the location of excitation to a receiver such as the driver. As the electrification of the mobility sector increases, gear mesh excitations contribute significantly to the noise of a vehicle, which was previously dominated by the combustion engine. The time-varying stiffness of the gear mesh is the main excitation in drivetrains and therefore needs to be optimized during product development. Elastic multi-body simulation (eMBS) has become an established method for modeling NVH characteristics of drivetrains. In order to evaluate the effect of gear mesh designs on a drivetrain’s excitations and therefore its NVH behavior, the time-varying stiffness of the gear mesh has to be incorporated into eMBS models. Today, this is either done by applying quasi-static transmission errors onto a model in frequency domain thereby neglecting the influence of the system’s oscillations back on the gear mesh excitations, e.g. by axial misalignment of the meshing shafts. Alternatively, analytical stiffness calculations are included in eMBS models in time-domain. However, these approaches are not yet validated. Also, incorporating the interaction between a flexible gear wheel and the gear mesh stiffness is paramount for NVH models of gears. Within previous works, a method for incorporating externally calculated gear mesh stiffnesses into eMBS has been proposed to overcome these challenges. The goal of this paper is to compare, validate and enhance the existing eMBS modeling methods of gear mesh stiffnesses with respect to their ability to model the dynamic behavior of gear boxes, defined by natural frequencies and critical operating points. A two-step approach consisting of quasi-static and dynamic validations is conducted. An improvement in accuracy between simulation and measurement of 10 dB is achieved compared to analytical eMBS methods.
... Another module, SubDyn, was developed for modeling slender substructures such as jackets. It uses a linear finite-element beam framework, together with a Craig-Bampton [35] dynamic system reduction and a static improvement method. This allows for a large reduction in the number of modes while retaining an accurate solution. ...
A coupled medium‐fidelity drivetrain model is developed and implemented in OpenFAST for a 10‐MW land‐based reference turbine. The implementation is verified against a fully coupled multibody wind turbine model, including a detailed drivetrain. The new model can simultaneously and accurately estimate main bearing loads and represent elastic bending of the drivetrain. It has low computational cost and is useful for early design phases, sensitivity analyses and complex systems like wind farms (where computational expense must be expended elsewhere). Here, the model is implemented for a monopile offshore wind turbine and used to investigate the sensitivity of main bearing basic rating life to different synthetic turbulence models. Large‐eddy simulations (LES) targeting stable, neutral, and unstable atmospheric conditions at below‐, near‐ and above‐rated wind speeds are used as a reference. The turbulence models recommended by the International Electrotechnical Commission, the Mann spectral tensor model, and the Kaimal spectral model with exponential coherence are fitted to the LES data. Additionally, a constrained turbulence generator, PyConTurb (short for Python Constrained Turbulence), based on LES data, is applied in the aero‐hydro‐servo‐elastic simulations. Taking PyConTurb as the baseline, the Kaimal model significantly underestimates fatigue of the downwind main bearing, with between 10% and 40% less damage. The Mann model also underestimates the downwind main bearing fatigue by up to 30%. The upwind main bearing damage is driven by mean loads, and differences between models are less significant, although the trends are similar. Reasons for these discrepancies are investigated and attributed to differences in spatial and temporal variations among the turbulence models.
... Three translational degrees of freedom per control point are used for the volumetric description of the problem. The Craig and Bampton method [48] is applied in order to reduce the total size of the system while keeping the contact degrees of freedom unchanged. ...
Shape optimization is an increasingly prevalent tool for designing and manufacturing mechanical systems with gradient-free nonlinear performance metrics. Uncertainty quantification is an essential part of the process as optimality can be called into question in the presence of unavoidable discrepancies between numerical designs and manufactured parts. This paper combines isogeometric analysis (IGA) and polynomial chaos expansions (PCE) towards shape optimization of a disc brake for noise minimization under uncertainties. The proposed approach sets robustness to manufacturing uncertainties as an optimization objective in order to directly obtain robust optimal solutions. IGA is chosen over other shape design alternatives for its absence of meshing approximations, which makes it potentially more suitable in the presence of uncertainties. PCE is used to quantify robustness through the variance of the output, in an attempt to alleviate the computational burden of uncertainty quantification. The studied application is a simplified disc brake system whose shape is modified to minimize undesirable squeal noise, which is quantified through complex eigenvalue analysis. The noise prediction model, PCE model, and a genetic algorithm are then combined for the purpose of searching for robust optimal solutions. Results show the capability to converge to a Pareto front of robust noise-minimizing disc brake shapes and overall high computational efficiency compared to Monte Carlo simulation for output variance estimation. Furthermore, our findings confirm the superiority of sparse PCE methods over the classical ordinary least squares PCE method for output variance quantification.
... The standard approach for deriving reduced component models from a given finite element model is component mode synthesis. The by far most popular technique is the Hurty-/Craig-Bampton method [25,26]. Here, the nodal displacements are approximated as a linear combination of static constraint modes and a set of fixed-interface normal modes. ...
Thin-walled structures clamped by friction joints, such as aircraft skin panels are exposed to bending-stretching coupling and frictional contact. We propose an original sub-structuring approach, where the system is divided into thin-walled and support regions, so that geometrically nonlinear behavior is relevant only in the former, and nonlinear contact behavior only in the latter. This permits to derive reduced component models, in principle, with available techniques. The Hurty-/Craig-Bampton method, combined with an interface reduction relying on an orthogonal polynomial series, is used to construct the reduction basis for each component. To model geometrically nonlinear behavior, implicit condensation is used, where an original, engineering-oriented proposition is made for the delicate scaling of the static load cases required to estimate the coefficients of the nonlinear terms. The proposed method is validated and its computational performance is assessed for the example of a plate with frictional clamping, using finite element analysis as reference. The numerical results shed light into an interesting mutual interaction: The extent of geometric hardening is limited by the reduced boundary stiffness when more sliding occurs in the clamping. On the other hand, the frictional dissipation is increased by the tangential loading induced by membrane stretching.
... Moment matching to approximate the inputs of the system and modal truncation to ensure approximately correct displacements of all surface nodes. The idea of combining different reduction methods to get appropriate reduced order models for a specific use case is not new but known from component mode synthesis, see [29,30]. ...
... The ROM is created using a reduced basis (RB) that extracts the main features of the full order model (FOM). Because the characteristics of the RB have a significant influence on the ROM, various studies have been conducted on how to construct it [5][6][7][8][9][10][11][12]. One of them is the condensation method, which constructs a ROM based on the relationship between the degrees of freedom (DOF) in the physical coordinates of the system. ...
In this paper, we propose a degree-of-freedom-based adaptive reduction method that ensures accuracy over a wide band. In the conventional dynamic condensation method, a single reduced model consisting of low-order modes is used throughout the analysis. This results in low accuracy in the high-frequency band because it does not reflect the characteristics of the frequencies. To address this issue, we implemented a reduced model for each frequency using a Taylor series. This method converts the transformation matrix into a frequency-independent form, which allows for a simple interpolation of the reduction model by updating the differences between frequencies. Numerical examples were adopted to examine the accuracy and efficiency of the proposed method.
... For large-scale problems, full-order solutions might not be computable in reasonable time frame; furthermore, in many computational mechanics applications, parametric variations induce topology changes that prevent the definition of a common HF discretization that is valid for all parameters. This limitation justifies the development of component-based (CB) MOR approaches [49]: CB MOR methods combine the monolithic PMOR strategies discussed in Chapter 2 with component mode synthesis (CMS) and substructing techniques [23,48] first appeared in the structural dynamics literature. CB-MOR techniques rely on the introduction of a library of archetype components for which a local reducedorder approximation and a local ROM are built during an offline stage; given a new configuration, we first instantiate components from the library to form the global system and then we estimate the global solution by gluing together the local ROMs. ...
In this habilitation dissertation, I review select contributions to model order reduction (MOR) of parametric systems that I have carried out since the completion of my PhD. Many engineering tasks, such as optimization, optimal control, and uncertainty quantification require simulations of complex systems for many different configurations and/or in real-time; MOR aims to reduce the marginal cost to solve parametric PDEs and ultimately enable the use of accurate three-dimensional models for real-time and many-query problems. My research focuses on the development of automated MOR techniques for systems governed by partial differential equations (PDEs). The dissertation is organized in three chapters. Firstly, I discuss my contributions to linear-subspace projection-based MOR --- this class of methods, now mature, has witnessed numerous advances over the past few decades and serves as the foundation for the methods discussed in the next two chapters. Second, I present work on nonlinear approximation methods based on coordinate transformations (registration), with emphasis on the application to advection-dominated problems. Third, I discuss the extension to the component-based setting, where MOR techniques are coupled with domain decomposition to handle parameter-induced topology changes and very large-scale systems.
... There are many other choices of local approximation spaces in localized MOR approaches. In DD methods reduced spaces on the interface or in the subdomains are for example chosen as the solutions of (local constrained) eigenvalue problems in component mode synthesis (CMS) [46,11,12,41] or (generalized) harmonic polynomials, plane waves, or local solutions of the PDE accounting for instance for highly heterogeneous coefficients in the Generalized Finite Element Method (GFEM) [7,6,10,9]. In the Discontinuous Enrichment Method (DEM) [31,32] local FE spaces are enriched by adding analytical or numerical free-space solutions of the homogeneous constantcoefficient counterpart of the considered PDE, while interelement continuity is weakly enforced via Lagrange multipliers. ...
In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.
... Methods were employed in many other contexts: transient dynamics [50], multifield problems (multiphysic problems such as porous media [56] and constrained problems such as incompressible flows [70,55]), Helmotz equations [21,37,36] and contact [27,26]. The use of domain decomposition methods in structural dynamic analysis is a rather old idea though it can now be confronted to well established methods in static analysis; the Craig-Bampton algorithm [17] is somehow the application of the primal strategy to such problems, the dual version of which was proposed in [90], moreover ideas like the adjunction of coarse problems enabled to improve these methods. ...
The modern design of industrial structures leads to very complex simulations characterized by nonlinearities, high heterogeneities, tortuous geometries... Whatever the modelization may be, such an analysis leads to the solution to a family of large ill-conditioned linear systems. In this paper we study strategies to efficiently solve to linear system based on non-overlapping domain decomposition methods. We present a review of most employed approaches and their strong connections. We outline their mechanical interpretations as well as the practical issues when willing to implement and use them. Numerical properties are illustrated by various assessments from academic to industrial problems. An hybrid approach, mainly designed for multifield problems, is also introduced as it provides a general framework of such approaches.
... Some of the earliest works of coupling structures in dynamics analysis were performed by Craig and Brampton [CB68] for Boeing in 1968. Their method derives from the principle of static condensation of stiffness matrices, which allows large finite element models to be reduced to lower order systems so that large systems of equations need not be solved at every timestep of a simulation. ...
This thesis presents the development of a detailed model for the dynamic analysis of landing gear and its integration into a multidisciplinary optimization (MDO) framework. The study demonstrates significant potential for mass savings and enhanced computational efficiency in landing gear design. The initial MDO model, applied to the Boeing 737 Max 8, shows a potential 6.8% mass reduction by integrating dynamic analysis into the design optimization process. This integration underscores the coupling between structural, kinematic, and dynamic disciplines.
Subsequent chapters refine the dynamic model to increase capability and fidelity. Enhancements include the modelling of a variable-diameter metering pin for precise damping control, improved gas spring computations accounting for oil compression and real gas behavior, and treating the landing gear as a flexible body responding to drag loads. These refinements lead to higher fidelity simulations and more accurate control over structural responses, gas pressures, and damping forces.
Further improvements focus on computational performance, achieving an 83% reduction in runtime of the dynamics model through implicit integration techniques and phase segregation. Additionally, a machine learning-based approach reduces the number of load cases required for structural analysis. Integrating the refined dynamic model into the MDO framework results in a 13% mass reduction compared to a structure sized with a metering pin optimized for efficiency a priori.
The thesis concludes by highlighting the critical importance of dynamic analysis in landing gear optimization by reinforcing the strong coupling between the structural, kinematic and dynamic aspects of the design.
... Therefore, the application of this approach to problems in which elastic bodies participate in macro motion but are subjected to only small deformations is often unjustified in terms of efficiency. The most effective method of model reduction is the Craig-Bampton method [3], commonly known as dynamic reduction or the coupled substructure method [31]. When deriving the equations of the dynamics of elastic bodies subject to global motion and small deformations described in [13; 15], the following additional approximation is introduced: the inertia of an elastic structure is concentrated in the nodes of its FEM model. ...
Efficient extraction of hard-to-recover hydrocarbon reserves requires impressive expenditures on the use of advanced technologies and energy-intensive equipment. Hydraulic fracturing technologies require the use of high-pressure pumps (105 and 138 MPa), and this determines the use of plunger pumps in these conditions. To relieve the radial load on the plunger seals, a crosshead layout of the drive part is used. The goal is to investigate crosshead (guide) load distribution throughout the crankcase and also the oil distribution in the crosshead-guide gap to find the most efficient design. The methods of simulation modeling of the kinematics of the crosshead and connecting rod group with the use of modal analysis were applied for the study. The analysis was performed taking into account the elastic properties of the crosshead, seal and connecting rod. The Reynolds-Stokes-Halerkin finite element method (R-SGFEM) with sliding boundary conditions was used to model and study hydrodynamic lubrication. A study was conducted for three types of oil channels on the crosshead: one longitudinal channel at the apex, one longitudinal channel and two transverse channels, one longitudinal channel and three transverse channels. Load distribution graphs were obtained for the contact area with respect to the oil-feeding hole in the guide. The obtained data make it possible to understand the nature of the crosshead sliding on the guide and evaluate the possibility of providing the hydrostatic mode in the friction knot at different types of the lubricant supply into the contact zone. The study showed one of the causes of increased wear of the bottom guide of the crosshead, which consists in insufficient non-drying ability of the lubricating layer and allowed to determine the parameters for further research and optimization of the node design.
In this paper, we introduce a nonlinear dynamic substructuring technique to efficiently evaluate nonlinear systems with localized nonlinearities in the frequency domain. A closed-form equation is derived from coupling the dynamics of substructures and nonlinear connections. The method requires the linear frequency response functions of the substructures, which can be calculated independently using reduced-order methods. Increasing the number of linear bases in the reduction method for substructures does not affect the number of nonlinear equations, unlike in component mode synthesis techniques. The performance of the method is evaluated through three case studies: a lumped parameter system with cubic nonlinearity, bars with a small gap (normal contact), and a plate with a couple of nonlinear energy sinks. The results demonstrate promising accuracy with significantly reduced computational cost.
The model reduction technique (MRT) is an integral part of the finite element model updating (FEMU) approach to address the issue of incompleteness in measurement. It basically condenses the size of a finite element (FE) model to fit with the available responses at limited degrees of freedom. The developments in MRTs and substructure coupling for structural health monitoring (SHM) applications have been enormous. The MRTs are partly discussed in the review articles on FEMU. However, no article is dedicated explicitly to MRTs in SHM applications. Thus, a review article on MRTs will likely augment the state-of-the-art developments of MRTs in FEMU for SHM applications. This review article synthesises the growing literature on different variants of MRTs in time and frequency domains. In doing so, the fundamentals of MRT, salient modifications on the basic MRTs to ease the computational efforts and understanding of its implementation and related developments are presented first. Further, the developments of various substructure coupling techniques used to reduce the order of large FE models are presented. The authors’ recently proposed improved MRTs are also briefly presented. Finally, the prospects and challenges in MRT and substructuring techniques are critically discussed. The review, in general, reveals that the developments in MRTs are gaining importance due to their excellent capability of handling incomplete measurements, indicating the relevance of reviewing the subject from time to time to update the latest developments.
Bladed disks always have variations between blades, known as mistuning, which is a large focus of research within the turbomachinery industry. When mistuning is present, the forced response shows localized amplification of blade responses. Due to the random nature of these variations, statistical analyses are conducted to understand sensitivity to mistuning. For high-dimensional models this is an intractable problem unless the model size is reduced. To this end, reduced order models have been developed that enabled projection of mistuning to reduce computational expenses. These methods include techniques for reducing systems with many types of mistuning including modulus and geometric variations.
This work creates a generalized model of mistuning (GMM) that incorporates mistuning into the blade or disk. The GMM method has many of the benefits of previous mistuning models by being compact and only requiring sector level models. GMM utilizes an augmented Craig-Bampton component mode synthesis to decouple the blade and disk. These models are then projected onto a reduced modal space with mistuning applied. The combination of the blade and disk reduction occurs through the projection of the interface onto special disk blade interface modes. Validation is performed for a few types of mistuning (e.g. stiffness, damping, geometric) to demonstrate the effectiveness of GMM.
Deviations in the bladed disk manufacturing, such as uneven thickness, surface roughness, or sinkholes in casted wheels, can cause geometric mistuning and result in vibration amplification, severely decreasing reliability. To study the dynamic characteristics of actual industrial bladed disks that have complex geometric shapes, it is essential to accurately and quickly forecast the vibration response of geometrically mistuned systems. This is still challenging due to the high dimensionality of the geometric mistuning parameters and the extreme sensitivity of vibration response to the random geometric mistuning parameters. This paper proposes a deep neural network framework for forecasting the vibration response of mistuned blade disks with high-dimensional geometric mistuning parameters. We constructed a geometrically mistuned bladed disk deep neural network (GMS-DNN) to model the relation between the geometric mistuning parameter matrix and the blade vibration response. This approach decouples the vibration equations of mistuned systems and substitutes a deep neural network for the coupling process. GMS-DNN adopts the transformer encoder to extract geometric mistuning parameter features and uses the blade-disk boundary response to represent the variation of geometric mistuning parameters for different blades to reduce the deep neural network input parameter dimensions. We verified the validity of the proposed method using geometric deviations from an actual machined industrial bladed disk. The results show that the R2 value of the predicted response is 0.99 for the unknown test data, while the error of the amplification factor of the actual vibration response is less than 0.01.
In the floating frame of reference formulation, the exact form of the inertia forces is derived using a continuum-based approach. This yields a mass matrix and quadratic velocity terms containing inertia shape integrals. To avoid these integrals, many implementations of the floating frame formulation approximate the inertia forces by defining the kinetic energy using the lumped finite element mass matrix. This work proposes an alternative approximation of the inertia forces based on the consistent finite element mass matrix for structural elements, addressing cases where the exact solutions available in literature for most solid elements are not applicable. The inertia forces are derived by defining the kinetic energy using the consistent finite element mass matrix or by using the inertia forces from the equation of motion of the corresponding linear finite element model. In this way, the inertia shape integrals are replaced by a readily available mass matrix. In comparison with the lumped approach, the proposed definition yields more accurate results for coarser meshes since a more realistic representation of the mass and inertia properties of the body is used. Furthermore, the proposed approach yields inertia forces similar to the exact continuum-based approach under the assumption of small deformations. If the influence of deformation on the mass matrix is significant or the quadratic velocity terms are important, mesh refinement is required to accurately represent the inertia forces. The accuracy of the proposed definition of the inertia forces is compared to the exact and lumped mass approaches through simulation of flexible systems.
High‐fidelity computational fluid dynamics (CFD) simulation usually carries a heavy computational burden, especially for parametric CFD simulations requiring multiple calculations. To address this challenge, researchers have developed reduced‐order modeling (ROM) to significantly decrease the computational burden by building a simplified model. This article proposes a hybrid method of weighted proper orthogonal decomposition and Kriging, a novel reduced‐order method. This method improves the accuracy of the reduced‐order model by assigning appropriate weights to the samples while estimating the specific design parameters. The main innovation of this work is the development of the optimized method for generating the weights. Firstly, the leave‐one‐out method is employed to divide the samples into the training set and test set, and the multivariate Gaussian distribution is used to convert the Euclidean distance between the training set and test set into weight. Then, we adopt the WPOD‐Kriging method to construct a reduced‐order model using the training set. This model is compared with the test set to obtain the error. By repeatedly resetting the training set and the test set, we receive multiple errors and average them to calculate the global error. This process involves an important parameter, which is the covariance matrix of multivariate Gaussian distribution. We can generate the optimal covariance matrix by minimizing the global error to achieve the optimized method for generating the weights. The efficacy of the WPOD‐Kriging method is validated through three parametric CFD simulations. Compared to other similar approaches, the proposed method offers a more accurate reduced‐order model.
Purpose
A transmission tower usually experiences bolt loosening under long-term alternating cyclic load, which may lead to collapse of the tower in extreme operating conditions. The paper aims to propose a data-driven identification method for bolt looseness of complicated tower structures based on reduced-order models and numerical simulations to perceive and evaluate the health state of a tower in operation.
Design/methodology/approach
The equivalent stiffnesses of three types of bolt joints under various loosening scenarios are numerically determined by three-dimensional finite element (FE) simulations. The order of the FE model of a tower structure with bolt loosening is reduced by means of the component modal synthesis method, and the dynamic responses of the reducer-order model under calibration loads are simulated and used to create the dataset. An identification model for bolt looseness of the tower structure based on convolutional neural networks driven by the acceleration sensors is constructed.
Findings
An identification model for bolt looseness of the tower structure based on convolutional neural networks driven by the acceleration sensors is constructed and the applicability of the model is investigated. It is shown that the proposed method has a high identification accuracy and strong robustness to data noise and data missing. Meanwhile, the method is less dependent on the number and location of sensors and is easier to apply in real transmission lines.
Originality/value
This paper proposes a data-driven identification method for bolt looseness of a complicated tower structure based on reduced-order models and numerical simulations. Non-linear relationships between equivalent stiffness of bolted joints and bolt preload depicting looseness are obtained and reduced-order model of tower structure with bolt looseness is established. Finally, this paper investigates applicability of identification model for bolt looseness.
This work presents a new formulation for flexible fluid-conveying pipe elements based on the widely used floating frame of reference formulation. The elements can be used as a tool for the analysis of flexible multibody systems that contain fluid-conveying pipes, as it is well known that the movement of the fluid can influence the behavior and stability of such systems. The pipe defines a control volume through which the fluid, which is considered to be a moving mass, axially flows. The velocity of a material point of the fluid is therefore a material derivative of its position, representing the large rigid movement and small elastic deformation of the pipe along with the velocity of the fluid with respect to the pipe. The equations of motion are derived through the principle of virtual work, spatial discretization by finite element interpolation functions, and model reduction. A simplification of the consistent equations of motion is proposed, which avoids the use of inertia shape integrals and reduces the effort required to implement the developed fluid-conveying pipe elements in existing multibody software. The developed elements are validated by simulation of a straight cantilevered pipe and a curved pipe constrained on one end by a hinge. A simulation of a concrete printing system illustrates the straightforward incorporation of the elements in larger multibody systems.
This paper introduces a new approach to modal synthesis by integrating simulations with real-scale experiments. The technique partitions the complete aircraft structure into substructures:one representing the aircraft itself and others representing the hanging substructure thatconnects to the wing. A method is presented for imposing new frequencies on the vibrationmodes of the hanging substructure. Subsequently, experimentally obtained frequencies for theclamped substructure’s roll, yaw, and pitch vibration modes are imposed to evaluate the benefitsin accurately predicting the overall aircraft behavior. As a result, the predicted frequency forthe roll vibration mode of the substructure in the aircraft increased from 7.6 Hz to 11.9 Hz,while the result from real-scale ground vibration test was 11.6 Hz.
Digital twins are considered to be one of the most promising technologies for both green transformation and digital transformation. To facilitate the early adoption of digital twin technology, it is crucial to reduce the development time of models that can simulate with low computational cost and high accuracy. However, verifying the accuracy of these models can be time-consuming due to the significant variations in boundary conditions caused by changes in environmental conditions, design requirements, and specifications of products. In this study, we proposed an evaluation method for evaluating the accuracy of reduced order models built through parametric model order reduction. We utilized the conceptual idea of the Monte Carlo method with Latin Hypercube Sampling to randomly and comprehensively generate boundary conditions. Based on these evaluation methods, we have developed an algorithm to determine the order of the projection matrix, which is obtained through singular value decomposition of the original projection matrix generated by parametric model order reduction techniques. We applied this algorithm to a 3D FEM model with linear parameter varying convective heat transfer. The results show that the simulation time of the reduced order model, with an accuracy of 0.1 ℃ or less, can be reduced to 1/200 compared to the original FEM model (full order model), and to 1/12 compared to the original reduced order model generated by conventional parametric model order reduction techniques. This significant reduction in simulation time would enable real-time simulations in the digital twins environment.
As China’s first asteroid exploration mission, the TianWen-2 spacecraft mainly focuses on surface sampling on the near-Earth asteroid 2016 HO3. Different from the usual sampling strategy of releasing a small lander or ejecting a projectile, the whole TianWen-2 spacecraft will be controlled to achieve touchdown and sampling. However, sampling on a low-gravity regolith-covered surface with unknown physical properties is particularly challenging and risky. In order to successfully implement an in situ exploration, a comprehensive understanding and evaluation of spacecraft dynamics and control behavior during sampling is essential for system design. In this paper, a system-level simulation method for the interaction between the spacecraft and the regolith surface is proposed. The discrete element method is adopted for the simulation of granular material dynamics. Parametric comparisons with test data from microgravity low-speed intrusion drop-tower experiments are implemented to calibrate and verify the physical parameters of the discrete element method simulation model. An explicit multibody dynamics model of the spacecraft is developed to describe dynamic characteristics such as orbit and attitude motion, solar array vibration, and sampler deformation. Finally, numerical simulations of the interaction between granular dynamics, multibody dynamics, and control are performed to study the dynamics and control behaviors of TianWen-2 sampling on an asteroid regolith surface.
In this study, a method is proposed for damage identification of large-scale of two-dimensional moment resisting steel frames by utilizing Craig-Bampton sub-structuring and Gray Wolf Optimizer (GWO). Initially, an index based on the Discrete Wavelet Transform (DWT) of the first mode shape variation is employed to identify the potentially damaged substructures. Subsequently, the global structure is partitioned into damaged and undamaged substructures, making the modal analysis of each substructure more computationally efficient, easier, and less time-consuming. An iterative GWO procedure is then conducted at a lower computational cost due to the reduction in the size of the decision variable vector, which corresponds to the number of elements in the damaged substructures. To validate the proposed method, three damage scenarios are examined, involving multiple damages and incomplete mode shape measurements contaminated with a 10% noise level, as well as frequencies affected by a 5% noise level. After applying the proposed method, the stiffness reduction of the suspected substructure elements from the previous stage is estimated with high accuracy based on the results of the GWO.
The paper presents a novel approach for reducing the co‐simulation interface representation between multiple large‐scale models. The methodology leverages model order reduction through component mode synthesis in some specific small deformation flexible multibody formulations that yield a constant transformation matrix between Cartesian coordinates and general multibody coordinates, such as the flexible natural coordinates formulation or the generalized component mode synthesis. The constant transformation matrix stemming from these techniques is further modified using modified Gram–Schmidt orthonormalization and the effective independence methodology to create a constant interface model reduction matrix. This matrix effectively connects a minimal set of interface nodes to the entire nodal domain, while simultaneously projecting the forces acting on the entire nodal domain onto the interface nodes. Notably, the proposed methodology scales the size of the required co‐simulation interface representation with the considered set of mode shapes rather than the size of the numerical finite element mesh. This co‐simulation interface model reduction strategy not only renders large distributed load models compatible with the Functional Mock‐Up Interface but also extends its applicability to any structural model beyond the flexible multibody scope, provided that deformations remain relatively small. Numerical validation with a simply supported beam, connected to springs at each node, demonstrates that the interface model reduction error is significantly smaller than the co‐simulation error. This suggests that substantial interface model reduction can be achieved without compromising accuracy. Moreover, additional numerical validation performed with a rotor‐drum model showcases the versatility and scalability of the proposed approach, particularly in addressing dynamic structural systems.
Friction-induced vibration emanating from aircraft braking system is a key issue in the design phase, due to the significant damage it can cause to the brake structure. Although the problem of unstable vibrations in aircraft braking systems has been studied by a number of researchers, the suitability of the mechanical modeling strategy for predicting instabilities remains an open problem. The need for relevant numerical models is therefore essential in order to be as predictive as possible during the design phase. Preliminary studies must therefore be carried out to validate or invalidate the modeling hypotheses traditionally used. Indeed the stability analysis of an aircraft braking system is performed in order to study a low-frequency instability. An industrial model is used, hence reducing the number of degrees of freedom (DoF) is of utmost importance in order to have reasonable computation times. When studying low-frequency phenomena, this can be achieved by neglecting the deformations of the disks. However, no current study has shown that this hypothesis is realistic. So the aim of this paper is to assess the effect of the rigidity hypothesis on the results predicted by the stability analysis. In order to do so, the stability analysis results of a model with rigid disks and one with non-rigid disks are compared, with a particular attention on the main instability phenomenon. It is found that considering rigid disks has a very limited influence on the frequency of the low-frequency eigenmodes, but it over-predicts the real part of the unstable eigenmode. Besides, a component mode synthesis (CMS) technique is shown to reduce significantly the size of the non-rigid disks model while ensuring a satisfying precision regarding eigenmodes prediction.
In aero-engines, small changes in the clamp position can sometimes significantly alter the vibration stress of the pipeline system, so there is an urgent need to study the influence of clamp position on pipeline vibration stress. Firstly, a dynamic modeling method of spatial pipeline system is proposed based on the finite element method, the modeling method can more accurately simulate the complex boundary constraints caused by multiple clamps and pipe fittings. Then, an improved dynamic substructure method (reduced-order method) is developed by combining the dynamic substructure method with the clamp position parametric model to improve the efficiency of the subsequent clamp position parameters influence analysis. Further, the rationality and efficiency of the reduced-order modeling approach are verified by numerical and experimental studies. Finally, the influence of the clamp position and the key parameters of different components (pipe body, clamps, fittings) on the pipeline vibration stress is investigated. The results show that when the main vibration region of the pipeline system coincides with the installation region of the clamp, the vibration stress can be significantly reduced by installing the clamp in the maximum modal displacement region of the corresponding pipe segment. The related modeling methods and conclusions can provide valuable references for the dynamics design of pipeline syste m in engineering practice.
Summary A matrix method of linear structural analysis is described for the calculation of stresses and deflections in an aircraft structure divided into a number of structural components (substructures). The necessity for dividing a structure into these components arises either from the requirement that different types of analysis have to be used on different components, or because the capacity of the digital computer is not adequate to cope with the analysis of the complete structure. In the present method each substructure is first analyzed separately, assuming that all common boundaries (joints) with the adjacent substructures are completely fixed; these boundaries are then relaxed simultaneously and the actual boundary displacements are determined from the equations of equilibrium of forces at the boundary joints. The substructures are then analyzed separately again under the action of specified external loading and the previously determined boundary displacements.
A method is developed for analyzing complex structural systems that can be divided into interconnected components. Displacements of the separate components are expressed in generalized coordinates that are defined by displacement modes. These are generated in three categories: rigid-body, "constraint," and "normal" modes. Rigid-body modes are convenient where displacements are denned in inertial space for dynamic analysis. "Constraint" modes are included to treat redundancies in the interconnection system. "Normal" modes define displacements relative to the connections. Generalized mass, stiffness, and damping matrices are determined for each component, as are generalized forces. The requirement of system continuity gives rise to equations of displacement compatibility at the connections. These serve as equations of constraint among the component coordinates and are used to construct a transformation relating component coordinates to system coordinates. This transformation is used to derive system properties and forces from component properties and forces. System equations of motion are formulated and solved to determine system response. Component responses are found using the transformation. Connection forces are computed from the component equations. Each component can then be isolated and treated separately.
A method is presented for calculating the natural frequencies and principal modes of free undamped vibration of a system with many degrees of freedom. It is particularly suitable for the kind of system obtained by replacing each component of a complex continuous vibrating system by an appropriate lumped mass model. The method proceeds by dividing the complete problem into two stages. In the first, certain sets of constraints are imposed on the system and certain of the principal modes of the constrained systems are found. In the second stage these modes are used in a Rayleigh-Ritz analysis 0f the whole system. It is shown that the method is more economical than the straightforward analysis, and it is suggested that the errors introduced by its use can be made negligible compared to those incurred by the use of lumped-mass models. The method may easily be programmed for a computer.