decoupling equations - application of specific cases of damping in

decoupling equations - application of specific cases of damping in

PROCEEDINGS OF THE INSTITUTE OF VEHICLES 4(104)/2015
Jerzy Osiński1
DECOUPLING EQUATIONS - APPLICATION OF SPECIFIC CASES
OF DAMPING IN THE CRASH SIMULATION
1. Introduction
The calculation methods are often used for specific cases of equation, allowing the
decoupling, which shortens the computation time. General principles of decoupling were
given by Professor Eugeniusz Kamiński at work[1], below is an excerpt from this work.
The coefficients of inertia are named as a, b are damping factors and c are coefficients of
elasticity.

 air rk i y  0dla y  k ,
i ,r


 bir rk i y  0  k  1,..n,
i ,r
 y  1,..n.
 cir rk i y  0 
i ,r

The above equations are nonlinear - so there aren't direct algorithms for solving these
equations. However there is a special case of nonlinear - solution exists if the number of
unknowns is equal to the number of equations. To determine is the number n(n-1)
distribution ratios and there are a 3/2 n(n-1) equations, so a condition of existence of a
solution is the need for one of the three systems will be linearly dependent on one of the
other two. It's used in the situation where damping factors are proportional to
coefficients of inertia or coefficients of elasticity.
2. Proportional damping – Rayleigh
2.1. Decoupling linear systems with constant coefficients
Consider the vibration system described by linear differential equations with constant
coefficients stored in the form (from this moment in work is used matrix notation arrays are indicated by bold font)
  Cq  Kq  Q(t)
Mq
(1)
with damping matrix proportional to the matrix of inertia and stiffness
C  M  K
(2)
Decoupling method of such equations is well known and often used - first step is to
solve the undamped eigenproblems and then make the substitution according to
q  Yz ,
1
Jerzy Osiński, DSc. PhD. Eng. Prof., Institute of Machine Design Fundamentals, Warsaw University
of Technology
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(3)
where Y it's an eigenvectors matrix(vibration form) of undamped system. After
substituting the transformation to the equation (1), we left multiply it by transposition
matrix Y
YTMYz  YTCYz  YTKYz  YTQ
(4)
after this operation, there is a system of coupled equations
z1  2h1 z1  12z1  Qz1 t 

zn  2hn zn  n2zn  Qzn t 
(5)
All equations (5) have known solutions. They can be designate, and then performing
multiplication (4), the resulting solutions are coordinates q, which have the physical
interpretation. Equations (5) are ordered with their eigenfrequencies, to calculation it
must be adopt some lower frequencies and skip equations with higher frequencies which
have negligible small values. Transformations (3) and (4) are also used for nonlinear
systems of equations and in systems with time dependent coefficients (parametric
excitation), where the main linear part meets the decoupling conditions - so we get
system corresponding to (5) in quasi-normal coordinate. The only feedbacks are in
nonlinear and parametric terms but similarly you can take to calculate only a few
equations with the lowest forms of vibration, what significantly reduces the computation
time. Such a method of analysis was used in works [2] and [3] to test the parametric
vibrations in constant load systems. Similar method was used to test discrete-continuous
systems, which have continuous part, fulfilling the conditions of decoupling, connected
to the discrete part containing parametric excitation and nonlinear dependencies. In this
case, we are introducing the transformation as
 P 0  ξ 
q  Yz  
 
 0 I  x 
(6)
where: P is a matrix of a continuous part of vibration form, I is a unit matrix, ξ
designate normal coordinates of continuous part, x indicate generalized coordinates
describing discreet part. Just like before we get partially decoupled equation system this method was used in monograph [4], articles [5], [6] and was describe in monograph
[7].
3. Extending the applicability of modal damping in the diagnosis of large objects
Extending the applicability of modal damping can be achieved by introducing the
conditions, for the first time set by T. K. Caughe [8], [9]
(M1C)(M1K)  (M1K)(M 1C)
(7)
Such conditions were used to determine the damping of composite roof panel from
Polonez Kombi car. Figure 1 shows the research stations and Figure 2 damping
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approximation (figures are from work [10], [11]). It was found that use of condition (7)
allows to get a damping meeting the conditions of decoupling and more accurately
corresponding to experimental results.
Fig. 1. Tests of roof panel from Polonez Kombi
Fig. 2. Values of damping coefficient get in different models: 1 - experimental values, 2
- proportional damping, 3,4 - damping meeting the conditions (7)
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4. Equations of parametric vibrations - the specific case of decoupling.
Suppose that equations system describing parametric vibrations can be represented in
form
  fC (t )Cq  f K (t )Kq  0
f M (t )Mq
(8)
where fM, fC i fK are scalar periodic functions of time and M, C, K are matrices of inertia,
damping and stiffness. Assume that the basic condition need to decoupling of equations
with constant coefficients are met, for example condition (2) or (7), after the
transformation (4), we get a decoupled equations system in the normal coordinates:
f M (t )1  fC (t )  2h11  f K (t )121  0



f M (t )N  fC (t )  2hN N  f K (t ) N2  N  0
(9)
It's obvious that system (4) consist of separate equation with periodically variable
coefficients, each of them has solutions like a parametric oscillation system with one
degree of freedom. Therefore, it follows an important conclusion that the system will not
have the combined parametric resonance, only the major and minor - a more detailed
description of this issue is given in work [12].
5. Application of Rayleigh damping models in a crash simulation
5.1. EXPLICIT simulation
Damping matrix in the form (2) is used in the FEM(Finite Element Method)
calculation - it's called in publications in this field a Rayleigh damping. Decoupling of
equation is not used in this calculations - describing of damping in this form allows to
reduce the number of values required to store - remains only two coefficients α and β
and the elements of the matrix C are calculated by multiplying the coefficients by the
elements of the matrices M and K. In short-term simulation of fast changing phenomena,
especially vehicle collision (impact with an obstacle or pedestrian), is used EXPLICIT
method (finite differences method with closed-form expression), wherein inertia matrix
is created by the lumped mass method and it's a diagonal matrix. Therefore the
calculation time is reduced, because the EXPLICIT method [13] requires repeated
inversion of an inertia matrix. In damped systems it's preferred to adoption only the first
part of the formula (2) with coefficient α - damping matrix is also diagonal, computation
time of damped system is similar to undamped system. Adoption of the second part of
the formula with coefficient β leads to a very long computation time - it's almost
impossible to do it. The author with his team has made a number of damping tests with
plastics and composite materials used in safety components of motor vehicles [14], [15],
[16], [17], [18], [19] and thereafter a simulation sample of Rayleigh damping with
coefficient α, using FEM (ABAQUS system). Value of coefficient α was determined
from condition of minimizing the difference between the area of the hysteresis loop
appointed numerically and experimentally.
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5.2. Damping in materials used in safety components of motor vehicles
Elastomers - hyperelastic materials (with very large deformations) - are used in the
suspension system in cars (shock absorbers, vibration damper, displacement limiter).
These materials are practically incompressible with damping strongly dependent on
temperature. In temperatures above 00 C they are traditionally considered as elastic
materials (damping is minimal), below the 00 C there is a considerably increase of
damping - the study are presented in [14], [15]. Figure 3 shows a comparison of the
hysteresis loop of the elastomer EPUNIT at two different temperatures. Cellular
polyurethane elastomer Cellasto has a very interesting features, they are dependent on
the load - to 50% it can be assume that it's a linear elastic material, above that damping
and stiffness significantly increase - e.g. test are shown in the Figure 4.
Fig. 3. The dependence = f() in compression of Epunit material type E2, v=1mm/s,
blue line - temperature +240C, red line temperature -250C [14], [15]
Fig. 4. The compressing of the Cellasto material type 550, v=0.1 mm/s, temperature
+240C, range of deformation 0÷50% and 0÷80% [15]
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Polypropylene is used in the pedestrian protection elements. It's a material with a
granular structures, which has high level of damping - an example is shown in Figure 5,
others studies are in works [16], [17]. Studies of soft foam used in the construction of car
seats was made in cooperation with FAURECIA [18].
Fig. 5. The hysteresis loop of compression polypropylene used in the pedestrian
protection element [16], [17]
Fig. 6. The hysteresis loop of the soft foam sample, temperature -15 ° C [18]
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Fig. 7. Steps of vehicle crash into the pole V=100 km/h [19]
6. Simulation of the car impact into the pole
The above presented method - EXPLICIT simulation of Rayleigh damping with
matrix proportional to matrix of inertia [19] - was used to simulate the vehicle impact to
the thin-walled steel pole (the ABAQUS system was used in this calculations). In Figure
7 are shown the successive steps of impact, the vehicle initial velocity was 27.78 m/s
(100 km/h). In Figure 8 and 9 is shown the deformation of an octagonal pole.
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Fig. 8. Pole after a collision
V = 35 km/h
Fig. 9. Pole after a collision
V = 100 km/h
The results of the simulation obtained in work [19] indicate that the lighting poles in
our country do not meet the standards of energy-absorbing.
7. Conclusions
During operations of technical facilities there is always some dissipation of energy.
Damping of vibrations is a phenomenon which in many cases have very high values and
is of great importance. Taking the damping into account in dynamic phenomenon
simulation making the calculation a lot complicated and extending the time need for
them. It's worth to use special cases of damping, leading to simplify the calculation.
Traditionally it was used decoupling of the equation - the paper presents the conditions
given by Professor Eugeniusz Kaminski - used in the author's works to calculate
parametric vibrations in the nonlinear discrete-continuous systems. Currently, the
Rayleigh damping is used to simplify calculation in the EXPLICIT algorithm. This
article shows the result of experimental studies on damping in materials used in the cars
safety components: elastomers, cellular polyurethane and soft foam. The paper presents
the results of simulate the vehicle impact to the thin-walled steel pole (the ABAQUS
system was used in this calculations), the successive steps of impact - the vehicle initial
velocity was 27.78 m/s (100 km/h) - and deformation of an octagonal pole. The results
of the simulation indicate that the lighting poles in our country do not meets the
standards of energy-absorbing.
References:
[1] Kamiński E.: Decoupling Equation of Mechanical Systems with viscoelastic
elements (Original title in Polish Rozprzęganie układów mechanicznych z
członami lepko-sprężystymi), IPPT Works 46/1970 Warsaw,
36
[2] Osiński J.: Parametric Vibration of single stage gearbox like systems with two
degree of freedom taking into consideration damping and constant loading
(Original title in Polish Drgania parametryczne modelu jednostopniowej przekładni
zębatej jako układu o dwóch stopniach swobody uwzględniającego tłumienie i stałe
obciążenie), PhD Dissertation, Warsaw University of Technology, 1978,
[3] Kamiński E., Osiński J.: Parametric Vibration of single stage gearbox taking into
consideration damping and constant loading (Original title in Polish Drgania
parametryczne jednostopniowej przekładni zębatej uwzględniającej tłumienie i
stałe obciążenie),. Archive of Machine Designs Fundamentals, Science Works of
Warsaw University of Technology, S. Mechanika, Z. 1/1981,
[4] Osiński J.: Parametric Vibration of discrete-continuous systems (Original title in
Polish Drgania parametryczne tłumionych układów dyskretno-ciągłych), Science
Works of Warsaw University of Technology, S. Mechanika, Z. 129, 1989,
[5] Osiński J.: Modeling and analysis of vibration of discrete-continuous systems with
parametric excitation under constant load Machine Dynamics Problems,
[6] Osiński J.: Parametric Vibration of Nonlinear Systems, Nonlinear Vibration
P25/1993,
[7] Damping of Vibration, Zbigniew Osiński, editor, A. A. Balkema Brookfield,
Rottherdam 1998,
[8] Caughe T. K.: Classical normal modes in damped linear dynamics, Journal of
Applied Mechanics, transaction of the ASME, Vol.27 p. 265-271, 1960,
[9] Caughe T. K., O’Kelly M. E.: Classical normal modes in damped linear dynamics,
Journal of Applied Mechanics, transaction of the ASME, Vol.32 p. 583-588, 1965,
[10] Hoszwa K.: Identyfication of viscoelastic features of composite structure (Original
title in Polish Identyfikacja właściwości lepkosprężystych struktury
kompozytowej), PhD Dissertation, Warsaw University of Technology, 2005,
[11] Osipiak M.: Simulation testing of viscoelastic features of composite structure
(Original title in Polish Badania symulacyjne właściwości lepkosprężystych
struktury kompozytowej), PhD Dissertation, Warsaw University of Technology,
2005,
[12] Mańkowski J., Osiński J., Żabicki A.: Particular case of Decoupling Parametric
Vibration Equation (Original title in Polish Szczególny przypadek rozprzęgania
drgań parametrycznych), Works of Institute of Machine Design Fundamentals,
Warsaw University of Technology z.21/2001,
[13] Timmel M., Kolling S.,Osterrieder P.,Du Bois P.A.: A finite element model for
impact simulation with laminated glass. International Journal of Impact
Engineering 34 (2007) 1465–1478,
[14] Jungowski A., Osiński J – Energy dissipation of an elastomeric friction damper in
sub-zero temperatures, MAINTENACE PROBLEMS 3/2014 p. 83-90,
[15] Żach P. – Structural Identification of viscoelastic features of hiperdeformable
materials (Original title in Polish Strukturalna identyfikacja właściwości sprężysto
– tłumiących materiałów hiperodkształcalnych), Biblioteka Problemów
Eksploatacji - Wydawnictwo Instytutu Technologii Eksploatacji w Radomiu, 2013,
[16] Osiński J., Rumianek P., Application of modified Ogden’s model for describing
features of composites with gas phase, Machine Dynamics Problems, 2/2012 p. 6475,
[17] Osiński J., Rumianek P., Simulation of energy dissipation during impacts with
hyperelastic elements, Machine Dynamics Problems, 2/2012 p. 77-83,
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[18] Dębniak M., Niesiobęcki D., Osiński J.: Stifness-strength analysis of metal-foam
structure (Original title in Polish Analiza wytrzymałościowo-sztywnościowa
struktury metalowo-piankowej), X Jubilee Science-Technology Conference Finite
Element Method Programs in Computer Aided of Analysis, Design and
Manufacturing, Kazimierz Dolny, 13-16.11.2007 r., p. 40-41,
[19] Chomacki B.: Design and Assessment of Passive Safety of Lightning Columns
with use of Finite Element Analysis (Original title in Polish Projektowanie i ocena
bezpieczeństwa biernego stalowych słupów oświetleniowych z wykorzystaniem
MES), MSc Dissertation, Warsaw University of Technology, 2015.
Abstract
The paper presents conditions of decoupling equation of linear systems with
constant coefficients given by Professor Eugeniusz Kamiński and using of damping with
matrix proportionate to stiffness and inertia matrix. It’s presented possibilities of using
partially decoupling equation in order to simplification calculation of parametric
vibration in nonlinear discrete-continuous systems. It’s features using of Rayleigh
damping – damping matrix proportionate to mass matrix in order to simplification
calculation algorithm EXPLICIT in Finite Element Method. It contains the results of
simulation of car impact to thin walled pole (calculation using ABAQUS system) with
initial velocity 27.78 m/s (100 km/h). Displacement of pole are presented too. The
results of the simulation indicate that the lighting poles in our country do not meets the
standards of energy-absorbing.
Keywords: damping of vibration, decoupling equation, EXPLICIT, Finite Element
Method
ROZPRZĘGANIE RÓWNAŃ – ZASTOSOWANIE SZCZEGÓLNYCH
PRZYPADKÓW TŁUMIENIA W SYMULACJI ZDERZEŃ
Streszczenie
W pracy przedstawiono warunki rozprzęgania układu liniowych równań
różniczkowych o stałych współczynnikach podane przez profesora Eugeniusza
Kamińskiego – zastosowanie tłumienia o składowych proporcjonalnych do bezwładności
i sztywności. Przedstawiono możliwość zastosowania częściowego rozprzęgania
równań w celu uproszczenia obliczeń drgań parametrycznych w nieliniowych układach
dyskretno-ciągłych. Przedstawiono również wykorzystanie tłumienia Rayleigha –
macierzy tłumienia proporcjonalnej do macierzy bezwładności dla uproszczenia
algorytmu obliczeń EXPLICIT w systemach Metody Elementów Skończonych.
Przedstawiono wyniki symulacji uderzenia pojazdu w stalowy słup cienkościenny
(obliczenia wykonano systemem ABAQUS) - kolejne etapy uderzenia pojazdu
poruszającego się z prędkością początkową 27,78 m/s (100 km/godz) w słup oraz
odkształcenia słupa o przekroju ośmiokątnym. Wyniki te wskazują, że słupy
oświetleniowe w naszym kraju nie spełniają wymagań norm w zakresie
energochłonności.
Słowa kluczowe: tłumienie drgań, rozprzęganie równań, EXPLICIT, Metoda
Elementów Skończonych
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