Discrete singular convolution–polynomial chaos expansion method for free vibration analysis of non-uniform uncertain beams

Abstract

This article enhances the discrete singular convolution method for free vibration analysis of non-uniform thin beams with variability in their geometrical and material properties such as thickness, specific volume (inverse of density) and Young’s modulus. The discrete singular convolution method solves the differential equation of motion of a structure with a high accuracy using a small number of discretisation points. The method uses polynomial chaos expansion to express these variabilities simulating uncertainty in a closed form. Non-uniformity is locally provided by changing the cross section and Young’s modulus of the beam along its length. In this context, firstly natural frequencies of deterministic uniform and non-uniform beams are predicted via the discrete singular convolution. These results are compared with finite element calculations and analytical solutions (if available) for the purpose of verification. Next, the uncertainty of the beam because of geometrical and material variabilities is modelled in a global manner by polynomial chaos expansion to predict probability distribution functions of the natural frequencies. Monte Carlo simulations are then performed for validation purpose. Results show that the proposed algorithm of the discrete singular convolution with polynomial chaos expansion is very accurate and also efficient, regarding computation cost, in handling non-uniform beams having material and geometrical variabilities. Therefore, it promises that it can be reliably applied to more complex structures having uncertain parameters.

Keywords

Non-uniform beam discrete singular convolution uncertainty polynomial chaos expansion

1. Introduction

In engineering, structures generally have different geometrical shapes, and sometimes, they are non-uniform along the structure for engineering reasons. Non-uniformity is generally provided to reinforce the regions with different stress distributions, especially for beams which are one of the most commonly used engineering structures. It is generally accomplished by changing either the mechanical or geometrical properties in a local manner along the structure. These types of structures are called as functionally graded materials in literature (Akgöz and Civalek, 2013; Alshorbagy et al., 2011; Sankar, 2001). However, the same structures or components manufactured by the same mass production line may exhibit different dynamic characteristics for an unpredicted reason. These variabilities are generally labelled as uncertainty. Uncertainty and variability are unavoidable because of various reasons such as material heterogeneity, production tolerances, manufacturing defects and environmental factors. A good example for variability is a work of Kompella and Bernhard (1993). Structure-borne sound at the driver position in 57 samples of Isuzu pick-up trucks, which were produced in the same production line, clearly showed variability. Many experiments and practical cases in industry showed that for a realistic design, uncertainty needs to be taken into account starting in the product design stage. Besides, it is also observed that higher frequencies are much more sensitive to these variabilities. This requirement leads to the development of some simulation methods for quantifying these variabilities. Methods used in uncertainty analysis may be categorised as probabilistic and non-probabilistic methods. Non-probabilistic methods are used in cases where the statistical properties of uncertain parameters are not known, but the limits of them are known. In such methods, only the limits of the uncertain response variable are obtained. When statistics of uncertain parameters are known or properly assumed, probabilistic methods such as the Monte Carlo simulation (Evans and Swartz, 2000; Rubinstein and Kroese, 2016), first- and second-order reliability methods (Hohenbichler and Rackwitz, 1989; Keane and Price, 1997), numerical integration–based methods (Evans, 1972; Rahman and Xu, 2004; Seo and Kwak, 2002) and spectral methods (Ghanem and Spanos, 2003; Lucor et al., 2004; Sepahvand, 2017; Sepahvand et al., 2007, 2010) can be appropriate methodologies to be selected. Among them, Monte Carlo simulations are the most widely used technique. However, it requires significant numerous experiments/simulation sets to determine response statistics, which makes it inefficient. Recently, spectral methods such as Karhunen–Loeve (KL) (Ghanem and Spanos, 2003) or polynomial chaos expansion (PCE) (Lucor et al., 2004; Sepahvand, 2017; Sepahvand et al., 2007, 2010), are preferable amongst the researchers because the variability is defined in a set of closed-form equations.

Beams are one of the most basic and thus most attracted structural elements in the research studies for the test and validation purpose of newly developed methods (Bailey, 1978; Elishakoff and Johnson, 2005; Korayem et al., 2012; Korayem and Homayooni, 2017; Wei, 2001b, 2001c; Wei et al., 2002a). For example Bailey (1978) developed a direct analytical solution, Elishakoff and Johnson (2005) introduced a closed-form solution procedure and all of them tested their techniques on non-uniform beams. Tan et al. (2016, 2018) demonstrated approaches for the free vibrations of cracked and non-cracked non-uniform beams. Nazemizadeh and Bakhtiari-Nejad (2015) investigated the quality factor of composite micro-/nano-beams using the non-local Euler–Bernoulli beam theory. Similarly, Wei (2001b, 2001c, 2002a) also tested his numerical approach, discrete singular convolution (DSC), on uniform bars and beams in his early studies.

The DSC method (Wei, 1999) promises great potential, especially in handling high frequency structural dynamic problems, because it has inherent global method accuracy and local method flexibility together with requiring a small number of discretisation points to define the geometrical domain. The method is based on the theory of distributions and wavelets. The DSC method solves the governing differential equation of motion of a structure with a high accuracy using a small number of discretisation points. Wei and his co-workers applied the DSC method to several different vibration problems (Wei, 2001a, 2001b, 2001c; Wei et al., 2001, 2002a, 2002b; Zhao et al., 2005). Seçgin and Sarigül (2008) adapted the DSC method to analyse vibration problems of laminated composite plates. Similarly, Civalek et al. (Baltacıoğlu et al., 2010, 2011; Civalek, 2007, 2013; Gürses et al., 2009) have made great contribution to the development of the DSC, especially for the analysis of laminated plates and nano-structures (Civalek, 2017; Gürses et al., 2012; Mercan and Civalek, 2016). Beside that Shokrollahi and Zayeri Baghlani Nejad (2014) investigated the natural frequencies of non-uniform bars having different combinations of cross sections via the DSC. Civalek (2008a, 2008b, 2009; Ersoy et al., 2009, 2010) applied the DSC to more complex non-uniform structures, that is shell and membranes. Seçgin (2013), Seçgin and Kara (2018), Seçgin et al. (2012) also combined the DSC with some uncertainty analysis methods. Monte Carlo simulation by using the DSC for thin isotropic and laminated plates is performed by Seçgin (2013), Seçgin et al. (2012), respectively, to estimate the statistical bounds of vibration via an extreme value model. In another study, Seçgin and Kara (2018) developed a closed-form solution methodology for the analysis of uncertain thin beams. In that study, they showed that the DSC presents a unique advantage because the characteristic matrix obtained via the DSC is independent of physical and mechanical properties of the considered structure. Besides, the DSC method is very efficient for the higher frequencies where the uncertainty dominates because the method uses a relatively small number of discretisation points.

As far as to the authors’ knowledge, there are a limited number of studies on uncertain and non-uniform structures in literature (2004; 2003). Impollonia and Sofi (2003), Falsone and Impollonia (2004) analysed static response of uncertain tapered cantilever beams via a novel response surface approach. Besides, there is no attempt yet to show the performance of the combined DSC and PCE for both uncertain and non-uniform structures. Therefore, this study performs such an attempt to test the methodology and applies the DSC method for more sophisticated structures.

In this study, the DSC and PCE are combined to analyse uncertain non-uniform beams. Non-uniformity is locally provided by a changing cross section and Young’s modulus of the beam along its length. However, the uncertainty is globally simulated by providing normally distributed variabilities in geometrical and material properties such as thickness, specific volume and Young’s modulus. Here, the non-uniform beam is modelled via the DSC, whereas uncertainty is defined by PCE in a closed form. In this context, the implementation procedure of the DSC and PCE is given in detail. Numerical analyses start with the prediction of the natural frequencies of deterministic uniform and non-uniform beams via the DSC. The results are verified by finite element calculations and analytical solutions (if possible). Then, the uncertainty of the beam because of geometrical and material variabilities is considered, and probability distribution functions of natural frequencies are determined by the discrete singular convolution–polynomial chaos expansion (DSC–PCE) combination. Monte Carlo simulations are then performed for validation purpose. It is shown that the DSC–PCE combination is very accurate as well as being efficient regarding computation cost.

2. Mathematical considerations

2.1. Stochastic partial differential equation of non-uniform thin beams

The homogeneous differential equation of motion for bending vibrations of an undamped thin beam (shown in Figure 1) with variable thickness and Young’s modulus along space is expressed as follows (Rao, 2011)

\frac{\partial^{2}}{\partial x^{2}} (E (x) I_{y} (x) \frac{\partial^{2} w (x, t)}{\partial x^{2}}) + ρ A (x) \frac{\partial^{2} w (x, t)}{\partial t^{2}} = 0

(1)

Here, x is the space variable, $ρ$ is the uniform mass density, w is the bending displacement in the z direction and E(x), $h (x)$ , $A (x)$ and $I_{y} (x)$ are the space-dependent Young’s modulus, thickness, cross-sectional area and area moment of inertia about the y axis, respectively. Assume that the beam has constant width b, the thickness and Young’s modulus have the form of $h (x) = h_{0} f_{h} (x)$ and $E (x) = E f_{E} (x)$ , where $h_{0}$ and E are the thickness and Young’s modulus at x = 0, $f_{h} (x)$ and $f_{E} (x)$ represent the space dependency of thickness and Young’s modulus, respectively. In this context, the separable solution of equation (1) can be rewritten for harmonic free vibration response $w (x, t, ϑ) = W (x, ϑ) \exp (j ω_{n} t)$ with stochastic parameters $ϑ = {ϑ_{1}, ϑ_{2}, ϑ_{3}}$ (W is the time-independent stochastic vibration amplitude, $ω_{n}$ is the natural frequency and $ϑ_{1}$ , $ϑ_{2}$ and $ϑ_{3}$ are the uncertain parameters affecting Young’s modulus, thickness and density, respectively, and $j = \sqrt{- 1}$ ) as follows

\begin{array}{l} \frac{1}{12} E (ϑ_{1}) h_{0}^{2} (ϑ_{2}) r (x, ϑ_{3}) [f_{E} f_{h}^{2} W^{(4)} (x, ϑ) + 2 (f_{E}^{(1)} f_{h}^{2} + 3 f_{E} f_{h}^{(1)} f_{h}) W^{(3)} (x, ϑ) \\ + (f_{E}^{(2)} f_{h}^{2} + 6 f_{E}^{(1)} f_{h}^{(1)} f_{h} + 6 f_{E} {(f_{h}^{(1)})}^{2} + 3 f_{E} f_{h}^{(2)} f_{h}) W^{(2)} (x, ϑ)}] - λ_{n} (ϑ) W (x, ϑ) = 0 \end{array}

(2)

Here, $r (x, ϑ_{3}) = 1 / ρ (x, ϑ_{3})$ is defined as the specific volume. The superscripts in parenthesis shows the order of differentiation with respect to space x and $λ_{n} = ω_{n}^{2}$ is the eigenvalue of the structure.

2.2. DSC

In the DSC algorithm, the computational domain of a 1D system is generally divided into three different parts: structural, left and right ghost (auxiliary) domains, as shown in Figure 2. In the computational domain, a function $f (x)$ and its nth order derivative can be approximated via discretised singular kernels of delta type (Wei, 2001c)

f^{(n)} (x_{i}) \approx \sum_{k = - M}^{M} C_{k}^{(n)} f (x_{i + k}), i = 0,1,2, \dots, M

(3)

Here, i represents the index of discretisation points in the structural domain and M stands for the number of ghost points. The function $f (x)$ is determined at uniformly distributed discretisation points. The term $C_{k}^{(n)}$ is the DSC kernel and can be expressed for the regularised Shannon delta kernel as (Wei, 2001c)

C_{k}^{(n)} = {{(\frac{d}{d x})}^{n} (\frac{\sin (α (x - x_{k}))}{α (x - x_{k})} \exp (\frac{- {(x - x_{k})}^{2}}{2 σ^{2}})) |}_{x = x_{i}}

(4)

where

α = π / Δ

is the Nyquist frequency,

σ = r Δ

Δ

represents the distance between the discretisation points and r is the regularisation coefficient which can be selected by trial and error to confirm accuracy and superscript n stands for the order of the derivative. One can follow the following steps to model a beam structure via the DSC:

The beam is discretised by using M + 1 number of structural and M number of ghost points as shown in Figure 2.

Equation (3) is written for each structural point.

Kernel coefficients $C_{k}^{(n)}$ are computed from equation (4).

Proper boundary condition implementation procedure is performed to get rid of the displacement in the ghost domains. Note that, this procedure is given in detail in Ref (Kara and Seçgin, 2019; Seçgin and Sarigül, 2008, 2009) for clamped, simply supported and free boundary conditions.

Next, equation (3) can be written in a matrix form as an eigenvalue problem

(E (ϑ_{1}) h^{2} (ϑ_{2}) r (ϑ_{3}) [D (X)] - λ_{n} (ϑ) [I]) {W (X, ϑ)} = 0

(5)

Here, $[D (X)]$ is the dynamic stiffness matrix, $X = {X_{0}, X_{1}, \dots, X_{M}}^{T}$ represents the vector for the discretised structural points and $W = {W_{0}, W_{1}, ..., W_{M}}^{T}$ is the corresponding bending displacement vector for the structural points. It should be noted that the dynamic stiffness matrix is independent of the amplitude of physical and mechanical properties (E, h₀ and r). Thus, in the DSC, once the matrix is calculated, one may then calculate the dynamic response and/or eigenvalues for different materials and thicknesses. To show the content of the dynamic stiffness matrix, the ith row of the matrix which belongs to (i−1)th structural point is given here as

\begin{array}{l} {D (X)}_{1 \times (M + 1)} = \frac{1}{12} {(f_{E} f_{h}^{2}) |}_{x = X_{i - 1}} {D^{(4)}} + \frac{1}{6} {(f_{E}^{(1)} f_{h}^{2} + 3 f_{E} f_{h}^{(1)} f_{h}) |}_{x = X_{i - 1}} {D^{(3)}} \\ + \frac{1}{12} {(f_{E}^{(2)} f_{h}^{2} + 6 f_{E}^{(1)} f_{h}^{(1)} f_{h} + 6 f_{E} {(f_{h}^{(1)})}^{2} + 3 f_{E} f_{h}^{(2)} f_{h}) |}_{x = X_{i - 1}} {D^{(2)}} \end{array}

(6)

Here,

{D^{(n)}}

is the ith row of the characteristic DSC matrix of differentiation order of n. Note that space-dependent functions and their derivatives are calculated at x = X_i-1.

2.3. PCE

In PCE, any uncertain variable Y can be expressed as the sum of orthogonal polynomials (Ghanem and Spanos, 2003)

\begin{array}{l} Y (ξ) = y_{0} ψ_{0} + \sum_{i_{1} = 1}^{\infty} y_{i_{1}} ψ_{1} (ξ_{i_{1}}) + \sum_{i_{1} = 1}^{\infty} \sum_{i_{2} = 1}^{i_{1}} y_{i_{1} i_{2}} ψ_{2} (ξ_{i_{1}}, ξ_{i_{2}}) + \sum_{i_{1} = 1}^{\infty} \sum_{i_{2} = 1}^{i_{1}} \sum_{i_{3} = 1}^{i_{2}} y_{i_{1} i_{2} i_{3}} ψ_{3} (ξ_{i_{1}}, ξ_{i_{2}}, ξ_{i_{3}}) + \dots \\ = \sum_{i = 0}^{\infty} y_{i} ψ_{i} (ξ) \end{array}

(7)

Here, $ξ = {ξ_{1}, ξ_{2}, \dots, ξ_{i_{n}}}$ is the vector containing stochastic parameters and y_i is deterministic coefficients of orthogonal polynomial basis of $ψ_{i} (\cdot)$ . Because polynomial terms are orthogonal to each other

〈 ψ_{i} \cdot ψ_{j} 〉 = {\begin{matrix} 0, for i \neq j \\ 〈 ψ_{i}^{2} 〉, for i = j \end{matrix}

(8)

Here, $〈〉$ represents the mean value. In numerical calculations, it is meaningful to cease infinite sum of series in equation (7) by a finite value of terms N_PC (Sepahvand et al., 2010)

N_{PC} = \frac{(m + p)!}{m! p!} - 1

(9)

Here, p shows the order of polynomial, m shows the number of uncertain parameter $(ξ_{i})$ to define uncertain variable (Y). In PCE, unknown deterministic coefficients of the polynomials can be determined by using a Galerkin projection which may be described as calculating the average of the considered equation and basis polynomial function multiplication (Sepahvand et al., 2010)

y_{i} = \frac{1}{〈 ψ_{i}^{2} 〉} \int_{Ω_{1}} \dots \int_{Ω_{m}} 〈 Y, ψ_{k} (ξ) 〉 d μ_{m} (ξ_{m}) \dots d μ_{1} (ξ_{1})

(10)

Note that $d μ (ξ) = μ (ξ) d ξ$ is the probability function of the random space $(Ω)$ of uncertain parameter $(ξ)$ and k is an integer starting from zero. Several types of polynomial basis functions $(ψ_{i})$ such as Laguerre polynomial, Jacobi polynomial and Legendre polynomial can be used in PCE, but Hermite polynomials are the most suitable one for the normal distribution (Ghanem and Spanos, 2003)

H_{i} (ξ) = {(- 1)}^{i} \exp (\frac{1}{2} ξ^{T} ξ) \frac{\partial^{i}}{\partial ξ_{i_{1}} \partial ξ_{i_{2}} \dots \partial ξ_{i_{n}}} \exp (- \frac{1}{2} ξ^{T} ξ)

(11)

In PCE, using a first-order polynomial is sufficient for representation of a normal distribution with Hermite polynomials with a single uncertain parameter, so PCE then reduces to KL expansion $(N_{P C} = 1)$ .

In this study, stochastic parameters given in Section 2.1 can be redefined as $ϑ_{i} = {ξ_{i, 1}, ξ_{i, 2}, \dots, ξ_{i, n}}$ . It is assumed that all uncertain variables depend on a single uncertain variable $(ϑ_{i} = ξ_{i, 1} = ξ_{i})$ with a normal distribution. Therefore, uncertain variables can be rewritten as follows

E (ϑ_{1}) = \sum_{i = 0}^{1} E_{i} H_{i} (ξ_{1}) = E_{0} + E_{1} ξ_{1}

(12)

h^{2} (ϑ_{2}) = \sum_{j = 0}^{1} h_{j}^{2} H_{j} (ξ_{2}) = h_{0}^{2} + h_{1}^{2} ξ_{2}

(13)

r (ϑ_{3}) = \sum_{k = 0}^{1} r_{k} H_{k} (ξ_{3}) = r_{0} + r_{1} ξ_{3}

(14)

Note that $E_{0}$ , $h_{0}$ and $r_{0}$ are the mean values of Young’s modulus and thickness at x = 0, and specific volume, respectively. $E_{1}$ , $h_{1}$ and $r_{1}$ are the standard deviations of those uncertain parameters. Equations (12)–(14) can be written in the vector form as

E (ϑ_{1}) = {\begin{matrix} E_{0} & E_{1} \end{matrix}} {\begin{matrix} 1 \\ ξ_{1} \end{matrix}}

(15)

h^{2} (ϑ_{2}) = {\begin{matrix} h_{0}^{2} & h_{1}^{2} \end{matrix}} {\begin{matrix} 1 \\ ξ_{2} \end{matrix}}

(16)

r (ϑ_{1}) = {\begin{matrix} r_{0} & r_{1} \end{matrix}} {\begin{matrix} 1 \\ ξ_{3} \end{matrix}}

(17)

Polynomial basis terms of the corresponding eigenvalues are written by using the tensor product of equations (15)–(17)

ψ_{k} (ξ_{1}, ξ_{2}, ξ_{3}) = {\begin{matrix} 1 \\ ξ_{1} \end{matrix}} \otimes {\begin{matrix} 1 \\ ξ_{2} \end{matrix}} \otimes {\begin{matrix} 1 \\ ξ_{3} \end{matrix}}

(18)

Here,

\otimes

denotes the tensor product. Because the dimension of the vector

ψ_{k} (ξ_{1}, ξ_{2}, ξ_{3})

8 \times 1

, the uncertain eigenvalues are described by the summation starting from 0 to

N_{λ} = 7

as follows

\begin{array}{l} λ_{n} = \sum_{k = 0}^{N_{λ}} λ_{n, k} ψ_{k} (ξ_{1}, ξ_{2}, ξ_{3}) \\ = λ_{n, 0} + λ_{n, 1} ξ_{1} + λ_{n, 3} ξ_{2} + λ_{n, 4} ξ_{3} + λ_{n, 4} ξ_{1} ξ_{2} + λ_{n, 5} ξ_{1} ξ_{3} + λ_{n, 6} ξ_{2} ξ_{3} + λ_{n, 7} ξ_{1} ξ_{2} ξ_{3} \end{array}

(19)

Substituting equations (12)–(14) and (19) into (5) leads to

[(E_{0} + E_{1} ξ_{1}) (h_{0}^{2} + h_{1}^{2} ξ_{2}) (r_{0} + r_{1} ξ_{3}) D (X) - λ_{n} (ξ_{1}, ξ_{2}, ξ_{3}) I] {W (X, ϑ)} = 0

(20)

Multiplying equation (20) with the uth term of the orthogonal basis of $ψ_{k}$ and applying the Galerkin projection yields the final form of the DSC–PCE-based eigenvalue equation of a non-uniform and uncertain thin beam

(\begin{array}{l} (E_{0} + E_{1} ξ_{1}) (h_{0}^{2} + h_{1}^{2} ξ_{2}) (r_{0} + r_{1} ξ_{3}) ψ_{u} (ξ_{1}, ξ_{2}, ξ_{3}) D (X) \\ - λ_{n, u} ψ_{u}^{2} (ξ_{1}, ξ_{2}, ξ_{3}) I \end{array}) {W (X, ϑ)} = 0

(21)

One can now calculate $λ_{n, u}$ by changing u starting from zero up to $N_{λ}$ in equation (21) by using any eigenvalue solver.

3. Numerical studies

In this section, free vibration analyses of a non-uniform and uncertain thin beam with simply supported boundary conditions are performed. Material and geometrical properties of the beam are given in Table 1. The study starts with a deterministic analysis for uniform and non-uniform beams. Then, stochastic analyses are performed for the non-uniform beam with variable material and geometrical properties by using the PCE-based DSC method. Both studies include a verification and convergence investigation. Numerical analyses are performed by a computer with Intel Core i5-3230m 4 × 2.6 GHz, 8 GB Ram, 64 Bit Win 10.

Table 1.

Material and geometrical properties of the beam.

Property (Unit)	Value
Young’ modulus (Pa)	3.25 E + 10
Density (kgm⁻³)	2500
Thickness at x = 0 (m)	0.25
Width (m)	0.3
Length (m)	15

3.1. Deterministic analysis

Here, firstly a uniform beam is considered. The first 10 natural frequencies are obtained via the DSC method for different numbers of structural points and via the finite element method (FEM) for different numbers of one-dimensional beam elements. The results are compared with analytical solutions in Table 2, and an optimum number of structural points for the DSC and number of elements for FEM are determined for further analyses. In this regard, relative differences of the DSC and FEM from analytical solutions are also presented in the table. The thickness is selected as 0.25 m in these calculations. Note that the maximum number of the DSC points and finite elements are decided here to achieve the same analytical results in a 3-digit floating accuracy.

Table 2.

Natural frequencies of the simply supported uniform beam (rad/s).

Mode seq.	Analy.	DSC	DSC	DSC	FEM	FEM	FEM	FEM
Mode seq.	(Rao, 2011)	N = 11	N = 21	N = 31	N = 20	N = 60	N = 120	N = 230
1	11.414	11.411	11.414	11.414	11.414	11.414	11.414	11.414
2	45.656	45.667	45.656	45.656	45.656	45.656	45.656	45.656
3	102.726	102.828	102.726	102.726	102.730	102.726	102.726	102.726
4	182.624	183.501	182.624	182.624	182.644	182.624	182.624	182.624
5	285.350	290.841	285.350	285.350	285.424	285.351	285.350	285.350
6	410.904	434.686	410.904	410.904	411.124	410.907	410.905	410.904
7	559.287	627.888	559.287	559.287	559.835	559.294	559.287	559.287
8	730.497	858.099	730.497	730.497	731.707	730.512	730.498	730.497
9	924.535	1059.665	924.538	924.535	926.960	924.566	924.537	924.535
10	1141.401	—	1141.427	1141.401	1145.906	1141.460	1141.405	1141.401
Error %
1	—	−2.78 × 10⁻²	−3.55 × 10⁻⁵	−4.39 × 10⁻⁷	4.22 × 10⁻⁵	4.92 × 10⁻⁷	2.80 × 10⁻⁶	8.27 × 10⁻⁶
2	—	2.35 × 10⁻²	2.16 × 10⁻⁶	2.71 × 10⁻⁸	6.75 × 10⁻⁴	8.35 × 10⁻⁶	5.94 × 10⁻⁷	1.01 × 10⁻⁶
3	—	9.94 × 10⁻²	−4.05 × 10⁻⁷	−5.27 × 10⁻⁹	3.40 × 10⁻³	4.22 × 10⁻⁵	2.70 × 10⁻⁶	5.67 × 10⁻⁷
4	—	4.80 × 10⁻¹	1.20 × 10⁻⁷	1.62 × 10⁻⁹	1.07 × 10⁻²	1.33 × 10⁻⁴	8.36 × 10⁻⁶	8.20 × 10⁻⁷
5	—	1.92 E + 00	−2.74 × 10⁻⁸	−6.42 × 10⁻¹⁰	2.60 × 10⁻²	3.26 × 10⁻⁴	2.04 × 10⁻⁵	1.61 × 10⁻⁶
6	—	5.79 E + 00	2.57 × 10⁻⁷	2.98 × 10⁻¹⁰	5.34 × 10⁻²	6.75 × 10⁻⁴	4.22 × 10⁻⁵	3.14 × 10⁻⁶
7	—	1.23 × 10¹	3.01 × 10⁻⁶	−1.51 × 10⁻¹⁰	9.81 × 10⁻²	1.25 × 10⁻³	7.83 × 10⁻⁵	5.82 × 10⁻⁶
8	—	1.75 × 10¹	3.26 × 10⁻⁵	8.31 × 10⁻¹¹	1.66 × 10⁻¹	2.13 × 10⁻³	1.33 × 10⁻⁴	9.90 × 10⁻⁶
9	—	1.46 × 10¹	2.97 × 10⁻⁴	−4.72 × 10⁻¹¹	2.62 × 10⁻¹	3.40 × 10⁻³	2.14 × 10⁻⁴	1.59 × 10⁻⁵
10	—	—	2.24 × 10⁻³	2.80 × 10⁻¹¹	3.95 × 10⁻¹	5.18 × 10⁻³	3.26 × 10⁻⁴	2.42 × 10⁻⁵

Note: DSC: discrete singular convolution; FEM: finite element method.

It is seen from Table 2 that the DSC with M + 1 = 31 accurately predicts the same analytical results, whereas the FEM achieves the same accuracy with 230 elements. For this discretisation points and number of elements, maximum relative differences are −4.39 × 10⁻⁷ and 2.42 × 10⁻⁵ for the DSC and FEM, respectively. Therefore, M + 1 = 31 for the DSC and N = 230 for the FEM are selected in the further analyses.

Next, analysis of a non-uniform beam (the beam with variable geometry and material properties along its length) is performed. Again 10 natural frequencies were determined for (i) the beam with non-uniform thickness, (ii) the beam with non-uniform Young’s modulus and (iii) the beam with non-uniform thickness and Young’s modulus. In the analyses, two different forms of the spatial variations are assumed for the thickness, that is $h_{0} f_{h} (x) = h_{0} {(1 + β x)}^{2}$ and $h_{0} f_{h} (x) = h_{0} (1 + β x)$ (non-uniformity parameter of thickness constant $β$ is selected as $- 0.0109$ ) and for Young’s modulus, that is $E f_{E} (x) = E (1 + 0.02 \cos (2 π x / l) + 0.01 \sin (4 π x / l))$ and $E f_{E} (x) = E {(1 - 0.02 x / l)}^{2}$ . Spatial variations of non-uniformities are plotted in Figure 3, and the natural frequency computations are tabulated in Table 3 together with the relative difference %.

Table 3.

Natural frequencies of the simply supported non-uniform beam (rad/s).

Form of thickness and Young’s modulus	$h (x) = h_{0} {(1 + β x)}^{2}$ $E (x) = E$			$h (x) = h_{0}$ $E (x) = E (\begin{array}{l} 1 + 0.02 \cos (2 π x / l) \\ + 0.01 \sin (4 π x / l) \end{array})$			$h (x) = h_{0} {(1 + β x)}^{2}$ $E (x) = E (\begin{array}{l} 1 + 0.02 \cos (2 π x / l) \\ + 0.01 \sin (4 π x / l) \end{array})$			$h (x) = h_{0} (1 + β x)$ $E (x) = E (\begin{array}{l} 1 + 0.02 \cos (2 π x / l) \\ + 0.01 \sin (4 π x / l) \end{array})$			$h (x) = h_{0} {(1 + β x)}^{2}$ $E (x) = E {(1 - 0.02 x / l)}^{2}$			$h (x) = h_{0} (1 + β x)$ $E (x) = E {(1 - 0.02 x / l)}^{2}$
Mode seq.	DSCM + 1 = 31	FEMN = 230	Rel. Diff. %	DSCM + 1 = 31	FEMN = 230	Rel. Diff. %	DSCM + 1 = 31	FEMN = 230	Rel. Diff. %	DSC M + 1 = 31	FEM N = 230	Rel. Diff. %	DSC M + 1 = 31	FEM N = 230	Rel. Diff. %	DSC M + 1 = 31	FEM N = 230	Rel. Diff. %
1	9.573	9.535	0.399	10.861	11.356	−4.354	9.143	9.489	−3.642	10.024	10.402	−3.633	9.458	9.433	0.262	10.399	10.346	0.516
2	38.669	38.353	0.823	45.631	45.652	−0.047	38.444	38.320	0.323	41.867	41.844	0.055	38.285	37.959	0.859	41.581	41.438	0.345
3	86.644	86.254	0.453	102.727	102.719	0.008	86.774	86.259	0.597	94.406	94.175	0.245	85.774	85.367	0.477	93.386	93.219	0.179
4	153.716	153.292	0.277	182.618	182.611	0.004	153.786	153.283	0.328	167.607	167.395	0.126	152.158	151.713	0.293	165.883	165.705	0.107
5	239.916	239.474	0.184	285.337	285.329	0.003	239.960	239.453	0.212	261.745	261.536	0.080	237.470	237.007	0.195	259.082	258.899	0.070
6	345.253	344.805	0.130	410.883	410.873	0.002	345.277	344.768	0.148	376.803	376.596	0.055	341.722	341.250	0.138	372.986	372.802	0.049
7	469.733	469.283	0.096	559.254	559.244	0.002	469.736	469.230	0.108	512.780	512.576	0.040	464.917	464.444	0.102	507.597	507.413	0.036
8	613.357	612.911	0.073	730.452	730.441	0.001	613.339	612.838	0.082	669.677	669.476	0.030	607.059	606.589	0.077	662.915	662.734	0.027
9	776.127	775.688	0.057	924.475	924.464	0.001	776.085	775.593	0.063	847.492	847.296	0.023	768.149	767.686	0.060	838.941	838.763	0.021
10	958.043	957.614	0.045	1141.325	1141.313	0.001	957.977	957.496	0.050	1046.226	1046.035	0.018	948.187	947.735	0.048	1035.674	1035.502	0.017

Note: DSC: discrete singular convolution; FEM: finite element method.

As inferred from Table 3, for different non-uniform structures, the DSC and FEM results are not exactly the same but are very consistent with each other. Average computational time is 2.87 s and 0.36 s for the FEM and the DSC, respectively. Besides, the consistency of the methods reduces for the more complex forms of Young’s modulus and thickness. Considering both the relative difference and the computational time, the DSC even with a very small number of discretisation points may represent non-uniform structures well and thus reliably used in further complex structures.

3.2. Stochastic analysis

In this part of the study, the effects of the geometrical and material variabilities on the natural frequencies are investigated using the DSC and PC expansion (DSC–PCE). In this manner, the beam thickness, Young’s Modulus and specific volume vary globally with a normal distribution, that is they are uncertain variables but their distributions are known. In the analyses, the standard deviations of the uncertain variables are 5% of their mean values given in Table 1. Seven uncertainty cases are considered for the uniform and non-uniform beams as shown in Table 4. The analysis starts by solving equation (21) eight times for the determination of each coefficient given in equation (19). Then, the distribution functions of the natural frequencies are determined using these coefficients with 10,000 samples noting that

ω_{n} = \sqrt{λ_{n}}

. In Figure 4, probability distribution functions of the non-dimensional natural frequency ratio,

(ω_{i} - E [ω_{i}]) / ω_{i}^{uniform})

), are compared with those obtained by Monte Carlo (MC) simulations performed on the DSC models. Here,

E [ω_{i}]

denotes the mean value of the ith mode. Because each of the probability distribution functions of the non-dimensional natural frequency ratio is the same for all i = 1, 2, …10, the probability distribution functions are represented here by a single figure. Note that 10,000 samples of uncertain variables are also used in MC simulations.

Table 4.

Uncertainty cases.

Non-uniformity	Form of the thickness, $h (x)$	Form of Young’s modulus, $E (x)$	Uncertain parameters: $E (ϑ_{1})$ , $h (ϑ_{2})$ , $r (ϑ_{3})$
Uniform	$h_{0}$	E	Case 1
Non-uniform	$h_{0} {(1 + β x)}^{2}$	E	Case 2
	$h_{0}$	$E (\begin{array}{l} 1 + 0.02 \cos (2 π x / l) \\ + 0.01 \sin (4 π x / l) \end{array})$	Case 3
	$h_{0} {(1 + β x)}^{2}$	$E (\begin{array}{l} 1 + 0.02 \cos (2 π x / l) \\ + 0.01 \sin (4 π x / l) \end{array})$	Case 4
	$h_{0} (1 + β x)$	$E (\begin{array}{l} 1 + 0.02 \cos (2 π x / l) \\ + 0.01 \sin (4 π x / l) \end{array})$	Case 5
	$h_{0} {(1 + β x)}^{2}$	$E {(1 - 0.02 x / l)}^{2}$	Case 6
	$h_{0} (1 + β x)$	$E {(1 - 0.02 x / l)}^{2}$	Case 7

It is seen from Figure 4 that the predictions of the DSC–PCE are quite consistent with those of MC simulations for all cases. Besides, average computation time for the DSC–PCE is 3.72 s whereas it is 8.52 for the Monte Carlo simulation. The results show that the DSC–PCE method is very accurate and efficient and therefore can be reliably used for uncertainty analysis of non-uniform structures.

4. Conclusions

In this study, a combination of DSC and PCE is introduced for non-uniform beams having geometrical and material variabilities. The DSC method is applied to model the non-uniform beam having local changes in Young’s Modulus and thickness along its length. The PCE is used to handle variabilities simulating uncertainty in a global manner. The DSC–PCE method is verified for uniform and non-uniform beams using the FEM and Monte Carlo simulations. It is shown that the DSC method is very straightforward in modelling non-uniformity along the spatial domain, and it is very accurate even with a relatively small number of discretisation points compared to the FEM. It is also shown that the combined DSC–PCE methodology is very efficient in computational cost and it quantifies quite well the uncertainty because of the geometrical and material parameters. The study promises that the present methodology is worth further development to apply it to more complex systems, especially for mid- and high-frequency analysis.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by “The Scientific and Technological Research Council of Turkey, TUBITAK” in the frame of TUBITAK 2219 programme.

ORCID iDs

Abdullah Seçgin

Murat Kara

Neil S Ferguson

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