Efficient Solution of Maggi’s Equations

2011 ◽  
Vol 7 (2) ◽  
Author(s):  
Javier García de Jalón ◽  
Alfonso Callejo ◽  
Andrés F. Hidalgo
Keyword(s):  
New York ◽  
Multibody Systems ◽  
Null Space ◽  
Mass Matrix ◽  
Real Life ◽  
Matrix Transformations ◽  
Step Method ◽  
Singular Mass Matrix ◽  
The Matrix ◽  
Constraint Enforcement

According to a recent paper (Laulusa and Bauchau, 2008, “Review of Classical Approaches for Constraint Enforcement in Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 3(1), 011004), Maggi’s formulation is a simple and stable way to solve the dynamic equations of constrained multibody systems. Among the difficulties of Maggi’s formulation, Laulusa and Bauchau quoted the need for an appropriate choice (and change, when necessary) of independent coordinates, as well as the high cost of computing and updating the basis of the tangent null space of constraint equations. In this paper, index-1 Lagrange’s equations are first considered, including the not-so-rare case of having a singular mass matrix and redundant constraints. The existence and uniqueness of solution for acceleration vector and Lagrange multipliers vector is studied in a very simple way. Then, following Von Schwerin (Von Schwerin, Multibody System Simulation. Numerical Methods, Algorithms and Software, Springer, New York, 1999), Maggi’s formulation is described as the most efficient way (in general) to solve these index-1 equations. Next, an improved double-step method, which implements the matrix transformations of Maggi’s formulation in an efficient way, is described. Finally, two large real-life examples are presented.

Efficient Solution of Maggi’s Equations

2011 ◽  
Author(s):  
Javier Garci´a de Jalo´n ◽  
Alfonso Callejo ◽  
Andre´s F. Hidalgo ◽  
Mari´a D. Gutie´rrez
Keyword(s):  
Null Space ◽  
Mass Matrix ◽  
Real Life ◽  
Matrix Transformations ◽  
Step Method ◽  
Double Step ◽  
Constraint Equations ◽  
Singular Mass Matrix ◽  
Acceleration Vector ◽  
The Matrix

According to a recent paper by Laulusa and Bauchau [1], Maggi’s formulation is a simple and stable way to solve the dynamic equations of constrained multibody systems. Among the difficulties of Maggi’s formulation, Laulusa and Bauchau quoted the need for an appropriate choice (and change, when necessary) of independent coordinates, as well as the high cost of computing and updating the basis of the tangent null space of constraint equations. In this paper, index-1 Lagrange’s equations are first considered, including the not-so-rare case of having a singular mass matrix and redundant constraints. The existence and uniqueness of solution for acceleration vector and Lagrange multipliers vector is studied in a very simple way. Then, following Von Schwerin [2], Maggi’s formulation is described as the most efficient way (globally speaking) to solve these index-1 equations. Next, an improved double-step method, which implements the matrix transformations of Maggi’s formulation in an efficient way, is described. Finally, two large real-life examples are presented.

scholarly journals Tunable characteristics of low-frequency bandgaps in two-dimensional multivibrator phononic crystal plates under prestrain

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Hai-Fei Zhu ◽  
Xiao-Wei Sun ◽  
Ting Song ◽  
Xiao-Dong Wen ◽  
Xi-Xuan Liu ◽  
...  
Keyword(s):  
Phononic Crystal ◽  
Real Life ◽  
Low Frequency ◽  
Acoustic Vibration ◽  
Crystal Plate ◽  
Frequency Noise ◽  
Two Dimensional ◽  
The Matrix ◽  
Noise Frequency ◽  
Realtime Control

AbstractIn view of the influence of variability of low-frequency noise frequency on noise prevention in real life, we present a novel two-dimensional tunable phononic crystal plate which is consisted of lead columns deposited in a silicone rubber plate with periodic holes and calculate its bandgap characteristics by finite element method. The low-frequency bandgap mechanism of the designed model is discussed simultaneously. Accordingly, the influence of geometric parameters of the phononic crystal plate on the bandgap characteristics is analyzed and the bandgap adjustability under prestretch strain is further studied. Results show that the new designed phononic crystal plate has lower bandgap starting frequency and wider bandwidth than the traditional single-sided structure, which is due to the coupling between the resonance mode of the scatterer and the long traveling wave in the matrix with the introduction of periodic holes. Applying prestretch strain to the matrix can realize active realtime control of low-frequency bandgap under slight deformation and broaden the low-frequency bandgap, which can be explained as the multiple bands tend to be flattened due to the localization degree of unit cell vibration increases with the rise of prestrain. The presented structure improves the realtime adjustability of sound isolation and vibration reduction frequency for phononic crystal in complex acoustic vibration environments.

scholarly journals Estimating Simultaneous Equation Models through an Entropy-Based Incremental Variational Bayes Learning Algorithm

Entropy ◽  
2021 ◽  
Vol 23 (4) ◽  
pp. 384
Author(s):  
Rocío Hernández-Sanjaime ◽  
Martín González ◽  
Antonio Peñalver ◽  
Jose J. López-Espín
Keyword(s):  
Statistical Theory ◽  
Learning Algorithm ◽  
Real Life ◽  
Simulated Data ◽  
Simultaneous Equation ◽  
Variational Bayes ◽  
Parameter Estimates ◽  
Step Method ◽  
The One ◽  
Simultaneous Equation Models

The presence of unaccounted heterogeneity in simultaneous equation models (SEMs) is frequently problematic in many real-life applications. Under the usual assumption of homogeneity, the model can be seriously misspecified, and it can potentially induce an important bias in the parameter estimates. This paper focuses on SEMs in which data are heterogeneous and tend to form clustering structures in the endogenous-variable dataset. Because the identification of different clusters is not straightforward, a two-step strategy that first forms groups among the endogenous observations and then uses the standard simultaneous equation scheme is provided. Methodologically, the proposed approach is based on a variational Bayes learning algorithm and does not need to be executed for varying numbers of groups in order to identify the one that adequately fits the data. We describe the statistical theory, evaluate the performance of the suggested algorithm by using simulated data, and apply the two-step method to a macroeconomic problem.

scholarly journals Estimating the number of usability problems affecting medical devices: modelling the discovery matrix

2020 ◽  
Vol 20 (1) ◽  
Author(s):  
Vincent Vandewalle ◽  
Alexandre Caron ◽  
Coralie Delettrez ◽  
Renaud Périchon ◽  
Sylvia Pelayo ◽  
...  
Keyword(s):  
Medical Devices ◽  
Market Access ◽  
Real Life ◽  
Usability Testing ◽  
Probability Of Detection ◽  
Usability Problems ◽  
Usability Problem ◽  
Wide Range ◽  
The Matrix ◽  
Problem Detection

Abstract Background Usability testing of medical devices are mandatory for market access. The testings’ goal is to identify usability problems that could cause harm to the user or limit the device’s effectiveness. In practice, human factor engineers study participants under actual conditions of use and list the problems encountered. This results in a binary discovery matrix in which each row corresponds to a participant, and each column corresponds to a usability problem. One of the main challenges in usability testing is estimating the total number of problems, in order to assess the completeness of the discovery process. Today’s margin-based methods fit the column sums to a binomial model of problem detection. However, the discovery matrix actually observed is truncated because of undiscovered problems, which corresponds to fitting the marginal sums without the zeros. Margin-based methods fail to overcome the bias related to truncation of the matrix. The objective of the present study was to develop and test a matrix-based method for estimating the total number of usability problems. Methods The matrix-based model was based on the full discovery matrix (including unobserved columns) and not solely on a summary of the data (e.g. the margins). This model also circumvents a drawback of margin-based methods by simultaneously estimating the model’s parameters and the total number of problems. Furthermore, the matrix-based method takes account of a heterogeneous probability of detection, which reflects a real-life setting. As suggested in the usability literature, we assumed that the probability of detection had a logit-normal distribution. Results We assessed the matrix-based method’s performance in a range of settings reflecting real-life usability testing and with heterogeneous probabilities of problem detection. In our simulations, the matrix-based method improved the estimation of the number of problems (in terms of bias, consistency, and coverage probability) in a wide range of settings. We also applied our method to five real datasets from usability testing. Conclusions Estimation models (and particularly matrix-based models) are of value in estimating and monitoring the detection process during usability testing. Matrix-based models have a solid mathematical grounding and, with a view to facilitating the decision-making process for both regulators and device manufacturers, should be incorporated into current standards.

scholarly journals A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process

2021 ◽  
Author(s):  
Michel Mandjes ◽  
Birgit Sollie
Keyword(s):  
Continuous Time ◽  
Real Life ◽  
Numerical Approach ◽  
Time Dependent ◽  
Death Process ◽  
Continuous Time Markov Chain ◽  
Likelihood Estimator ◽  
Real Life Data ◽  
The Matrix ◽  
Birth Death

AbstractThis paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

Use of Integrated Viscous Dampers to Control Wind Induced Vibrations in Tall Buildings

2019 ◽  
Author(s):  
Ian Ashcroft ◽  
Melissa Burton ◽  
David Farnsworth
Keyword(s):  
New York ◽  
Real Life ◽  
Tall Buildings ◽  
Cost Effective ◽  
Design Stage ◽  
Viscous Dampers ◽  
High Performing ◽  
Space Efficiency ◽  
Wind Events ◽  
Wind Comfort

<p>The tall building world is seeing a trend pushing building heights and slenderness ratios to levels previously unseen. The design of these buildings for both strength and serviceability is typically governed by the dynamic response of the building to wind. Comfort of building occupants during relatively low return period wind events is a key challenge, and engineers are increasingly turning to damping technologies to limit building accelerations rather than adding stiffness or mass. Large tuned mass dampers (TMDs) are a commonly used solution.</p><p>This paper suggests that integrating viscous dampers within a tall building’s structure can deliver a cost- effective alternative to TMDs, delivering high performing buildings with additional benefits in terms of robustness and space efficiency.</p><p>Two case studies are presented. Firstly, measured data from a tower in New York with viscous dampers integrated into the structure is provided, comparing design stage predictions to real-life performance. Furthermore, a case study for a super-slender tower is described, demonstrating the potential for enhanced performance and significant cost and space savings using integrated damping.</p>

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

scholarly journals On Domain of Nörlund Sequences

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 268 ◽  
Author(s):  
Kuddusi Kayaduman ◽  
Fevzi Yaşar
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Sequence Space ◽  
Convergent Series ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Nörlund Matrix ◽  
Inclusion Relations ◽  
The Matrix ◽  
New Space

In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space and and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α­, β­, γ­duals, and characterized their matrix transformations on this space and into this space.

CASE STUDY OF THE CONVERGENCY OF NONLINEAR PERTURBATION SERIES: MORSE–FESHBACH NONLINEAR SERIES

1999 ◽  
Vol 14 (13) ◽  
pp. 2103-2115 ◽  
Author(s):  
BISWANATH RATH
Keyword(s):  
New York ◽  
Energy Levels ◽  
Perturbation Series ◽  
Nonlinear Perturbation ◽  
Theoretical Physics ◽  
The Matrix ◽  
Diagonalization Method ◽  
Matrix Diagonalization ◽  
Divergent Behavior

We study the divergent behavior of the Morse–Feshbach nonlinear perturbation series (MFNS) [P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953)] for producing convergent energy levels using the ground state of a quartic anharmonic oscillator (AHO) in the strong coupling limit. Numerical calculations have been done up to tenth order. Further comparison of the MFNS convergent result has been made with the matrix diagonalization method.

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