A close-form solution applied to the free vibration of the Euler–Bernoulli beam with edge cracks

Abstract

This paper investigates the free vibration of a homogeneous Euler–Bernoulli beam with multiple transverse cracks under non-symmetric boundary conditions. The differential equation is formulated by introducing Dirac’s delta function into the uniform flexural stiffness, and the close-form solution of mode shapes is then derived by applying the Laplace transform technique. The proposed method is validated against existing experimental method for damaged cantilever beams. With the validated method, a parametric study is performed to study the effect of crack numbers, damage parameters and crack locations on the natural frequencies and mode shapes for three non-symmetric boundary conditions (pinned–clamped, clamped–free shear and pinned–free shear).

Appendices

Appendix 1: Schwarz’s distribution theory

Theorem 1

Assume that a function f(x) is n times continuously differentiable. Then, we have:

$$\begin{aligned} f(x)\delta ^{(n)}(x-x_0 )= & {} (-1)^{n}f^{(n)}(x_0 )\delta (x-x_0 )+(-1)^{n-1}nf^{(n-1)}(x_0 )\delta ^{(1)}(x-x_0 )\nonumber \\&+\,(-1)^{n-2}\frac{n(n-1)}{2!}f^{(n-2)}(x_0 )\delta ^{(2)}(x-x_0 )+\cdots +f(x_0 )\delta ^{(n)}(x-x_0 ) \end{aligned}$$

(26)

Corollary 1

$$\begin{aligned} \left[ {f(x)H(x-x_0 )} \right] ^{(n)}= & {} f^{(n)}(x)H(x-x_0 )+f^{(n-1)}(x_0 )\delta (x-x_0 )+f^{(n-2)}(x_0 )\delta ^{(1)}(x-x_0 )+\cdots \nonumber \\&+\,f(x_0 )\delta ^{(n-1)}(x-x_0 ) \end{aligned}$$

(27)

Definition 1

The Laplace transform of the Heaviside function, the delta function and their distributional derivatives can be defined as:

$$\begin{aligned}&L\left\{ {H(x-x_0 )} \right\} =\frac{1}{s}\hbox {e}^{-sx_0 } \end{aligned}$$

(28)

$$\begin{aligned}&L\left\{ {\delta (x-x_0 )} \right\} =\hbox {e}^{-sx_0 } \end{aligned}$$

(29)

$$\begin{aligned}&L\left\{ {\delta ^{(k)}(x-x_0 )} \right\} =s^{k}\hbox {e}^{-sx_0 } \end{aligned}$$

(30)

Theorem 2

Let $f^{(k)}(x)$ be continuous with respect to all x, then

$$\begin{aligned} L\left\{ {f^{(k)}(x)} \right\} =s^{k}F(s)-f(0)s^{k-1}-f^{(1)}(0)s^{k-2}-\cdot \cdot \cdot -f^{(k-1)}(0) \end{aligned}$$

(31)

Appendix 2: Relationship between the damage parameter and the equivalent rotational stiffness

The relation between damage parameter $\lambda _i$ and the equivalent rotational stiffness $K_{\mathrm{eq},i}$ is given by

$$\begin{aligned} K_{\mathrm{eq},i} =\frac{E_0 I_0 }{\lambda _i } \end{aligned}$$

(32)

In the case of a beam of a rectangular cross section of width b and height h, the equivalent rotational stiffness $K_{\mathrm{eq},i}$ can be obtained by using the strain energy density function [29]:

$$\begin{aligned} K_{\mathrm{eq},i} =\frac{E_0 I_0 }{h}\frac{1}{C(\beta )} \end{aligned}$$

(33)

where $\beta =\frac{d}{h}$ is the ratio of the crack depth d and the cross section height h, and $C(\beta )$ is the dimensionless local compliance which for a single-sided open crack can be expressed in the following form:

$$\begin{aligned} C(\beta )=a_0 \sum _{n=1}^{10} {a_n \beta ^{n}} \end{aligned}$$

(34)

As the strain energy density function should be confirmed through the experiment, five polynomial formulas are proposed due to different measure results of the strain energy density concentration in the vicinity of the crack tip. Variation of formulas are reflected in the diversity of the coefficients $a_0$ and $a_n$, which can be seen in Table 3.

Table 3 Variation of the coefficients $a_0$ and $a_n$ in five polynomial formulas

Appendix 3: Mechanism of modal curvature ${\phi }''(x)$ in characterizing damage

According to the close-form solution presented in Eq. (10), modal curvature ${\phi }''(x)$can be obtained as follows:

$$\begin{aligned} {\phi }''(x)= & {} \frac{S_1 (\alpha x)}{\alpha }{\phi }'''(0)+S_0 (\alpha x){\phi }''(0)+\alpha S_3 (\alpha x){\phi }'(0)+\alpha ^{2}S_2 (\alpha x)\phi (0) \nonumber \\&+\,\alpha \sum _{i=1}^n {\gamma _i {\phi }''(x_{0i} )} S_3 (\alpha (x-x_{0i} ))H(x-x_{0i} )+\sum _{i=1}^n {\gamma _i {\phi }''(x_{0i} )} \delta (x-x_{0i} ) \end{aligned}$$

(35)

Similar to the proof of $M(x_{0j} )$, the modal curvature ${\phi }''(x_{0j} )$ at the cracked cross sections $x_{oj}$ can be derived by utilizing the integral property of Dirac’s delta:

$$\begin{aligned} {\phi }''(x_{0j} )=\int _{-\infty }^{+\infty } {{\phi }''(x)\delta (x-x_{0j} )} \hbox {d}x \end{aligned}$$

(36)

Then substituting Eq. (35) into Eq. (36), we can obtain the expression of modal curvature as follows:

$$\begin{aligned} {\phi }''(x_{0j} )=\frac{1}{1-A\gamma _j }\left[ {\begin{array}{l} \frac{S_1 (\alpha x_{0j} )}{\alpha }{\phi }'''(0)+S_0 (\alpha x_{0j} ){\phi }''(0)+\alpha S_3 (\alpha x_{0j} ){\phi }'(0) \\ +\,\alpha ^{2}S_2 (\alpha x_{0j} )\phi (0)+\alpha \sum _{i=1}^n {\gamma _i {\phi }''(x_{0i} )} S_3 (\alpha (x_{0j} -x_{0i} ))H(x_{0j} -x_{0i} ) \\ \end{array}} \right] \end{aligned}$$

(37)

Due to the non-negativity of $A\gamma _j$ and $1-A\gamma _j$, the value of $\frac{1}{1-A\gamma _j }$ ranges from 1 to $\infty$, which can give rise to an abrupt change when the spatial variable x is incredibly close to the crack position ($x_{0j} )$, thus can significantly magnify damage information. And the magnification factor is higher when the damage is more severe. Thus, the modal curvature brings the damage undetected in the mode shape into prominence.