System Mass Matrix Research Articles

Robosample: A software using constrained molecular dynamics coupled with Gibbs sampling to investigate macromolecules.

An ongoing challenge for the simulation of macromolecules is their complex potential energy surface which is a result of high dimensionality and chemical complexity. The surface contains high energy barriers usually difficult to overcome by existing simulation methods. We present here Robosample, a software combining the advantages of robot mechanics for fast constrained simulations and of Gibbs sampling for achieving ergodicity. While energy calculation is the main computational bottleneck for macromolecule simulations, a big part of the work can be short-circuited by increasing the trajectory integration timestep. Usually this is achieved by modifying the system's mass matrix. The way this is implemented in Robosample is by using constraints in the framework of reduced coordinates. A good example is torsional dynamics, which transforms the system in Bond-Angle-Torsion coordinates and then ignores the motion of the bonds and the angles. Robosample sampling method is Hamiltonian Monte Carlo coupled with Gibbs sampling (GCHMC), with random variables divided among molecular degrees of freedom. With GCHMC, certain molecule segments are considered rigid and sub-sampled, while focusing on soft degrees of freedom as Gibbs blocks. Ergodicity is achieved by interleaving fully flexible Cartesian dynamics Monte Carlo proposals. The validity of the method was first proven on small systems correctly reproducing alanine dipeptide free energy landscape and showed to increase sampling efficiency of macrocycle containing small molecules which are notoriously difficult to simulate. The software was subsequently extended to include spherical and cylindrical robotic joints and proved to be efficient on medium sized biological molecules, such as glycans. Now we have reached a stage where we can successfully sample big macromolecules such as antibodies on graphical processing units.

Dynamics analysis of a slender ball-screw feed system considering the changes of the worktable position

For a slender ball-screw feed system, the position of the screw–nut joints will change greatly when the moving worktable moves along its whole stroke. Therefore, the compression/tension and torsion stiffness of the screw-shaft on both sides of the screw–nut joints will change. Thus, the system transmission stiffness will also vary and affect its dynamics. In this article, considering the changes of the worktable position and using hybrid element method, a dynamic model was established for the slender ball-screw feed system, and a calculation method of the system stiffness and mass matrix was provided. The experiments on the slender ball-screw feed system with different positions were also performed to verify the proposed dynamic model. Finally, the influences of length, screw-tension force, pitch, nominal diameter of screw and rated dynamic load of screw-nut joints on the slender ball-screw feed system’s natural frequency along its whole stroke were studied. The results showed that the slender ball-screw feed system possesses obvious position-dependent variable dynamics along its whole stroke; the ratio of length to diameter of the screw ( Lf/ d0) would be better designed smaller than 50 in order to obtain the relative uniformity of the system dynamics when manufacturing large machine tools.

An analytical workbench for system level multibody dynamics

There has been considerable focus in the research community on developing accurate models, as well as on fast algorithms for solving the equations of motion of multibody systems required for simulating the dynamics of such systems. This paper focuses on the less explored complementary topic of evaluating system level dynamics properties of multibody systems. Examples of such dynamics properties are the system mass matrix, Jacobians and sensitivities of these quantities. These system level quantities manifest the dynamical properties of the system and are important for design, optimization and control. While such system level quantities are often used in theory to describe the underlying mathematical physics of the systems, due to their complexity there is a lack of a systematic methods for computing them when desired. In this paper we describe a computational workbench framework that provides a bridge between theory and the computation of such system level quantities. This workbench framework builds upon the Spatial Operator Algebra (SOA) that has been used for analysis and algorithm development for multibody dynamics. Mathematical operator expressions from the SOA can be transcribed literally to the workbench command line to allow the easy evaluation of complex dynamics quantities. We use a specific Python/C++ implementation of a workbench called PyCraft to illustrate the structure and use of such a workbench. Several examples illustrating operator-based analysis and corresponding PyCraft-based computation are included.

High-speed BEV Reducer NVH Performance Optimization and Experimentation

To solve the common rattle and whine issues in high-speed BEV reducer, this article aims to calculate and analyze the NVH performance of the reducer by using MASTA. By optimizing gear macro geometry and micro geometry to reduce the transmission error of gear pair which plays a decisive role in the common rattle and whine. By using MASTA, import the macro and micro geometry parameters of the gear pairs of the reducer. And using ABAQUS/CAE to preprocessing reducer housing, differential housing etc. and import them to MASTA. And then, assembly those components together to simulate the reducer as precise as possible. The system stiffness matrix and mass matrix are calculated which is strongly concerned to the NVH performance of the reducer. This article analyzed the assembly vibration model by calculating the assembly stiffness matrix and mass matrix and optimized the gear pairs’ parameters to reduce the transmission error of the gear pairs. And compared the optimized reducer with the original reduce in the same conditions, to make sure that the optimization is work. And so get the experience to control the NVH performance from the research. And according to these experiences, instruct engineers to design reducers that meet the harshness requirements of NVH to optimize the performance of speed reducer assembly.

Preliminary study of vibration characteristics of intelligent beam with the influence of SMA's restoring force

In order to solve the problem of noise and comfort caused by vibration during automobile driving, this paper proposes to use shape memory alloy wire as driving element and simplified beam structure to form intelligent beam composite structure, and use the restoring force generated by shape memory alloy wire to realize vibration control of the structure. several macroscopic phenomenological constitutive models of NiTi alloy were summarized and analyzed. On the basis of the Liang-Rogers one-dimensional constitutive model, Simulating the response force changing process of the SMA wire by useing MATLAB. According to the oretical analysis, we can obtain the system stiffness matrix and mass matrix, establish the dynamic model of smart beam composite structure, and analyze the interaction between shape memory composite wire prestress and natural frequency. The numerical simulation method is used to establish the simulation model and study the influence of the size and position parameters of the intelligent beam structure and the restoring force on its vibration characteristics. The results show that the natural frequency of the beam decreases and the natural frequency of the SMA wire increases with the increasing restoring force of SMA wire. As the distance from the SMA wire to the neutral axis increases, the natural frequency of the system increases, revealing the mechanism of vibration control of the shape memory alloy smart beam composite structure.

Musculoskeletal dynamics simulation using shape-varying muscle mass models

Recent studies have shown that the constant inertia assumption made in typical muscle dynamic models can lead to significant discrepancies in accuracy of simulation or inverse dynamics. This paper proposes a general framework for musculoskeletal dynamic simulation that takes into account changes in muscle inertia that occur during movement. We first develop a general shape-varying muscle mass model in which muscle deformations are modeled via linear volume-preserving transformations, and derive a corresponding muscle mass matrix and Jacobian in a Lagrangian setting. A dynamic musculoskeletal model is then constructed, in which each muscle is segmented into multiple segments that are each modeled using our earlier muscle deformation model. Depending on the extent of muscle segmentation, the musculoskeletal dynamics can be simulated to arbitrary resolution. To improve the computational efficiency of the simulation, we propose a spline-based dynamics algorithm consisting of an offline and online computation stage. In the offline stage, a parametrized B-spline surface on the space \(\mathcal{P}(n)\) of n×n symmetric positive-definite matrices is constructed so as to fit a set of sampled values of the system mass matrix. In the online computation stage, given an arbitrary configuration, the mass matrix for that configuration is obtained as a weighted average of the nearest sampled values (i.e., the control points of the B-spline surface). The Coriolis forces are evaluated directly from the partial derivatives of the B-spline approximation of the mass matrix. Our method ensures that the online computational costs effectively remain fixed independently of the system dimension or complexity. Detailed case studies involving planar arms with multiple shape-varying muscles attached demonstrate the feasibility and computational advantages of our proposed method for musculoskeletal modeling and simulation.

Dynamic analysis of non-planar coupled shear walls with stiffening beams using Continuous Connection Method

In this paper, the dynamic analysis of non-planar non-symmetrical coupled shear walls on a rigid foundation has been considered. The analysis deals with coupled shear walls having a finite number of stiffening beams, whose properties vary from region to region along the height. In the analysis, Continuous Connection Method (CCM) and Vlasov׳s theory of thin-walled beams are employed to find the stiffness matrix of the structure. The system mass matrix has been found in the form of lumped masses at the heights where the unit forces have been applied. Following the free vibration analysis, uncoupled stiffness, damping and mass matrices have been found employing the mode superposition method. A time-history analysis has been carried out using Newmark numerical integration method to find the system displacement vector for every time step. Finally, a computer program has been prepared in Fortran language and an asymmetrical example has been solved. The results have been verified via comparisons with those of the SAP2000 structural analysis program using frame method and a perfect match has been observed.

An analysis and design framework for robust control of a multi-axis active vibration isolation system with unknown payload

This paper presents a framework for model-based analysis of robust stability and performance for a multi-axis active vibration isolation system with constant but unknown payload and subject to modelling errors associated with structural flexibility. The theoretical treatment involves a linear time-invariant system subject to real parameter uncertainty associated with the unknown payload. A set of performance indices are formulated based on generalized () and measures. A method for stability/performance verification is then developed using a parameter-dependent Lyapunov function that incorporates the kinetic energy of the uncertain payload mass. This allows nonconservative bounds on the performance indices to be established via numerical solution of a corresponding set of matrix inequalities. The approach is especially suitable, and computationally efficient, for multi-degree-of-freedom systems as the overall (symmetric positive-definite) properties of the system mass matrix are accounted for without involving information for each scalar parameter. The associated LMIs can therefore be solved in polynomial time with respect to the number of unknown parameters. Numerical examples for the case of sky-hook damping control and multi-objective control are provided that demonstrate the effectiveness of the method as a tool for model-based controller evaluation and multi-objective optimization.

A radial interpolation finite element method for 2D homogeneous and multifluid coupling acoustic problems

In the acoustics Finite Element Method (FEM), the typical problems of the four-node isoparametric element are low accuracy and high sensitivity to the quality of mesh in the finite element mesh model. Therefore, a Radial Interpolation Finite Element Method (RIFEM), whose shape function is constructed by using the meshless radial interpolation technique and the partition-of-unity method, is proposed for the 2D homogeneous and multifluid coupling acoustic problems. In acoustic RIFEM, to maintain the integral accuracy of the sound pressure derivatives and apply the accurate boundary conditions easily, the acoustic system stiffness matrix and the vectors of the boundary integrals are constructed by using the bilinear shape function. On the contrary, to improve the interpolation accuracy of the approximated sound pressure function, the acoustic system mass matrix is constructed by the shape function of RIFEM using the four-node isoparametric element. Numerical examples on a 2D homogeneous acoustic tube, the 2D homogeneous acoustic cavity of a car and a 2D multifluid coupling tube verify that RIFEM achieves higher accuracy, when compared with the linear FEM and the smoothed FEM.

Finite Element Modeling of Seismic Response of Field Fabricated Liquefied Natural Gas (LNG) Spherical Storage Vessels

All real physical structures behave dynamically when subjected to loads or displacements. This research paper, therefore, presents seismic response of field fabricated liquefied natural gas spherical storage vessels using finite element analysis. The seismic analysis procedure used represents a practical approach in quantifying the response of spherical storage vessel with its content when it is subjected to seismic loading. In the finite element method approach, six degrees of freedom per node is used for legs/column of the spherical storage tanks. Lumped mass procedure is employed to determine system mass matrix of the structure. Computer programme code is developed for the resulting matrix equation form finite element analysis of the structure using FORTRAN 90 programming language. The modeling of the seismic load utilizes the ground acceleration curve of a site. From the results of the modal analysis, the system is uncoupled thereby gives way to the application of Newmark’s method. Newmark’s method as one of the widely used time-step approach for the seismic response is applied. The developed programme coding is validated with analytical results (P > 0.5). It shows that the approach in this research work can be successfully used in determine the stability of large spherical storage vessels against seismic loadings when base acceleration spectral of the site are known. This approach gives better results than the static-force approach which gives conservative results. While the approach used in this research treats seismic loads as time event, static-force approach assumed that the full ground force due to seismic motion is applied instantaneously.

Free Vibration Analysis of Asymmetric-Plan Shear Wall and Core Buildings Using One-dimensional Finite Element

In this study, a method is presented for free vibration analysis of multi-storey asymmetric structures whose bearing system is composed of shear walls and cores. The multi-storey structure was modeled as an equivalent cantilever beam in the study. The equivalent cantilever beam in the method was divided into as many finite elements as the number of the storeys. In the study first of all, a differential equation belonging to the equivalent cantilever beam was written for ith storey by using the calculation model in the continuous system. Then, a stiffness matrix for ith storey was obtained using the solution of the written differential equation. The masses in the model were applied on the storey levels. Angular frequencies were obtained using the system mass matrix and system stiffness matrix formed by storey stiffness matrices. At the end of the study, a sample taken from literature was solved and the results were evaluated to test the convenience of the method.

Controllable velocity projection for constraint stabilization in multibody dynamics

The direct numerical solution of the index-3 algebraic-differential equation system (DAE) associated with the constrained dynamics of a multibody system poses several computational difficulties mainly related to stability. Specially in long simulations, the instability is related to the drift of the solution from the velocity constraint manifold. One of the techniques proposed in the literature to overcome this problem is the velocity projection, which is a post-stabilization method that brings the solution back to the invariant manifold. This projection introduce a nonnegative artificial dissipation of energy when performed with the system mass matrix, which can affect the long term quality of the solution. This paper proposes a novel controllable velocity projection procedure capable of meeting certain requirements in the projected solution; namely, a maximum constraint value or a maximum dissipated energy. Several issues related to the efficiency of the implementation are discussed and two numerical simulations are presented. These simulations illustrate the performance of the proposed methodology and provide interesting insights about the relevance of the accuracy of the projection in the stabilization effect.

Vibration, Buckling and Dynamic Stability of a Cracked Cylindrical Shell with Time-Varying Rotating Speed

Vibration, linear elastic buckling and dynamic stability behaviors of a cracked cylindrical shell with time-varying rotating speed are analyzed. Finite element method is used to obtain the system mass and stiffness matrix, and Bolotin's method is applied to explore the dynamic stability region. The effects of constant rotating speed, crack length and orientation, and length-diameter ratio of the cylindrical shell on the free-vibration and buckling behaviors are investigated. The stability characteristics of the cracked shell are also researched and the influences of crack length, crack orientation, rotating speed basic value, steady load factor, dynamic load factor and damping ratio are considered. Numerical examples show that crack length and orientation can affect the vibration, buckling and dynamic stability behavior of cylindrical shell significantly.

Graph Theoretic Foundations of Multibody Dynamics Part I: Structural Properties.

This is the first part of two papers that use concepts from graph theory to obtain a deeper understanding of the mathematical foundations of multibody dynamics. The key contribution is the development of a unifying framework that shows that key analytical results and computational algorithms in multibody dynamics are a direct consequence of structural properties and require minimal assumptions about the specific nature of the underlying multibody system. This first part focuses on identifying the abstract graph theoretic structural properties of spatial operator techniques in multibody dynamics. The second part paper exploits these structural properties to develop a broad spectrum of analytical results and computational algorithms.Towards this, we begin with the notion of graph adjacency matrices and generalize it to define block-weighted adjacency (BWA) matrices and their 1-resolvents. Previously developed spatial operators are shown to be special cases of such BWA matrices and their 1-resolvents. These properties are shown to hold broadly for serial and tree topology multibody systems. Specializations of the BWA and 1-resolvent matrices are referred to as spatial kernel operators (SKO) and spatial propagation operators (SPO). These operators and their special properties provide the foundation for the analytical and algorithmic techniques developed in the companion paper.We also use the graph theory concepts to study the topology induced sparsity structure of these operators and the system mass matrix. Similarity transformations of these operators are also studied. While the detailed development is done for the case of rigid-link multibody systems, the extension of these techniques to a broader class of systems (e.g. deformable links) are illustrated.

A new model for dry and lubricated cylindrical joints with clearance in spatial flexible multibody systems

A new approach to model and analyze flexible spatial multibody systems with clearance of cylindrical joints is presented in this work. The flexible parts are modeled by using absolute nodal coordinate formulation (ANCF)-based elements, while the rigid parts are described by employing the natural coordinate formulation (NCF), which can lead to a constant system mass matrix for the derived system equations of motion. In a simple way, a cylindrical joint with clearance is composed of two main elements, that is, a journal inside a bearing. Additionally, a lubricant fluid can exist between these two mechanical elements to reduce the friction and wear and increase the system’s life. For the case in which the joint is modeled as a dry contact pair, a technique using a continuous approach for the evaluation of the contact force is applied, where the energy dissipation in the form of hysteresis damping is considered. Furthermore, the frictional forces developed in those contacts are evaluated by using a modified Coulomb’s friction law. For the lubricated case, the hydrodynamic theory for dynamically loaded journal bearings is used to compute the forces generated by lubrication actions. The lubricated model is based on the Reynolds equation developed for the case of journal bearings with length-to-diameter ratios up to 1. Using this approach, the misalignment of the journal inside the bearing can be studied. Finally, two demonstrative examples of application are used to provide results that support the discussion and show the validity of the proposed methodologies.

Dynamic analysis and control of multibody systems using nonlinear combination methods

In this paper, the computer computation of nonlinear combination methods in deformable multibody dynamics is described. The organization of the computer program, which is used to automatically construct and numerically solve the system of loosely coupled equations of motion expressed in terms of the Cartesian, Euler Angles and joint coordinates, is discussed. The main processor computer program consists of three main modules: constraint module, mass module and force module. The constraint module is used for the computation of the relationship among translational, rotational and joint accelerations. The mass module is used for the computation of the system mass matrix as well as the nonlinear Coriolis and centrifugal forces associated with the Cartesian, Euler Angles and modal coordinates. The force module is used to evaluate the generalized external and modal forces associated with the translational, rotational, joint and modal coordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting sparse system of loosely coupled equations. Elimination of the modal coordinates leads to small number of equations that can be solved for joint accelerations. The joint and the modal accelerations are integrated forward in time using explicit numerical integration methods. The translational/rotational displacements and velocities can then be determined using the kinematic relationships. The multibody manipulator and the slider-crank mechanical systems are used as two examples to demonstrate the use of the computational procedure discussed in this paper. Copyright © 2010 John Wiley & Sons, Ltd.

System identification of coupled heave–pitch motion of ships with forward speed in random ocean waves

Ship dynamics in ocean waves involve frequency-dependent added mass and radiation damping which can be estimated either by potential theory-based calculations or by experiments. With uniform forward speed, the added mass and damping matrices become asymmetric. In this paper, we attempt to use a system identification approach, specifically the reverse multiple input–single output (R-MISO) method, for coupled heave–pitch response (two degrees of freedom) of a ship moving with uniform forward speed in random ocean waves. The system mass matrix has both frequency-independent and frequency-dependent components, whereas its damping matrix has only frequency-dependent components. The frequency-dependent components of the mass and damping matrices are asymmetric. The excitation force and moment due to linear monochromatic waves which act on the system are assumed to be known and can be either calculated or obtained from experiments. For numerical illustration, a ship whose hydrodynamic behaviour has been computed by strip theory has been considered. The motion, as well as the loading, is simulated assuming Pierson–Moskowitz spectrum in conjunction with response amplitude operators from the strip theory code, and these results are analysed by the R-MISO method yielding frequency-dependent asymmetric coupled added mass and radiation damping coefficients satisfactorily.

TWO-LINK FLEXIBLE MANIPULATOR MODELLING AND TIP TRAJECTORY TRACKING BASED ON THE ABSOLUTE NODAL COORDINATE METHOD

It has been demonstrated that the absolute nodal coordinate formulation (ANCF) proposed recently in literature can be used to exactly describe the flexible multibody system unlike traditional methods such as the floating coordinate method and assumed mode method. Therefore, in this paper a new dynamic modeling technique for a two-link flexible manipulator based on absolute nodal coordinate method is proposed. The link shear effect was taken into account by using the 2D ANCF shear beam element. The resulting state equation can be explicitly described by generalized coordinate since the system mass matrix is constant in the ANCF framework. The proposed method is validated through the two-link flexible manipulator tip circle and square trajectory tracking control simulations by using a simple PD controller. To improve computational efficiency, the invariant matrix method and the Broyden quasi-Newton method are introduced. To improve the tracking accuracy, different PD parameters in different simulation periods are used. The simulation results indicate that modeling and controlling the flexible manipulator based on the ANCF is effective.

Master Dynamic Stability Formula for Structural Members Subjected to Periodic Loads

G = system geometric stiffness matrix K = system stiffness matrix M = system mass matrix N = axial or in-plane compressive periodic load Ncr = buckling load Ns = constant part of N Nt = periodic part of N = Ns=Ncr = Nt=Ncr f g = as given in Eq. (7) f 1g = eigenvector of the dynamic stability problem f 2g = eigenvector of the free vibration problem f 3g = eigenvector of the buckling problem = radian frequency of Nt = as defined in Eq. (12) = =! ! = natural radian frequency

Dynamic response-based characterization of ring-based vibratory angular rate sensors

Dynamic response behaviour of a rotating ring is investigated in order to better understand the achievable performance improvements as well as system limitations. For this purpose, the governing equations that represent the transverse as well as the tangential in-plane motion of a rotating ring are derived via the Hamilton’s principle. These equations are then discretized to represent a two-degree-of-freedom time-varying gyroscopic system. The asymmetry effects are considered important and are included by considering mass mismatch in the system mass matrix. In order to predict dynamic behaviour of a ring system subjected to external excitation and body rotation, time and frequency response analyses are performed. The natural frequency variations due to the gyroscopic coupling presented in the system are first characterized for varying input angular rates. The effects of system parameters such as damping and mass mismatch on the sensor sensitivity and operating range are quantified via suitable time and frequency response analyses.