Fuzzy Stochastic Vibrations of Double-Beam Complex System as Model Sandwich Beam with Uncertain Parameters

Abstract

The dynamic behavior of a double Euler-Bernoulli beam system with uncertain parameters (fuzzy random variables) under a fuzzy stochastic excitation and axial compression is being considered. The beams are identical and parallel, one is above the other, and they are continuously coupled by a linear two-parameter (Pasternak subsoil) elastic element. This double Euler-Bernoulli beam system can be also treated as a theoretical model of a sandwich beam. The load process is fuzzy random both in space and time. The top beam carries a fuzzy stochastic load. The solution of the problem was found thanks to the fuzzy random dynamic influence function. The aim of the paper is to find the solution for the membership function of the probabilistic characteristics of the response of the structure.

1. Introduction

Uncertainty modeling in computational mechanics has received significant attention in recent years. Dynamic analysis of structures often involves two kinds of uncertainty. One of them is referred to as aleatory uncertainty (randomness, stochastic variability) and the other as epistemic uncertainty which describes, among others, imprecision, vagueness, ambiguity, and lack of the data. The random variability is described by the use of the probability theory and the imprecision by the use of fuzzy sets. Very often sufficient statistical data is not available; in this case a fuzzy function (fuzzy process) or a fuzzy random variable (fuzzy stochastic process) is suitable for the modeling purposes. The basis of the probabilistic methods is primarily statistical data while in the theory of fuzzy sets we refer also to the human intuition and experience. Using fuzzy random variables or fuzzy stochastic processes we combine these two approaches. The classical problem of the dynamic response of structures under stochastic excitation has been presented in many monographs. In most cases it is assumed that the parameters of the structure are deterministic. On the other hand, the structural parameters like geometry characteristics and material and damping properties might be uncertain to some extent. Their uncertainty may have a strong influence on the reliability of a structure in the dynamic context. The idea of fuzzy set theory was initiated by Zadeh [1]. The concept of fuzzy random variables was introduced by Kwakernaak [2] and Puri and Ralescu [3] and combines both randomness and imprecision. The application of the uncertain forecasting in engineering and computational mechanics based on fuzzy stochastic processes is presented in the monograph [4]. Various types of structures and girders like beams, plates, shells, and frames have been considered. An important technological extension of a single string, beam, or plate is a double-string, double-beam, or string-beam system. Various aspects of the dynamic response of a double-string and double-beam system have been considered by Oniszczuk [5–9]. Free and forced vibrations of a double-beam system have been considered among others in [10, 11]. The problem of the vibration and buckling of a double-beam system under compressive axial loading is presented in [12]. Vibrations of a complex system under moving force have been studied in [13–15].

In this paper the dynamic behavior of a double Euler-Bernoulli beam system with uncertain parameters (fuzzy random variables) under a fuzzy stochastic excitation and axial compression is being considered. The beams are identical, parallel, one above the other and continuously coupled by a linear two-parameter (Pasternak subsoil) elastic element. This double Euler-Bernoulli beam system can be also treated as a theoretical model of a sandwich beam, in which the shear and compression of core are taken into account. The load process is fuzzy random both in space and time. The solution of the problem was found thanks to the fuzzy random dynamic influence function [16–18]. The aim of the paper is to find the solution for the membership function of the probabilistic characteristics of the response of the structure. The probabilistic characteristics of the response of the structure are sought in the form of the first two probabilistic moments, that is, the expected value and the correlation (covariance) function. The stochastic response of structure with random and uncertain parameters has been considered, among others, in [18–27]. Fuzzy stochastic finite element method based a spectral approach to analyze complex engineering structures under dynamic excitation has been presented in [28].

2. Formulation of the Problem: General Solution

Let us consider the vibrations of a connected double Euler-Bernoulli beam continuous system connected by two-parameters elastic element under axial compression 𝑁 excited as is shown Figure 1.

Figure 1 

Double-beam system.

The differential equations of motion of the beams system have the form 𝐸₀𝐼𝜕⁴𝑤₁(𝑥,𝑡)𝜕𝑥⁴𝜕+𝑁²𝑤₁(𝑥,𝑡)𝜕𝑥²−𝑘₁𝜕²𝑤₁(𝑥,𝑡)𝜕𝑥²−𝜕²𝑤₂(𝑥,𝑡)𝜕𝑥²+𝑘₀𝑤₁(𝑥,𝑡)−𝑤₂(𝑥,𝑡)+𝑐𝜕𝑤₁(𝑥,𝑡)𝜕𝜕𝑡+𝑚²𝑤₁(𝑥,𝑡)𝜕𝑡²(),𝐸=𝑝𝑥,𝑡₀𝐼𝜕⁴𝑤₂(𝑥,𝑡)𝜕𝑥⁴𝜕+𝑁²𝑤₂(𝑥,𝑡)𝜕𝑥²−𝑘₁𝜕²𝑤₂(𝑥,𝑡)𝜕𝑥²−𝜕²𝑤₁(𝑥,𝑡)𝜕𝑥²+𝑘₀𝑤₂(𝑥,𝑡)−𝑤₁(𝑥,𝑡)+𝑐𝜕𝑤₂(𝑥,𝑡)𝜕𝜕𝑡+𝑚²𝑤₂(𝑥,𝑡)𝜕𝑡²=0, (1)(2) where 𝐸₀𝐼 is the flexural rigidity of the beam, 𝐸₀ is Young’s modulus of elasticity, 𝐼 is the moment of inertia of the cross-section area of the beam, 𝑚 is the mass of the beam, 𝑐 is the damping coefficient, 𝑘₀ is the elastic stiffness (Winkler’s parameter), and 𝑘₁ is the shear stiffness of the elastic element.

The boundary conditions of the simply supported beam are 𝑤₁(0,𝑡)=𝑤₁()𝜕𝐿,𝑡=0,₂𝑤₁(𝑥,𝑡)𝜕𝑥²_𝑥=0=𝜕²𝑤₁(𝑥,𝑡)𝜕𝑥²_𝑥=𝐿𝑤=0,₂(𝑥,0)=𝑤₂()𝜕𝑥,𝐿=0,₂𝑤₂(𝑥,𝑡)𝜕𝑥²_𝑥=0=𝜕²𝑤₂(𝑥,𝑡)𝜕𝑥²_𝑥=𝐿=0. (3) The above double-beam complex system can be treated also as a theoretical model of a sandwich beam which consists of two bendable outer layers and a sheared middle layer, which is also compressed.

The following system parameters are assumed to be uncertain: the Young’s modulus 𝐸₀=𝐸_0𝛼(𝑒), the mass of the beam 𝑚=𝑚_𝛼(𝑒), the damping coefficient 𝑐=𝑐_𝛼(𝑒), and the parameters of the elastic element 𝑘₀=𝑘_0𝛼(𝑒), 𝑘₁=𝑘_1𝛼(𝑒). The symbol 𝑒 denotes an elementary event, and for simplicity we will skip it in the paper. The symbol 𝛼 denotes the fuzziness of the parameters. The uncertain parameters of the system are assumed to be fuzzy random variables. The load process 𝑝(𝑥,𝑡)=𝑝_𝛼(𝑒,𝑥,𝑡) is assumed to be a fuzzy stochastic process. The responses of the system 𝑤₁(𝑥,𝑡)=𝑤_1𝛼(𝑒,𝑥,𝑡) and 𝑤₂(𝑥,𝑡)=𝑤_2𝛼(𝑒,𝑥,𝑡) are also fuzzy stochastic processes as well. The aim is to find the solution for the membership function of the probabilistic characteristics of the response, 𝑤_1𝛼(𝑥,𝑡) and 𝑤_2𝛼(𝑥,𝑡) of the double-beam complex system. The probabilistic characteristics of the response of the structure are sought in the form of the first two probabilistic moments, that is, the expected value and the correlation (covariance) function.

In order to decouple (1) and (2), let us introduce two new functions 𝑤_𝐼𝛼(𝑥,𝑡)=𝑤_1𝛼(𝑥,𝑡)+𝑤_2𝛼(),𝑤𝑥,𝑡_𝐼𝐼𝛼(𝑥,𝑡)=𝑤_1𝛼(𝑥,𝑡)−𝑤_2𝛼(𝑥,𝑡). (4)

From (1) and (2), we obtain two new differential equations for functions 𝑤_𝐼𝛼(𝑥,𝑡) and 𝑤_𝐼𝐼𝛼(𝑥,𝑡): 𝐸_0𝛼𝐼𝜕⁴𝑤_𝐼𝛼(𝑥,𝑡)𝜕𝑥⁴𝜕+𝑁²𝑤_𝐼𝛼(𝑥,𝑡)𝜕𝑥²+𝑐_𝛼𝜕𝑤_𝐼𝛼(𝑥,𝑡)𝜕𝑡+𝑚_𝛼𝜕²𝑤_𝐼𝛼(𝑥,𝑡)𝜕𝑡²=𝑝_𝛼(),𝐸𝑥,𝑡_0𝛼𝐼𝜕⁴𝑤_𝐼𝐼𝛼(𝑥,𝑡)𝜕𝑥⁴+𝑁−2𝑘_1𝛼𝜕²𝑤_𝐼𝐼𝛼(𝑥,𝑡)𝜕𝑥²+2𝑘_0𝛼𝑤_𝐼𝐼𝛼(𝑥,𝑡)+𝑐_𝛼𝜕𝑤_𝐼𝐼𝛼(𝑥,𝑡)𝜕𝑡+𝑚_𝛼𝜕²𝑤_𝐼𝐼𝛼(𝑥,𝑡)𝜕𝑡²=𝑝_𝛼(𝑥,𝑡). (5)(6)

Equation (5) describes vibrations of a single beam, while (6) describes vibrations of a single beam resting on an elastic Pasternak support with parameters 2𝑘₀ and 2𝑘₁. From (4), it follows that 𝑤_1𝛼()=𝑤𝑥,𝑡^𝐼𝛼(𝑥,𝑡)+𝑤_𝐼𝐼𝛼(𝑥,𝑡)2,𝑤_2𝛼()=𝑤𝑥,𝑡^𝐼𝛼(𝑥,𝑡)−𝑤_𝐼𝐼𝛼(𝑥,𝑡)2. (7)

In the case when the parameters of (5) and (6) are random, we can solve the problem if the right-hand side is deterministic. To overcome these difficulties, we introduce the space-time fuzzy random dynamic influence functions 𝐻_𝐼𝛼(𝑥,𝑡) and 𝐻_𝐼𝐼𝛼(𝑥,𝑡) (FRDIF) which satisfy the following equations with a deterministic right-hand side: 𝐸_0𝛼𝐼𝜕⁴𝐻_𝐼𝛼(𝑥,𝜉,𝑡)𝜕𝑥⁴𝜕+𝑁²𝐻_𝐼𝛼(𝑥,𝜉,𝑡)𝜕𝑥²+𝑐_𝛼𝜕𝐻_𝐼𝛼(𝑥,𝜉,𝑡)𝜕𝑡+𝑚_𝛼𝜕²𝐻_𝐼𝛼(𝑥,𝜉,𝑡)𝜕𝑡²(𝑡)𝛿(),𝐸=𝛿𝑥−𝜉_0𝛼𝐼𝜕⁴𝐻_𝐼𝐼𝛼(𝑥,𝜉,𝑡)𝜕𝑥⁴+𝑁−2𝑘_1𝛼𝜕²𝐻_𝐼𝐼𝛼(𝑥,𝜉,𝑡)𝜕𝑥²+2𝑘_0𝛼𝐻_𝐼𝐼𝛼(𝑥,𝜉,𝑡)+𝑐_𝛼𝜕𝐻_𝐼𝐼𝛼(𝑥,𝜉,𝑡)𝜕𝑡+𝑚_𝛼𝜕²𝐻_𝐼𝐼𝛼(𝑥,𝜉,𝑡)𝜕𝑡²=𝛿(𝑡)𝛿(𝑥−𝜉). (8)

After solving (8) one obtains 𝐻_1𝛼()=𝐻𝑥,𝜉,𝑡^𝐼𝛼(𝑥,𝜉,𝑡)+𝐻_𝐼𝐼𝛼(𝑥,𝜉,𝑡)2,𝐻_2𝛼()=𝐻𝑥,𝜉,𝑡^𝐼𝛼(𝑥,𝜉,𝑡)−𝐻_𝐼𝐼𝛼(𝑥,𝜉,𝑡)2. (9)

If the FRDIF 𝐻_1𝛼(𝑥,𝜉,𝑡) and 𝐻_2𝛼(𝑥,𝜉,𝑡) are known, the response of the system 𝑤_1𝛼(𝑥,𝑡) and 𝑤_2𝛼(𝑥,𝑡) can be presented in the following form: 𝑤_𝑖𝛼(𝑥,𝑡)=^𝐿₀^𝑡_𝑡0𝐻_𝑖𝛼(𝑥,𝜉,𝑡−𝜏)𝑝_𝛼(𝜉,𝜏)𝑑𝜏𝑑𝜉, (10) where 𝑖=1,2, and if 𝑡₀=0, then one considers transition vibrations, and for 𝑡₀=−∞, one considers steady-state vibration case.

Thus, in order to determine the probabilistic characteristics of the displacement of the beams, one can apply the expectancy operator to (10) and consequently obtain the expected values as follows: 𝐸𝑤_𝑖𝛼(𝑥,𝑡)_𝛼=^𝐿₀^𝑡_𝑡0𝐸𝐻_𝑖𝛼(𝑥,𝜉,𝑡−𝜏)𝐸𝑝_𝛼(𝜉,𝜏)𝑑𝜏𝑑𝜉. (11) Using 𝛼-level optimization procedure [4] for arbitrary 𝛼=𝛼_𝑘∈[0,1] or the max-min operator in the extension principle [1], the smallest and the largest expected values at an established point 𝑥 and time 𝑡 can be found.

Due to relationship (11), one obtains 𝐸𝑤_𝑖𝛼(𝑥,𝑡)_𝛼𝑙=min^𝐿₀^𝑡_𝑡0𝐸𝐻_𝑖𝛼(𝑥,𝜉,𝑡−𝜏)𝐸𝑝_𝛼(,𝐸𝑤𝜉,𝜏)𝑑𝜏𝑑𝜉_𝑖𝛼(𝑥,𝑡)_𝛼𝑟=max^𝐿₀^𝑡_𝑡0𝐸𝐻_𝑖𝛼(𝑥,𝜉,𝑡−𝜏)𝐸𝑝_𝛼(𝜉,𝜏)𝑑𝜏𝑑𝜉. (12)

Taking into account (10), the covariance functions of the beams displacement on the fuzziness level 𝛼 have the form 𝐶_𝑤𝑖𝛼𝑥₁,𝑥₂,𝑡₁,𝑡₂=^𝐿₀^𝐿₀^𝑡1_𝑡0^𝑡2_𝑡0𝐸𝐻_𝑖𝛼𝑥₁,𝜉₁,𝑡₁−𝜏₁𝐻_𝑖𝛼𝑥₂,𝜉₂,𝑡₂−𝜏₂×𝐶_{(𝑝𝑝)𝛼}𝜉₁,𝜉₂,𝜏₁,𝜏₂𝑑𝜏₁𝑑𝜏₂𝑑𝜉₁𝑑𝜉₂+^𝐿₀^𝐿₀^𝑡1_𝑡0^𝑡2_𝑡0𝐶_{(𝐻𝑖𝐻𝑖)𝛼}𝑥₁,𝑥₂,𝜉₁,𝜉₂,𝑡₁−𝜏₁,𝑡₂−𝜏₂×𝐸𝑝_𝛼𝜉₁,𝜏₁×𝐸𝑝_𝛼𝜉₂,𝜏₂𝑑𝜏₁𝑑𝜏₂𝑑𝜉₁𝑑𝜉₂, (13) where 𝐶_{(𝐻𝑖𝐻𝑗)𝛼}𝑥₁,𝑥₂,𝜉₁,𝜉₂,𝑡₁,𝑡₂=𝐸𝐻_𝑖𝛼𝑥₁,𝜉₁,𝑡₁𝐻_𝑗𝛼𝑥₂,𝜉₂,𝑡_2𝛼−𝐸𝐻_𝑖𝛼𝑥₁,𝜉₁,𝑡_1𝛼𝐸𝐻_𝑗𝛼𝑥₂,𝜉₂,𝑡_2𝛼,𝐶_{(𝑝𝑝)𝛼}𝜉₁,𝜉₂,𝜏₁,𝜏₂=𝐸𝑝_𝛼𝜉₁,𝜏₁𝑝_𝛼𝜉₂,𝜏_2𝛼−𝐸𝑝_𝛼𝜉₁,𝜏_1𝛼𝐸𝑝_𝛼𝜉₂,𝜏_2𝛼. (14)

The lower and upper endpoints of the covariance could be defined using (13) as 𝐶_{𝑤𝑖𝛼𝑙}𝑥₁,𝑥₂,𝑡₁,𝑡₂=min^𝐿₀^𝐿₀^𝑡1_𝑡0^𝑡2_𝑡0𝐸𝐻_𝑖𝛼𝑥₁,𝜉₁,𝑡₁−𝜏₁×𝐻_𝑖𝛼𝑥₂,𝜉₂,𝑡₂−𝜏₂𝐶_{(𝑝𝑝)𝛼}×𝜉₁,𝜉₂,𝜏₁,𝜏₂𝑑𝜏₁𝑑𝜏₂𝑑𝜉₁𝑑𝜉₂+^𝐿₀^𝐿₀^𝑡1_𝑡0^𝑡2_𝑡0𝐶_{(𝐻𝑖𝐻𝑖)𝛼}×𝑥₁,𝑥₂,𝜉₁,𝜉₂,𝑡₁−𝜏₁,𝑡₂−𝜏₂×𝐸𝑝_𝛼𝜉₁,𝜏₁,×𝐸𝑝_𝛼𝜉₂,𝜏₂,𝑑𝜏₁𝑑𝜏₂𝑑𝜉₁𝑑𝜉₂,𝐶_{𝑤𝑖𝛼𝑟}𝑥₁,𝑥₂,𝑡₁,𝑡₂=max^𝐿₀^𝐿₀^𝑡1_𝑡0^𝑡2_𝑡0𝐸𝐻_𝑖𝛼𝑥₁,𝜉₁,𝑡₁−𝜏₁×𝐻_𝑖𝛼𝑥₂,𝜉₂,𝑡₂−𝜏₂𝐶_{(𝑝𝑝)𝛼}×𝜉₁,𝜉₂,𝜏₁,𝜏₂𝑑𝜏₁𝑑𝜏₂𝑑𝜉₁𝑑𝜉₂+^𝐿₀^𝐿₀^𝑡1_𝑡0^𝑡2_𝑡0𝐶_{(𝐻𝑖𝐻𝑖)𝛼}×𝑥₁,𝑥₂,𝜉₁,𝜉₂,𝑡₁−𝜏₁,𝑡₂−𝜏₂×𝐸𝑝_𝛼𝜉₁,𝜏₁,×𝐸𝑝_𝛼𝜉₂,𝜏₂,𝑑𝜏₁𝑑𝜏₂𝑑𝜉₁𝑑𝜉₂. (15)(16)

In order to find the expected values 𝐸[𝐻_𝑖𝛼(𝑥,𝜉,𝑡)] and 𝐸[𝐻_𝑖𝛼(𝑥₁,𝜉₁,𝑡₁)𝐻_𝑖𝛼(𝑥₂,𝜉₂,𝑡₂)], we can use the perturbation method or Monte Carlo method. In this paper the perturbation method has been used for a particular solution of the expected values and variances and is presented in Section 3 for the response of the double-beam system.

3. Particular Solutions

The solutions of (8) for boundary conditions (3) are assumed to be in the form of the sine series: 𝐻_𝐽𝛼(𝑥,𝜉,𝑡)=^∞_𝑛=1𝑦_𝐽𝑛𝛼(𝑡,𝜉)sin𝑛𝜋𝑥𝐿, (17) where 𝐽=𝐼,𝐼𝐼.

By substituting expression (17) into (8) and using orthogonalization method, one obtains the following set of uncoupled ordinary differential equations: 𝑦̈_𝐽𝑛𝛼(𝑡,𝜉)+2𝛽_𝛼𝑦̇_𝐽𝑛𝛼(𝑡,𝜉)+𝜔²_𝐽𝑛𝛼𝑦_𝐽𝑛𝛼()=2𝑡,𝜉𝛿(𝑡)𝑚𝐿sin𝑛𝜋𝜉𝐿, (18) where, for 𝐽=𝐼, 𝜔²_𝐽𝑛𝛼=𝜔²_𝐼𝑛𝛼=(𝑛𝜋/𝐿)²[𝐸_0𝛼𝐼(𝑛𝜋/𝐿)²−𝑁]/𝑚 and for 𝐽=𝐼𝐼, 𝜔²_𝐽𝑛𝛼=𝜔²_{𝐼𝐼𝑛𝛼}=((𝑛𝜋/𝐿)⁴𝐸_0𝛼𝐼−(𝑛𝜋/𝐿)²(𝑁−2𝑘_1𝛼)+2𝑘_0𝛼)/𝑚 and 2𝛽_𝛼=𝑐_𝛼/𝑚_𝛼.

The dots denote differentiation with respect to the time. These functions fulfill the initial conditions 𝑦_𝐽𝑛𝛼(0,𝜉)=0,𝑦̇_𝐽𝑛𝛼(0,𝜉)=0. (19) The solution of (18) taking into account the initial conditions (19) has the form 𝑦_𝐽𝑛𝛼()=2𝑡,𝜉𝑚𝐿𝜔_𝐽𝑛𝛼𝑒^{−𝛽𝛼𝑡}sin𝜔_𝐽𝑛𝛼𝑡sin𝑛𝜋𝜉𝐿, (20) where 𝜔²_𝐽𝑛𝛼=𝜔²_𝐽𝑛𝛼−𝛽²_𝛼.

Taking into account (9), (17), and (20) we have 𝐻_1𝛼()=1𝑥,𝜉,𝑡𝑒𝑚𝐿^{−𝛽𝛼𝑡∞}_𝑛=1sin𝜔_𝐼𝑛𝛼𝑡𝜔_𝐼𝑛𝛼+sin𝜔_{𝐼𝐼𝑛𝛼}𝑡𝜔_{𝐼𝐼𝑛𝛼}sin𝑛𝜋𝜉𝐿sin𝑛𝜋𝑥𝐿,𝐻_2𝛼()=1𝑥,𝜉,𝑡𝑒𝑚𝐿^{−𝛽𝛼𝑡∞}_𝑛=1sin𝜔_𝐼𝑛𝛼𝑡𝜔_𝐼𝑛𝛼−sin𝜔_{𝐼𝐼𝑛𝛼}𝑡𝜔_{𝐼𝐼𝑛𝛼}sin𝑛𝜋𝜉𝐿sin𝑛𝜋𝑥𝐿. (21)

Let us consider the steady-state vibration (𝑡₀=−∞), assuming that the excitation load is a fuzzy weakly stationary “white noise” stochastic process both in time and space. In this case we have 𝐸[𝑝_𝛼(𝑥,𝑡)]_𝛼=𝐸[𝑝_𝛼]_𝛼=const. and 𝐶_{(𝑝𝑝)𝛼}(𝑥₁,𝑥₂,𝑡₁,𝑡₂)=𝜎²_(𝑝)𝛼𝛿(𝑡₁−𝑡₂)𝛿(𝑥₁−𝑥₂). We assume that the Young’s modulus 𝐸_0𝛼 is a fuzzy random variable. The other system parameters are assumed to be deterministic. In order to find the probabilistic characteristics the function of the random variables has been expanded into Taylor series around the mean value and restricted to three items of the expansion. The expected value for steady-state case is equal to 𝐸𝑤_1𝛼(=𝐸𝑝𝑥,∞)_𝛼𝛼𝐿^∞_𝑛=11−(−1)^𝑛×1𝑛𝜋/𝐿𝐸𝐸_0𝛼𝐼(𝑛𝜋/𝐿)⁴−𝑁(𝑛𝜋/𝐿)²×()1+𝑛𝜋/𝐿⁸𝐼²𝜎²_𝐸0𝐸𝐸_0𝛼𝐼(𝑛𝜋/𝐿)⁴−𝑁(𝑛𝜋/𝐿)²²+1𝐸𝐸_0𝛼𝐼(𝑛𝜋/𝐿)⁴−𝑁−2𝑘₁(𝑛𝜋/𝐿)²+2𝑘₀×1+𝑛𝜋𝐿⁸𝐼²𝜎²_𝐸0×𝐸𝐸_0𝛼𝐼𝑛𝜋𝐿⁴−𝑁−2𝑘₁𝑛𝜋𝐿²+2𝑘₀²⁻¹sin𝑛𝜋𝑥𝐿,𝐸𝑤_2𝛼(=𝐸𝑝𝑥,∞)_𝛼𝛼𝐿^∞_𝑛=11−(−1)^𝑛×1𝑛𝜋/𝐿𝐸𝐸_0𝛼𝐼(𝑛𝜋/𝐿)⁴−𝑁(𝑛𝜋/𝐿)²×()1+𝑛𝜋/𝐿⁸𝐼²𝜎²_𝐸0𝐸𝐸_0𝛼𝐼(𝑛𝜋/𝐿)⁴−𝑁(𝑛𝜋/𝐿)²²−1𝐸𝐸_0𝛼𝐼(𝑛𝜋/𝐿)⁴−𝑁−2𝑘₁(𝑛𝜋/𝐿)²+2𝑘₀×1+𝑛𝜋𝐿⁸𝐼²𝜎²_𝐸0×𝐸𝐸_0𝛼𝐼𝑛𝜋𝐿⁴−𝑁−2𝑘₁𝑛𝜋𝐿²+2𝑘₀²⁻¹sin𝑛𝜋𝑥𝐿, (22) where 𝜎²_𝐸0is the variance of the Young modulus.

Formula (13) gives the variance of the system displacement for the steady-state vibrations (𝑡→∞) in the form 𝜎²_𝑤𝑖𝛼(𝑥,∞)=𝐶_𝑤𝑖𝛼(𝑥,𝑥,∞,∞)=𝜎²_𝑝𝛼^𝐿₀^∞₀𝐸𝐻²_𝑖𝛼(𝑥,𝜉,𝜏)𝑑𝜏𝑑𝜉+𝐸²𝑝_𝛼𝛼×^𝐿₀^𝐿₀^∞₀^∞₀𝐶_{(𝐻𝑖𝐻𝑖)𝛼}×𝑥,𝑥,𝜉₁,𝜉₂,𝜏₁,𝜏₂𝑑𝜏₁𝑑𝜏₂𝑑𝜉₁𝑑𝜉₂, (23) where 𝜎²_𝑝𝛼^𝐿₀^∞₀𝐸𝐻²_1𝛼(=𝜎𝑥,𝜉,𝜏)𝑑𝜏𝑑𝜉_2𝑝𝛼𝐿2(𝑚𝐿)²×^∞_𝑛=1𝐸𝐽_𝐼𝑛𝛼+𝐽_{𝐼𝐼𝑛𝛼}+𝐽_{𝐼,𝐼𝐼𝑛𝛼}sin²𝑛𝜋𝑥𝐿,𝜎²_𝑝𝛼^𝐿₀^∞₀𝐸𝐻²_2𝛼(=𝜎𝑥,𝜉,𝜏)𝑑𝜏𝑑𝜉_2𝑝𝛼𝐿2(𝑚𝐿)²×^∞_𝑛=1𝐸𝐽_𝐼𝑛𝛼+𝐽_{𝐼𝐼𝑛𝛼}−𝐽_{𝐼,𝐼𝐼𝑛𝛼}sin²𝑛𝜋𝑥𝐿,𝐽_𝐼𝑛𝛼=1𝐸14𝛽𝜔²_𝐼𝑛𝛼=𝑚4𝛽(𝑛𝜋/𝐿)⁴𝐸𝐸_0𝛼𝐼−(𝑛𝜋/𝐿)²𝑁×1+𝑛𝜋𝐿⁸𝐼²𝑣²_𝐸0𝛼×𝑛𝜋𝐿⁴𝐼−𝑛𝜋𝐿²𝑁𝐸𝐸_0𝛼²⁻¹,𝐽_{𝐼𝐼𝑛𝛼}=1𝐸14𝛽𝜔²_{𝐼𝐼𝑛𝛼}=𝑚4𝛽(𝑛𝜋/𝐿)⁴𝐸𝐸_0𝛼𝐼−(𝑛𝜋/𝐿)²𝑁−2𝑘₁+2𝑘₀×1+𝑛𝜋𝐿⁸𝐼²𝑣²_𝐸0𝛼×𝑛𝜋𝐿⁴𝐸𝐸_0𝛼𝐼−𝑛𝜋𝐿²×(𝑁−2𝑘₁)𝐸𝐸_0𝛼+2𝑘₀𝐸𝐸_0𝛼²⁻¹,𝐽_{𝐼,𝐼𝐼𝑛𝛼}1=8𝛽𝐸2𝛽²+𝜔²_𝐼𝑛𝛼+𝜔²_{𝐼𝐼𝑛𝛼}²−4𝜔²_𝐼𝑛𝛼𝜔²_{𝐼𝐼𝑛𝛼}. (24) The numerical calculations have been done under the following assumptions.(i)The mean values 𝐸[𝐸_0𝛼] and 𝐸[𝑝_𝛼] are fuzzy numbers with triangular membership function (Figures 2 and 3),(ii)The variation coefficients 𝑣_𝐸0 are constant and do not depend on the level of fuzziness.(iii)Other quantities describing the structure are deterministic.

(a)

(b)

Figure 2 

Membership function of the mean value for 𝑁=0.

(a)

(b)

Figure 3 

Membership function of the variance for 𝑁=0.

The following values of the parameters are used in the following numerical calculations: 𝐿=10m, 𝐸[𝐸₀₁]=1⋅10¹⁰Nm⁻², 𝐸[𝐸_00𝑙]=0,8⋅10¹⁰Nm⁻², 𝐸[𝐸_00𝑟]=1,2⋅10¹⁰Nm⁻², 𝐼=3,8⋅10⁻⁴m⁴, 𝑘₀=1⋅10⁵Nm², 𝑘₁=2⋅10⁴Nm², 𝑁=0 and 𝑁=0,4(𝜋²/𝐿²)𝐸[𝐸₀₁]𝐼, 𝑣_𝐸0=0,2, 𝐸[𝑝₁]=const., 𝐸[𝑝_0𝑙]=0,8𝐸[𝑝₁], and 𝐸[𝑝_0𝑟]=1,2𝐸[𝑝₁].

In (23) it has been assumed that 𝐸²[𝑝_𝛼]_𝛼=0.

In Figures 2, 3, 4, 5, 6, and 7 the membership of the expected value and variance of the beams response are presented.

(a)

(b)

(c)

(d)

(e)

Figure 4 

Expected value and variance of the response of the beams.

(a)

(b)

Figure 5 

Membership function of the mean value for 𝑁=0,4(𝜋2/𝐿2)𝐸[𝐸01]𝐼.

(a)

(b)

Figure 6 

Membership function of the mean variance for 𝑁=0,4(𝜋2/𝐿2)𝐸[𝐸01]𝐼.

(a)

(b)

(c)

(d)

(e)

Figure 7 

Expected value and variance of the response of the beams.

4. Fuzzy Stochastic Moving Load

Let us notice that the general solution presented in Section 2 can also be used, after some modifications, if the double-beam system is loaded by a fuzzy stochastic load moving with velocity 𝑣 (Figure 8).

Figure 8 

The double-beam system loaded by fuzzy stochastic moving load.

In this case the right-hand side in (1) is equal to 𝑝_𝛼(𝑥,𝑡)=𝑝_𝛼(𝑥−𝑣𝑡).

The FRDIF 𝐻_𝑖𝛼(𝑥,𝑡) (𝑖=1,2) in solutions (10)–(16) should be replaced by fuzzy random dynamic moving influence function (FRDMIF) 𝐻_𝑖𝑚𝛼(𝑥,𝜉,𝑡) which satisfies, for 0≤𝑡≤𝐿/𝑣, the equations 𝐸_0𝛼𝐼𝜕⁴𝐻_Im𝛼(𝑥,𝑡)𝜕𝑥⁴𝜕+𝑁²𝐻_Im𝛼(𝑥,𝑡)𝜕𝑥²+𝑐_𝛼𝜕𝐻_Im𝛼(𝑥,𝑡)𝜕𝑡+𝑚_𝛼𝜕²𝐻_Im𝛼(𝑥,𝑡)𝜕𝑡²(),𝐸=𝛿𝑥−𝑣𝑡_0𝛼𝐼𝜕⁴𝐻_𝐼Im𝛼(𝑥,𝑡)𝜕𝑥⁴+𝑁−2𝑘_1𝛼𝜕²𝐻_𝐼Im𝛼(𝑥,𝑡)𝜕𝑥²+2𝑘_0𝛼𝐻_𝐼Im𝛼(𝑥,𝑡)+𝑐_𝛼𝜕𝐻_𝐼Im𝛼(𝑥,𝑡)𝜕𝑡+𝑚_𝛼𝜕²𝐻_𝐼Im𝛼(𝑥,𝑡)𝜕𝑡²(),𝐻=𝛿𝑥−𝑣𝑡_1𝑚𝛼()=𝐻𝑥,𝑡^Im𝛼(𝑥,𝑡)+𝐻_𝐼Im𝛼(𝑥,𝑡)2,𝐻_2𝑚𝛼()=𝐻𝑥,𝑡^Im𝛼(𝑥,𝑡)−𝐻_𝐼Im𝛼(𝑥,𝑡)2. (25)

The fuzzy random dynamic moving influence functions have the forms 𝐻_Im𝛼()𝐿𝑥,𝑡=2³𝐸𝐼^∞_0𝛼𝑛=1sin𝑛𝜋(𝑣𝑡/𝐿)sin𝑛𝜋(𝑥/𝐿)(𝑛𝜋)²(𝑛𝜋)²−𝑁₀−𝜂²𝐿−2³𝐸𝐼_0𝛼𝜂×^∞_𝑛=1sin𝑛𝜋(𝑛𝜋)²−𝑁₀/𝜂(𝑣𝑡/𝐿)sin𝑛𝜋(𝑥/𝐿)(𝑛𝜋)²(𝑛𝜋)²−𝑁₀−𝜂²(𝑛𝜋)²−𝑁₀,𝐻_𝐼Im𝛼()𝐿𝑥,𝑡=2³𝐸𝐼^∞_0𝛼𝑛=1sin𝑛𝜋(𝑣𝑡/𝐿)sin𝑛𝜋(𝑥/𝐿)(𝑛𝜋)⁴−(𝑛𝜋)²𝑁₀−2𝜅₁+𝜂²+2𝜅₀𝐿−2³𝐸𝐼_0𝛼𝜂×^∞_𝑛=1(𝑛𝜋)sin(𝑛𝜋)⁴−(𝑛𝜋)²𝑁₀−2𝜅₁+2𝜅₀𝜂𝑣𝑡𝐿𝑥×sin𝑛𝜋𝐿×(𝑛𝜋)⁴−(𝑛𝜋)²𝑁₀−2𝜅₁+𝜂²+2𝜅₀×(𝑛𝜋)⁴−(𝑛𝜋)²𝑁₀−2𝜅₁+2𝜅₀⁻¹, (26)(27) where 𝑁₀=𝑁𝐿²/𝐸𝐼_0𝛼, 𝜅₀=𝑘₀𝐿⁴/𝐸𝐼_0𝛼, 𝜅₁=𝑘₁𝐿²/𝐸𝐼_0𝛼, and 𝜂²=𝑚𝑣²𝐿²/𝐸𝐼_0𝛼.

The general solutions for moving fuzzy stochastic load after modification of (10)–(16) have the forms 𝑤_𝑖𝑚𝛼(𝑥,𝑡)=^𝑡₀𝐻_𝑖𝑚𝛼(𝑥,𝑡−𝜏)𝑝_𝛼(𝜏)𝑑𝜏, (28) where 𝑖=1,2, 0≤𝑡≤𝐿/𝑣, 𝐸𝑤_𝑖𝑚𝛼(𝑥,𝑡)_𝛼=^𝑡₀𝐸𝐻_𝑖𝑚𝛼(𝑥,𝑡−𝜏)𝐸𝑝_𝛼(𝜏𝐸𝑤)𝑑𝜏,_𝑖𝑚𝛼(𝑥,𝑡)_𝛼𝑙=min^𝑡₀𝐸𝐻_𝑖𝑚𝛼(𝑥,𝑡−𝜏)𝐸𝑝_𝛼(𝜏,𝐸𝑤)𝑑𝜏_𝑖𝑚𝛼(𝑥,𝑡)_𝛼𝑟=max^𝑡₀𝐸𝐻_𝑖𝑚𝛼(𝑥,𝑡−𝜏)𝐸𝑝_𝛼(𝜏,𝐶)𝑑𝜏_{𝑤𝑖𝑚𝛼}𝑥₁,𝑥₂,𝑡₁,𝑡₂=^𝑡1₀^𝑡2₀𝐸𝐻_𝑖𝑚𝛼𝑥₁,𝑡₁−𝜏₁𝐻_𝑖𝑚𝛼𝑥₂,𝑡₂−𝜏₂×𝐶_{(𝑝𝑝)𝛼}𝜏₁,𝜏₂𝑑𝜏₁𝑑𝜏₂+^𝑡1₀^𝑡2₀𝐶_{(𝐻𝑖𝐻𝑖)𝛼}𝑥₁,𝑥₂,𝑡₁−𝜏₁,𝑡₂−𝜏₂×𝐸𝑝_𝛼𝜉₁,𝜏₁,𝐸𝑝_𝛼𝜉₂,𝜏₂,𝑑𝜏₁𝑑𝜏₂, (29)(30)(31)(32) where 𝐶_{(𝐻𝑖𝐻𝑗)𝛼}𝑥₁,𝑥₂,𝑡₁,𝑡₂=𝐸𝐻_𝑖𝑚𝛼𝑥₁,𝑡₁𝐻_𝑗𝑚𝛼𝑥₂,𝑡_2𝛼−𝐸𝐻_𝑖𝑚𝛼𝑥₁,𝑡_1𝛼𝐸𝐻_𝑗𝑚𝛼𝑥₂,𝑡_2𝛼,𝐶_{(𝑝𝑝)𝛼}𝜏₁,𝜏₂=𝐸𝑝_𝛼𝜏₁𝑝_𝛼𝜏_2𝛼−𝐸𝑝_𝛼𝜏_1𝛼𝐸𝑝_𝛼𝜏_2𝛼. (33)

The lower and upper endpoints of the covariance could be defined using (32) as 𝐶_{𝑤𝑖𝛼𝑚𝑙}𝑥₁,𝑥₂,𝑡₁,𝑡₂=min^𝑡1₀^𝑡2₀𝐸𝐻_𝑖𝑚𝛼𝑥₁,𝑡₁−𝜏₁𝐻_𝑖𝑚𝛼𝑥₂,𝑡₂−𝜏₂×𝐶_{(𝑝𝑝)𝛼}𝜏₁,𝜏₂𝑑𝜏₁𝑑𝜏₂+^𝑡1₀^𝑡2₀𝐶_{(𝐻𝑖𝐻𝑖)𝛼}𝑥₁,𝑥₂,𝑡₁−𝜏₁,𝑡₂−𝜏₂×𝐸𝑝_𝛼𝜉₁,𝜏₁,×𝐸𝑝_𝛼𝜉₂,𝜏₂,𝑑𝜏₁𝑑𝜏₂,𝐶_{𝑤𝑖𝛼𝑚𝑟}𝑥₁,𝑥₂,𝑡₁,𝑡₂=max^𝑡1₀^𝑡2₀𝐸𝐻_𝑖𝑚𝛼𝑥₁,𝑡₁−𝜏₁𝐻_𝑖𝑚𝛼𝑥₂,𝑡₂−𝜏₂×𝐶_{(𝑝𝑝)𝛼}𝜏₁,𝜏₂𝑑𝜏₁𝑑𝜏₂+^𝑡1₀^𝑡2₀𝐶_{(𝐻𝑖𝐻𝑖)𝛼}𝑥₁,𝑥₂,𝑡₁−𝜏₁,𝑡₂−𝜏₂×𝐸𝑝_𝛼𝜏₁𝐸𝑝_𝛼𝜏₂𝑑𝜏₁𝑑𝜏₂, (34) where 0≤𝑡₁ and 𝑡₂≤𝐿/𝑣.

For steady-state solutions and it should be assumed that 𝑡₁=𝑡₂=𝐿/𝑣.

5. Conclusion

In the paper, the dynamic behavior of a double Euler-Bernoulli beam complex system with uncertain parameters (fuzzy random variables) under a fuzzy stochastic excitation and axial compression has been studied. The load process is fuzzy random both in space and time. In order to find the solution for the membership function of the probabilistic characteristics of the response of the structure, the idea of the fuzzy random dynamic influence function has been used. The probabilistic characteristics of the response of the structure are sought in the form of the first two probabilistic moments, that is, the expected value and the correlation (covariance) function. This double Euler-Bernoulli beam system can be also treated as a theoretical model of a sandwich beam. The algorithm (similar to the one presented in the paper for the double Euler-Bernoulli beam complex system) can be used in stochastic dynamic analysis of other complex systems like strings, beams, plates, and so on, with uncertain parameters.

Acknowledgment

The second author gratefully acknowledges the support of the Polish National Science Centre under Grant number ID: 147573.