Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section

Abstract

In this manuscript, static and free vibration responses on Euler–Bernoulli beams with a Koch snowflake cross-section are studied. By applying the finite element method, the transversal displacement in static load condition, natural frequencies, and vibration modes are solved and validated using Matlab. For each case presented, the transversal displacement and natural frequency are analyzed as a Hausdorff dimension function. It is found that the maximum displacement increases as the Hausdorff dimension increases, with the relationship $y_{max}=k^{0.79ln{d}_{\mathcal{H}}+0.37}$, being k the iteration number of pre-fractal. The natural frequencies increase as $\omega \sim M^{2.51}$, whereas the bending stiffness is expressed as $EI=1165.4ln(d_{\mathcal{H}}+k)$. Numerical examples are given in order to discuss the mechanical implications.

1. Introduction

The Euler–Bernoulli bending beam theory is a widely used framework in engineering science, and its applications range from construction, textile, automobiles, ships, and aircraft industries to antenna design, nano-electromechanical systems, and various items of use in daily life.

This theory is based on three assumptions [1]: (i) there are not deformations in the plane of the cross-section because the cross-section is infinitely rigid in its own plane, (ii) the cross-section of a beam remains flat after deformation occurs, and (iii) the cross-section remains normal to the deformed axis of the beam. Its geometrical interpretation can be seen in detail in many references such as [1,2,3].

The mechanical behavior of classical beams has been described using numerical [4,5] and theoretical methods [6,7], where their static and dynamic responses are studied with ordinary calculus. Accordingly, structural studies on beams with a variable cross-section [8], and in beams that have discontinuities [9,10] or that are cracked [2,11], have been reported. Additionally, it has been demonstrated that almost always the natural and man-made structures that have a complex architectural design exhibit statistical-scale invariance over many length scales [12,13,14,15].

In this context, Euler–Bernoulli beams with complex geometry have been solved using fractional calculus [16,17,18]. Meanwhile, solutions of the Euler–Bernoulli self-similar beam equation using fractal continuum calculus [19,20], a product-like fractal measure [21], and fractal calculus [22] were recently introduced. Furthermore, the framework of the equivalent continuum was applied to obtain analytical solutions for vibrating fractal composite rods and beams, which use ordinary calculus on pre-fractals [23].

Based on the Koch snowflake in [24], a solidification process was developed in a thermal storage system with a Koch snowflake inner cross section, and it was found that higher fractal iteration leads to the acceleration of the solidification process due to enhanced thermal penetration depth. Other authors [25] proposed a Koch Snowflake shaped microstrip terahertz antenna for superwideband spatial diversity application and discovered that interport isolation for the MIMO antenna is less than −23 dB across the wideband of operation, whereas the values of the MIMO performance parameters are best. Rostami et al. [26] analyzed a Timoshenko beam, also with a Koch snowflake cross-section, using different boundary conditions for variable properties, and it was found out that beams with fractal geometry have the highest natural frequency compared with the ones with Euclidean geometry. It is worth highlighting that natural frequencies (eigenvalues) and the shape mode (eigenfunctions) problem in a fractal region defined by the Koch snowflake geometry were numerically solved in [27,28].

In these fractal models, however, there is not a relationship that describes the influence of fractal parameters as the Hausdorff dimension and the iteration numbers of the Koch snowflake. Accordingly, the goal of this work is to determinate the static and free vibration responses as the Hausdorff dimension function using a generalized Koch snowflake in the range $1<d_{\mathcal{H}}<2$ on Euler–Bernoulli beams with a fractal cross section by applying ordinary calculus.

The rest of the paper is outlined as follows: In Section 2, we review the mathematical aspects, whereas Section 3 is devoted to developing the main subject of this work. Section 4 presents a discussion of physical implications, and Section 5 concludes the paper.

2. Problem Statement and Formulation

In this section, we review and define the basic tools needed in the paper.

2.1. Dynamic Beam Equation

The governing differential equation for a vibration of a Euler–Bernoulli beam within the domain $x\in [0,1]$ is defined as

$$\begin{array}{c}\hfill \frac{{\partial }^2}{\partial x^2}\left[EI\left(x\right)\frac{{\partial }^{2}v(x,t)}{\partial x^2}\right]+\rho A\frac{{\partial }^{2}v(x,t)}{\partial t^2}=p(x,t)\end{array}$$

(1)

where $EI\left(x\right)$ is the bending stiffness, $\rho A$ is the mass per unit length, and $v(x,t)$ is the transversal displacement. Note that the moment-curvature relation is given by $M=EI\left({\partial }^{2}v/\partial x^2\right)$, called the bending moment, so $V=EI\left({\partial }^{3}v/\partial x^3\right)$ is the transverse shear force and $\partial V/\partial x=p(x,t)$ is the transversal load per unit length. For the free vibration analysis, $p(x,t)=0$, whereas for the static bending analysis the second term in Equation (1) is neglected.

2.2. Finite Element Formulation

Applying Galerkin’s method [29] to Equation (1), the residual equation has the following form:

$$\begin{array}{c}\hfill {\int }_0^L{W}_i\left[\frac{{\partial }^2}{\partial x^2}\left(EI\left(x\right)\frac{{\partial }^{2}v}{\partial x^2}\right)+\rho A\frac{{\partial }^{2}v}{\partial t^2}\right]dx=0,\end{array}$$

(2)

where $W_i=W_i\left(x\right)$ is the weight function. For the discretization of the domain, the beam is partitioned in elements with 4 degrees of freedom, defined as shown in Figure 1, and using the following $N_i$ cubic shape functions:

$$N_1=1-\frac{3x^2}{L^2}+\frac{2x^3}{L^3}$$

$$N_2=x-\frac{2x^2}{L}+\frac{x^3}{L^2}$$

$$N_3=\frac{3x^2}{L^2}-\frac{2x^3}{L^3}$$

$$N_4=-\frac{x^2}{L}+\frac{x^3}{L^2}$$

Then, Equation (2) can be rewritten in discrete form for n elements using the same shape functions for the weight functions, $W_i=N_i$ and $v=\left[N\right]\left\{u\right\}$; the residual takes the form since there is not applied load ($p(x,t)=0$), and it is a free vibration problem:

$$\begin{array}{c}\hfill \sum _{i=1}^n{\int }_0^{\ell }{\left[N\right]}^T\left[EI\frac{{\partial }^4\left(\left[N\right]\left\{u\right\}\right)}{\partial x^4}+\rho A\frac{{\partial }^4\left(\left[N\right]\left\{u\right\}\right)}{\partial t^4}\right]dx=0,\end{array}$$

(3)

where $\left\{u\right\}={[v_i,{\theta }_i,v_j,{\theta }_j]}^T$ is the vector of nodal values and l is the length of element. Integrating by parts the first integrand of Equation (3)

$$\begin{array}{c}\hfill \sum _{i=1}^n{\int }_0^{\ell }{\left[B\right]}^{T}EI\left[B\right]\left\{u\right\}dx+{\left[{\left[N\right]}^{T}EI\frac{{\partial }^3\left(\left[N\right]\left\{u\right\}\right)}{\partial x^3}-\frac{\partial {\left[N\right]}^{T}EI\left[B\right]\left\{u\right\}}{\partial x}\right]}_0^{\ell }+{\int }_0^{\ell }{\left[N\right]}^T\rho A\left(\left[N\right]\frac{{\partial }^2\left\{u\right\}}{\partial t^2}\right)dx=0,\end{array}$$

(4)

where $\left[B\right]={\partial }^2\left[N\right]/\partial x^2$, then Equation (4) is as follows:

$$\begin{array}{c}\hfill \sum _{i=1}^n{\int }_0^{\ell }\left({\left[B\right]}^{T}EI\left[B\right]\left\{u\right\}+\rho A{\left[N\right]}^T\left[N\right]\left\{\ddot{u}\right\}\right)dx+{\left[{\left[N\right]}^{T}V-\frac{\partial {\left[N\right]}^{T}M}{\partial x}\right]}_0^{\ell }=0,\end{array}$$

(5)

where $\left\{\ddot{u}\right\}={\partial }^2\left\{u\right\}/\partial t^2$ and V is the transversal shear load, whereas M is the bending moment. The defining element stiffness matrix $K^e={\int }_0^{\ell }{\left[B\right]}^{T}EI\left[B\right]dx$, the element mass matrix $M^e={\int }_0^{\ell }\rho A{\left[N\right]}^T\left[N\right]dx$, and the element load vector $F^e={[{\left[N\right]}^{T}V-\partial \left({\left[N\right]}^{T}M\right)/\partial x]}_0^{\ell }$. Assembling the matrices and load vectors for the n elements of the beam, we have the following dynamic equation:

$$\begin{array}{c}\hfill \left[K\right]\left\{U\right\}+\left[M\right]\left\{\ddot{U}\right\}=\left\{F\right\}.\end{array}$$

(6)

Note that for the free vibration analysis load vector, $F=0$. So, the eigenvalue problem is defined as follows:

$$\begin{array}{c}\hfill \left(\left[K\right]-{\omega }^2\left[M\right]\right)\left\{x\right\}=0\end{array}$$

(7)

where $\omega =\left({\beta }_{n}L\right){(EI/\rho A{L}^4)}^{1/2}$ is the angular natural frequency in radians per second and $\left\{x\right\}$ is the mode shape.

2.3. Koch Snowflakes-like cross Section

In previous publications, beams with fractal cross section have been studied, specifically with Sierpinski’s carpet-like, where the beam also is a fractal [20]. However, this manuscript is devoted to analyze the response of Euclidean beams with a fractal cross section, whose fractal geometry is Koch snowflake-like [26]. It is well-known that the Hausdorff dimension of the classical Koch snowflake is given by $d_{\mathcal{H}}=2log2/log3$. In order to obtain a series of analyses including all possibles areas of a cross section with Hausdorff dimensions $1<d_{\mathcal{H}}<2$, a generic Koch snowflake with a Hausdorff dimension is defined as [30]:

$$\begin{array}{c}\hfill d_{\mathcal{H}}=\frac{2log2}{log\Omega }\end{array}$$

(8)

where $\Omega =2\left[1+sin(\gamma /2)\right]$ is a scale factor, and $\gamma$ is the indentation angle. If $\gamma ={60}^{\circ }$ and the initiator is a line segment that corresponds to the classical Koch curve, but if the initiator is an equilateral triangle, the fractal curve is called a Koch snowflake. In Figure 2 is sketched, the cross sections are used.

3. Numerical Examples

This section is devoted to application examples of concepts reviewed in the behold section.

3.1. Bending of a Cantilever Beam

The bending analysis has been carried out on a cantilever beam Figure 1a subjected to a concentrated stationary load applied at its free end. The length of the beams for all cases was $L=1$ m, and the cross section is the Koch snowflake with the side of the equilateral triangle of the initiator $a=0.02$ m. The indentation angles of the Koch snowflake cross section used in the analysis are $\gamma ={20}^{\circ },{40}^{\circ },{60}^{\circ },{80}^{\circ },{90}^{\circ },{100}^{\circ }$, whose Hausdorff dimension can be seen in Figure 2. The material is AISI 1020 steel with elastic modulus $E=200$ GPa, Poisson’s ratio equal to $0.29$, and mass density $\rho =7900$ kg/m${}^3$. The concentrated applied load is $F=50N$.

The boundary conditions for the fixed support at $x=0$ are null displacement $v\left(0\right)=0$ and null rotation $\partial v\left(0\right)/\partial x=0$.

3.2. Modal Analysis of a Cantilever Beam

The boundary conditions for the fixed support at $x=0$ are null displacement $v(0,t)=0$ and null rotation $\partial v(0,t)/\partial x=0$, and there is not applied load, $F=0$.

4. Results and Discussion

In the case of a beam with a fractal cross section such as the Koch snowflake, the moment of inertia with respect to the bending axis presents an increase in its value with each iteration. Figure 3a shows the transversal displacement for a cross section with Hausdorff dimension $d_{\mathcal{H}}=1.26$ for iterations $k=1,2,3,4$. Moreover, it can be seen that the deflection decreases as the flexural stiffness $EI$ increases with each iteration as

$$\begin{array}{c}\hfill EI\sim 1165.4ln(d_{\mathcal{H}}+k),\end{array}$$

(9)

which was obtained by non-linear regression, whereas the analytical maximum displacement is described by (Figure 3b)

$$\begin{array}{c}\hfill y_{max}\sim k^{0.79ln\left(d_{\mathcal{H}}\right)+0.37},\end{array}$$

(10)

The free vibration analysis of the cantilever beam has been carried out by obtaining the first four natural frequencies and their respective modes of vibration, as shown in (Figure 4a). The behavior of the natural frequency ($\omega$) with respect to each iteration of the cross section of the Koch snowflake is shown in (Figure 4b). A mathematical relationship has been obtained, and it approximates the natural frequency values for each iteration of the cross section as Hausdorff dimension function:

$$\begin{array}{c}\hfill {\omega }_n={\left({\beta }_{n}L\right)}^2{\left(\frac{1165.4ln(d_{\mathcal{H}}+k)}{\rho A{L}^4}\right)}^{1/2},\end{array}$$

(11)

where ${\beta }_{n}L=(2n-1)\pi /2$ and the numerator of second fraction is the flexural stiffness given in Equation (9). Specifically, $\omega$ can be computed using $\omega \sim M^{2.515}$ (where M is the mode shape) once that natural frequencies have been determined by Equation (11) (see Figure 4b).

These results can be used to evaluate variations in the design of soft robot arms modelling such as the Euler–Bernoulli beam [5] in order to know quantitatively and methodically address questions of soft-arm capability with other materials, as well as to study the effects of snowflake hierarchies of tunable locally resonant metamaterials [31]. Several engineering applications of snowflake geometry have been reported as, for example, wave propagation in snowflakes plates [32], antenna design for Wi-Fi applications [33,34], the thermal transfer analysis of tubes with extended surface with fractal design [35], and experimental studies on divertor with snowflakes-like configurations in nuclear reactor was carried out in [36].

5. Conclusions

A generalized study of static and dynamic response of the Euler–Bernoulli beam with complex cross section type Koch’s snowflake fractal was carried out. The governing equations of the beam were solved by applying the finite element method. So, the transversal displacement, rotation, mode shapes, and frequencies on the beam were obtained using Matlab. The effects of the fractal parameters on the mechanical properties of beams were found, and the conclusions and outlooks are as follows:

i

The cross-sectional area and its inertia change when the iteration does. This implies that when the iteration $k_i$ increases, the flexural rigidity $EI$ increases as the cross-section Hausdorff dimension decreases see Equation (9).

ii

As the iteration of the cross-section increases, the maximum displacement decreases. It can be described by the power law expression presented in Equation (10) and depicted in Figure 3.

iii

The natural frequency grows with the analytical expression given in Equation (11) for all Hausdorff dimensions of Koch’s snowflake, as is shown in Figure 4.

iv

The beam solution with the classical cross-section is obtained for the first iteration of the Koch’s snowflake fractal.

v

In an upcoming report, the dynamic responses of fractal beams with the Euler–Bernoulli theory will be given using the fractal calculus framework [13,19,20].