A Computational Investigation of the Disk-Housing Impacts of Accelerating Rotors Supported by Hydrodynamic Bearings

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Jaroslav Zapoměl ◽  
Petr Ferfecki
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Angular Speed ◽  
Contact Forces ◽  
Computational Time ◽  
Radial Clearance ◽  
Lagrange Equations ◽  
Time Step ◽  
Hydrodynamic Bearings ◽  
Nonlinear Force

As the radial clearance between disks and the casing of rotating machines is usually very narrow, excessive lateral vibration of accelerating rotors passing critical speeds can produce impacts between the disks and the housing. The computer modeling method is an important tool for investigating such events. In the developed procedure, the shaft is flexible and the disks are absolutely rigid. The hydrodynamic bearings and the impacts are implemented in the mathematical model by means of nonlinear force couplings. Most of the publications and computer codes from the field of rotor dynamics are referred only in the case when the rotor turns at a constant angular speed and in simple cases of disk-housing impacts. Moreover, if the disks turning at variable speed are investigated, the resulting form of the equations of motion derived by different authors slightly differs and the differences depend on the method used for their derivation. Therefore, particular emphasis in this article is given to the derivation of the motion equations of a continuous rotor turning with variable revolutions to explain the mentioned differences, to develop a computer algorithm enabling the investigation of cases when impacts between an arbitrary number of disks and the stationary part take place, and to analyze the mutual interaction between the impacts and the fluid film bearings. The Hertz theory is applied to determine the contact forces. Calculation of the hydrodynamic forces acting on the bearings is based on solving the Reynolds equation and taking cavitation into account. Lagrange equations of the second kind and the principle of virtual work are used to derive equations of motion. The Runge–Kutta method with an adaptive time step is applied for their solution. The applicability of the developed procedure was tested by computer simulations. The results show that it can be used for the modeling of complex rotor systems and that the short computational time enables carrying out calculations for a number of design and operation parameters.

Numerical Computation of Differential-Algebraic Equations for Non-Linear Dynamics of Multibody Systems Involving Contact Forces

1999 ◽  
Vol 123 (2) ◽  
pp. 272-281 ◽  
Author(s):  
B. Fox ◽  
L. S. Jennings ◽  
A. Y. Zomaya
Keyword(s):  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Differential Algebraic Equations ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Lagrange Equations ◽  
Constraint Equations

The well known Euler-Lagrange equations of motion for constrained variational problems are derived using the principle of virtual work. These equations are used in the modelling of multibody systems and result in differential-algebraic equations of high index. Here they concern an N-link pendulum, a heavy aircraft towing truck and a heavy off-highway track vehicle. The differential-algebraic equation is cast as an ordinary differential equation through differentiation of the constraint equations. The resulting system is computed using the integration routine LSODAR, the Euler and fourth order Runge-Kutta methods. The difficulty to integrate this system is revealed to be the result of many highly oscillatory forces of large magnitude acting on many bodies simultaneously. Constraint compliance is analyzed for the three different integration methods and the drift of the constraint equations for the three different systems is shown to be influenced by nonlinear contact forces.

A COMPARISON OF DIFFERENT SOLUTION ALGORITHMS FOR THE NUMERICAL ANALYSIS OF VEHICLE–BRIDGE INTERACTION

2014 ◽  
Vol 14 (02) ◽  
pp. 1350065 ◽  
Author(s):  
K. LIU ◽  
N. ZHANG ◽  
H. XIA ◽  
G. DE ROECK
Keyword(s):  
Numerical Analysis ◽  
Time Domain ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Good Choice ◽  
Contact Forces ◽  
Time Dependent ◽  
Time Step ◽  
Solution Algorithms ◽  
Coupled Equation

The interaction between a bridge and a train moving on the bridge is a coupled dynamic problem. The equations of motion of the bridge and the vehicle are coupled by the time dependent contact forces. At each time step, the motion of the bridge influences the forces transferred to the vehicle and this, in turn, changes the forces acting on the bridge. In this paper, a comparison of three different time domain solution algorithms for the coupled equation of motion of the train–bridge system is presented. Guidelines are given for a good choice of the time step.

The Recent Developments of Molecular Dynamics Simulation

2013 ◽  
Vol 444-445 ◽  
pp. 1370-1373
Author(s):  
Wen Hai Gai ◽  
Ran Guo ◽  
Yuan Yuan Liu
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulation ◽  
Rapid Development ◽  
Equations Of Motion ◽  
Dynamics Simulation ◽  
Impact Factors ◽  
Lagrange Equations ◽  
Time Step ◽  
Basic Principles ◽  
Recent Developments

Based on the development of nanomaterials and the research on performance parameters of materials, molecular dynamics simulation has been rapid development and application. It is widely found that the material's physical, mechanical and other properties are both closely related to its macroscopic state and microstructure [. In order to explore and understand the nature of the material properties we need to analyze various impact factors including macroscopic, mesoscopic and microscopic. This paper describes the basic concepts and methods of molecular dynamics. The contents are comprised of time step, formulas such as Lagrange equations of motion and Hamiltonian equations of motion. The basic principles and recent developments of molecular dynamics were reviewed.

scholarly journals Formulation of Equations of Motion for a Simply Supported Bridge under a Moving Railway Freight Vehicle

2007 ◽  
Vol 14 (6) ◽  
pp. 429-446 ◽  
Author(s):  
Ping Lou ◽  
Qing-yuan Zeng
Keyword(s):  
Equations Of Motion ◽  
Contact Forces ◽  
Dynamic Responses ◽  
Railway Track ◽  
Time Step ◽  
Bogie Frame ◽  
Simply Supported ◽  
Car Body ◽  
Railway Freight ◽  
Interaction System

Based on energy approach, the equations of motion in matrix form for the railway freight vehicle-bridge interaction system are derived, in which the dynamic contact forces between vehicle and bridge are considered as internal forces. The freight vehicle is modelled as a multi-rigid-body system, which comprises one car body, two bogie frames and four wheelsets. The bogie frame is linked with the car body through spring-dashpot suspension systems, and the bogie frame is rigidly linked with wheelsets. The bridge deck, together with railway track resting on bridge, is modelled as a simply supported Bernoulli-Euler beam and its deflection is described by superimposing modes. The direct time integration method is applied to obtain the dynamic response of the vehicle-bridge interaction system at each time step. A computer program has been developed for analyzing this system. The correctness of the proposed procedure is confirmed by one numerical example. The effect of different beam mode numbers and various surface irregularities of beam on the dynamic responses of the vehicle-bridge interaction system are investigated.

A Versatile 3D Vehicle-Track-Bridge Element for Dynamic Analysis of the Railway Bridges under Moving Train Loads

2019 ◽  
Vol 19 (04) ◽  
pp. 1950050 ◽  
Author(s):  
Xiang Xiao ◽  
Wei-Xin Ren
Keyword(s):  
Virtual Work ◽  
Interaction Analysis ◽  
Equations Of Motion ◽  
Time Integration ◽  
Contact Forces ◽  
Dynamic Responses ◽  
Railway Bridges ◽  
Beam Elements ◽  
Track Bridge ◽  
Moving Train

There has been a growing interest to carry out the vehicle–track–bridge (VTB) dynamic interaction analysis using 2D or 3D finite elements based on simplified wheel–rail relationships. The simplified or elastic wheel–rail contact relationships, however, cannot consider the lateral contact forces and geometric shapes of the wheel and rails, and even the occasional jump of wheels from the rails. This does not guarantee a reliable analysis for the safety running of trains over bridges. To consider the wheel–rail constraint and contact forces, this paper proposes a versatile 3D VTB element, consisting of a vehicle, eight rail beam elements, four bridge beam elements, and continuous springs as well as the dampers between the rail and bridge girder. With the 3D VTB element matrices formulated, a procedure for assembling the interaction matrices of the 3D VTB element is presented based on the virtual work principle. The global equations of motion of the VTB interaction system are established accordingly, which can be solved by time integration methods to obtain the dynamic responses of the vehicle, track and bridge, as well as the stability and safety indices of the moving train. Finally, an illustrative example is used to verify the proposed the versatile 3D VTB element for the dynamic interactive analysis of railway bridges under moving train loads.

Model and Method for a Time-Efficient Analysis of Lateral Drillstring Dynamics

2015 ◽  
Author(s):  
Ilja Gorelik ◽  
Marcus Neubauer ◽  
Jörg Wallaschek ◽  
Oliver Höhn
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Element Model ◽  
Steady State Solution ◽  
Contact Forces ◽  
Lumped Parameter Model ◽  
Computational Time ◽  
Operational Parameters ◽  
Real Time Analysis ◽  
Inclined Boreholes

The lumped parameter model and numerical method proposed in this paper aim at gaining a better understanding of the mechanisms leading to harmful lateral drillstring vibrations in inclined boreholes. The shooting method is applied to the equations of motion in order to skip transients and to arrive quickly at a steady state solution. In combination with a sequential continuation technique, parameter maps are generated that show regions where harmful vibrations can be avoided. Comparisons to a finite element model show that the steady state is predicted accurately, while consuming only a fraction of the computational time. The proposed model is experimentally validated on a test rig. Special emphasis is put upon the evaluation of contact forces and the frequency content of the signals. The presented investigations create the basis for real-time analysis of drillstring dynamics and can be used to give recommendations to adjust operational parameters.

A Modified Numerical Substructure Method for Dynamic Analysis of Vehicle–Track–Bridge Systems

2020 ◽  
Vol 20 (12) ◽  
pp. 2050134
Author(s):  
Yongdou Liu ◽  
Quan Gu
Keyword(s):  
Large Scale ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Interaction Force ◽  
Small Scale ◽  
Computational Accuracy ◽  
Substructure Method ◽  
Dynamic Substructuring ◽  
Nonlinear Force ◽  
Track Bridge

This paper presents a modified numerical substructure method for simulating the dynamic response of vehicle–track–bridge (VTB) systems. The method can be used to analyze large-scale VTB systems accurately and efficiently. Based on the principle of virtual work, the equations of motion are derived for two separate subsystems, i.e. a small-scale of finely modeled VTB substructure and a coarsely meshed large main bridge subsystem using different level of refinement. Different from the conventional dynamic substructuring approaches, the bridge spans close to the vehicle are modeled in both the main and substructure models, and the contradiction of repeatedly modeling is solved using a “nonlinear force corrector”. A special wheel–rail interaction (WRI) element is used to simulate the fast-moving interaction force between the vehicle and rail. In this way, the two models remain unchanged while the vehicle moves forward, and the computational accuracy is the same as the large-scale purely refined model, while the efficiency is significantly improved, particularly, for the large-scale long VTB systems. Two examples of realistic VTB systems with either smooth or un-smooth rails are used to verify the proposed method. The results demonstrate that the presented method has remarkable advantages of computational efficiency and accuracy, providing a practically useful tool for analysis of large-scale VTB systems.

A Methodology to Detect the Precise Instant of Contact in Multibody Dynamics

2011 ◽  
Author(s):  
Paulo Flores
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Time Step ◽  
Dynamic Simulations ◽  
Integration Algorithm ◽  
Current Time ◽  
Impact Time ◽  
System Equations ◽  
The Impact

The main purpose of this work is to present a general and comprehensive approach to automatically adjust the time step for the contact and non contact periods in multibody dynamics. The basic idea of the described methodology is to ensure that the first impact within a multibody system does not occur with a large value for relative bodies’ penetration in order to avoid the artificially large contact forces associated. The detection of the instant of contact takes place when the distance between two bodies change the sign between two discrete moments in time. In fact, in theory, the contact starts when this distance is zero, or a very small value to prevent the round-off errors. Thus, during the numerical solution of the system equations of motion if the first penetration is below this small value previously specified, then the current time is taken as the impact time. On the other hand, if the first penetration is larger than the specified tolerance, then the current time step is beyond the impact time. In this case, integration algorithm is forced to go back and take a smaller time step until a step can be taken within the acceptable tolerance. The main features of this approach are the easiness to implement and the good computational efficiency. In addition, it can easily deal with the transitions between non contact and contact cases in multibody dynamics. Finally, results obtained from dynamic simulations are presented and discussed to study the validity of the methodology proposed in this work.

Investigation of GPU Use in Conjunction With DCA-Based Articulated Multibody Systems Simulation

2015 ◽  
Author(s):  
Jeremy J. Laflin ◽  
Kurt S. Anderson ◽  
Michael Hans
Keyword(s):  
Multibody Systems ◽  
Graphics Processing Units ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Time ◽  
Test Case ◽  
Time Step ◽  
Dynamic Simulations ◽  
Computational Performance ◽  
Compute Time

Since computational performance is critically important for simulations to be used as an effective tool to study and design dynamic systems, the computing performance gains offered by Graphics Processing Units (GPUs) cannot be ignored. Since the GPU is designed to execute a very large number of simultaneous tasks (nominally Single Instruction Multi-Data (SIMD)), recursive algorithms in general, such as the DCA, are not well suited to be executed on GPU-type architecture. This is because each level of recursion is dependent on the previous level. However, there are some ways that the GPU can be leveraged to increase computational performance when using the DCA to form and solve the equations of motion for articulated multibody systems with a very large number of degrees-of-freedom. Computational performance of dynamic simulations is highly dependent on the nature of the underlying formulation and the number of generalized coordinates used to characterize the system. Therefore, algorithms that scale in a more desirable (lower order) fashion with the number of degrees-of-freedom are generally preferred when dealing with large (N > 10) systems. However, the utility of using simulations as a scientific tool is directly related to actual compute time. The DCA, and other top performing methods, have demonstrated the desirable property of the required compute time scaling linearly with (O(n)) with the number of degrees-of-freedom (n) and sublinearly (O(logn) performance when implemented in parallel. However for the DCA, total compute time could be further reduced by exploiting the large number of independent operations involved in the first few levels of recursion. A simple chain-type pendulum example is used to explore the feasibility of using the GPU to execute the assembly and disassembly operations for the levels of recursion that contain enough bodies for this process to be computationally advantageous. A multi-core CPU is used to perform the operations in parallel using Open MP for the remaining levels. The number of levels of recursion that utilizes the GPU is varied from zero to all levels. The data corresponding to zero utilization of the GPU provides the reference compute-time in which the assembly and disassembly operations necessary at each level are performed in parallel using Open MP. The computational time required to simulate the system for one time-step where the GPU is utilized for various levels of recursion is compared to the reference compute time also varying the number of bodies in the system. A decrease in the compute-time when using the GPU is demonstrated relative to the reference compute-time even for systems of moderate size n < 1000 for arrangements using the GPU. This is a lower number of bodies than was expected for this test case and confirms that the GPU can bring significant increases in computational efficiency for large systems, while preserving the attractive sub-linear scalability (w.r.t. compute time) of the DCA.

Dynamic Behavior of Cracked Cantilevered Pipes Conveying Fluid

2010 ◽  
Author(s):  
Fengchun Cai ◽  
Fenggang Zang ◽  
Xianhui Ye
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Relative Depth ◽  
Pipe Conveying Fluid ◽  
Lagrange Equations ◽  
Geometrical Discontinuity ◽  
Cantilevered Pipe ◽  
Work Done ◽  
Discontinuity Conditions ◽  
Conveying Fluid

In this article, the effects of the non-propagating open cracks on the dynamic behaviors of a cantilevered pipe conveying fluid are studied. The model divides the pipe into a number of segments from the crack sections and assembles all segments each by each by a rotational spring which has no mass. The stiffness of the spring is obtained through linear fracture mechanics. In order to obtain the modal functions which satisfy the boundary conditions and geometrical discontinuity conditions at the crack’s location, a simple approach is used. That is adding polynomial functions to the modal functions of the uncracked beam. The equations of motion for the cracked cantilevered pipe conveying fluid is derived based on the extended Lagrange equations for systems containing non-material volumes. Not only the virtual work done by the discharged fluid, but also that done by the fluid at the crack position due to the geometrical discontinuity conditions are considered in the present equations of motion. In this article, several numerical examples are given. The comparisons of solutions of the present equations with that of model in existence show that the present work is better. The influences of the relative depth, the position ratio of the cracks, the flow velocity on the eigenvalues are depicted.

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