Novel method for analyzing the behavior of composite beams with non-smooth interfaces

Abstract

An almost exact solution is derived for the forced vibration of a composite beam with periodically varying non-smooth interface through a moderate weak-form formulation. The material property of a non-uniform beam is characterized by its flexural rigidity function R(x). In the novel method, R(x) is relaxed to be an integrable function rather than a ${\mathcal{C}}^2$ smooth function in the usual approach. The R(x)-orthogonal bases in the linear span of all boundary functions are derived such that the second-order derivatives of the bases elements are orthogonal with respect to the weight function R(x). When the deflection of the beam is expressed in terms of the bases, the expansion coefficients can be determined exactly in closed form owing to the R(x)-orthogonality of the bases. The solution obtained is almost exact, since its accuracy can be up to the order $10^{-15}$. This powerful method is used to analyze the forced vibration behavior of composite beams with three different periodic interfaces.

Appendices

Appendix 1

In this appendix we derive Eq. (11). Multiplying Eq. (3) by $\phi (x)$ and integrating the resultant from $x=0$ to $x=L$, yield

$$\begin{aligned}&\int _0^L [R(x)y''(x)]''\phi _j(x)dx\nonumber \\&=\int _0^L q(x)\phi _j(x)dx. \end{aligned}$$

(24)

Integrating the integral term in the left-hand side by parts two times and using Eqs. (8) and (12), we have

$$\begin{aligned}&\int _0^L [R(x)y''(x)]''\phi _j(x)dx\nonumber \\&\quad =\left. [R(x)y''(x)]'\phi _j(x)\right| _0^L\nonumber \\&\qquad -\,\int _0^L [R(x)y''(x)]'\phi _j'(x)dx \nonumber \\&\quad =-\int _0^L [R(x)y''(x)]'\phi _j'(x)dx\,{\text {due}}\;{\text {to}}~~\nonumber \\&\qquad \phi _j(0)=\phi _j(L)=0 \nonumber \\&\quad =-\left. R(x)y''(x)\phi _j'(x)\right| _0^L\nonumber \\&\qquad +\,\int _0^L R(x)y''(x)\phi _j''(x)dx \nonumber \\&\quad =\int _0^L R(x)y''(x)\phi _j''(x)dx\nonumber \\&\qquad {\text {due}}\;{\text {to}}\,y''(0)=y''(L)=0. \end{aligned}$$

(25)

Inserting it into Eq. (24) we can derive Eq. (11) for the simple beam equation.

For the clamped–clamped beam we have

$$\begin{aligned}&\int _0^L [R(x)y''(x)]''\phi _j(x)dx\nonumber \\&\quad =\left. [R(x)y''(x)]'\phi _j(x)\right| _0^L\nonumber \\&\qquad -\,\int _0^L [R(x)y''(x)]'\phi _j'(x)dx \nonumber \\&\quad =-\int _0^L [R(x)y''(x)]'\phi _j'(x)dx~~\nonumber \\&\qquad {\text {due}}\;{\text {to}}\;\phi _j(0)=\phi _j(L)=0 \nonumber \\&\quad =-\left. R(x)y''(x)\phi _j'(x)\right| _0^L\nonumber \\&\qquad +\,\int _0^L R(x)y''(x)\phi _j''(x)dx \nonumber \\&\quad =\int _0^L R(x)y''(x)\phi _j''(x)dx~~\nonumber \\&\qquad {\text {due}}\;{\text {to}}\;\phi _j'(0)=\phi _j'(L)=0. \end{aligned}$$

(26)

Inserting it into Eq. (24) we can derive Eq. (11) for the clamped–clamped beam equation.

Similarly, for the cantilevered beam we have

$$\begin{aligned}&\int _0^L [R(x)y''(x)]''\phi _j(x)dx\nonumber \\&\quad =\left. [R(x)y''(x)]'\phi _j(x)\right| _0^L\nonumber \\&\qquad -\,\int _0^L [R(x)y''(x)]'\phi _j'(x)dx \nonumber \\&\quad =-\int _0^L [R(x)y''(x)]'\phi _j'(x)dx\nonumber \\&\qquad {\text {due}}\;{\text {to}}\;\phi _j(0)=0,~ y''(L)=0,~y'''(L)=0 \nonumber \\&\quad =-\left. R(x)y''(x)\phi _j'(x)\right| _0^L\nonumber \\&\qquad +\,\int _0^L R(x)y''(x)\phi _j''(x)dx \nonumber \\&\quad =\int _0^L R(x)y''(x)\phi _j''(x)dx~~\nonumber \\&\qquad {\text {due}}\;{\text {to}}\;\phi _j'(0)=0,~y''(L)=0. \end{aligned}$$

(27)

Inserting it into Eq. (24) we can derive Eq. (11) for the cantilevered beam equation.

Appendix 2

In this appendix we derive Eqs. (15)–(17). Let

$$\begin{aligned} B_j(x)=x^{j+3}+ax^{j+2}+bx^{j+1},~~j \ge 2, \end{aligned}$$

(28)

be a boundary function for the simple beam, where a and b are to be determined such that $B_j(x)$ satisfies all the boundary conditions in Eq. (8). It is obvious that $B_j(x)$ satisfies $B_j(0)=0$ and $B_j''(0)=0$ due to $j\ge 2$. To satisfy $B_j(L)=0$ and $B_j''(L)=0$ we can derive

$$\begin{aligned}&L^{j+3}+aL^{j+2}+bL^{j+1}=0, \end{aligned}$$

(29)

$$\begin{aligned}&(j+3)(j+2)L^{j+1}+a(j+2)(j+1)L^j\nonumber \\&\quad +\,b(j+1)jL^{j-1}=0, \end{aligned}$$

(30)

which can be solved to

$$\begin{aligned} a=-\frac{2j+3}{j+1}L,~~b=\frac{j+2}{j+1}L^2. \end{aligned}$$

(31)

Inserting a and b into Eq. (28) we can derive Eq. (15).

Let

$$\begin{aligned} B_j(x)=x^{j+3}+ax^{j+2}+bx^{j+1},~~j \ge 1, \end{aligned}$$

(32)

be a boundary function for the clamped–clamped beam. It is obvious that $B_j(x)$ satisfies $B_j(0)=0$ and $B_j'(0)=0$ due to $j\ge 1$. To satisfy $B_j(L)=0$ and $B_j'(L)=0$ we can derive

$$\begin{aligned}&L^{j+3}+aL^{j+2}+bL^{j+1}=0, \end{aligned}$$

(33)

$$\begin{aligned}&(j+3)L^{j+2}+a(j+2)L^{j+1}\nonumber \\&\quad +\,b(j+1)L^j=0, \end{aligned}$$

(34)

which can be solved to

$$\begin{aligned} a=-2L,~~b=L^2. \end{aligned}$$

(35)

Inserting a and b into Eq. (32) we can derive Eq. (16).

Finally, let

$$\begin{aligned} B_j(x)=x^{j+3}+ax^{j+2}+bx^{j+1},~~j \ge 1, \end{aligned}$$

(36)

be a boundary function for the cantilevered beam. It is obvious that $B_j(x)$ satisfies $B_j(0)=0$ and $B_j'(0)=0$ due to $j\ge 1$. To satisfy $B_j''(L)=0$ and $B_j'''(L)=0$ we can derive

$$\begin{aligned}&(j+3)(j+2)L^{j+1}+a(j+2)(j+1)L^j\nonumber \\&\quad +\,b(j+1)jL^{j-1}=0, \end{aligned}$$

(37)

$$\begin{aligned}&(j+3)(j+2)(j+1)L^j+a(j+2)(j+1)jL^{j-1}\nonumber \\&\quad +\,b(j+1)j(j-1)L^{j-2}=0, \end{aligned}$$

(38)

which can be solved to

$$\begin{aligned} a & = -\frac{2j+3}{j+1}L,\nonumber \\ b & = \frac{(j+2)(j+3)}{j(j+1)}L^2. \end{aligned}$$

(39)

Inserting a and b into Eq. (36) we can derive Eq. (17).