Time Response and Natural Frequency Analysis for Structures with Interval Parameters

. This paper deals with the evaluations of lower and upper bounds of dynamic responses and natural frequencies for structures with uncertain interval parameters. The structural physical parameters, geometric dimensions and initial conditions can be considered as interval parameters. The modified Chebyshev interval method (MCIM) is presented to handle the uncertainties. In comparison with the Chebyshev interval method (CIM), MICM provides with tighter bounds of the time responses and natural frequencies without a significant increase in computation time. Monte Carlo method is also used to provide with exact bounds. MICM, CIM and Monte Carlo method are applied for structures with interval parameters, and the efficiency and accuracy of CIM and MCIM are verified.


Introduction
Probabilistic uncertainty and imprecision in structural parameters and in environmental conditions and loads are challenging phenomena in engineering analyses [1]. For instance, variations of inertial, stiffness and damping properties due to uncertainties may significantly affect the dynamics characteristics of structures. Therefore, it is of significant importance to estimate the effects of geometrical and material uncertainties on structural dynamics. The most common approach to structures with uncertainties is to model the structural parameters as random or probabilistic variables. Under these circumstances, the probabilistic approaches demand the probability density functions of the structural parameters. The common probabilistic approaches include generalized polynomial chaos method [2,3] and Monte Carlo, etc. Unfortunately, the knowledge of probability density function is not always available. As alternative tools, the non-probabilistic approaches, such as fuzzy finite element analysis [4,5], interval model [6] and convex model [7,8], have been extensively used for handling uncertainties arising in engineering problems. Moens and Hannss [9] presented a general overview of the state-of-art in interval and fuzzy finite element analysis. The interval model is widely used when only the bounds of uncertain parameters are required. Moore [10] and Alefeld [11] done the pioneering work in the field of interval analysis. When the uncertain parameters are modelled as interval parameters, the mass, stiffness and damping matrices of the structure are also interval matrices; the time responses and eigenvalues (natural frequencies) are solved by so-called interval analysis methods. Both time response problem and eigenvalue problem have attracted much research attention in the last decades. The major challenge or objective for interval analysis is the evaluation of the narrowest intervals enclosing all possible solutions. The evaluation of natural frequencies or eigenvalues is crucial in vibration analysis of structures in the presence of uncertainty. Qiu et al. [12] introduced the eigenvalue inclusion principle to determine the lower and upper bounds on the eigenvalues due to interval parameters. Chen et al. [13] applied matrix perturbation theory for evaluating the bounds of eigenvalue of structures. Wang et al. [14] presented a modified interval perturbation finite element method for structural design, in which the interval matrix and vector were expanded by the Taylor series. Gao [15] presented the interval factor method for the natural frequency and mode shape analysis of truss structures with interval parameters. Li et al. [16] divided the structural eigenvalue problem with interval parameters into a series of QB (quadratic programming with box constraints) problems by the use of information of the second-order partial derivatives of eigenvalues. Sofi et al. [17] presented an efficient procedure to seek the bounds of the eigenvalues, where interval uncertainties were handled following the improved interval analysis via extra unitary interval [18,19]. The common drawback of the aforementioned approaches is the calculation of the derivatives of the mass and stiffness matrices w.r.t. interval variables. It is a rather difficult and time consuming task to calculate the derivatives, especially for complex structures with multi uncertain variables. The uncertainty analysis of time response of structures has received relatively less attention in comparison with evaluation of eigenvalues. The common interval analysis methods include mathematical programming methods [20][21][22][23], perturbation methods [24][25][26] and polynomial series expansion methods [6,27], etc. Qiu and Wang [24,25] presented first-order perturbation methods to estimate the range of the dynamic response of structures. Xia et al. [26] presented a perturbation method based on the vertex solution theorem for the first-order derivation of the dynamic response from its central value and avoided interval extension/overestimation problems. The main disadvantage of perturbation methods is the requirement to calculate the first derivatives of the system responses w.r.t. uncertain variables. To overcome this disadvantage, Wu et al. [6] presented the Chebyshev interval method for multibody dynamics to achieve shaper and tighter bounds for meaningful solutions. The Chebyshev interval method has been widely used for multibody systems [6,28,29] and vehicle dynamic system [30], etc. In comparison with most conventional interval models, Chebyshev interval method can reduce the interval overestimation and it doesn't require derivatives of system responses w.r.t. uncertain variables. In this paper, the Chebyshev interval method (CIM) is modified to determine the upper and lower bounds of time responses and natural frequencies of structures in the presence of uncertainties. The modified Chebyshev interval method (MCIM) achieves more accurate interval bounds and the computation time is not significantly increased. The Monte Carlo method is used to provide with the exact interval bounds although it is time consuming. In Section 2, primary knowledge about interval analysis is reviewed. In Section 3, time response and eigenvalue problems for structures with interval parameters are formulated. In Section 4, the CIM is briefly reviewed and MCIM is presented. In Section 5, numerical examples are used to show the efficiency of CIM and MCIM. In Section 6, conclusions throughout the paper are drawn.

Preliminaries
An interval parameter represents a range of uncertainty. For each interval parameter, there are two numbers that represent the lower and upper bounds of the parameter. A real interval   x in real set R is defined as  ,   min  ,  ,  ,  , max  ,  ,  , min / , / , / , / ,max / , / , / , /

Eq. (2) can be rewritten in a detailed manner as
Note that the division operation is not defined if . Several interval variables can form an interval vector. For instance, vector   x includes k interval variables: , , ⋯ , and each element has an interval bound, i.e.
A function containing one or more interval variables is called an interval function. To provide the exact bounds of interval function is the major objective in interval analysis. However, interval function has some overestimation caused by the intrinsic wrapping effect [31], meaning that the bounds of interval function are overestimated. Therefore, an interval inclusion function is more easily to be obtained. The interval inclusion function It is not necessary to distinguish the interval function and interval inclusion function, because interval models or methods generally overestimate the bounds of the interval function. The interval methods used in this paper, CIM and MCIM have small overestimations.

Problem formulation
The equation of motion for a linear uncertain structure with n degrees of freedom is formulated as where

Interval analysis methods
This section introduces the basic idea of CIM briefly. For details, the readers can refer to [6]. Then MCIM is presented and introduced in details.

Chebyshev interval method (CIM)
A function defined in   , C a b can be approximated by the truncated Chebyshev series of degree n The Chebyshev polynomials Chebyshev polynomial on   , a b is also defined by Eq.
is the weighting function.
The Chebyshev interval function is defined in terms of For multidimensional problems, the original function , , ⋯ , is approximated by multidimensional Chebyshev polynomials of degree n, such that Similarly, the multi-dimensional coefficients is calculated by  

Modified Chebyshev interval method (MCIM)
The process of constructing Chebyshev approximation polynomial in MCIM is the same as that in CIM. In MCIM, the least squares method is used to calculate the coefficients, this process would be the traditional RSM. Transform Eq.
In general, the original function such as Eq. (5) and (6) is extremely complicated; using Monte Carlo is time consuming. However, is a polynomial model which is much less complicated than the original function, so using Monte Carlo is not time consuming.

Comparison of CIM and MCIM
This section offers qualitative comparison of CIM and MCIM.
(1) Both CIM and MCIM are based on Chebyshev polynomial approximation (Eq.10 and Eq.14). It is noted that both CIM and MCIM do not require the derivatives of the system responses w.r.t interval variables, which is an advantage over perturbation methods and Taylor series based methods.
(2) Both CIM and MCIM need to calculate the coefficients of the polynomials. The difference is that CIM uses multiple integral (Eq.12) to calculate the coefficients while MCIM uses least squares (Eq.15). The authors have found that the coefficients obtained by multiple integral and least squares method are nearly the same. For the problems with large number of interval variables, the least squares method is easier to be implemented.
(3) After getting the Chebyshev polynomial approximation formulation, interval function is obtained using the properties of Chebyshev polynomials in CIM (Eq.13), while using Monte Carlo method in MCIM (Eq.18). Thus CIM introduces overestimation and MCIM does not. It can be seen in the numerical examples that MCIM is more accurate than CIM.

12-bar truss structure
The first example concerns a plane truss structure which is modelled with 8 nodes and 12 elements (Fig.1). The The truss structure is solved for a period of 0.1s with a step size of 0.0001s using CIM, MCIM and Monte Carlo, respectively. Fourth-order Chebyshev polynomial is used for each interval variable, so 5 sampling points (zero points of fifth-order Chebyshev polynomial) is used for each interval variable. Since there are 3 interval variables, the total number of sampling points is 5 3 =125 for CIM and MCIM. For the Monte Carlo method, 1000 samplings are used. The time responses of the truss structure are shown in Fig.2 to 8. Fig.2 to 5 shows the horizontal and vertical displacements of nodes 7 and 8 under uncertain dynamic loads. Fig.6 to 8 shows the stresses at the truss elements. It is observed from Fig.2 to 8 that each time response has upper and lower bounds due to the interval parameters. Monte Carlo obtains the exact bounds without overestimations. CIM obtains slightly wider response intervals than Monte Carlo due to the inherent wrapping effects as mentioned in Section 2. In comparison with CIM, MCIM obtains fairly exact response intervals, meaning that MCIM controls the wrapping effects better.  Table 1 shows the lower and upper bounds of the first 5 natural frequencies (eigenvalues). Since the eigenvalues are calculated through algebraic operations, the three methods obtain nearly the same natural frequency bounds. Particularly, MCIM obtains the exact results as the Monte Carlo does. Moreover, the width of natural frequency bound increases with the increasing modal order. It should be noted that although 3 interval parameters are considered in this example, only the Young's modulus and material density affect the bounds of eigenvalues. Table 2 shows the total calculation times for time responses and eigenvalues. It is observed that Monte Carlo is the most time consuming and CIM is the least time consuming. MCIM costs about 5s longer than CIM for calculating the bounds of Chebyshev polynomials, but costs much less time than Monte Carlo.

Cantilever beam
The second example concerns a cantilever Euler-Bernoulli beam uniformly discretized into 10 elements, as shown in Fig.9 Fig.10 to 13 shows the time responses of the truss structure. Fig.10 shows the dynamic deflection of the cantilever beam (vertical displacement of node 11) under uncertain dynamic loads. Fig.11 shows the shear force of element 1 under uncertain dynamic loads. Fig.12 and 13 show the bending moments of element 1 at the left end and the right end. It is observed that each time response has upper and lower bounds due to the interval parameters. In theory, Monte Carlo provides with the exact bounds without overestimations. CIM obtains slightly wider response intervals than Monte Carlo due to the wrapping effects as mentioned in Section 2. MCIM obtains the nearly the same exact results with Monte Carlo. However, the overestimations of CIM are very small and can be accepted.

Conclusions
An interval analysis method termed as modified Chebyshev interval method (MCIM) is presented to estimate the time responses and eigenvalues of structures with interval parameters. MCIM is developed based on the Chebyshev interval method (CIM). MCIM, CIM and Monte Carlo method are applied for solving two structures in presence of interval parameters. These three methods are compared in terms of overestimation and computation time. In theory, Monte Carlo method provides with the exact bounds without any overestimation, but costs a long computation time. CIM costs much less time than Monte Carlo method, and introduces a slight of overestimations. MCIM can also provide with the exact results, but costs a little longer computation time than CIM. Even so, the computation time of MCIM is much shorter than that of Monte Carlo method. The overall conclusion is that both MCIM and CIM are effective and accurate interval analysis methods. It should be noted that both MCIM and CIM do not demand the derivatives of the system outputs w.r.t interval variables. So MCIM and CIM are suitable for complex systems where the calculation of derivatives is difficult. MCIM and CIM can also be used for black-box problems where the mathematical models of the systems are unknown.