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A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system
Accepted Manuscript A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system A. Guerine, A. El Hami, L. Walha, T. Fakhfakh, M. Haddar PII:
S0997-7538(16)30025-0
DOI:
10.1016/j.euromechsol.2016.03.007
Reference:
EJMSOL 3296
To appear in:
European Journal of Mechanics / A Solids
Received Date: 20 October 2014 Revised Date:
31 January 2016
Accepted Date: 14 March 2016
Please cite this article as: Guerine, A., El Hami, A., Walha, L., Fakhfakh, T., Haddar, M., A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system, European Journal of Mechanics / A Solids (2016), doi: 10.1016/j.euromechsol.2016.03.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system A. Guerine1,2 *, A. El Hami1 , L. Walha2, T. Fakhfakh2, M. Haddar2 Laboratory Optimization and Reliability in Structural Mechanics LOFIMS, Mechanical Engineering Department, National Institute of Applied Sciences of Rouen, BP 08-76801 Saint Etienne du Rouvray Cedex, France 2
Mechanics, Modelling and Manufacturing Laboratory LA2MP, Mechanical Engineering Department, National School of Engineers of Sfax, BP 1173-3038 Sfax, Tunisia
E-mail: * [email protected] Abstract
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1
In this paper, we propose a new method for taking into account uncertainties occurring due to gear friction,
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based on the projection on polynomial chaos. The new method is used to determine the dynamic response of a spur gear system with uncertainty associated to friction coefficient on the teeth contact. The simulation results
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are obtained by the polynomial chaos method for dynamic analysis under uncertainty. The proposed method is an efficient probabilistic tool for uncertainty propagation. The polynomial chaos results are compared with Monte Carlo simulations.
Key Words. Uncertainty, Friction coefficient, Spur gear system, Chaos Polynomial method. 1. Introduction
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The study and analysis of the dynamic behavior with nonlinear systems is a major interest in the industrial sector. Thus they allow overcoming the areas of instability and reducing vibration levels. Indeed, the negative consequences that may result from the instability of systems require designers to develop the most rigorous solution. This passes through a detailed study and analysis of the dynamic behavior of these systems before considering their actual implementation.
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Several parametric studies have shown the great sensitivity of the dynamic behavior of gear systems. However, these parameters admit strong dispersions. Therefore, it becomes necessary to take into account these
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uncertainties to ensure the robustness of the analysis (Guerine et al., 2015a; Guerine et al., 2015b). Also there are several studies in reliability for vibration structures taking into account the uncertainties (Abo Al-kheer et al., 2011; Mohsine and El Hami, 2010; El Hami et al., 2009; Radi and El Hami, 2007; El Hami and Radi, 1996; El Hami et al., 1993).
The mechanisms of transmission by gear tooth contact are characterized by the presence of friction coefficient that affects the vibration and noise of these systems. Parameter estimation is an important problem, because many parameters simply cannot be measured physically with good accuracy, such us the friction coefficient, especially in real time application (Blanchard et al., 2009, 2010). The coefficient of friction is a very important factor for designing, operating, and maintaining the gear transmission. Indeed, the accurate estimation of this coefficient is difficult due to the effects of various uncertain parameters, e.g., materials of gears, roughness and contact patch size, etc. However, the friction coefficient admits a strong dispersion (Nechak et al., 2011; Lee et al., 2012). Therefore, it becomes necessary to take into
ACCEPTED MANUSCRIPT account these uncertainties in order to ensure the robustness of the analysis. A study of the nonlinear dynamic behavior will help to analyze stability and to predict the vibration levels according to the parameters variations. Several methods are proposed in the literature. Monte Carlo (MC) simulation is a well-known technique in this field (Fishman, 1996). It can give the entire probability density function of any system variable, but it is often too costly since a great number of samples are required for reasonable accuracy. Parallel simulation (Papadrakakis and Papadopoulos, 1999) and proper orthogonal decomposition (Lindsley and Beran, 2005) are
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some solutions proposed to circumvent the computational difficulties of the MC method. Polynomial Chaos Expansion (PCE) is presented in the literatures a more efficient probabilistic tool for uncertainty propagation. It was first introduced by Wiener and launched by Ghanem and Spanos who used Hermite orthogonal polynomials to model stochastic processes with Gaussian random variables (Li and Ghanem, 1998).
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Polynomial Chaos (PC) gives a mathematical framework to separate the stochastic components of a system response from the deterministic ones. It used to compute the deterministic components called stochastic modes in an intrusive and non-intrusive manner while random components are concentrated in the polynomial basis
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used. The Polynomial Chaos (PC) method has been shown to be considerably more efficient than Monte Carlo in the simulation of systems with a small number of uncertain parameters (Blanchard et al., 2009; Sandu et al., 2006).
The main originality of the present paper is that the uncertainty of the gear friction system in the dynamic behavior study of one stage gear system is taken into account. The main objective is to investigate of the capabilities of the new method to determine the dynamic response of a spur gear system subject to uncertain friction coefficient. So, an eight degree of freedom system modelling the dynamic behavior of a spur gear system
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is considered. The modelling of a one stage spur gear system is presented in Section 2. The modelling of friction coefficient is presented in Section 3. In the next section, the theoretical basis of the polynomial chaos is presented. In Section 5, the equations of motion for the eight degrees of freedom are presented. Numerical results are presented in Section 6. Finally in Section 7, to conclude, some comments are made based on the study
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carried out in this paper.
2. One stage spur gear system modelling
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The dynamic model of the one stage gear system is represented on Fig. 1. This model is composed of two blocks. x Every block is supported by flexible bearing which the bending stiffness is k1 and the traction-compression y y x stiffness is k1 for the first block, k 2 and k 2 for the second block, respectively.
The wheels (11) and (22) characterize the drive and the respectively the driven gears. The two shafts (1) and (2) θ θ admit some torsional stiffness k1 and k 2 .
Angular displacements of every wheel are noticed by θ(1,1) , θ(1,2) , θ(2,1) and θ(2,2) . Besides, the linear displacements of the bearing noted by x1 and y1 for the first block, x 2 and y 2 for the second block, are measured in the plan which is orthogonal to the wheels axis of rotation (Kahraman et al., 2007).
ACCEPTED MANUSCRIPT θ(1,1)
θ(1,2)
ur Y1 ur X1
ur Y
k1x k1y
k (t)
Plan of action ur Y2
ur Z
ur X
θ( 2,1)
k 2x
θ( 2,2 )
ur X2 k 2y
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k1θ
k θ2
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Fig. 1. Model of the one stage gear system
ur ur Fig. 2 defines a reference frame O, X, Y and the position of the wheels of the one stage gear system. α is the
)
pressure angle of gearmesh contact.
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(
ur Y
ur Y
Cm 12
α
21
ur X
ur X
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11
22
line of action
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Fig. 2. Position of the wheels of the one stage gear system The teeth deflection is expressed along the line of action, and it can be written as:
δ ( t ) = s α ( x1 − x 2 ) + cα ( y1 − y 2 ) + r(b1,2) θ(1,2) − r(b2,1) θ( 2,1)
(1)
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b b Where s α = sin ( α ) and cα = cos ( α ) and r(1,2) , r(2,1) represent the base gears radius.
Generally the gear mesh stiffness variation k(t) is modelled by a sinusoid wave or by a square wave form. The later is the most representative of the real phenomenon and is represented on Fig. 3. The gear mesh stiffness variation can be decomposed in two components: an average component noted by kc, and a time variant one noted by kv(t) (Walha et al., 2009). k ( t ) = kc + kv(t)
The time component of the mesh stiffness is defined by the following periodic form (Fig. 4).
(2)
ACCEPTED MANUSCRIPT k(t) (εα - 1) Te kM kc
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km (2 - εα)Te
Time (s)
Te
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Fig. 3. Modelling of the mesh stiffness variation k(t) The extreme values of the mesh stiffness variation are defined by: kc 2 εα
and k M = −k m
2 − εα εα − 1
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km = −
(3)
ε α and Te represent respectively the contact ratio and mesh period corresponding to the two gearmeshes contacts. kv(t)
(ε
α
)
− 1 Te
kM
km
α
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0
( 2 − ε ) Te
Time (s)
Te
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Fig. 4. Modelling of the mesh stiffness variation kv(t) 3. Friction coefficient modelling
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The instantaneous coefficient of friction is defined as the ratio between the measured friction force Ff and the normal force Fn. In the case of the gear system, the number of components of the friction force is equal to the number of pair of teeth in contact. The modeling of the friction forces is usually made based on the Coulomb law. According to this model the friction coefficient is assumed constant. In the dynamic model, the friction can be introduced by two friction torques applied on the gears (12) and (21) (Fig. 5).
ξ1(t) 12
Ff ( t ) 22
11
21
P
ξ2(t) Fig. 5. Friction modeling following line of action
ACCEPTED MANUSCRIPT On the first spur gear (12), the friction torque is expressed by:
Cf12 ( t ) = Ff ( t ) . ξ1 ( t )
(4)
Also, on the second spur gear (21), the friction torque is expressed by:
Cf21 ( t ) = Ff ( t ) . ξ2 ( t )
(5)
ξ1(t) and ξ2(t) are the time varying length between the pitch point P and the corresponding center point of the
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gear (12) and (21) respectively (Fig. 5). The friction force on the pitch point P is defined by the sum of two components which corresponding to case of one or two pair of teeth in contact following the time. So, the first friction force corresponding to the first pair in contact is defined by: Ff 1 (t) = µ.k1 (t).δ(t)
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While the second force at the second pair in contact is defined by: Ff 0 (t) = µ.k 0 (t).δ(t)
(6)
(7)
δ(t) = Lδ .{Q(t)} Lδ is defined by: Lδ = [s α
− sα
cα
− cα
b 0 r(1,2) − r(b2,1)
0]
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δ(t) is the deflection along the gear mesh contact, it can be written by: (Kahraman et al., 2007) (8)
(9)
{Q(t)} is the vector of the model generalized coordinates, it is in the form: 1
y 1 x 2 y 2 θ (1,1) θ (1,2 ) θ ( 2 ,1) θ ( 2 ,2 ) ]T
4. Polynomial chaos method
(10)
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{Q(t)} = [x
In this section, we propose to apply a method based on the polynomial chaos to the treatment of uncertainties in
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the gear friction. This method consists in projecting the stochastic desired solutions on a basis of orthogonal polynomials in which the variables are Gaussian orthonormal (Guerine et al., 2015b; Dessombz, 2000). The properties of the base polynomial are used to generate a linear system of equations by means of projection. The
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resolution of this system led to an expansion of the solution on the polynomial basis, which can be used to calculate the moments of the random solution. With this method, we can easily calculate the dynamic response of a mechanical system.
Let us consider a multi-degrees of freedom linear system with mass and stiffness matrices [ M T ] and [ K T ] respectively. The equations of motion describing the forced vibration of a linear system are:
[ M ] {&&u }( t ) + [ K ] {u }( t ) = {f }( t ) T
T
T
T
T
Where {u T } is the nodal displacement vector and {f T } is the external excitation. The chaotic polynomials ψ m corresponding to the multidimensional Hermite polynomials obtained by the formula (12):
(11)
ACCEPTED MANUSCRIPT ψ m (α1 ,...,α P ) = ( −1) e P
1T {α}{α } 2
1T − {α }{α }
∂ P e 2 ∂α1 ...∂α P
(12)
Where {α } is the vector grouping the random variables:
{α }
T
= α1 ...α P
(13)
[M ] = [M ]
% + M T
[K ] = [K ]
% + K T
T
T 0
T
T 0
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The random matrices mass and stiffness [ M T ] and [ K T ] of the mechanical system can be written as: (14)
(15)
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% and K % The matrices [ M T ]0 and [ KT ]0 are deterministic matrices, the matrices M T T correspond to the random part of the mass and stiffness matrices.
% M % T and KT are rewritten from an expression of type Karhunen-Loeve (Ghanem and Spanos, 1991) in the
% M T = % K T =
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following form: P
∑ [M ] α T
p
p =1 P
∑ [K
T
p =1
]
p
p
αp
(17)
(16)
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Where α p are independent Gaussian centered reduced which may correspond to the first polynomial ψ p , while the matrices [M T ] p and [K T ] p are deterministic. We pose α 0 = 1 , we can write then: T
T
p =0 P
[K ] = ∑[K ] T
T
p =0
p
p
αp
αp
(18)
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P
[M ] = ∑ [M ]
(19)
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In the same way, we can write for {f T } : P
{f } = ∑ {f } T
T
p =0
p
αp
(20)
The dynamic response is obtained by solving the following equation knowing that the initial conditions are predefined: K eq {u T }( t + ∆t ) = {Feq }
(21)
Where: K eq = [ K T ] + a 0 [ M T ]
(22)
{F } = {f }( t + ∆t ) + [ M ] ( a {u }( t ) + a {u& }( t ) + a {&&u }( t ) ) eq
Where:
T
T
0
T
1
T
2
T
(23)
ACCEPTED MANUSCRIPT 1 B 1 , a1 = and a 2 = A ∆t A ∆t A ∆t 2
a0 =
(24)
A and B are the parameters of Newmark.
{u }( t + ∆ t ) T
is decomposed on polynomials which the P variables are Gaussian orthonormal.
{u }( t + ∆t ) = ∑ ({u }( t + ∆t ) ) ψ ({α } N
T
T
n
n
n =0
i
P
i =1
)
(25)
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Where N is the polynomial chaos order. K eq and {Feq } are written in yields the following form:
∑ [K ] T
p =0
{Feq } =
α p + a0 p
∑ ({f }( t + ∆t )) P
T
p =0
P
∑ [M ] T
p=0
αp + p
αp = p
P
∑ [M ] T
p =0
P
∑ K p =0
eq 2
α p p
(
) ∑ {F } α
α p a 0 ({u T }( t ) )0 + a1 ({u& T }( t ) )0 + a 2 ({&& u T }( t ) )0 = p
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P
K eq =
(26)
P
eq 2
p =0
p
(27)
p
Substituting Eqs. (25), (26) and (27) into Eq. (21) and forcing the residual to be orthogonal to the space spanned
P
N
p =0
n =0
∑∑
K eq 2 {u T }n α p ψ n ψ m = p
∑ {F } P
eq 2
p =0
p
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by the polynomial chaos ψ m yield the following system of linear equation:
αp ψm
m = 0, 1, . . . , N,
(28)
Where . . denotes the inner product defined by the mathematical expectation operator This algebraic equation can be rewritten in a more compact matrix form as: (0 N)
[ D]( ) ij
L
( 0) ({u T }( t + ∆t ) ) {f } 0 M M ( j ) M ({u T }( t + ∆t ) ) j = {f } O M M N ( NN ) [ D] ({u T }( t + ∆t ) ) N {f }( )
[ D]
L
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[ D]( 00) O M [ D]( N0)
[ D]
=
{f }
=
( j)
P
∑ K
eq 2
p =0
∑ {F }
α p ψ i ψ j p
P
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( ij )
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Where:
eq 2
p= 0
p
(29)
(30)
αp ψ j
(31)
Of the fact that of the orthogonality of the polynomials, most terms α p ψ n ψ m are zero. Indeed, we have:
ψi ψ j
i≠ j
=0
(32)
After resolution of the algebraic system (29), the mean values and the variances of the dynamic response are given by the following relationships: E {u T } = ({u T }( t + ∆t ) )0
Var {u T } =
(33)
∑ (({u }( t + ∆t ) ) ) (ψ ) N
2
T
n =1
n
j
2
(34)
ACCEPTED MANUSCRIPT 5. Equations of motion The differential equations describing the dynamic behavior of our system (Fig. 1) are obtained using Lagrange
Where Θ1 = ( θ(1,1) − θ(1,2) ) and Θ 2 = ( θ( 2,1) − θ( 2,2) )
Cf12p (t) = µ.k p (t).δ(t).ξ1p (t) p p p Cf 21 (t) = µ.k (t).δ(t).ξ2 (t) p represents the case of the pair 0 or 1.
6. Numerical simulation
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The torque friction components are expressed by:
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m1&& x1 + k1x x1 + s α k ( t ) Lδ {Q} = ( Ff1 − Ff0 ) s α m1&& y1 + k1y y1 + cα k ( t ) Lδ {Q} = − ( Ff1 − Ff0 ) cα m 2 && x 2 + k 2x x 2 − s α k ( t ) Lδ {Q} = − ( Ff1 − Ff0 ) s α m 2 && y 2 + k 2y y 2 − cα k ( t ) Lδ {Q} = ( Ff1 − Ff0 ) cα θ(1,1) + k1θ Θ1 = Cm I(1,1) && θ b δ 1 0 && I(1,2) θ(1,2) − k1 Θ1 + r(1,2) k ( t ) L {Q} = Cf12 ( t ) − Cf12 ( t ) θ b δ 1 0 && I( 2,1) θ( 2,1) + k 2 Θ2 − r( 2,1) k ( t ) L {Q} = − Cf 21 ( t ) + Cf 21 ( t ) θ && I( 2,2) θ( 2,2) + k 2 Θ 2 = 0
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formalism. These equations are represented as follows
(35)
(36)
(37)
summarized in the table 1.
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The technological and dimensional features of the one stage gear system in the presence of friction are
Table 1 System parameters
Motor torque
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Bearing stiffness
Cm=200 N.m k1x = k 2x =107 N/m
k1θ = k θ2 =105 Nm/rad
Number of teeth
Z(12)=40 ; Z(21)=50
Module of teeth
m=4*10-3 m
Contact ratio
εα = 1.7341
Pressure angle
α=20°
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Torsional stiffness of shaft
k1y = k 2y =107 N/m
6.1. Stability analysis For a designer, the main objective of a robust stability analysis is to define with certainty that the system studied is stable for an uncertain parameter defined within an uncertain interval (such as the dispersion interval of the friction coefficient). This parametric study consists in calculating the eigenvalues of the linearized system at each value of the uncertain parameter, then stability is analyzed by testing the sign of the real parts of the eigenvalues obtained (Nechak et al., 2011). This procedure becomes difficult for nonlinear systems of a higher
ACCEPTED MANUSCRIPT order, since the calculation of the corresponding eigenvalues goes through the resolution of characteristic equations of higher order (Fisher and Bhattacharya, 2008; Sinou and Jézéquel, 2007; Nechak et al., 2011). The evolution of the system eigenvalues is plotted against the values of the friction coefficient in Figs. 6.1 and 6.2. The imaginary parts of these eigenvalues represent the instability frequencies. It appears that at values of friction coefficient which belong to [0,0.109], the real part of the eigenvalues is negative. The system is then stable.
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µc
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Furthermore, the frequencies associated with the eigenvalues of the system are discrete.
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Fig. 6.1. Evolution of the real part of eigenvalues according to the friction coefficient values
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µc
Fig. 6.2. Evolution of the imaginary part of eigenvalues according to the friction coefficient values
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The answer is composed of oscillations of constant level. When the coefficient of friction increases, the frequencies tend to be closer to become equal to a certain value of the coefficient of friction, which we call by the following critical µ c . In this case, the coefficient of friction is equal to review 0.109 and corresponds to the value at which the eigenvalues coupling. Beyond this value, the eingenvalues possess a positive real part. The system is unstable. Note that the overall response of the system is imposed by the unstable mode dominates rapidly in amplitude on the stable mode. This therefore results in increasing the frequency characterizing the unstable mode oscillations.
6.2. Dynamic behavior with friction Fig. 7 represents the temporal fluctuation of the resultant displacement of the first bearing. We notice that the presence of friction affects little the linear displacements of the bearing and the amplitudes augment slightly
ACCEPTED MANUSCRIPT without changing the form of the signal. Indeed, the friction causes the amplifications of the amplitudes of all degrees of freedom. Figs. 8 and 9 represent the effects of friction on the dynamic behavior of the wheels. We notice that the friction reduces the angular displacements of the wheels. Its effects increase over time, when the displacement also increases.
- -: µ=0.08
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―: µ=0
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Fig. 7. Temporal fluctuation of the resultant displacement of the first bearing
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— :θ(1,1) - - - :θ(1,2) -.-. :θ(2,1) … : θ(2,2)
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Fig. 8. Temporal fluctuations of the angular displacement of the wheels (µ=0)
— :θ(1,1) - - - :θ(1,2) -.-. :θ(2,1) … : θ(2,2)
Fig. 9. Temporal fluctuations of the angular displacement of the wheels (µ=0.08)
ACCEPTED MANUSCRIPT 6.3. Study with Polynomial Chaos In this section numerical results are presented for the new method for which the formulations are derived in the section 4. The polynomial chaos results are compared with Monte Carlo simulations with 100000 simulations. The friction coefficient is supposed random variable and defined as follow: µ = µ0 + σµ ξ
(38)
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Where ξ is a zero mean value Gaussian random variable, µ 0 is the mean value and σ µ is the standard deviation of this parameter.
The mean value and the variance of the dynamic component of the linear displacement of the first bearing in two directions x and y have been calculated by the polynomial chaos method. The obtained results are compared with those given by the Monte Carlo simulations with 100000 simulations. The results are plotted in Figs. 10 and 12
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for σµ =2% and in Figs. 11 and 13 for σµ =5%.
MC …. PC (N=2)
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MC …. PC (N=2)
Fig. 11.1. Mean value of x1(t) σµ =5%
MC …. PC (N=2)
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MC …. PC (N=2)
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Fig. 10.1. Mean value of x1(t) σµ = 2%
Fig. 10.2. Variance of x1(t) σµ = 2%
Fig. 11.2. Variance of x1(t) σµ = 5%
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ACCEPTED MANUSCRIPT
MC …. PC (N=2)
MC …. PC (N=2)
Fig. 13.1. Mean value of y1(t) σµ = 5%
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Fig. 12.1. Mean value of y1(t) σµ = 2%
MC …. PC (N=2)
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MC …. PC (N=2)
Fig. 12.2. Variance of y1(t) σµ = 2%
Fig. 13.2. Variance of y1(t) σµ = 5%
These figures show that the obtained solutions oscillate around the Monte Carlo simulations reference solution. It can be seen that for small standard deviation σµ =2%, the polynomial chaos solutions in second order
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polynomial provides a very good accuracy as compared with the Monte Carlo simulations. When the standard
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deviation increases the error increases.
ـــ …. -.-.
MC PC (N=2) PC (N=4) PC (N=8)
Fig. 14.1. Mean value of x2(t) σµ = 10%
ـــ …. -.-.
MC PC (N=2) PC (N=4) PC (N=8)
Fig. 14.2. Variance of x2(t) σµ = 10%
ACCEPTED MANUSCRIPT ـــ …. -.-.
MC PC (N=2) PC (N=4) PC (N=8)
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MC PC (N=2) PC (N=4) PC (N=8)
Fig. 15.1. Mean value of y2(t) σµ = 10%
Fig. 15.2. Variance of y2(t) σµ = 10%
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ـــ …. -.-.
The mean value and the variance of the dynamic component of the linear displacement of the second bearing in
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two directions x and y are presented in Figs. 14 and 15 for σµ =10%.
The polynomial chaos results are compared with Monte Carlo simulation with 100000 simulations. It is evident from these figures that N=2 case clearly does not have enough chaos terms to represent the output. As N increases, the results seem to become better, and with N=8, the dynamic response of the linear displacement of the second bearing with polynomial chaos values almost exactly match with the Monte Carlo simulation results.
7. Conclusion
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An approach based on the polynomial chaos method has been proposed to study the dynamic behavior of one stage gear system was modeled by eight degrees of freedom in the presence of friction coefficient between teeth that admits some dispersion. Therefore, it becomes necessary to take this uncertainty into account in the stability analysis of the gear system to ensure robust predictions of stable and instable behaviors. The polynomial chaos
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method has been used to determine the dynamic response of this system. The efficiency of the proposed method was compared with the Monte Carlo simulation. The main results of the present study show that the polynomial chaos may be an efficient tool to take into account the dispersions of the friction coefficient in the dynamic
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behavior study of a spur gear system. An interesting perspective is to apply this method to a system with higher degree of freedom like epicyclic gear system. Further work in this context is in progress.
Nomenclature
Cm : motor torque (N.m)
Cf12 , Cf 21 : friction torque on the spur gear (21) and (21) (N.m)
Ff ( t ) : friction force Ff0 ( t ) , Ff1 ( t ) : friction force of one and two pair of teeth in contact k 1x , k 1y , k 2x , k 2y : traction-compression stiffness of the first and second bearing (N/m) k1θ , k θ2 : torsional stiffness of the shaft 1 and 2 (N.m/rad)
ACCEPTED MANUSCRIPT k ( t ) : gear mesh stiffness
kc : mean component of mesh stiffness kv ( t ) : time mesh stiffness varying component k 0 ( t ) , k1 ( t ) : mesh stiffness of the one and two pair of teeth in contact
r(b1,2) , r(b2,1) : basic radius of the wheel (21) and (21)
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{Q} : generalized coordinate’s vector θ(1,1) , θ(1,2) , θ( 2,1) , θ( 2,2) : angular displacement of the wheel (11) ,(12),(21) and (22) (°) x1 , y1 , x2 , y2 : translational displacement of the first and second bearing (m) µ : friction coefficient
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µc : critical friction coefficient
α : pressure angle (°)
ε α : contact ratio
[ M ] , [ K ] : mass and stiffness matrices {f } : external force vector T
T
T
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δ : teeth deflection (m)
% % M T , K T : random mass and stiffness matrices α p : random variables
[M ] , [K ] T 0
T
0
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ψ m (α p ) : multidimensional orthogonal polynomials chaos : average of mass and stiffness matrices
.. : iner product defined by the mathematical expectation operator
EP
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Blanchard, E., Sandu, A., Sandu, C., 2009. Parameter estimation for mechanical systems via an explicit representation of uncertainty. International Journal for Computer-Aided Engineering Computation. 26, 541569.
Blanchard, E., Sandu, A., Sandu, C., 2010. Polynomial chaos-based parameter estimation methods applied to vehicle system. Proc. Of the Institution of Mechanical Engineers Part K: J. of Multi-Body Dynamics. 224, 59-81. Dessombz, O., 2000. Analyse dynamique de structures comportant des paramètres incertains. Ph.D. Thesis, Ecole Centrale de Lyon, Lyon, France. El Hami, A., Lallement, G., Minotti, P., Cogan, S., 1993. Methods that combine finite group theory with component mode synthesis in the analysis of repetitive structures. International Journal Computers & Structures. 48, 975-982.
ACCEPTED MANUSCRIPT El Hami, A., Radi, B., 1996. Some decomposition methods in the analysis of repetitive structures. International Journal Computers & Structures. 58, 973-980. El Hami, A., Radi, B., Cherouat, A., 2009. The frictional contact of the shaping of the composite fabric. International Journal of Mathematical and Computer Modelling. 49(7-8), 1337-1349. Fisher, J., Bhattacharya, R., 2008. Stability analysis of stochastic systems using polynomial chaos. American Control Conference, June 11-13, USA, 4250-4255.
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Fishman, G.S., 1996. Monte Carlo, Concepts, Algorithms and Applications, first Ed. Springer-Verlag. Ghanem, R.G., Spanos, P.D., 1991. Stochastic Finite Elements: A Spectral Approach, revised Ed. Springer Verlag.
Guerine, A., El Hami, A., Walha, L., Fakhfakh, T., Haddar, M., 2015a. A perturbation approach for the dynamic analysis of one stage gear system with uncertain parameters. Mechanism and Machine Theory. 92, 113-126.
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Guerine, A., El Hami, A., Fakhfakh, T., Haddar, M., 2015b. A polynomial chaos method to the analysis of the dynamic behavior of spur gear system. Structural Engineering and Mechanics, An International Journal. 53, 819-831.
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Jakerman, J.D., Roberts, S. G., 2009. Stochastic galerkin and collocation methods for quantifying uncertainties in differential equation: a review. ANZIAM J. 50, 815-830.
Kahraman, A., Lim, J., Ding, H., 2007. A Dynamic Model of a Spur Gear Pair with Friction. 12th IFToMM World Congress, Besançon (France), June18-21.
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Nonlinear Random Vibration. Probabilistic Engineering Mechanics. 13, 125–136. Lindsley, N.J., Beran, P.S., 2005. Increased efficiency in the stochastic interrogation of an uncertain nonlinear aeroelastic system. International Forum on Aeroelasticity and Structural Dynamics, Munich, Germany, June. Mohsine, A., El Hami, A., 2010. A Robust Study of Reliability-Based Optimisation Methods under Eigen-
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Nagy, Z. K., Braatz, R.D., 2006. Distributional uncertainty analysis using power series and polynomial chaos
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expansions. Journal Process Control. 17, 229-240. Nechak, L., Berger, S., Aubry, E., 2011. A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems. European Journal of Mechanics - A/Solids. 30, 594-607. Papadrakakis, M., Papadopoulos, V., 1999. Parallel solution methods for stochastic finite element analysis using Monte Carlo simulation. Comput. Methods Appl. Mech. Eng. 168, 305-320. Radi, B., El Hami, A., 2007. Reliability analysis of the metal forming process. International Journal of Mathematical and Computer Modelling. 45(3-4), 431-439. Sandu, C., Sandu, A., Ahmadian, M., 2006. Modeling multibody dynamic systems with uncertainties. Part II: Numerical Applications, Multibody system Dynamic. 15, 241-262. Sinou, J. J., Jézéquel, L., 2007. Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping. European Journal of Mechanics - A/solids. 26, 106-122.
ACCEPTED MANUSCRIPT Vaishya, M., Singh, R., 2001. Sliding Friction-Induced Non-Linearity and Parametric Effects in Gear Dynamics. Journal of Sound and Vibration. 248(4), 671-694. Vaishya, M., Singh, R., 2003. Strategies for Modeling Friction in Gear Dynamics. Journal of mechanical Design. 125, 383-393. Walha, L., Fakhfakh, T., Haddar, M., 2009. Nonlinear dynamics of a two-stage gear system with mesh stiffness fluctuation, bearing flexibility and backlash. Mechanism and Machine Theory. 44, 1058–1069.
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A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system A. Guerine1,2 *, A. El Hami1, L. Walha2, T. Fakhfakh2, M. Haddar2 Laboratory Optimization and Reliability in Structural Mechanics LOFIMS, Mechanical Engineering Department, National Institute of Applied Sciences of Rouen, BP 08-76801 Saint Etienne du Rouvray Cedex, France 2
Mechanics, Modelling and Manufacturing Laboratory LA2MP, Mechanical Engineering Department, National School of Engineers of Sfax, BP 1173-3038 Sfax, Tunisia
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1
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Dear Prof. Peter Eberhard,
The manuscript has been resubmitted to “European Journal of Mechanics - A/Solids”. We
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believe that the manuscript has been improved now and we look forward to your positive response. Yours sincerely
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A.GUERINE
S0997-7538(16)30025-0
DOI:
10.1016/j.euromechsol.2016.03.007
Reference:
EJMSOL 3296
To appear in:
European Journal of Mechanics / A Solids
Received Date: 20 October 2014 Revised Date:
31 January 2016
Accepted Date: 14 March 2016
Please cite this article as: Guerine, A., El Hami, A., Walha, L., Fakhfakh, T., Haddar, M., A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system, European Journal of Mechanics / A Solids (2016), doi: 10.1016/j.euromechsol.2016.03.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system A. Guerine1,2 *, A. El Hami1 , L. Walha2, T. Fakhfakh2, M. Haddar2 Laboratory Optimization and Reliability in Structural Mechanics LOFIMS, Mechanical Engineering Department, National Institute of Applied Sciences of Rouen, BP 08-76801 Saint Etienne du Rouvray Cedex, France 2
Mechanics, Modelling and Manufacturing Laboratory LA2MP, Mechanical Engineering Department, National School of Engineers of Sfax, BP 1173-3038 Sfax, Tunisia
E-mail: * [email protected] Abstract
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1
In this paper, we propose a new method for taking into account uncertainties occurring due to gear friction,
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based on the projection on polynomial chaos. The new method is used to determine the dynamic response of a spur gear system with uncertainty associated to friction coefficient on the teeth contact. The simulation results
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are obtained by the polynomial chaos method for dynamic analysis under uncertainty. The proposed method is an efficient probabilistic tool for uncertainty propagation. The polynomial chaos results are compared with Monte Carlo simulations.
Key Words. Uncertainty, Friction coefficient, Spur gear system, Chaos Polynomial method. 1. Introduction
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The study and analysis of the dynamic behavior with nonlinear systems is a major interest in the industrial sector. Thus they allow overcoming the areas of instability and reducing vibration levels. Indeed, the negative consequences that may result from the instability of systems require designers to develop the most rigorous solution. This passes through a detailed study and analysis of the dynamic behavior of these systems before considering their actual implementation.
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Several parametric studies have shown the great sensitivity of the dynamic behavior of gear systems. However, these parameters admit strong dispersions. Therefore, it becomes necessary to take into account these
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uncertainties to ensure the robustness of the analysis (Guerine et al., 2015a; Guerine et al., 2015b). Also there are several studies in reliability for vibration structures taking into account the uncertainties (Abo Al-kheer et al., 2011; Mohsine and El Hami, 2010; El Hami et al., 2009; Radi and El Hami, 2007; El Hami and Radi, 1996; El Hami et al., 1993).
The mechanisms of transmission by gear tooth contact are characterized by the presence of friction coefficient that affects the vibration and noise of these systems. Parameter estimation is an important problem, because many parameters simply cannot be measured physically with good accuracy, such us the friction coefficient, especially in real time application (Blanchard et al., 2009, 2010). The coefficient of friction is a very important factor for designing, operating, and maintaining the gear transmission. Indeed, the accurate estimation of this coefficient is difficult due to the effects of various uncertain parameters, e.g., materials of gears, roughness and contact patch size, etc. However, the friction coefficient admits a strong dispersion (Nechak et al., 2011; Lee et al., 2012). Therefore, it becomes necessary to take into
ACCEPTED MANUSCRIPT account these uncertainties in order to ensure the robustness of the analysis. A study of the nonlinear dynamic behavior will help to analyze stability and to predict the vibration levels according to the parameters variations. Several methods are proposed in the literature. Monte Carlo (MC) simulation is a well-known technique in this field (Fishman, 1996). It can give the entire probability density function of any system variable, but it is often too costly since a great number of samples are required for reasonable accuracy. Parallel simulation (Papadrakakis and Papadopoulos, 1999) and proper orthogonal decomposition (Lindsley and Beran, 2005) are
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some solutions proposed to circumvent the computational difficulties of the MC method. Polynomial Chaos Expansion (PCE) is presented in the literatures a more efficient probabilistic tool for uncertainty propagation. It was first introduced by Wiener and launched by Ghanem and Spanos who used Hermite orthogonal polynomials to model stochastic processes with Gaussian random variables (Li and Ghanem, 1998).
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Polynomial Chaos (PC) gives a mathematical framework to separate the stochastic components of a system response from the deterministic ones. It used to compute the deterministic components called stochastic modes in an intrusive and non-intrusive manner while random components are concentrated in the polynomial basis
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used. The Polynomial Chaos (PC) method has been shown to be considerably more efficient than Monte Carlo in the simulation of systems with a small number of uncertain parameters (Blanchard et al., 2009; Sandu et al., 2006).
The main originality of the present paper is that the uncertainty of the gear friction system in the dynamic behavior study of one stage gear system is taken into account. The main objective is to investigate of the capabilities of the new method to determine the dynamic response of a spur gear system subject to uncertain friction coefficient. So, an eight degree of freedom system modelling the dynamic behavior of a spur gear system
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is considered. The modelling of a one stage spur gear system is presented in Section 2. The modelling of friction coefficient is presented in Section 3. In the next section, the theoretical basis of the polynomial chaos is presented. In Section 5, the equations of motion for the eight degrees of freedom are presented. Numerical results are presented in Section 6. Finally in Section 7, to conclude, some comments are made based on the study
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carried out in this paper.
2. One stage spur gear system modelling
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The dynamic model of the one stage gear system is represented on Fig. 1. This model is composed of two blocks. x Every block is supported by flexible bearing which the bending stiffness is k1 and the traction-compression y y x stiffness is k1 for the first block, k 2 and k 2 for the second block, respectively.
The wheels (11) and (22) characterize the drive and the respectively the driven gears. The two shafts (1) and (2) θ θ admit some torsional stiffness k1 and k 2 .
Angular displacements of every wheel are noticed by θ(1,1) , θ(1,2) , θ(2,1) and θ(2,2) . Besides, the linear displacements of the bearing noted by x1 and y1 for the first block, x 2 and y 2 for the second block, are measured in the plan which is orthogonal to the wheels axis of rotation (Kahraman et al., 2007).
ACCEPTED MANUSCRIPT θ(1,1)
θ(1,2)
ur Y1 ur X1
ur Y
k1x k1y
k (t)
Plan of action ur Y2
ur Z
ur X
θ( 2,1)
k 2x
θ( 2,2 )
ur X2 k 2y
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k1θ
k θ2
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Fig. 1. Model of the one stage gear system
ur ur Fig. 2 defines a reference frame O, X, Y and the position of the wheels of the one stage gear system. α is the
)
pressure angle of gearmesh contact.
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(
ur Y
ur Y
Cm 12
α
21
ur X
ur X
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11
22
line of action
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Fig. 2. Position of the wheels of the one stage gear system The teeth deflection is expressed along the line of action, and it can be written as:
δ ( t ) = s α ( x1 − x 2 ) + cα ( y1 − y 2 ) + r(b1,2) θ(1,2) − r(b2,1) θ( 2,1)
(1)
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b b Where s α = sin ( α ) and cα = cos ( α ) and r(1,2) , r(2,1) represent the base gears radius.
Generally the gear mesh stiffness variation k(t) is modelled by a sinusoid wave or by a square wave form. The later is the most representative of the real phenomenon and is represented on Fig. 3. The gear mesh stiffness variation can be decomposed in two components: an average component noted by kc, and a time variant one noted by kv(t) (Walha et al., 2009). k ( t ) = kc + kv(t)
The time component of the mesh stiffness is defined by the following periodic form (Fig. 4).
(2)
ACCEPTED MANUSCRIPT k(t) (εα - 1) Te kM kc
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km (2 - εα)Te
Time (s)
Te
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Fig. 3. Modelling of the mesh stiffness variation k(t) The extreme values of the mesh stiffness variation are defined by: kc 2 εα
and k M = −k m
2 − εα εα − 1
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km = −
(3)
ε α and Te represent respectively the contact ratio and mesh period corresponding to the two gearmeshes contacts. kv(t)
(ε
α
)
− 1 Te
kM
km
α
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0
( 2 − ε ) Te
Time (s)
Te
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Fig. 4. Modelling of the mesh stiffness variation kv(t) 3. Friction coefficient modelling
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The instantaneous coefficient of friction is defined as the ratio between the measured friction force Ff and the normal force Fn. In the case of the gear system, the number of components of the friction force is equal to the number of pair of teeth in contact. The modeling of the friction forces is usually made based on the Coulomb law. According to this model the friction coefficient is assumed constant. In the dynamic model, the friction can be introduced by two friction torques applied on the gears (12) and (21) (Fig. 5).
ξ1(t) 12
Ff ( t ) 22
11
21
P
ξ2(t) Fig. 5. Friction modeling following line of action
ACCEPTED MANUSCRIPT On the first spur gear (12), the friction torque is expressed by:
Cf12 ( t ) = Ff ( t ) . ξ1 ( t )
(4)
Also, on the second spur gear (21), the friction torque is expressed by:
Cf21 ( t ) = Ff ( t ) . ξ2 ( t )
(5)
ξ1(t) and ξ2(t) are the time varying length between the pitch point P and the corresponding center point of the
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gear (12) and (21) respectively (Fig. 5). The friction force on the pitch point P is defined by the sum of two components which corresponding to case of one or two pair of teeth in contact following the time. So, the first friction force corresponding to the first pair in contact is defined by: Ff 1 (t) = µ.k1 (t).δ(t)
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While the second force at the second pair in contact is defined by: Ff 0 (t) = µ.k 0 (t).δ(t)
(6)
(7)
δ(t) = Lδ .{Q(t)} Lδ is defined by: Lδ = [s α
− sα
cα
− cα
b 0 r(1,2) − r(b2,1)
0]
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δ(t) is the deflection along the gear mesh contact, it can be written by: (Kahraman et al., 2007) (8)
(9)
{Q(t)} is the vector of the model generalized coordinates, it is in the form: 1
y 1 x 2 y 2 θ (1,1) θ (1,2 ) θ ( 2 ,1) θ ( 2 ,2 ) ]T
4. Polynomial chaos method
(10)
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{Q(t)} = [x
In this section, we propose to apply a method based on the polynomial chaos to the treatment of uncertainties in
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the gear friction. This method consists in projecting the stochastic desired solutions on a basis of orthogonal polynomials in which the variables are Gaussian orthonormal (Guerine et al., 2015b; Dessombz, 2000). The properties of the base polynomial are used to generate a linear system of equations by means of projection. The
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resolution of this system led to an expansion of the solution on the polynomial basis, which can be used to calculate the moments of the random solution. With this method, we can easily calculate the dynamic response of a mechanical system.
Let us consider a multi-degrees of freedom linear system with mass and stiffness matrices [ M T ] and [ K T ] respectively. The equations of motion describing the forced vibration of a linear system are:
[ M ] {&&u }( t ) + [ K ] {u }( t ) = {f }( t ) T
T
T
T
T
Where {u T } is the nodal displacement vector and {f T } is the external excitation. The chaotic polynomials ψ m corresponding to the multidimensional Hermite polynomials obtained by the formula (12):
(11)
ACCEPTED MANUSCRIPT ψ m (α1 ,...,α P ) = ( −1) e P
1T {α}{α } 2
1T − {α }{α }
∂ P e 2 ∂α1 ...∂α P
(12)
Where {α } is the vector grouping the random variables:
{α }
T
= α1 ...α P
(13)
[M ] = [M ]
% + M T
[K ] = [K ]
% + K T
T
T 0
T
T 0
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The random matrices mass and stiffness [ M T ] and [ K T ] of the mechanical system can be written as: (14)
(15)
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% and K % The matrices [ M T ]0 and [ KT ]0 are deterministic matrices, the matrices M T T correspond to the random part of the mass and stiffness matrices.
% M % T and KT are rewritten from an expression of type Karhunen-Loeve (Ghanem and Spanos, 1991) in the
% M T = % K T =
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following form: P
∑ [M ] α T
p
p =1 P
∑ [K
T
p =1
]
p
p
αp
(17)
(16)
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Where α p are independent Gaussian centered reduced which may correspond to the first polynomial ψ p , while the matrices [M T ] p and [K T ] p are deterministic. We pose α 0 = 1 , we can write then: T
T
p =0 P
[K ] = ∑[K ] T
T
p =0
p
p
αp
αp
(18)
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P
[M ] = ∑ [M ]
(19)
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In the same way, we can write for {f T } : P
{f } = ∑ {f } T
T
p =0
p
αp
(20)
The dynamic response is obtained by solving the following equation knowing that the initial conditions are predefined: K eq {u T }( t + ∆t ) = {Feq }
(21)
Where: K eq = [ K T ] + a 0 [ M T ]
(22)
{F } = {f }( t + ∆t ) + [ M ] ( a {u }( t ) + a {u& }( t ) + a {&&u }( t ) ) eq
Where:
T
T
0
T
1
T
2
T
(23)
ACCEPTED MANUSCRIPT 1 B 1 , a1 = and a 2 = A ∆t A ∆t A ∆t 2
a0 =
(24)
A and B are the parameters of Newmark.
{u }( t + ∆ t ) T
is decomposed on polynomials which the P variables are Gaussian orthonormal.
{u }( t + ∆t ) = ∑ ({u }( t + ∆t ) ) ψ ({α } N
T
T
n
n
n =0
i
P
i =1
)
(25)
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Where N is the polynomial chaos order. K eq and {Feq } are written in yields the following form:
∑ [K ] T
p =0
{Feq } =
α p + a0 p
∑ ({f }( t + ∆t )) P
T
p =0
P
∑ [M ] T
p=0
αp + p
αp = p
P
∑ [M ] T
p =0
P
∑ K p =0
eq 2
α p p
(
) ∑ {F } α
α p a 0 ({u T }( t ) )0 + a1 ({u& T }( t ) )0 + a 2 ({&& u T }( t ) )0 = p
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P
K eq =
(26)
P
eq 2
p =0
p
(27)
p
Substituting Eqs. (25), (26) and (27) into Eq. (21) and forcing the residual to be orthogonal to the space spanned
P
N
p =0
n =0
∑∑
K eq 2 {u T }n α p ψ n ψ m = p
∑ {F } P
eq 2
p =0
p
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by the polynomial chaos ψ m yield the following system of linear equation:
αp ψm
m = 0, 1, . . . , N,
(28)
Where . . denotes the inner product defined by the mathematical expectation operator This algebraic equation can be rewritten in a more compact matrix form as: (0 N)
[ D]( ) ij
L
( 0) ({u T }( t + ∆t ) ) {f } 0 M M ( j ) M ({u T }( t + ∆t ) ) j = {f } O M M N ( NN ) [ D] ({u T }( t + ∆t ) ) N {f }( )
[ D]
L
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[ D]( 00) O M [ D]( N0)
[ D]
=
{f }
=
( j)
P
∑ K
eq 2
p =0
∑ {F }
α p ψ i ψ j p
P
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( ij )
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Where:
eq 2
p= 0
p
(29)
(30)
αp ψ j
(31)
Of the fact that of the orthogonality of the polynomials, most terms α p ψ n ψ m are zero. Indeed, we have:
ψi ψ j
i≠ j
=0
(32)
After resolution of the algebraic system (29), the mean values and the variances of the dynamic response are given by the following relationships: E {u T } = ({u T }( t + ∆t ) )0
Var {u T } =
(33)
∑ (({u }( t + ∆t ) ) ) (ψ ) N
2
T
n =1
n
j
2
(34)
ACCEPTED MANUSCRIPT 5. Equations of motion The differential equations describing the dynamic behavior of our system (Fig. 1) are obtained using Lagrange
Where Θ1 = ( θ(1,1) − θ(1,2) ) and Θ 2 = ( θ( 2,1) − θ( 2,2) )
Cf12p (t) = µ.k p (t).δ(t).ξ1p (t) p p p Cf 21 (t) = µ.k (t).δ(t).ξ2 (t) p represents the case of the pair 0 or 1.
6. Numerical simulation
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The torque friction components are expressed by:
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m1&& x1 + k1x x1 + s α k ( t ) Lδ {Q} = ( Ff1 − Ff0 ) s α m1&& y1 + k1y y1 + cα k ( t ) Lδ {Q} = − ( Ff1 − Ff0 ) cα m 2 && x 2 + k 2x x 2 − s α k ( t ) Lδ {Q} = − ( Ff1 − Ff0 ) s α m 2 && y 2 + k 2y y 2 − cα k ( t ) Lδ {Q} = ( Ff1 − Ff0 ) cα θ(1,1) + k1θ Θ1 = Cm I(1,1) && θ b δ 1 0 && I(1,2) θ(1,2) − k1 Θ1 + r(1,2) k ( t ) L {Q} = Cf12 ( t ) − Cf12 ( t ) θ b δ 1 0 && I( 2,1) θ( 2,1) + k 2 Θ2 − r( 2,1) k ( t ) L {Q} = − Cf 21 ( t ) + Cf 21 ( t ) θ && I( 2,2) θ( 2,2) + k 2 Θ 2 = 0
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formalism. These equations are represented as follows
(35)
(36)
(37)
summarized in the table 1.
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The technological and dimensional features of the one stage gear system in the presence of friction are
Table 1 System parameters
Motor torque
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Bearing stiffness
Cm=200 N.m k1x = k 2x =107 N/m
k1θ = k θ2 =105 Nm/rad
Number of teeth
Z(12)=40 ; Z(21)=50
Module of teeth
m=4*10-3 m
Contact ratio
εα = 1.7341
Pressure angle
α=20°
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Torsional stiffness of shaft
k1y = k 2y =107 N/m
6.1. Stability analysis For a designer, the main objective of a robust stability analysis is to define with certainty that the system studied is stable for an uncertain parameter defined within an uncertain interval (such as the dispersion interval of the friction coefficient). This parametric study consists in calculating the eigenvalues of the linearized system at each value of the uncertain parameter, then stability is analyzed by testing the sign of the real parts of the eigenvalues obtained (Nechak et al., 2011). This procedure becomes difficult for nonlinear systems of a higher
ACCEPTED MANUSCRIPT order, since the calculation of the corresponding eigenvalues goes through the resolution of characteristic equations of higher order (Fisher and Bhattacharya, 2008; Sinou and Jézéquel, 2007; Nechak et al., 2011). The evolution of the system eigenvalues is plotted against the values of the friction coefficient in Figs. 6.1 and 6.2. The imaginary parts of these eigenvalues represent the instability frequencies. It appears that at values of friction coefficient which belong to [0,0.109], the real part of the eigenvalues is negative. The system is then stable.
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µc
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Furthermore, the frequencies associated with the eigenvalues of the system are discrete.
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Fig. 6.1. Evolution of the real part of eigenvalues according to the friction coefficient values
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µc
Fig. 6.2. Evolution of the imaginary part of eigenvalues according to the friction coefficient values
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The answer is composed of oscillations of constant level. When the coefficient of friction increases, the frequencies tend to be closer to become equal to a certain value of the coefficient of friction, which we call by the following critical µ c . In this case, the coefficient of friction is equal to review 0.109 and corresponds to the value at which the eigenvalues coupling. Beyond this value, the eingenvalues possess a positive real part. The system is unstable. Note that the overall response of the system is imposed by the unstable mode dominates rapidly in amplitude on the stable mode. This therefore results in increasing the frequency characterizing the unstable mode oscillations.
6.2. Dynamic behavior with friction Fig. 7 represents the temporal fluctuation of the resultant displacement of the first bearing. We notice that the presence of friction affects little the linear displacements of the bearing and the amplitudes augment slightly
ACCEPTED MANUSCRIPT without changing the form of the signal. Indeed, the friction causes the amplifications of the amplitudes of all degrees of freedom. Figs. 8 and 9 represent the effects of friction on the dynamic behavior of the wheels. We notice that the friction reduces the angular displacements of the wheels. Its effects increase over time, when the displacement also increases.
- -: µ=0.08
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―: µ=0
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Fig. 7. Temporal fluctuation of the resultant displacement of the first bearing
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— :θ(1,1) - - - :θ(1,2) -.-. :θ(2,1) … : θ(2,2)
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Fig. 8. Temporal fluctuations of the angular displacement of the wheels (µ=0)
— :θ(1,1) - - - :θ(1,2) -.-. :θ(2,1) … : θ(2,2)
Fig. 9. Temporal fluctuations of the angular displacement of the wheels (µ=0.08)
ACCEPTED MANUSCRIPT 6.3. Study with Polynomial Chaos In this section numerical results are presented for the new method for which the formulations are derived in the section 4. The polynomial chaos results are compared with Monte Carlo simulations with 100000 simulations. The friction coefficient is supposed random variable and defined as follow: µ = µ0 + σµ ξ
(38)
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Where ξ is a zero mean value Gaussian random variable, µ 0 is the mean value and σ µ is the standard deviation of this parameter.
The mean value and the variance of the dynamic component of the linear displacement of the first bearing in two directions x and y have been calculated by the polynomial chaos method. The obtained results are compared with those given by the Monte Carlo simulations with 100000 simulations. The results are plotted in Figs. 10 and 12
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for σµ =2% and in Figs. 11 and 13 for σµ =5%.
MC …. PC (N=2)
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MC …. PC (N=2)
Fig. 11.1. Mean value of x1(t) σµ =5%
MC …. PC (N=2)
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MC …. PC (N=2)
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Fig. 10.1. Mean value of x1(t) σµ = 2%
Fig. 10.2. Variance of x1(t) σµ = 2%
Fig. 11.2. Variance of x1(t) σµ = 5%
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MC …. PC (N=2)
MC …. PC (N=2)
Fig. 13.1. Mean value of y1(t) σµ = 5%
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Fig. 12.1. Mean value of y1(t) σµ = 2%
MC …. PC (N=2)
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MC …. PC (N=2)
Fig. 12.2. Variance of y1(t) σµ = 2%
Fig. 13.2. Variance of y1(t) σµ = 5%
These figures show that the obtained solutions oscillate around the Monte Carlo simulations reference solution. It can be seen that for small standard deviation σµ =2%, the polynomial chaos solutions in second order
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polynomial provides a very good accuracy as compared with the Monte Carlo simulations. When the standard
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deviation increases the error increases.
ـــ …. -.-.
MC PC (N=2) PC (N=4) PC (N=8)
Fig. 14.1. Mean value of x2(t) σµ = 10%
ـــ …. -.-.
MC PC (N=2) PC (N=4) PC (N=8)
Fig. 14.2. Variance of x2(t) σµ = 10%
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MC PC (N=2) PC (N=4) PC (N=8)
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MC PC (N=2) PC (N=4) PC (N=8)
Fig. 15.1. Mean value of y2(t) σµ = 10%
Fig. 15.2. Variance of y2(t) σµ = 10%
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ـــ …. -.-.
The mean value and the variance of the dynamic component of the linear displacement of the second bearing in
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two directions x and y are presented in Figs. 14 and 15 for σµ =10%.
The polynomial chaos results are compared with Monte Carlo simulation with 100000 simulations. It is evident from these figures that N=2 case clearly does not have enough chaos terms to represent the output. As N increases, the results seem to become better, and with N=8, the dynamic response of the linear displacement of the second bearing with polynomial chaos values almost exactly match with the Monte Carlo simulation results.
7. Conclusion
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An approach based on the polynomial chaos method has been proposed to study the dynamic behavior of one stage gear system was modeled by eight degrees of freedom in the presence of friction coefficient between teeth that admits some dispersion. Therefore, it becomes necessary to take this uncertainty into account in the stability analysis of the gear system to ensure robust predictions of stable and instable behaviors. The polynomial chaos
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method has been used to determine the dynamic response of this system. The efficiency of the proposed method was compared with the Monte Carlo simulation. The main results of the present study show that the polynomial chaos may be an efficient tool to take into account the dispersions of the friction coefficient in the dynamic
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behavior study of a spur gear system. An interesting perspective is to apply this method to a system with higher degree of freedom like epicyclic gear system. Further work in this context is in progress.
Nomenclature
Cm : motor torque (N.m)
Cf12 , Cf 21 : friction torque on the spur gear (21) and (21) (N.m)
Ff ( t ) : friction force Ff0 ( t ) , Ff1 ( t ) : friction force of one and two pair of teeth in contact k 1x , k 1y , k 2x , k 2y : traction-compression stiffness of the first and second bearing (N/m) k1θ , k θ2 : torsional stiffness of the shaft 1 and 2 (N.m/rad)
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kc : mean component of mesh stiffness kv ( t ) : time mesh stiffness varying component k 0 ( t ) , k1 ( t ) : mesh stiffness of the one and two pair of teeth in contact
r(b1,2) , r(b2,1) : basic radius of the wheel (21) and (21)
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{Q} : generalized coordinate’s vector θ(1,1) , θ(1,2) , θ( 2,1) , θ( 2,2) : angular displacement of the wheel (11) ,(12),(21) and (22) (°) x1 , y1 , x2 , y2 : translational displacement of the first and second bearing (m) µ : friction coefficient
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µc : critical friction coefficient
α : pressure angle (°)
ε α : contact ratio
[ M ] , [ K ] : mass and stiffness matrices {f } : external force vector T
T
T
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δ : teeth deflection (m)
% % M T , K T : random mass and stiffness matrices α p : random variables
[M ] , [K ] T 0
T
0
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ψ m (α p ) : multidimensional orthogonal polynomials chaos : average of mass and stiffness matrices
.. : iner product defined by the mathematical expectation operator
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A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system A. Guerine1,2 *, A. El Hami1, L. Walha2, T. Fakhfakh2, M. Haddar2 Laboratory Optimization and Reliability in Structural Mechanics LOFIMS, Mechanical Engineering Department, National Institute of Applied Sciences of Rouen, BP 08-76801 Saint Etienne du Rouvray Cedex, France 2
Mechanics, Modelling and Manufacturing Laboratory LA2MP, Mechanical Engineering Department, National School of Engineers of Sfax, BP 1173-3038 Sfax, Tunisia
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1
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Dear Prof. Peter Eberhard,
The manuscript has been resubmitted to “European Journal of Mechanics - A/Solids”. We
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believe that the manuscript has been improved now and we look forward to your positive response. Yours sincerely
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A.GUERINE