Dynamic Substructure Method Research Articles

A composite experimental dynamic substructuring method based on partitioned algorithms and localized Lagrange multipliers

Abstract Successful online hybrid (numerical/physical) dynamic substructuring simulations have shown their potential in enabling realistic dynamic analysis of almost any type of non-linear structural system (e.g., an as-built/isolated viaduct, a petrochemical piping system subjected to non-stationary seismic loading, etc.). Moreover, owing to faster and more accurate testing equipment, a number of different offline experimental substructuring methods, operating both in time (e.g. the impulse-based substructuring) and frequency domains (i.e. the Lagrange multiplier frequency-based substructuring), have been employed in mechanical engineering to examine dynamic substructure coupling. Numerous studies have dealt with the above-mentioned methods and with consequent uncertainty propagation issues, either associated with experimental errors or modelling assumptions. Nonetheless, a limited number of publications have systematically cross-examined the performance of the various Experimental Dynamic Substructuring (EDS) methods and the possibility of their exploitation in a complementary way to expedite a hybrid experiment/numerical simulation. From this perspective, this paper performs a comparative uncertainty propagation analysis of three EDS algorithms for coupling physical and numerical subdomains with a dual assembly approach based on localized Lagrange multipliers. The main results and comparisons are based on a series of Monte Carlo simulations carried out on a five-DoF linear/non-linear chain-like systems that include typical aleatoric uncertainties emerging from measurement errors and excitation loads. In addition, we propose a new Composite-EDS (C-EDS) method to fuse both online and offline algorithms into a unique simulator. Capitalizing from the results of a more complex case study composed of a coupled isolated tank-piping system, we provide a feasible way to employ the C-EDS method when nonlinearities and multi-point constraints are present in the emulated system.

Compliant assembly variation analysis of sheet metal with shape errors based on primitive deformation patterns

In the compliant assembly of sheet metal, the performance of the product is highly related to the shape errors of surface. Therefore, variation analysis is generally required to reveal the influence principle of the components’ manufacturing variations on the surface shape errors of the product. The traditional compliant assembly variation analysis methods were used to build a variation propagation model based on characteristic points between parts and product without considering shape errors. In this paper, a new method based on primitive deformation patterns considering shape errors is proposed. The primitive deformation patterns of part can be obtained by natural mode analysis of ideal part, and the primitive deformation patterns of product can be calculated by the dynamic substructure method. The initial shape errors of part are decomposed into the individual contributions of primitive deformation patterns. Considering the force equilibrium relationship in assembly process, a variation propagation model is built based on the primitive deformation patterns between parts and product. This model reveals variation propagation in assembly process by the basic element of dimension error field (deformation patterns), which is convenient for evaluating the assembly quality. A case study on a panel parts assembly process is presented to demonstrate the proposed variation analysis method. The results show the effectiveness and accuracy of the proposed method compared with the method of finite element analysis conducted in commercial software ABAQUS.

Frequency Domain Updating of Thin-Walled Workpiece Dynamics Using Reduced Order Substructuring Method in Machining

The structural dynamics of thin-walled parts vary as the material is removed during machining. This paper presents a new, computationally efficient reduced order dynamic substructuring method to predict the frequency response function (FRF) of the workpiece as the material is removed along the toolpath. The contribution of the removed mass to the dynamics of the workpiece is canceled by adding a fictitious substructure having the opposite dynamics of the removed material. The equations of motion of the workpiece are updated, and workpiece FRFs are evaluated by solving the hybrid set of assembled equations of motion in frequency domain as the tool removes the material between two consecutive dynamics update steps. The orders of the initial workpiece structure and the removed substructures are reduced using a model order reduction method with a newly introduced automatic master set selection criterion. The reduced order FRF update model is validated with peripheral milling tests and FRF measurements on a plate-shaped workpiece. It is shown that the proposed model provides ∼20 times faster FRF predictions than the full order finite element (FE) model.

Effect of Soil-Structure Interaction on Seismic Performance of Long-Span Bridge Tested by Dynamic Substructuring Method

Because of the limitations of testing facilities and techniques, the seismic performance of soil-structure interaction (SSI) system can only be tested in a quite small scale model in laboratory. Especially for long-span bridge, a smaller tested model is required when SSI phenomenon is considered in the physical test. The scale effect resulting from the small scale model is always coupled with the dynamic performance, so that the seismic performance of bridge considering SSI effect cannot be uncovered accurately by the traditional testing method. This paper presented the implementation of real-time dynamic substructuring (RTDS), involving the combined use of shake table array and computational engines for the seismic simulation of SSI. In RTDS system, the bridge with soil-foundation system is divided into physical and numerical substructures, in which the bridge is seen as physical substructures and the remaining part is seen as numerical substructures. The interface response between the physical and numerical substructures is imposed by shake table and resulting reaction force is fed back to the computational engine. The unique aspect of the method is to simulate the SSI systems subjected to multisupport excitation in terms of a larger physical model. The substructuring strategy and the control performance associated with the real-time substructuring testing for SSI were performed. And the influence of SSI on a long-span bridge was tested by this novel testing method.

HYBRID DYNAMIC MODELING OF SHEARER'S DRUM DRIVING SYSTEM AND THE INFLUENCE OF HOUSING TOPOLOGY ON THE DYNAMIC CHARACTERISTICS OF GEAR

The drum shearer is one of the main equipment of the long-wall mining system that has been widely used in coal mining for decades. Influenced by large fluctuation and intensive impact of drum load, the drum driving system is a weak part of the drum shearer due to the large dynamic deformation on cutting unit housing and changes of gear meshing state. This study is concerned with hybrid finite element/lumped parameter dynamic modeling of the housing-transmission coupled system. The influence of housing topological optimization on the dynamic characteristics of gear is also analyzed through the hybrid dynamic model. In order to model the drum driving system, it is subdivided into two substructure: housing and transmission system, the housing is modeled by finite element method while the transmission system is modeled by lumped parameter method. Then the dynamic substructuring method (DSM) is used to develop a hybrid dynamic model, taking elastic coupling between the housing and transmission system into consideration. The housing topological optimization is conducted through the Optistruct solver in Hypermesh, aiming at increasing the natural frequency of housing. The effect of topological optimization on reducing the gears' equivalent mesh misalignment is validated through dynamic analysis under both random and step load.

Implementation of a Simplified Method in Design of Hysteretic Dampers for Isolated Highway Bridges

AbstractUsing seismic isolation systems for highway bridges modifies the structure’s principal vibration modes and effectively reduces the seismic base shear conveyed from the superstructure to the substructure. However, for some low-damping rubber isolation bearings, large displacements can be a problem. Supplemental hysteretic dampers can be introduced into the base-isolated bridge, which might nevertheless increase the structure base shear, and the merit of adding dampers has to be evaluated properly. In this paper, a simplified method was implemented for the design of a low-cost hysteretic damper, and the resulting isolator-damper system was tested experimentally. The design method used is based on an equivalent linearization approach. A full-scale elastomeric isolation bearing was characterized and used in the design of a hysteretic damper. Both the isolator and the damper went through cyclic testing and real-time dynamic substructuring (RTDS) methods to verify the capacity of the method to design ba...

Analysis of geometric dispersion effect of impact-induced transient waves in composite rod using dynamic substructure method

Wave propagation in a composite rod is encountered in many contact-impact events. During the travelling of the impact-induced waves along the rod, geometric dispersion is possible when the diameter of the rod is of the same order as the wavelength. The aim of this paper is to develop an efficient method, namely dynamic substructure method to analyze the geometric dispersion of the waves in the problem of the finite-length composite rod coaxially struck by a moving rigid mass. For analyzing the geometric dispersion, a new rod element is presented through taking the lateral kinetic energy of the rod into account. The transient dynamic equation expressed by the reduced modal coordinates is derived by the Lagrange equation and the fixed-interface mode synthesis method. An adhesive contact model is used to account for contact constraint. The applications of the proposed method are demonstrated using 1) compress trapezoidal impulse and 2) impact-induced waves propagate in the large-diameter rod. In order to validate the feasibility of the proposed method, the comparison is also presented between the LS-DYNA3D solutions and the substructure solutions. It indicates that the proposed method is sufficiently accurate to compute the time history of contact force and to describe the geometric dispersion characteristics such as the attenuation of the amplitude of wavefront and the increasing of rise time. The error of the 2nd peak value of the contact force Fc2 is less than 5.0%, while the time cost of the proposed method are only 1.8% of those of the LS-DYNA3D program. Furthermore, numerical results show that the geometric dispersion results in “succession collisions” phenomenon become more complicated, and it experiences three times of the transitions from contact to separation.

An efficient Laplace transform-wave packet method hybrid with substructure technique

By virtue of substructure technique, the wave packet method can be applied to efficiently evaluate the thermal conduction of each phonon mode in the large periodic harmonic lattice. The substructures are connected by the inter-belts due to the long-range interatomic interaction. A parallel dynamic substructure method is presented in the Laplace transform domain, without any approximation in the condensation process. The validity and efficiency of the proposed method is demonstrated by the one- and three-dimensional cases. By launching a wide band wave packet, the whole given phonon branch can be studied by a one-time calculation. Defined as the ratio between square of Fourier transform of the transmitted and incident wave packets, the resulted transmission coefficient agrees with that obtained by the atomistic Green’s function method. Moreover the time evolution of the wave field can be given by the method, and the multilevel substructure technique can also be used.

A frequency-based substructuring approach to efficiently model position-dependent dynamics in machine tools

Structural deformations of machine tool components that translate and/or rotate relative to each other to realize tool motion results in tool point dynamics varying along the tool path. These changing dynamics interact with the cutting process and the control loop of the drives to limit machine performance, making it necessary to virtually characterize these interactions such as to guide design decisions. To facilitate rapid evaluation of these varying dynamics, this paper describes a generalized frequency-based substructuring approach that combines the position-invariant component level receptances at the contacting interfaces between substructures to obtain the position-dependent tool point response. Receptances at the contacting interfaces are approximated by projecting them to a point to facilitate a multiple point receptance coupling formulation. Complete machine behavior is represented by just a few sets of receptances, making the model computationally more efficient than full-order finite element models and other dynamic substructuring methods. Position-dependent dynamic behavior for a representative three axis milling machine is simulated and numerically verified. Rapid investigations of the varying dynamics assist in virtually characterizing machine performance before eventual prototyping.

An Impulse Based Substructuring method for coupling impulse response functions and finite element models

In the field of Dynamic Substructuring (DS), large and complex structures are divided into several smaller and simpler components. The linear substructures are subsequently described in their dominant dynamics and reassembled, allowing one to compute the coupled dynamic behavior. DS methods are often classified into two distinct families, the Frequency Based Substructuring (FBS) methods and Component Mode Synthesis (CMS) techniques. In the former substructures are assembled whose dynamics are described in terms of frequency response functions (FRFs) and the latter are used to reduce and assemble the substructure finite element (FE) models. Lately a new substructuring method has been proposed, one that does not fit the framework of the FBS and CMS methods. The method, named Impulse Based Substructuring (IBS), was first used to obtain the coupled response of a system by assembling its component impulse response functions (IRFs). In this paper the IBS method is extended, thereby allowing one to determine the coupled behavior of structures that are composed of both substructure FE models and substructure IRFs. The method can be regarded as an extension to the normal time integration methods used for obtaining the time responses of FE models. As the linear substructures (described in their IRFs) are fully condensed on the interface of the FE model, one can significantly reduce the computational cost required for time integrating otherwise large FE models. However, as the linear (IRF) domains are exactly accounted for, the IBS method can be seen as a dynamic condensation on the interface, but not as a reduction method in the classical sense. Nonetheless, one can regard IRFs as a sort of “superelements in time” and the IBS method can therefore serve as an attractive alternative to CMS methods in case these are not available in the applied FE modeling programs, or if a high spectral bandwidth of the substructure is required. The method proposed in this work is based on the generalized-α time integration scheme and it is analytically proven that it can be applied in such a way that the simulation results are identical to the responses obtained from a monolithic integration of the full system, thereby guaranteeing its stability and accuracy. The method is demonstrated using a numerical test case, where a wind turbine FE model is coupled with a the IRFs of a marine foundation.

A New Method to Extract Substructures for the Multi-Shaft Transmission

Dynamic substructure method has been widely used in analyzing the torsional vibration characteristics of multi-shaft transmission. However, the existing literatures extract substructures of the multi-shaft transmission by treating its gears and shaft as several rotors, resulting in complex displacement coordination relationships and making it difficult to build dynamic model for the whole system. In this paper, a new method to extract substructures of the multi-shaft transmission is proposed when analyzing its torsional vibration characteristics using the dynamic substructure method. The proposed method avoids analyzing the displacement coordination relationships for the gear meshing, making it easier and more effective to establish the dynamic model for multi-shaft transmission.

Dynamic substructure analysis of stress waves generated by impacts on non-uniform rod structures

The dynamic substructure method (DSM) has high computational efficiency, and has been successfully applied in analyzing the modal responses of complex flexible bodies and in computing the dynamic responses of flexible bodies undergoing strong motion. However, the use of this method to analyze the propagation of stress waves generated by impacts has yet to be validated. The object of this paper is to propose a dynamic substructure computational procedure for the analysis of impact-induced stress waves in a non-uniform flexible structure. The procedure is established based on fixed-interface modal synthesis theory, finite element theory and a local contact model. The application of the procedure is demonstrated using composite rods subjected to longitudinal impacts. The governing equations during the contact and separation phases are solved by the Newmark-β method, and then the stress waveforms are obtained. In order to verify the accuracy of the procedure, the numerical solutions are compared with the analytical solutions obtained by method of characteristics. Furthermore, the ‘succession collision’ phenomenon resulting from the elastic wave effect is captured. All investigations showed that DSM was sufficiently accurate to analyze wave motion problems in complicated mechanical structures.

Elimination of Multi-Points Support Effects in Modal Testing

The measured Frequency Response Functions (FRFs) often have errors due to the support effects in free-free modal testing. In [1], elimination of support effects from the measured FRFs has been investigated and both stiffness and damping have been considered in support conditions. However, the method is only applicable for single point support occasion while the test structure is usually supported from more than one location in a practical testing. Therefore, this paper further investigates this method to extend its application to multi-points support conditions. Firstly, general formulas of removing the multipoint support effects from the measured FRFs are derived based on the dynamic substructure method. Then a simulated modal test of free-free beam with two points suspension is proposed to verify this method. Finally, the performance of the method is investigated with simulated noisy data. It is shown that the proposed method is effective in removing support stiffness as well as support damping from the measured FRFs as long as the measured data are not heavily contaminated by the noise.

Elimination of Support Effects in Modal Testing

In modal testing, the measured Frequency Response Functions (FRFs) are often inaccurate due to the adverse mechanical effects, such as mass-loading effects of transducers, shaker-structure interaction and the support effects. This paper deals with the elimination of the support effects from measured FRFs using dynamic substructure method. Both stiffness and damping are considered in the support effects. The validity of the method is proved in a simulated modal test. It is shown that with the given support conditions the affected FRFs can be corrected provided that some additional FRFs concerned with the support point are also measured. Finally, performance of this method is assessed with noisy measured data. Simulation shows that 1% white noise is acceptable in the proposed correction method.

MODELLING AND ANALYSIS OF A PIEZOELECTRIC ANNULAR STATOR STRUCTURE FOR A NEW IN-SITU DYNAMIC BALANCE REGULATOR

This paper presents the modelling and analysis of a piezoelectric annular stator structure for a new in-situ dynamic balance regulator which is driven by the in-plane bending vibration of the stator. The mathematical model for the in-plane vibration is first developed using an analytical method to determine the theoretical excitation frequency. The excitation modes and structural parameters are then analysed to optimise the performance of the stator. To improve the efficiency of the design and to achieve precise control, a semi-analytical electromechanical coupling model is developed using the dynamic substructure method and the finite element method. The model is verified in preliminary tests.

Analysis of Nonlinear Dynamic Accuracy on RV Transmission System <sup></sup>

RV (Rotate Vector) transmission is a new precision transmission system. In order to improve its accuracy, we study the RV transmission system. It is researched in comprehensive factors including displacement errors, elastic deformation (static transmission error, design transmission error), gear meshing errors, backlash of gear, time-varying mesh stiffness, mesh damping, bearing stiffness, torsional stiffness of input shaft, etc. The mathematical and mechanical model of dynamic transmission accuracy is established by the concentrated mass method and the dynamic substructure method. Then, the meshing force of each part is analyzed in RV reducer. The motion differential equation of RV drive system is obtained, which lays the foundation for the calculation and analysis of the transmission error.

Aeroelastic Stability of a Slender Missile with Constant Thrust

The transverse vibration, dynamic stability and flutter characteristics of a flexible missile with constant thrust are discussed. In the vibration analysis, the missile body is idealized as an un-uniform beam and the end rocket thrust is considered as constant force. The rudder is connected to the body, using the dynamic sub-structure method. Following the establishment of mathematic model based on the extended Hamilton's principle, the finite element method is introduced in the simulation of model for dynamic analysis. The first order piston theory is used in the calculation of supersonic unsteady aerodynamics. Then the flutter characteristics of flexible missile are analyzed. The results indicate that the transverse vibration characteristics of missile can be influenced in consideration of the constant thrust, which results in the dynamic instability and reduction of critical flutter speed.

Dynamic Modeling and Bifurcation Analysis of Hypersonic Aircraft

In this paper, the hypersonic aircraft model is divided into several substructures according to its structure and shape characteristics. Furthermore, the hysteresis nonlinearity of wing-fuselage connection which is caused by relaxation and friction is within consideration. The third-order piston theory and strip theory is employed to obtain the distribution of nonlinear aerodynamic force which is applied upon the wings. By means of free interface modal synthesis of dynamic substructure method, the nonlinear dynamic model of hypersonic aircraft is obtained. Then, through numerical simulation method, this paper researches on the model’s dynamic response and bifurcation characteristics, which is influenced by different hysteresis nonlinear structure parameters. In the end, it is showed that the parameter of hysteresis nonlinearity has a great impact on response and bifurcation characteristics of the system.

Approximate eigensolution of locally modified regular structures using a substructuring technique

A structure is called regular if its model can be considered as a product graph. A great computational advantage of regular structures, especially in the context of free vibration analysis, is their capacity for decomposition. In the present paper, the idea of decomposition is extended to the analysis of practical structural systems emerging as slight perturbations of the regular ones. A free interface dynamic substructuring method is employed to reformulate the eigenproblem as an assemblage of a base model and the modification part. The proposed method is very efficient in estimating a few lower eigenpairs of large-scale modified regular structures.

Dynamic substructure method for elastic fractal structures

We investigate the fractal properties on the distribution of natural frequencies of elastic structures. We use the multi-level-multi-scale dynamic substructure method so that we need only to eliminate a few nodes in advancing a fractal level. We shall give four numerical examples including the von Koch curves and the L-system. The natural frequencies and the dynamic power laws are presented. Example 4 has a fractal dimension that is a function of fractal level and has intermittence power laws discovered here for the first time.