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MODELING FLEXIBLE BODIES WITH ABSOLUTE NODE COORDINATES IN MULTIBODY SYSTEM DYNAMICS

Authors:

Abstract

Equations of motion of flexible bodies in absolute coordinates. Implementation in Universal Mechanism software. Verification. Application in well drilling dynamics, flexible tire, air spring bellows and spring models.
Санкт-Петербург 2023
MODELING FLEXIBLE BODIES WITH ABSOLUTE
NODE COORDINATES IN MULTIBODY SYSTEM
DYNAMICS
D. POGORELOV, A. RODIKOV
Bryansk State Technical University
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg
Contents
Equations of motion of flexible body in absolute coordinates
Verification of equations
Application examples
Modeling of well drilling process
Dynamic model of flexible tire
Dynamic model of air spring bellows
Modeling of spring dynamics
Conclusions
2
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg
Equations of motion of flexible body in absolute coordinates
3
Methods for simulation of flexible bodies in multibody system dynamics
1. Craig-Bampton method
M. Bampton, R. Craig // Coupling of Substructures for Dynamic Analyses. AIAA Journal. 6:7 1968. 1313-1319.
Separation of the motion of flexible body points into arbitrary spatial movements as a rigid body and small relative elastic displacements
decomposed into static and dynamic modes.
Advantage: small number of degrees of freedom.
Disadvantage: large flexible displacements cannot be modelled.
2. ANCF (Absolute nodal coordinate formulation)
A. A. Shabana // Flexible multibody dynamics: review of past and recent developments. Multibody System Dynamics. 1. 1997. 189-222.
Coordinates of nodes relative to the inertial frame are used (absolute coordinates).
Advantage: large flexible displacements are allowed.
Disadvantages:
- Big number od degrees of freedom;
- Significant CPU expenses on calculation of elements of nonlinear equations of motion.
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 4
Equations of motion of flexible body in absolute coordinates
An example of use the ANCF method for simulation of flexible body dynamics with large flexible
displacements
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 5
ri
SCi
SCF
u
rf
i
SCO
r
Equations of motion of flexible body in absolute coordinates
SCO inertial system of coordinates
SCF floating system of coordinates that performs arbitrary spatial
motion
SCi Node isystem of coordinates
Single finite element: beam, plate (triangle, quadrilateral, trapezoidal), solid
Basic ideas of the proposed method for generation of dynamic equations for flexible body in absolute nodal coordinates
1. Equations of motion for each of the finite element are derived according to the Craig-Bamption method.
2. Transition from Craig-Bampton variables to absolute nodal coordinates is executed.
3. The equations of motion of a flexible body are generated using equations for separate elements in a standard FEM
manner.
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 6
Equations of motion of flexible body in absolute coordinates
Craig-Bampton variables for single finite element
1. Floating frame position (SCF, 6 coordinates)
2. Coordinates corresponding to the natural modes of a free finite element (6n-6 coordinates, where n is the
number of nodes)
Examples of bending modes for a trapezoidal finite element
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 7
Equations of motion of flexible body in absolute coordinates
Absolute nodal coordinates, which are used for generation of equations
For each of the nodes: three Cartesian coordinates of origin of the nodal system of coordinates as well as three
angle of its orientation (6n coordinates, 24 coordinates for the four-node plate FE).
The number of Craig-Bampton coordinates and absolute nodal coordinates are the same. They are connected by
a system of nonlinear algebraic equations, which must be solved at each step of integration of the equations of
motion.
....1
),()()(
),)((
00
0
ni
ififfii
riifffi
qHAπAπA
qHρπArr
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 8
Equations of motion of flexible body in absolute coordinates
).(
,
~~ ,
~~~
f
T
f
Tq
T
f
T
rq
T
ffqffc
cffrqfcf
m
mm
ωVHqΩqεMHaMH
JωωqHMJεaρ
ρωωqHMερa
Equation of motion for a free finite element in the Craig-Bamption variables
).(,
,
1wMkDkMDDM
kMw
ff
TT
The equations are converted to the absolute nodal coordinates using linear transformation
Derivation of equations of motion is published in the paper
D. Pogorelov, A. Rodikov // The trapezoidal finite element in absolute coordinates for dynamic modeling of
automotive tire and air spring bellows. Part 1: Equations of motion. Transport Problems. 16:2 2021. 141-152.
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 9
Contents
Equations of motion of flexible body in absolute coordinates
Verification of equations
Application examples
Modeling of well drilling process
Dynamic model of flexible tire
Dynamic model of air spring bellows
Modeling of spring dynamics
Conclusions
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 10
Verification of equations
Verification of the equations is published in the paper
D. Pogorelov, A. Rodikov // The trapezoidal finite element in absolute coordinates for dynamic modeling of automotive
tire and air spring bellows. Part 2: Verification. Transport Problems. 16:3 2021. 5-16.
1. Linearized equations: computation of natural frequencies for different plates and shells, comparison with theoretical
and experimental results of other authors
Ring Conical shell 1 Conical shell 2
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 11
Verification of equations
2. Large static deflections of plates:
a) Large deflections of a square plate with clamped edges under uniform normal pressure
b) Large deflection of an annular plate with a rigid body attached to the inner edge. The rigid body is loaded in the center
with a concentrated force
Comparison of the theoretical Levy’s solution
with computation for different discretization.
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 12
Verification of equations
3. Large oscillations of a rectangular thin plate with an attached rigid body
Y
X
Z
Comparison of computations (solid lines) with experiment (markers) for
motion of the rigid body attachment points along three Cartesian axes
Experimental results are published in the paper
Wan-Suk Yoo, Jeong-Han Lee, Su-Jin Park, Jeong-Hyun Sohn, Dmitry Pogorelov, Oleg Dmitrochenko. Large deflection analysis of a
thin plate: Computer Simulations and Experiments. Multibody System Dynamics. 2004. Vol. 11. P. 185208.
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 13
Contents
Equations of motion of flexible body in absolute coordinates
Verification of equations
Application examples
Modeling of well drilling process
Dynamic model of flexible tire
Dynamic model of air spring bellows
Modeling of spring dynamics
Conclusions
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 14
Modeling of well drilling process
This is the first practical implementation of the described method: the use of a beam finite element to simulate the
dynamics of long drill strings and bottom hole assemblies in curved wells.
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 15
Dynamic model of flexible tire
This is one of the most promising areas for using the developed method: the creation of a dynamic model of a flexible tire
Finite element models of different tires
Yellow points correspond to contacts of the tire with the road
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 16
Dynamic model of flexible tire
Tire model features
1. The tire is modeled as a shell using a trapezoidal finite element
2. The finite element model takes into account the multilayer nature of the
tire material (L.P. Kollar, G.S. Springer, Mechanics of Composite
Structures. Cambridge University Press, Cambridge, 2003)
3. To contact the tire with the road surface, a brush model is implemented,
in which the contact surface of the tire is represented by a set of elastic
inertia-free bristles. One end of a bristle is connected to the final element
and the other can come into contact with the supporting surface. The
blistle realizes elastic-dissipative interaction in the radial direction and
frictional one the tangent one. An example of solving a static normal
contact problem
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 17
Dynamic model of flexible tire
Main problems to be solved
1. Computation of curves for fitting simplified tire models, which can be used in simulation of road vehicle dynamics (TMEasy,
FIALA, Pacejka tire models)
- Dependence of lateral tire force and aligning moment on side slip
- Dependence of longitudinal tire force on the longitudinal slip
2. Use the full FE tire model for simulation of vehicle dynamics in non-standard situations
- hitting a curb, hitting a stone, driving over a damaged part of the road, etc.;
- Movement of vehicle over the soil with sinkage.
3. Tire wear prediction
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 18
Tire: Dunlop P195-65R15
Tire parameter source:
Hall, Wayne (2003) Finite element modelling and
simulation for a 'smart' tyre. PhD thesis, University
of Warwick.
Marker corresponds to experiment.
Solid lines correspond to computations according
to different models of multilayer tire material.
Dynamic model of flexible tire
Comparison with static load test: deflection vs load
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 19
Comparison with tire test rig data
Dynamic model of flexible tire
Tire: Dunlop P195-70R15
Test results were provided by the Department of Automobile Transport, INRTU, Prof. Fedotov A.I.
Side force (kN) vs lateral slip angle (deg)
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 20
Comparison with tire test rig data
Dynamic model of flexible tire
Comparison of simulation results (solid lines,
speed 3.8 m/s) with test rig data (markers, speed
3.8, 6.8, 12 m/s).
Number of degrees of freedom: 3906
CPU expenses for one simulation step: 10 ms
Side force (kN) vs lateral slip angle (deg)
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 21
Friction model
Dynamic model of flexible tire
Stribeck model Influence of friction model parameters
Side force (kN) vs lateral slip angle (deg)
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 22
Self-excited oscillations for large side slip
Dynamic model of flexible tire
К. Г. Шаршуков, С. С. Капралов . МЕТОДИКА ОПРЕДЕЛЕНИЯ
ХАРАКТЕРИСТИК БОКОВОГО СЦЕПЛЕНИЯ ШИНЫ И РАСЧЕТ ИХ
ОЦЕНОЧНЫХ ПАРАМЕТРОВ, ПОЛУЧЕННЫХ В СТЕНДОВЫХ
УСЛОВИЯХ. Вестник Сибирской государственной автомобильно-
дорожной академии. 2014. № 4(38). С.44-47.
Side force (kN) vs lateral slip angle (deg)Experimental results
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 23
Dynamic model of flexible tire
Simulation with Universal Mechanism software. Screenshots: dynamics of a car with a flexible tire.
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg
Dynamic model of air spring bellows
24
Pneumatic suspensions of vehicles
Passenger rail transport
Road vehicles (buses, trucks, passenger cars)
Monorail trains
Maglev trains
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg
Dynamic model of air spring bellows
25
Finite element model of an air spring bellows using trapezoidal elements.
Number of d.o.f.: 5700
Average CPU time for one simulation step: 8.7 ms
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 26
Dynamic model of air spring bellows
Calculation of shear force depending on
displacement
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 27
Calculation of vertical force depending on
displacement
Dynamic model of air spring bellows
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 28
Torsional instability
Dynamic model of air spring bellows
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 29
Modeling of spring dynamics
The spring model is developed with the beam finite element.
Applications:
- Incorporating a full finite element spring model into a dynamic multibody model
for some types of studies
- Calculation of stiffness coefficients versus longitudinal load/deflection curves for
use in massless spring models
- Generation of models taking into account the lower natural frequencies of the
spring using the Craig-Bampton method
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 30
Modeling of spring dynamics
where , , , ,

Spring buckling
100 КЭ
200 КЭ
400 КЭ
XIII All-Russian Congress on Theoretical and Applied Mechanics >> August 21-25, 2023 >> St. Petersburg 31
Conclusions
The proposed method provides an accurate, reliable and fast tool for modeling the dynamics of elastic bodies subject
to large elastic displacements and small deformations
Further development of the method is planned with the aim of using it to simulate dynamics
Rope and belt transmissions
Belt conveyors
Leaf springs
Rope systems
Railway catenary
Cable cars
And so on…
Санкт-Петербург 2023
Contacts
pogorelov@umlab.ru
+7 (903) 260 08 18
Professor Dmitry Pogorelov
Bryansk state technical university
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