Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects Adrian Sandu*, Corina Sandu§, and Mehdi Ahmadian§ Virginia Polytechnic Institute and State University *Computer Science Department,
[email protected] § Mechanical Engineering Department, {csandu, ahmadian}@vt.edu
Abstract This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”. The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions. The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties. Keywords: multibody system, uncertainty, polynomial chaos, stochastic ODE, stochastic DAE.
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Introduction Practical mechanical systems often operate with some degree of uncertainty. The uncertainties can result from poorly known or variable parameters (e.g., variation in suspension stiffness and damping characteristics), from uncertain inputs (e.g., soil properties in vehicle-terrain interaction), or from rapidly changing forcings that can be best described in a stochastic framework (e.g., rough terrain profile). For realistic predictions of the system behavior and performance multibody dynamic models must account for these uncertainties. Current methods used to formally assess uncertainties include Monte Carlo simulations, and linear and nonlinear approximations of the system response. Monte Carlo simulations are costly, and the accuracy of the estimated statistical properties improves with only the square root of the number of runs. Perturbation, statistical linearization, and nonlinear approximation methods compute several moments of the uncertainty distribution of the solution, but do not capture essential features of the nonlinear dynamics (e.g., as revealed by power spectral density). This study applies the generalized polynomial chaos theory to formally assess the uncertainty in multibody dynamic systems. We investigate the computational aspects of incorporating various types of uncertainties into multibody dynamic system models, and illustrate the methodology presented on practical examples related to vehicle dynamics and mobility in off-road conditions. The approach used is to extend the model along the stochastic dimension to explicitly parameterize the uncertainty distribution. Polynomial chaos offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest. For such systems, the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The methods discussed in this study allow the quantification of uncertainties and enable the simulations of multibody systems to produce results with “error bars”, similar to the way the experimental results are often presented. Moreover, the proposed methodology allows the quantification of uncertainties in both time and frequency domains. Polynomial chaos has been successfully applied in structural mechanics and in fluid mechanics studies. To our knowledge, this is the first application to multibody dynamics. It applies polynomial chaos to differential algebraic equations (DAEs) for constrained systems. It proposes direct statistical collocation as an alternative to the Galerkin method, and establishes its equivalence to a surface response approach. It presents a computationally efficient way to treat polynomial nonlinearities based on splitting the multidimensional integrals. The study is organized in two parts. Part I (the current paper) presents the theoretical and computational aspects of the study. After providing a background on methods currently available to treat uncertainties in multibody dynamic systems, and the generalized polynomial chaos expansion, the paper discusses modeling of parametric and external sources of uncertainty in mechanical 2
systems. Next, it introduces the basis functions used for the representation of the stochastic dimension. This is followed by presenting the stochastic ordinary differential equation formulation of multibody dynamics, and the stochastic differential algebraic formulation. The paper continues with the analysis of the uncertainty in model results in both time and frequency domains. Finally, the conclusions of this study are presented. The Legendre, Jacobi, and Hermite polynomials are given in the appendix. Part II is the complementary paper [1] which presents numerical results and discussions for representative case studies.
Background In this section we review the equations governing multibody dynamics, discuss the methods currently employed to treat uncertainties in such systems, and give an overview of the polynomial chaos approach. Polynomial chaos has been used extensively to model uncertainties in structural mechanics and in fluids, but to our knowledge it has not been previously applied to multibody dynamic simulations. Formulations of Multibody Dynamic Equations The dynamics of a multibody system can be described in local or in global coordinates, in Cartesian or in generalized coordinates. The dynamics of an unconstrained mechanical system [2] can be described by a set of simultaneous fist order differential equations (ODE): y& = v , v& = F (t , y, v; p ) , y t 0 = y 0 , t 0 ≤ t ≤ t F (1) d d Here y ∈ ℜ are the generalized positions, v ∈ ℜ are the generalized velocities, v& ∈ ℜd are the generalized accelerations, and p ∈ ℜ e is a vector of system parameters. The dot notation represents derivative with respect to time. Using the same notation as in Eq. (1), one can represent a constrained mechanical system in Cartesian coordinates [3] by a system of index-3 differential algebraic equations (DAE): ⎧ y& = v ⎪ T (2) ⎨ M ( y )v& = F (t , y , v ) + ψ y λ ⎪ψ ( y ) = 0 ⎩ We consider holonomic position constraints denoted by ψ : ℜ d → ℜ c ; M : ℜ d → ℜ d ×d is the generalized mass matrix, F : ℜ × ℜd × ℜd → ℜd are the external generalized forces and torques, λ ∈ ℜc represent the Lagrange multipliers, and ψ Ty λ represent the constraint forces. Partial derivatives are
( )
denoted by subscripts; e.g., ψ y is the Jacobian of ψ with respect to y . By differentiating the holonomic position constraints twice with respect to time, one obtains the constraint equations for velocity and acceleration: ψ y ( y) v = 0 (3) ψ y ( y ) v& =τ ( y, v)
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Replacing the position constraints in Eq. (2) by the acceleration constraints from Eq. (3) leads to the index-1 DAE formulation, which is convenient for computational purposes [3]: ⎡ M ( y ) ψ Ty ( y )⎤ ⎡ v& ⎤ ⎡ F (t , y, v)⎤ &y = v, = (4) ⎢ ⎥ 0 ⎦⎥ ⎢⎣λ ⎥⎦ ⎢⎣ τ ( y, v) ⎥⎦ ⎣⎢ψ y ( y ) The system in Eq. (4) is mathematically equivalent with the one given by Eq. (2), however a numerical scheme applied to Eq. (4) will lead to increasing errors in the position and velocity constraints [4]. To alleviate this drift-off, the y n , v~n ) is projected at each time step onto the coordinate numerical solution ( ~ constraint manifold by solving the following nonlinear system for y n : M ( ~y )( y − ~y ) +ψ T ( ~ y )η = 0 , ψ ( y ) = 0 (5) n
n
n
y
n
n
Similarly, the computed velocity is projected onto the velocity constraint manifold by solving the following linear system for vn : M ( y n )(vn − v~n ) +ψ Ty ( y n )η = 0 , ψ y ( y n ) vn = 0 (6) In Eq. (6) η ∈ ℜ c is another set of Lagrange multipliers. The projected values ( y n , vn ) are the new numerical solution at t n . Methods Available for Treating Uncertainties The traditional modeling approach assumes an ideal input with precisely defined parameters which determine the value of the output. When some parameter values or external forcings are not known, or cannot be accurately represented, the probabilistic framework is more appropriate. In this framework one models uncertain input parameters, and formulates the dynamic model to reflect the propagation of uncertainty in the output. Some of the commonly-used methods that are adapted for solving systems with uncertainties are now reviewed. A very general approach is to solve the Fokker-Plank (FP) equation which governs the evolution of uncertainty distribution under system dynamics [5,6]. The method works in a high-dimensional probability space, and is not a practical computational tool for systems of interest. Monte Carlo approach has been used extensively in dynamic models. An ensemble of runs is performed with each member using a different set of parameters drawn from the corresponding uncertainty distribution [7-9]. The statistical properties of the outputs are obtained from the ensemble of simulation results. The approach is computationally expensive as the estimation of the variance converges with the inverse square root of the number of runs. Related, more economical approaches are Latin Hypercube Sampling [10] and Bayesian Monte Carlo [11-12]. The perturbation (sensitivity) approach uses first and higher order sensitivity coefficients to derive low order moments of the simulation uncertainty [13-15]. This approach is useful when uncertainties are small, and behave like perturbations of the model [16,17].
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The Neumann series expansion of the governing stochastic operator is described in [18]. Application of this technique to large nonlinear systems is, however, difficult. Response approximation methods build simplified models to capture the relationship between the uncertain inputs and uncertain outputs. Statistical linearization [19,20] and quadratization [21] produce models with the correct statistical moments. However, they do not provide any information on power spectral density of the response, which is important to analyzing uncertain mechanical systems. Finite order Volterra series [22] and nonlinear methods [23] were used to describe the input-output relation of nonlinear systems with stochastic excitations. These methods are difficult to apply to systems with multiple degrees of freedom. The deterministic equivalent model [24,25] and the stochastic response surface approach [26-29] use polynomial chaos representations of the inputs and outputs, and determine the coefficients of the output model through a collocation approach based on a small number of model runs. Stochastic averaging [30-32] and non-Gaussian closure techniques [33,34] pursue information regarding the statistical moments of the results. Stochastic averaging has proven to be effective for deriving approximate solutions for weekly damped systems. Its application to nonlinear dynamic systems, which are often moderately to heavily damped, has proven difficult. The approach employed by [35,36] in the context of finite elements for solid mechanics uses a spectral approximation the uncertainty which allows high order representations. The fundamental ideas stem from Wiener's homogeneous chaos theory [37]. Comprehensive work was done by Ghanem and co-workers for modeling uncertainty in other applications including nonlinear vibrations, fluids, porous media, etc. [21,35,36,38-43]. Karniadakis and co-workers introduced the concept of generalized polynomial chaos, and studied extensively its use to model uncertainties in fluids, including applications such as advection and diffusion, turbulence, and flow-structure interactions [44-55]. Keese presents a comprehensive review of recent methods for the numerical solution of stochastic partial differential equations [56]. Polynomial chaos offers a tractable computational approach for the large, nonlinear multibody models of engineering systems of interest. For such systems the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). Generalized Polynomial Chaos Expansion Polynomial chaoses [37,57] are generalizations of polynomials to the case where the independent variables are themselves measurable functions (in this paper random variables). The fundamental idea is that random processes of interest can be approximated (with arbitrary accuracy) by sums of orthogonal polynomial chaoses of random independent variables. These sums offer a computationally attractive approach to representing the state of physical systems operating under uncertainty.
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Second order random processes are processes with finite variance; from a physical point of view they have finite energy. A second order random process X (θ ) , viewed as a function of the random event θ , (0 < θ < 1) , can be expanded in terms of orthogonal polynomial chaoses as [35,36]: ∞
X (θ ) = ∑ c j φ j (ξ (θ ))
(
(7)
j =1
)
Here φ i ξ i1 Kξ in are generalized Askey-Wiener polynomial chaoses of order n(i ) ,
(
)
in terms of the multi-dimensional random variable ξ = ξ i1 Kξ in . For Gaussian random variables the basis are Hermite polynomials, for uniformly distributed random variables the basis are Legendre polynomials, for beta distributed random variables the basis are Jacobi polynomials, and for gamma distributed random variables the basis are Laguerre polynomials [46.47]. The basis chaos polynomials form a complete orthogonal basis for the Hilbert space of square integrable random variables, φ i , φ j = 0 for i ≠ j (8) This orthogonality relation holds with respect to the ensemble average inner product, f , g = ∫ f (ξ ) g (ξ ) w(ξ ) dξ (9)
Here w(ξ ) is the joint probability density of the random variables ξ . The series in Eq. (7) converges to any random process in L2 sense [57]. In practice, a truncated expansion is used, S
X = ∑ c jφ j (ξ )
(10)
j =1
This means that we consider a finite number n of random variables ξ = (ξ1 Kξ n ) , and sum polynomials only up to a maximal order P . The total number of terms S = (n + P )! (n! P!) increases rapidly with the number of stochastic parameters n and the order of the polynomial chaos P . Note that statistical bilinearization, for example, also constructs a system that is considerably larger than the original [22]. In Eq. (10), and for the remaining of this paper, we drop the explicit dependence of the random variable on the event θ .
Modeling Sources of Uncertainty in Mechanical Systems Dynamic systems are often affected by multiple sources of uncertainty. The main classes are parametric uncertainty and uncertain external excitations. In this section we discuss several approaches to model these uncertainties. Parametric Uncertainty Uncertain parameters in mechanical systems take values between well defined bounds. Consequently, they cannot be accurately represented as normal random variables, since the Gaussian distribution has infinite support. We will focus on probability densities with finite support, namely the uniform and beta distributions.
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These distributions, defined on the finite interval ξ ∈ [−1,1] , have the following probability density functions (PDF): 1. Uniform probability distribution: w(ξ ) = 1
(11)
2
2. Beta probability distribution:
w(ξ ) =
∞ Γ(a + b + 2 ) a b ( ) ( ) ( ) t x −1e −t dt ⋅ 1 − ξ 1 + ξ , where Γ x = a +b +1 ∫ 0 2 Γ(a + 1) Γ(b + 1)
(12)
Here Γ(x) denotes the gamma function. The beta distribution has two parameters, a and b , which define the shape of the distribution, as illustrated in Fig. 1. For a = b = 0 the beta distribution reduces to the uniform probability distribution.
pdf
1.2
1.2
uniform
1.2
beta(1,1)
0.9
0.9
0.9
0.6
0.6
0.6
0.3
0.3
0.3
0 −1
−0.5
0 ξ
0.5
1
(a) Uniform distribution
0 −1
−0.5
0 ξ
0.5
(b) Beta distribution
1
beta(3,1)
0 −1
−0.5
0 ξ
0.5
1
(c) Beta distribution
Figure 1. Examples of distributions with finite support Parametric Representation of Stochastic Forcing An important aspect in the study of vehicle behavior over rough terrain is modeling the uncertainty in the terrain profile. More generally, an important aspect in the study of mechanical systems is the representation of uncertain external forcings. In this section, the external forcing depends on x (space) instead of t (time) to intuitively represent terrain variation. The discussion, however, is general and can be directly applied to any time dependent stochastic forcing function. The terrain profile is considered a random process z ( x ) defined over the spatial domain x ∈ D . The random terrain height can be represented as a mean height z ( x ) plus a sum of deterministic shapes g k (x ) multiplied by random amplitudes ξ k , n
z (x ) = z (x ) + ∑ ξ k g k (x )
(13)
k =1
The shape functions are linearly independent, e.g., can be chosen from an 2 orthonormal base of the set of square integrable functions L ([0, D ]) .If the
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random amplitudes ξ k are assumed to be independent identically distributed random variables, with zero mean and variances σk, the forcing covariance function is: n
R ( x1 , x2 ) = z ( x1 ) − z ( x1 ), z ( x2 ) − z ( x2 ) = ∑ σ k2 g k ( x1 )g k ( x2 )
(14)
k =1
Similarly, if the terrain profile z (x, θ ) has known covariance R( x1 , x2 ) , then the random profile can be represented by the Karhunen-Loeve (KL) expansion [35,36,46-52]: ∞
z ( x ) = z ( x ) + ∑ λk ξ k g k ( x )
(15)
k =1
Here ξ k is an independent set of random variables of mean 0 and variance 1. The shape functions g k ( x ) are the eigenvectors, and λk the eigenvalues of the covariance function: (16) ∫ R(x1 , x2 ) g k (x2 ) dx2 = λk g k (x1 ) D
A practical representation of the stochastic process is obtained by truncating the KL series in Eq. (14) to n terms, based on the relative magnitude of the eigenvalues, such that λ n+1
