scholarly journals Numerical Integration of Multibody Dynamic Systems Involving Nonholonomic Equality Constraints

2021 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Panagiotis Passas ◽  
Elias Paraskevopoulos
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Coupled System ◽  
Weak Form ◽  
Integration Scheme ◽  
Motion Constraints ◽  
Multibody Dynamic ◽  
Time Integration Scheme

Abstract This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton’s second law and appears as a coupled system of strongly nonlinear second order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated to the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state of the art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.

Numerical Integration of a New Set of Equations of Motion for Mechanical Systems With Scleronomic Constraints

2015 ◽  
Author(s):  
Nikolaos Potosakis ◽  
Elias Paraskevopoulos ◽  
Sotirios Natsiavas
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Weak Form ◽  
Integration Scheme ◽  
Weak Formulation ◽  
First Order ◽  
Motion Constraints ◽  
Time Integration Scheme

Some new theoretical and numerical results are presented on the dynamic response of a class of mechanical systems with equality motion constraints. At the beginning, the equations of motion of the corresponding unconstrained system are presented, first in strong and then in a weak form. Next, the formulation is extended to systems with holonomic and/or nonholonomic constraints. The formulation is based on a new set of equations of motion, represented by a system of second order ordinary differential equations (ODEs) in both the coordinates and the Lagrange multipliers associated to the motion constraints. Moreover, the position, velocity and momentum type quantities are assumed to be independent, forming a three field set of equations. The weak formulation developed was first used to cast the equations of motion as a set of first order ODEs in the coordinates and the corresponding momenta. Then, the same formulation was also employed as a basis for producing a suitable time integration scheme for the systems examined. The validity and efficiency of this scheme was tested and illustrated by applying it to a number of characteristic example systems.

Plain and Full Floating Bearing Simulations With Rigid Shaft Dynamics

2007 ◽  
Author(s):  
Ian McLuckie ◽  
Scott Barrett
Keyword(s):  
System Development ◽  
Equations Of Motion ◽  
Time Integration ◽  
Forced Response ◽  
Design Parameters ◽  
Integration Scheme ◽  
Damping Coefficients ◽  
Time Integration Scheme ◽  
Combined Solution ◽  
Stiffness And Damping

This paper shows a promising predictive bearing model that can be used to reduce turbocharger bearing system development times. Turbocharger development is normally done by varying design parameters such as bearing geometry in a very time consuming experimentation process. Full Floating Bearings (FFB) are used in most automotive turbochargers and, due to emissions regulations, there has been a push towards downsizing engines and applying turbo charging to generate optimized engine solutions for both gasoline and diesel applications. In this paper the turbocharger rotor is regarded as being rigid, and the equations of motion are solved using the Bulirsch Stoer time integration scheme. These equations are solved simultaneously with the bearing model which is used also to determine nonlinear stiffness and damping coefficients. The bearings are solved using a Rigid Hydro Dynamic (RHD) Finite Difference Successive Over Relaxation (SOR) scheme of Reynolds equation that includes both rotational and squeeze velocity terms. However the solver can also consider bearing and rotor elasticity in a Multi-Body Dynamic (MBD) and Elasto-Hydro Dynamic (EHD) combined solution. Two bearing types have been studied, a plain grooved (PGB) and a full floating bearing (FFB) for comparative purposes. The mathematical models used are generic and suitable for whole engine bearing studies. The results in this paper show they are suitable for determining the onset of turbocharger bearing instability, and also the means by which bearing instability may be suppressed. The current study has investigated forced response with the combined effects of gravity and unbalance. It is worth noting that the effects of both housing excitation and aerodynamic excitation from the compressor and turbine can be easily accommodated, and will be the subject of a future paper. Other topics introduced here that will be explored further in the future include the effect of bearing and rotor flexibility in the MBD and EHD solution and the use of automatically generated stiffness and damping coefficients for any bearing geometry.

A Newly Developed Algorithm for Treating the Coupled Dynamic Response of Robotic Fixtureless Assembly

1996 ◽  
Author(s):  
Genady Shagal ◽  
Shaker A. Meguid
Keyword(s):  
Finite Element ◽  
Dynamic Response ◽  
Equations Of Motion ◽  
Time Integration ◽  
Integration Scheme ◽  
The Finite Element Method ◽  
Coupled Equations ◽  
Time Integration Scheme ◽  
Implicit Approach ◽  
Cooperating Robots

Abstract The coupled dynamic response of two cooperating robots handling two flexible payloads for the purpose of fixtureless assembly and manufacturing is treated using a new algorithm. In this algorithm, the equations describing the dynamics of the system are obtained using Lagrange’s method for the rigid robot links and the finite element method for the flexible payloads. A new time integration scheme is developed to treat the coupled equations of motion of the rigid links for a given displacement of the flexible payloads. The finite element equations of the flexible payloads are then treated using an implicit approach. The new algorithm was verified using simplified examples and was later used to examine the dynamic response of two cooperating robot arms manipulating flexible payloads which are typical of the automotive industry.

Random Wave Response of Double Pendulum Articulated Off-Shore Tower

2003 ◽  
Author(s):  
Nazrul Islam ◽  
Suhail Ahmad
Keyword(s):  
Equations Of Motion ◽  
Linear Equations ◽  
Time Integration ◽  
Power Spectra ◽  
Integration Scheme ◽  
Random Wave ◽  
Double Pendulum ◽  
Non Linear ◽  
Time Histories ◽  
Time Integration Scheme

Present study investigates the non-linear dynamic behavior of Double Hinged Articulated Tower (DHAT) under long crested random Sea and directional random sea. The non-linearities due to time wise variation of submergence, buoyancy, added mass, instantaneous tower orientation and resulting hydrodynamic loading have been taken into account for modeling the forcing functions of equation of motion which is derived by Largrangian approach. A long crested random sea has been modeled by Monte-Carlo Simulation using P-M spectrum. The non-linear equations of motion are solved by an iterative time integration scheme using Newmark’s β integration scheme. Various important parameters such as heel angles, deck displacements, base share for double hinged articulated tower under long and short crested random sea are compared and presented in the form of time-histories and their respective PSDFs. Statistical studies of random time histories have been carried out and important characteristics like mean, maxima, minima, standard deviations etc. have been analyzed. The dynamic behaviors have been investigated in detail in terms of various parametric combinations. Effect of current, and significant wave height are also studied. Sub and super harmonic excitations are highlighted through power spectra. A multi-hinged articulated tower is found to be economical and suitable for various offshore activities in adverse environmental and deep sea conditions.

Finite Element Simulation of Axisymmetric Elastic and Electroelastic Wave Propagation Using Local-Domain Wave Packet Enrichment

2021 ◽  
Vol 144 (2) ◽  
Author(s):  
Amit Kumar ◽  
Santosh Kapuria
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Wave Packet ◽  
Equations Of Motion ◽  
Time Integration ◽  
Coupled System ◽  
Integration Scheme ◽  
Local Domain ◽  
Complex Wave ◽  
Wave Modes

Abstract A local-domain wave packet enriched multiphysics finite element (FE) formulation is employed for accurately solving axisymmetric wave propagation problems in elastic and piezoelastic media, involving complex wave modes and sharp jumps at the wavefronts, which pose challenges to the conventional FE solutions. The conventional Lagrangian interpolations for the displacement and electric potential fields are enriched with the element-domain sinusoidal functions that satisfy the partition of unity condition. The extended Hamilton’s principle is employed to derive the coupled system of equations of motion which is solved using the simple Newmark-β direct time integration scheme without resorting to any remeshing near the wavefronts or post-processing. The performance of the enrichment is assessed for the axisymmetric problems of impact waves in elastic and piezoelectric cylinders and elastic half-space, bulk and Rayleigh waves in the semi-infinite elastic domain and ultrasonic Lamb wave actuation and propagation in plate-piezoelectric transducer system. The element shows significant improvement in the computational efficiency and accuracy over the conventional FE for all problems, including those involving multiple complex wave modes and sharp discontinuities in the fields at the wavefronts.

scholarly journals Structural modal interaction of a four degree-of-freedom bladed disk and casing model

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Mathias Legrand ◽  
Christophe Pierre ◽  
Bernard Peseux
Keyword(s):  
Traveling Waves ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Specific Interaction ◽  
Time Integration ◽  
Harmonic Balance Method ◽  
Integration Scheme ◽  
Nodal Diameter ◽  
Bladed Disk ◽  
Time Integration Scheme

Consideration is given to a very specific interaction phenomenon that may occur in turbomachines due to radial rub between a bladed disk and surrounding casing. These two structures, featuring rotational periodicity and axisymmetry, respectively, share the same type of eigenshapes, also termed nodal diameter traveling waves. Higher efficiency requirements leading to reduced clearance between blade-tips and casing together with the rotation of the bladed disk increase the possibility of interaction between these traveling waves through direct contact. By definition, large amplitudes as well as structural failure may be expected. A very simple two-dimensional model of outer casing and bladed disk is introduced in order to predict the occurrence of such phenomenon in terms of rotational velocity. In order to consider traveling wave motions, each structure is represented by its two nd-nodal diameter standing modes. Equations of motion are solved first using an explicit time integration scheme in conjunction with the Lagrange multiplier method, which accounts for the contact constraints, and then by the harmonic balance method (HBM). While both methods yield identical results that exhibit two distinct zones of completely different behaviors of the system, HBM is much less computationally expensive.

scholarly journals Study of Component Mode Synthesis Methods in a Rotor-Stator Interaction Case

2007 ◽  
Author(s):  
Alain Batailly ◽  
Mathias Legrand ◽  
Patrice Cartraud ◽  
Christophe Pierre ◽  
Jean-Pierre Lombard
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Component Mode Synthesis ◽  
Integration Scheme ◽  
Synthesis Methods ◽  
Specific Contact ◽  
Time Integration Scheme ◽  
Rotor Stator Interaction ◽  
Fixed Interface

The study of rotor-stator interactions between blade-tips and outer casings through direct contact in modern turbomachines is very time-consuming if the classical finite element method is used. In order to improve the knowledge over these interaction phenomena, faster methods have to be applied. The construction of reduced-order models using component mode synthesis methods generally allows for dramatic increase in computational efficiency. Two of these methods, namely a fixed interface method and a free interface methods are considered in an original manner to reduce the size of a realistic two-dimensional model. They are then compared in a very specific contact case-study. The equations of motion are solved using an explicit time integration scheme with the Lagrange multiplier method where friction is accounted for. The primary goal of the present study is to investigate the general behavior of such approaches in the presence of contact nonlinearities. It will be shown that in our contact case, a good accuracy can be obtained from a reduced models with very limited number of modes.

An Efficient Numerical Method for Dynamic Interaction Analysis of High-Speed Shinkansen Vehicles, Rail, and Bridge

1994 ◽  
Author(s):  
Makoto Tanabe ◽  
Hajime Wakui ◽  
Nobuyuki Matsumoto
Keyword(s):  
Finite Element ◽  
Numerical Method ◽  
High Speed ◽  
Interaction Analysis ◽  
Equations Of Motion ◽  
Time Integration ◽  
Integration Scheme ◽  
Element Formulation ◽  
Exact Time ◽  
Time Integration Scheme

Abstract This paper describes a finite element formulation to solve for the combined dynamic behavior of Shinkansen (bullet train) vehicles, irregular rails, and bridges. A mechanical model for interactions between a wheel and an irregular rail is discussed. The bridge is modeled by use of various finite elements. An efficient numerical method, based on modal analysis and exact time integration, is described for solving the nonlinear equations of motion of the Shinkansen vehicle and bridge. The convergence of the exact time integration scheme is discussed and compared with a previous numerical time integration scheme. A finite element computer program has been developed to analyze the dynamic response of Shinkansen vehicles operating at high speed over irregular rails and a bridge. Numerical examples are presented to demonstrate the effectiveness and validity of the present approach.

scholarly journals Aircraft Engine Structural Rotor-Stator Modal Interaction

2005 ◽  
Author(s):  
Mathias Legrand ◽  
Se´bastien Roques ◽  
Bernard Peseux ◽  
Christophe Pierre
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Aircraft Engine ◽  
Travelling Wave ◽  
Integration Scheme ◽  
Frequency Domain Method ◽  
Vibration Modes ◽  
Jet Engines ◽  
Bladed Disk ◽  
Time Integration Scheme

In modern turbo machines such as aircraft jet engines, contact between the casing and bladed disk may occur through a variety of mechanisms: coincidence of vibration modes, thermal deformation of the casing, rotor imbalance, etc. These nonlinear interactions may result in severe damage to both structures and it is important to understand the physical mechanisms that cause them and the circumstances under which they occur. In this study, we focus on the phenomenon of interaction caused by modal coincidence. A simple two-dimensional model of the casing and bladed disk structures is introduced in order to predict the occurrence of the interaction phenomenon versus the rotation speed of the rotor. Each structure is represented in terms of its two k-nodal diameter vibration modes, which are characteristic of axi-symmetric structures and allow for travelling wave motions that may interact through direct contact. The equations of motion are solved using an explicit time integration scheme in conjunction with the Lagrange multiplier method where friction is considered. Results of the numerical tool and theory show good agreement in the prediction of rotational speed to be avoided. To conclude, the mathematical statements of a multi-frequency domain-method are proposed. This method is to be used to circumvent numerical issues inherent to time-marching procedures.

Dynamic Modeling of Viscoelastic Body in Multibody System

2011 ◽  
Vol 105-107 ◽  
pp. 587-594
Author(s):  
Da Zhi Cao ◽  
Zhi Hua Zhao ◽  
Ge Xue Ren
Keyword(s):  
Fractional Derivative ◽  
Multibody System ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Time Integration ◽  
Three Dimensional ◽  
Viscoelastic Body ◽  
Integration Scheme ◽  
Large Rotation ◽  
Time Integration Scheme

Dynamic equations of viscoelastic bodies with fractional constitutive are derived base on the principle of virtual work and the theory of continuum mechanics. The three-dimensional fractional derivative viscoelastic constitutive model is implemented into the flexible multibody system (FMBS), using the 3D solid element based on the absolute nodal coordinate formulation (ANCF), which can exactly describe the geometric nonlinearities due to large rotation and large deformation. The BDF time integration scheme in conjunction with the Grünwald approximation of fractional derivative and the Newton-Raphson algorithm are used to solve the equations of motion. Several numerical examples are presented to demonstrate the use of the modeling procedure presented in this investigation and the effects of parameters in the fractional derivative model.

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