Nonlinear vibrations of large structures with uncertain ... - SAGE Journals

Research Article

Nonlinear vibrations of large structures with uncertain parameters

Advances in Mechanical Engineering 2017, Vol. 9(7) 1–11 Ó The Author(s) 2017 DOI: 10.1177/1687814017709663 journals.sagepub.com/home/ade

Mohammed Lamrhari1, Driss Sarsri2, Lahcen Azrar3, Miloud Rahmoune1 and Khalid Sbai1

Abstract The effects of uncertainties on the nonlinear dynamics of complex structures remain poorly mastered and most methods deal with the linear case. This article deals with a model of a large and complex structure with uncertain parameters for the nonlinear dynamic case, and the reduction in the model discretized by the finite element method is obtained by reducing the degrees of freedom in the numerical model. This is achieved by the development of the unknown displacement vector on the basis of the eigenmodes; a particular attention is paid to the calculation of the nonlinear stiffness coefficients of the model. The method combines the stochastic finite element methods with a modal reduction class based on sub-structuring the component mode synthesis method. The reference method is the Monte Carlo simulation which consists in making several simulations for different values of the uncertain parameters. The simulation of complex and nonlinear structures is costly in terms of memory and computation time. To solve this problem, the perturbation method combined with the component mode synthesis reduction method significantly reduces the computational cost by preserving the physical content of the original structure. The numerical integration by the Newmark schema is used; the first statistical moments (mean and variance) of the nonlinear dynamic response are computed. Numerical simulations illustrate the accuracy and effectiveness of the proposed methodology. Keywords Nonlinear dynamic, uncertain parameters, finite element method, component mode synthesis, perturbation methods

Date received: 19 October 2016; accepted: 13 February 2017 Academic Editor: Luı´s Godinho

Introduction In a robust design process, the determination of the variability of the nonlinear dynamic response of a large and complex structure is essential. Uncertainties come from the tolerances of manufacturing, the boundary conditions, and the external excitations. These structures are largely used in the fields of aerospace, automotive, civil engineering, and so on. The uncertainty of the physical parameters, nonlinearity, and complexity of the structure require the development of a complete mathematical approach for predicting the dynamic behavior variability. In addition, several methods have been developed in the literature to take account of uncertain parameters in the nonlinear dynamic response.

The reference method is the Monte Carlo simulation.1 This method allows a statistical evaluation based on a large number of deterministic analyses by considering different values of uncertain parameters. However, it requires the generation of big size samples 1

Laboratoire d’Etudes des Mate´riaux Avance´s et Applications, FS–EST, Moulay Ismail University, Meknes, Morocco 2 Laboratoire des Technologies Innovantes, ENSA, Abdelmalek Essaadi University, Te´touan, Morocco 3 Department of Applied Mathematics & Info, ENSET, Mohammed V University, Rabat, Morocco Corresponding author: Mohammed Lamrhari, Laboratoire d’Etudes des Mate´riaux Avance´s et Applications, FS–EST, Moulay Ismail University, Meknes 50040, Morocco. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

2 and then generates a prohibitory time computing. This approach is thus very costly. Also, perturbation methods are widely used to calculate the first moments (mean and standard deviation) of dynamic response whose uncertain variables vary slightly. These techniques are based on the Taylor series development of the response around its mean. This allows the direct determination of the variability of the response according to the physical parameters (mechanical and geometrical) randomly. Indeed, perturbation methods based on a development in Taylor series of second order2 and Neumann expansion method3 are generally efficient. Another development in the first order4 gives a similar result to the previous developments with a reduced time computing. Recently, in the field of thermal conduction with uncertain parameters,5–7 combination of a probabilistic study based on the perturbation method with a numerical resolution based on the finite difference method is applied.5 A prediction of the temperature field with random and fuzzy parameters of the properties of materials is studied; a numerical technique called fuzzy stochastic finite element method based on the combination of the perturbation theory with the moment methods is adopted.6 Furthermore, another form of development is a polynomial chaos expansion (PCE).8,9 The stochastic solution may be expanded in terms of the polynomial chaos basis whose elements are obtained from orthogonal polynomial.10 The properties of this polynomial basis are used to generate a system of deterministic equations. The resolution of this system is used to determine the variability of the response. However, nonlinear dynamics, the large number of degrees of freedom (DOFs) due to the mesh of a large structure, and higher order developing for modeling uncertainty induced a considerable increase in deterministic equations. One way to solve this problem is the reduction by component mode synthesis (CMS) method proposed in the literature.11–18 This method allows condensing the large number of DOFs into a small number using the generalized coordinates. FA Lulf et al.19 proposed a comparative study of different bases for reduction in nonlinear dynamic structures. Thus, in the CMS method, the overall structure is divided into sub-structures, each of which is analyzed independently in order to obtain the corresponding solution. These solutions are combined to obtain the overall solution of the structure by imposing constraints on the interfaces. The different methods are classified according to CMS interface: fixed interface,11 free interface,12,13 or hybrid interface.14,16 Recently, D Sarsri et al.20,21 developed an approach coupling CMS reduction method and developing uncertainty by a PCE to calculate the frequency transfer

Advances in Mechanical Engineering functions and response temporal for linear stochastic structures. J Sinou et al.22 proposed for simple structures, requiring no reduction, a technique taking into account the uncertainties in nonlinear models by combining the method of harmonic balance method (HBM) and developing uncertainty by a PCE. This method is based on a formulation of nonlinear dynamic problem in which the physical parameters, nonlinear forces, and the excitation force are considered randomly. In another work, D Sarsri and L Azrar23 used the CMS method coupled with the perturbation method to calculate the stochastic modes of large finite element models with uncertain parameters for the linear problems. This work is an extension to the nonlinear problems. The aim is to estimate the stochastic nonlinear dynamic response for a large structure with a minimum computational cost. To do this, we develop a methodological approach for calculating the temporal response of a large structure with uncertain parameters. This approach is based on coupling of the perturbation method and the reduction (CMS) method. First, we develop the nonlinear dynamic equations considering geometrical nonlinearity. The resolution of the nonlinear dynamic problem by the finite element method is adopted. Then, the temporal integration by Newmark scheme is developed. Second, we take completely the random phenomena using the perturbation method. The method of stochastic finite element is used. Various types of CMS interface method are used to optimally reduce the model size. The first moments of the nonlinear dynamic response of the reduced system are compared with the entire system. Several numerical simulations have shown the accuracy and efficiency of procedures and methodologies developed.

Modeling of nonlinearity The discretization by the finite element method of a linear structure gives the following matrix system _ + ½Kfug = fFext g ½Mf€ug + ½Cfug

ð1Þ

where ½M is the mass matrix, ½C is the damping matrix, [K] is the stiffness matrix, fug is the displacement vector, and fFext g is the external force vector. In this system there may be nonlinear phenomena due to the following: 

Intermittent contact or friction called nonlinearity of contact modeled by nonlinear force  fFnl ðfugÞg =

k1 fug if fug.0 k2 fug if fug\0

Lamrhari et al. 

3

Large displacement for thin structures called geometric nonlinearity modeled by nonlinear force fFnl (fug)g = ½Knl fugp , p essentially quadratic or cubic.

The nonlinear system modeling the nonlinear dynamics of a structure discretized by the finite element method is given by

In the CMS methods, the physical displacements of the sub-structure SS(i) are expressed as a linear combination of the sub-structure modes. After some algebraic transformations, a set of Ritz vectors Q is obtained and the displacement vector of each sub-structure can be expressed as ( (i)

(i)

fug = ½Q

_ + ½Kfug + fFnl ðfugÞg = fFext g ð2Þ ½Mf€ ug + ½Cfug where fFnl (fug)g is the nonlinear force vector. In this study, we used a cubic geometric nonlinearity.

CMS reduction method

_ i + ½Ki fugi + fFnl gi = fFe gi ugi + ½Ci fug ½Mi f€ i

ð3Þ

i

where ½M , ½C , and ½K are, respectively, the mass matrix, the damping matrix, and the stiffness matrix for sub-structures SS(i). The displacement vector fugi is partitioned into a vector fuj gi , called interface DOF and fuin gi is the vector of internal DOF i



fug =

uj uin

fFe gi = Fej

ð7Þ

In the free interface method, the displacements of each sub-structure are expressed as 

      (i)   (i)  (i)  jr + ca ja u(i) = F(i) h(i) + c(i) r

ð8Þ

where ½F(i)  is the matrix of truncated undamped normal modes of the sub-structure SS (i) with a free interface as boundary condition. ½c(i) r  is the matrix of rigid body modes for an unconstrained sub-structure SS (i) with a free interface. ½c(i) a  is the matrix of attachment modes associated with the interface, which are the static deformation shapes of SS (i) obtained by applying successively a unit force to one coordinate of the interface   (i) n (i) o Fj ð9Þ c(i) a = G  (i) I where fFj(i) g = j and ½G(i)  is the residual flexibil0 ity matrix. The expression of ½G(i)  depends on the nature of the problem. If the sub-structure is statically determined (i.e. no rigid body modes), then 



  1 G(i) = K (i)

else 

i ð4Þ

   1  (i)  A G(i) = t A(i) Kc(i)

with

The external force vector fFe g is composed of vectors fFej gi and fFee gi , called interface force and external applied force i

= ½Q(i) fuc g(i)

where h(i) p are the generalized coordinates. The matrix ½Q(i) is defined according to the method of substructuring used (fixed or free interface15).

i



)

1. Free interface method

The CMS method consists in using simultaneously a sub-structuring technique and a reduction method. The large and complex structure is partitioned into substructures. Each sub-structure is represented by a reduced basis composed of the normal modes and the interface modes. We present the theoretical bases of the CMS method. Initially, the eigenmodes and the interface static deformations are given for each sub-structure. Then, the overall system is projected on these bases taking into account the interface couplings between the sub-structures, after the reduced system is solved. Finally, the complete system solution is reconstituted. The finite element model of the entire structure is partitioned into N sub-structures SS(i) (i = 1, ., N). The equations of motion for each nonlinear substructure SS(i) are

i

u(i) j h(i) p

+ fFee gi

ð5Þ

The nonlinear force vector fFnl gi is composed of vectors fFnlj gi and fFnle gi , called interface force and external nonlinear force  i fFnl gi = Fnlj + fFnle gi

ð6Þ



    A(i) = ½ I   u(i) t u(i)

and t



   u(i) M (i) u(i) = ½ I 

where ½I is an unit matrix. ½u(i)  is a matrix of rigid modes. ½Kc(i)  is a stiffness matrix obtained by fixing arbitrary DOF to make the structure isostatic and replacing the corresponding part of the initial stiffness matrix by zero.

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Advances in Mechanical Engineering

To preserve the interface DOF, the following partition is used "



# (i)

 Fj F(i) = F(i) in " (i) # cr j  (i)  cr = c(i) r " (i)in # c  (i)  aj ca = (i) ca in

o h i1 n o h i1 u(i) ja ðiÞ = c(i)  c(i) j aj aj h i  h i1 h i  F(i) c(i)(i) h(i)  c(i) j(i) j r aj rj

" ½Q =

1

1

(i) (i) (i) (i) (i) c(i) a caj cr  ca caj crj (i)

F 

where ½bi is the matrix of localization or of geometrical connectivity of the SS(i) sub-structure. It makes possible to locate the DOF of each sub-structure SS(i) in the global DOF of the assembled structure. They are the Boolean matrices whose elements are 0 or 1. A transformation matrix can be defined for each sub-structure SS(i) by

fugðiÞ = ½Z i fuc g

ð10Þ

½Mc f€uc g + ½Cc fu_ c g + ½Kc fuc g N   X i t ½Z i Fnlj + fFnle gi +

ð11Þ

i=1

2. Fixed interface method = In the fixed interface method, the displacements of each sub-structure are expressed as 

     n (i) o uj u(i) = F(i) h(i) + c(i) c

N X

t

½Z i

Fej

i

+ fFee gi



where N X

½ Mc  =

The matrix ½Q is given by



ð18Þ

i=1

ð12Þ

i

t

½Z i ½ M i ½Z i

i=1

  (i) ½Qi = c(i) c F

ð13Þ

where ½F(i)  is the matrix of truncated undamped normal modes of the sub-structure SS (i) with a fixed interface as boundary condition. ½c(i) c  is the matrix of the constrained mode associated with the interface, which is the static deformation shape of SS (i) obtained by imposing successively a unit displacement on one interface while holding the remaining interface coordinates fixed. The conservation of interface DOF allows assembling these matrices as in the ordinary finite element methods. Let us denote the vector of independent displacements of the assembled structure by fuc g 8 (1) hp > > > > < . .. fuc g = > (N ) > > > hp : uj

ð17Þ

Inserting equation (17) into equation (1) and multiply on the right by t ½Zi , using the sum for all sub-structures, the following equation is obtained

#

(i)1 (i) c(i) a c aj Fj

ð16Þ

where ½Qi is given by the considered CMS method. The displacement vector fug(i) is then given by

The matrix Q is then given by i

ð15Þ

½Z i = ½Qi ½bi

Using this partition, one obtains n

fuc gi = ½bi fuc g

9 > > > > = > > > > ;

ð14Þ

The compatibility of interface displacements of the assembled structure is obtained by writing for each substructure SS(i) the following relation

½Cc  =

N X

t

½Z i ½C i ½Z i

t

½Z i ½ K i ½Z i

ð19Þ

i=1

½Kc  =

N X i=1

Using the interface DOF compatibility of displacements, it can easily be shown that N X

t

 i ½Z i Fej = 0

ð20Þ

i=1

Finally, the reduced equation of motion can be written as follows ½Mc f€uc g + ½Cc fu_ c g + ½Kc fuc g N   X i t + ½Z i Fnlj + fFnle gi i=1

=

N X i=1

t

½Z i fFee gi

ð21Þ

Lamrhari et al.

5

Stochastic perturbation method The perturbation method is largely employed in the field of the stochastic finite elements. It is based on an approximation the random variables by their development in Taylor series around their average value. These developments are truncated at the second order. The perturbation method must obey the conditions of existence and validity, in particular the reduced field of variation of the random variables. We present an extension of this method for the nonlinear dynamic systems with uncertain parameters. Let us assume that for each sub-structure, the mass matrix ½Mi , the dumped matrix ½Ci , the stiffness matrix ½Ki , and the external force vector fFe gi are related to a vector of the random variables qi (i = 1, ., I). Thus, the condensed mass ½Mc , dumped ½Cc , and stiffness ½Kc  matrices are related to same vector of the random variables. In the time domain, the resulting reduced stochastic differential system equation (21) has to be solved. The first two moments of time response (average and variance) will be calculated using the second-order perturbation method. One defines the vector of the average parameters qi and the quantity dqi = qi  qi . All the matrices and vector in equation (21) are random and are expanded through second-order Taylor series as follows

0

n

np

½Kc  = ½Kc  + ½Kc  dqn + ½Kc  dqn dqp

t

½Z (i) ½ X 0(i) ½Z (i)

t

½Z (i) ½ X n(i) ½Z (i)

i=1 N X

½Xc n =

ð23Þ

i=1

½Xc np =

N X

t

½Z (i) ½ X np(i) ½Z (i)

i=1

where ½Xc  can take the symbol ½Mc , ½Cc , or ½Kc . The unknown vector displacement, velocity, and acceleration are also developed through Taylor series as follows fuc g = fuc g0 + fuc gn dqn + fuc gnp dqn dqp fu_ c g = fu_ c g0 + fu_ c gn dqn + fu_ c gnp dqn dqp 0

n

ð24Þ

np

f€uc g = f€uc g + f€uc g dqn + f€uc g dqn dqp And the vector nonlinear force is given as n  o fFnl ðfugÞg = Fnl fug0 + fugn dqn + fugnp dqn dqp n  o fFnl ðfugÞg = Fnl fug0   ∂Fnl  + fugn dqn + fugnp dqn dqp ð25Þ ∂u Then n  o fFnl g0 = Fnl fug0

½Mc  = ½Mc 0 + ½Mc n dqn + ½Mc np dqn dqp ½Cc  = ½Cc 0 + ½Cc n dqn + ½Cc np dqn dqp

N X

½Xc 0 =

ð22Þ

½Fe  = ½Fe 0 + ½Fe n dqn + ½Fe np dqn dqp where [.]0, [.]n, and [.]np are deterministic matrices corresponding to the zero-, the first-, and the second-order partial derivatives with respect to the random parameter qi and given by

∂Fnl ðfugn Þ ∂u ∂Fnl ðfugnp Þ fFnl gnp = ∂u fFnl gn =

The condensed nonlinear forces are N X

fFnl g0c =

t

½Z ðiÞ fFnl g0(i)

t

½Z ðiÞ fFnl gn(i)

i=1 0

½ X  = ½ X ðqÞjq = q

∂½ X ðqÞ

n ½X  = ∂qn q = q

1 ∂2 ½ X ðqÞ

np ½X  = 2 ∂qn ∂qp q = q Indicial notations are used, with indices n, p running over the sequence 1, 2, ., I as well as the repeated indices summation. For structures with small uncertainties, one can assume that the transformation matrix ½Z is deterministic. The zero-, first-, and second-order derivatives of the condensed mass ½Mc , dumped ½Cc , and stiffness ½Kc  matrices are given by

N X

fFnl gnc =

ð26Þ

i=1 N X

fFnl gnp c =

t

½Z ðiÞ fFnl gnp(i)

i=1

The condensed external forces are N X

fFe g0c =

t

½Z (i) fFe g0(i)

t

½Z (i) fFe gn(i)

i=1 N X

fFe gnc =

i=1

fFe gnp c =

N X i=1

t

½Z (i) fFe gnp(i)

ð27Þ

6

Advances in Mechanical Engineering d a(Dt)2 1 a2 = aðDtÞ d a4 =  1 a

Substituting these developments into equation (21) and writing the terms of same order on gets the following differential systems:

a0 =

Zero-order equation ½ Mc  0 f € uc g0 + ½Cc 0 fu_ c g0 + ½Kc 0 fuc g0 + fFnl g0c = fFe g0c

a6 = (Dt)(1  d)

ð28Þ

a7 = (Dt)d

Based on these notations, the following equations are resulted:

First-order equation ½Mc 0 f€ uc gn + ½Cc 0 fu_ c gn + ½Kc 0 fuc gn + ½Mc n f€uc g0

Zero-order equation

+ ½Cc n fu_ c g0 + ½Kc n fuc g0 + fFnl gnc = fFe gnc



ð29Þ Second-order equation

0  0 Keq fuc ðt + DtÞg0 = Feq

ð32Þ

with 

½ Mc  0 f € uc gnp + ½Cc 0 fu_ c gnp + ½Kc 0 fuc gnp + ½Mc np f€uc g0 + ½Cc np fu_ c g0 + ½Kc np fuc g0 + 2½Mc n f€uc gp

Keq

0

= ½Kc 0 + ½Knlc 0 + a0 ½Mc 0 + a1 ½Cc 0 t

½Z (i) ½Knl 0(i) ½Z (i)

i=1

np + fFnl gnp c = fFe gc

½Knl 0(i) =

ð30Þ



The temporal response from time 0 to time T of equations (28)–(30) is required. The time T is subdivided into n intervals Dt = T =n, and the numerical solution is obtained at times tr = r:Dt r 2 IN and 0  r  n Assume that the solutions at times t are known and that the solution at time (t + Dt) is required next. According to the Newmark method, the following assumption is used at time (t + Dt)

∂fFnl gi

∂u fug = fug0(i)

0 Feq = fFe (t + Dt)g0c + ½Mc 0  a0 fuc (t)g0 + a2 fu_ c (t)g0 + a3 f€uc (t)g0  + ½Cc 0 a1 fuc (t)g0 + a4 fu_ c (t)g0 + a5 f€uc (t)g0

Stochastic temporal response

First-order equation 

n  n Keq fuc (t + Dt)gn = Feq

with

fuc (t + Dt)gx  fuc (t)gx uc (t + Dt)g = f€ a(Dt)2 x



fu_ c (t)gx ð1  2aÞf€uc (t)gx  a(Dt) 2a x

N X

½Knlc 0 =

+ 2½Cc n fu_ c gp + 2½Kc n fuc gp



d a(Dt) 1 a3 = 1 2a

 (Dt) d a5 = 1 2 a

a1 =

Keq

n

= ½Kc n + ½Knlc n + a0 ½Mc n + a1 ½Cc n

½Knlc n =

N X

t

½Z (i) ½Knl n(i) ½Z (i)

i=1

x

dðfuc (t + Dt)g  fuc (t)g Þ fu_ c (t + Dt)g = a(Dt)



 d d x  1 fu_ c (t)g  (Dt) 1  f€uc (t)gx a 2a x

ð31Þ where x can take 0, n, and np values. In which, the two parameters a and d verify d  1=2 and a  (d + 0:5)=4 in order to get accurate and stable solution. The following notations are used

½Knl  

n(i)

∂fFnl gi

= ∂u fug = fugn(i)

n Feq = fFe (t + Dt)gnc + ½Mc 0 ða0 fuc (t)gn + a2 fu_ c (t)gn + a3 f€uc (t)gn Þ

+ ½Cc 0 ða1 fuc (t)gn + a4 fu_ c (t)gn + a5 f€uc (t)gn Þ  ½Mc n f€uc (t + Dt)g0 + ½Cc n fu_ c (t + Dt)g0 + ½Kc n fuc (t + Dt)g0 + fFnl (t + Dt)gnc Second-order equation

ð33Þ

Lamrhari et al. 

7 Keq

np

 np fuc (t + Dt)gnp = Feq

ð34Þ

with  np Keq = ½Kc np + ½Knlc np + a0 ½Mc np + a1 ½Cc np ½Knlc np =

N X

t

 = ðfu(t + Dt)gn fu(t + Dt)gp Þcov qn , qp  + ðfu(t + Dt)gn fu(t + Dt)gpq ÞE qn , qp , qq    + fu(t + Dt)gnp fu(t + Dt)gql E qn , qp E qq , ql   + E qn , qq E qp , ql

½Z (i) ½Knl np(i) ½Z (i)

i=1

½Knl np(i) = 

np

∂fFnl gi

∂u fug = fugnp(i)

0 = fFe (t + Dt)gnp c + ½Mc  ða0 fuc (t)gnp + a2 fu_ c (t)gnp + a3 f€uc (t)gnp Þ

Feq

0

np

ð37Þ

np

np

+ ½Cc  ða1 fuc (t)g + a4 fu_ c (t)g + a5 f€uc (t)g Þ ða1 fuc (t)gnp + a4 fu_ c (t)gnp + a5 f€uc (t)gnp Þ np

np

0

0

 ½Mc  f€ uc (t + Dt)g  ½Cc  fu_ c (t + Dt)g  ½Kc np fuc (t + Dt)g0 n  fFnl (t + Dt)gnp uc (t + Dt)gp c 2½Mc  f€  2½Cc n fu_ c (t + Dt)gp  2½Kc n fuc (t + Dt)gp  fFnl (t + Dt)gnc

The solution of the problem is obtained by successively solving the following algebraic equations 0

fuc (t + Dt)g =



0 1 

0

Keq Feq   n 1 n Feq fuc (t + Dt)gn = Keq  np 1  np fuc (t + Dt)gnp = Keq Feq 

ð35Þ

The derivative of the physical displacements of each sub-structure is then obtained by fu(t + Dt)g0(i) = ½Z (i) fuc (t + Dt)g0 fu(t + Dt)gn(i) = ½Z (i) fuc (t + Dt)gn np(i)

fu(t + Dt)g

(i)

ð36Þ

np

= ½Z  fuc (t + Dt)g

The mean and the variance values of displacement u(t + Dt) are given by

Figure 1. Decomposed structure.

Eðfu(t + Dt)gÞ = fu(t + Dt)g0  1 + fu(t + Dt)gnp cov qn , qp 2 Covðfu(t + Dt)gÞ

If the correlation between the random parameters is not considered, equation (37) can be simplified as Eðfu(t + Dt)gÞ = fu(t + Dt)g0 1 + fu(t + Dt)gnn varðqn Þ 2 2 Varðfu(t + Dt)gÞ = ðfu(t + Dt)gn Þ Varðqn Þ  1 2 + ðfu(t + Dt)gnp Þ Varðqn ÞVar qp 4

ð38Þ

Numerical example For nonlinear discrete systems with stochastic parameters, some benchmark tests are elaborated to demonstrate the efficiency of the methodological approach. The presented method can be applied to continuous or discrete systems. In this article, we restrict ourselves to show the applicability and effectiveness of these methods for the dynamic analysis of nonlinear discrete systems with N DOFs. A nonlinear dynamic system consisting of 20 masses connected by 21 springs nonlinearly is shown in Figure 1. This structure will be divided into two sub-structures SS(1) with 11 internal DOFs and SS(2) with 8 internal DOFs, and 1-DOF of junction the mass m/2. The starting equation 20-DOF will be condensed and will bring to the resolution of a 10-DOF equation, divided into 1 junction DOF, 5 modes free or fixed interfaces of SS(1) and 4 modes free or fixed interfaces for SS(2).

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Advances in Mechanical Engineering

Figure 2. The temporal displacement response for m(12).

Figure 4. The temporal displacement response for m(18).

Figure 3. The temporal displacement response for m(1).

The following characteristics are considered:   

Masses: m1 = m2 =    = m20 = 2 kg; Linear stiffness: k1 = k3 =    = k41 = 50 N=m; Nonlinear cubic stiffness: k2 = k4 =    = k40 = k42 = 10 N=m3 .

The initial conditions are fuc g = f0, 0, 0, 0, 0, 0:5, 0, 0, 0, 0g fu_ c g = f0, 0, 0, 0, 0, 0, 0, 0, 0, 0g To illustrate the steps of the previously presented method, one begins by writing the vibration of the overall system of equations and those subsystems. The time responses of the entire system (without reduction) and the system decreased with a fixed interface and free interface (CMS) method are shown in

Figure 5. (a) The mean of temporal response for m(12) and (b) the variance of temporal response for m(12).

Figures 2–4 which correspond, respectively, to temporal displacements of the mass (12) which corresponds to DOF junction, mass (1) of the sub-structure SS(1),

Lamrhari et al.

9

Figure 6. (a) The mean of temporal response for m(1) and (b) the variance of temporal response for m(1).

Figure 7. (a) The mean of temporal response for m(18) and (b) the variance of temporal response for m(18).

and the mass (18) of the sub-structure SS(2). We can see that the different methods of modal synthesis provide very similar results. Other numerical results were obtained for different types of loading and modal decomposition. Adaptation to the assembled continuous systems is straightforward using the finite element method. In this study, it has been chosen to investigate the effects of uncertainties by considering mass uncertain parameters. The mass parameter is supposed to be a random variable and defined as follows: m = m0 (1 + sm qm ) where sm is a zero mean value Gaussian random variable, m0 = mii = 1, ..., 20 = 2 kg is the mean value, and qm = 2% is the standard deviation of this parameter. First, the mean and variance of the magnitude of displacement have been computed by the perturbation method for the entire system (without reduction). The obtained results are compared with those given by the direct Monte Carlo simulation with 700 samples. The obtained results are plotted in Figures 5–7 which correspond, respectively, to

temporal displacements of the mass (12) which corresponds to DOF junction, mass (1) of the sub-structure SS(1), and the mass (18) of the sub-structure SS(2). Finally, we used the approach based on the coupling of the perturbation method with CMS condensation method. This approach allows reducing the size of the problem and the computational cost. The mean and variance of displacement have been shown in Figures 8–11. We can see that the different methods provide very similar results.

Conclusion The main aim of this work is to provide the variability of the transient solution of a large and complex structure by considering geometric nonlinearities. We have achieved this by implementing an integrated approach, the coupling perturbation method, CMS reduction method, and temporal integration. The perturbation method was used to model the uncertain parameters by a development in Taylor series of second order. We

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Advances in Mechanical Engineering

Figure 8. The mean of temporal response for m(1), Monte Carlo simulation with 700 samples, and perturbation with whole structure and with free and fixed interface (CMS) methods, qm = 2%.

Figure 10. The mean of temporal response for m(12), Monte Carlo simulation with 700 samples, and perturbation with whole structure and with free and fixed interface (CMS) methods, qm = 2%.

Figure 9. The variance of temporal response for m(1), Monte Carlo simulation with 700 samples, and perturbation with whole structure and with free and fixed interface (CMS) methods, qm = 2%.

Figure 11. The variance of temporal response for m(12), Monte Carlo simulation with 700 samples, and perturbation with whole structure and with free and fixed interface (CMS) methods, qm = 2%.

developed the CMS method in the nonlinear case for reducing the finite element model. The implementation of the temporal integration by Newmark schema has allowed us to establish the variability of the solution for nonlinear reduced model with uncertain parameters. We could solve the problem of calculating the tangent matrix. The numerical tests show the accuracy of the results and minimization of cost calculation, thus validating this approach.

Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) received no financial support for the research, authorship, and/or publication of this article.

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