Free vibrations of rotationally restrained nonhomogeneous circular beams by means of the Green function

Abstract

A one-dimensional linear model is developed to tackle the free vibrations of nonhomogeneous circular beams. The material distribution can depend symmetrically on the cross-sectional coordinates. It can be either continuous (functionally graded materials) or constant in the layers that constitute the beam (multi-layered material). The end supports are identical rotationally restrained pins by means of linear rotational springs. The extensibility of the beam centerline is incorporated into the mechanical model. We determine the Green function matrix of the related problem in closed form. With this in hand, a numerical technique (based on the Fredholm theory) is used to tackle the vibratory problem. Thus, it is possible not only to get the eigenfrequencies but also to find out how the spring stiffness affects the dynamic behavior. Graphical results contribute to the better understanding of the issue.

Appendix: The Green function matrix for rotationally restrained circular beams

1.1 Solutions for the matrices ${{\mathbf {B}}}_i$

To determine the elements of the unknown matrices ${{\mathbf {B}}}_i$ we have to solve equation system (25). Substituting the functions ${{\mathbf {Y}}}_i$ and the discontinuities $\overset{2}{P}{}^{-1}_{11}$ and $\overset{4}{P}{}^{-1}_{22}$ into (25) we get the following linear equations:

$$\begin{aligned}&\left[ \begin{array}{cccccc} \cos \gamma &{} -\sin \gamma &{} -\sin \gamma +\gamma \cos \gamma &{} (1+{{\mathcal {M}}})\gamma &{} -\cos \gamma -\gamma \sin \gamma &{} 1\\ \sin \gamma &{} \cos \gamma &{} \gamma \sin \gamma &{} -{{\mathcal {M}}} &{} \gamma \cos \gamma &{} 0\\ -\sin \gamma &{} -\cos \gamma &{} -\gamma \sin \gamma &{} 1+{{\mathcal {M}}} &{} -\gamma \cos \gamma &{} 0\\ \cos \gamma &{} -\sin \gamma &{} \gamma \cos \gamma +\sin \gamma &{} 0 &{} -\gamma \sin \gamma +\cos \gamma &{} 0\\ -\sin \gamma &{} -\cos \gamma &{} -\gamma \sin \gamma +2\cos \gamma &{} 0 &{} -\gamma \cos \gamma -2\sin \gamma &{} 0\\ -\cos \gamma &{} \sin \gamma &{} -\gamma \cos \gamma -3\sin \gamma &{} 0 &{} \gamma \sin \gamma -3\cos \gamma &{} 0 \end{array} \right] \nonumber \\&\quad \times \left[ \begin{array}{cc} \overset{1}{B}_{11} &{} \overset{1}{B}_{12}\\ \overset{2}{B}_{11} &{} \overset{2}{B}_{12}\\ \overset{3}{B}_{11} &{} \overset{3}{B}_{12}\\ \overset{3}{B}_{21} &{} \overset{3}{B}_{22}\\ \overset{4}{B}_{11} &{} \overset{4}{B}_{12}\\ \overset{4}{B}_{21} &{} \overset{4}{B}_{22} \end{array} \right] =\left[ \begin{array}{cc} 0 &{} 0\\ 0 &{} 0\\ \dfrac{1}{2{{\mathcal {M}}}} &{} 0\\ 0 &{} 0\\ 0 &{} 0\\ 0 &{} -\dfrac{1}{2} \end{array}\right] . \end{aligned}$$

(39)

The closed-form solutions are given below:

$$\begin{aligned} \begin{array}{l} \overset{1}{B}_{11}=\dfrac{1}{2}\sin \gamma -\dfrac{1}{4}\gamma \cos \gamma \\ \overset{2}{B}_{11}=\dfrac{1}{4}\gamma \sin \gamma +\dfrac{1}{2}\cos \gamma \\ \overset{3}{B}_{11}=\dfrac{1}{4}\cos \gamma \\ \overset{3}{B}_{21}=\dfrac{1}{2{{\mathcal {M}}}}\\ \overset{4}{B}_{11}=-\dfrac{1}{4}\sin \gamma \\ \overset{4}{B}_{21}=-\dfrac{1}{2}\left( 1+{{\mathcal {M}}}\right) \dfrac{\gamma }{{{\mathcal {M}}}}\end{array} \qquad {{\text {and}}}\qquad \begin{array}{l} \overset{1}{B}_{12}=-\dfrac{1}{4}\cos \gamma -\dfrac{1}{4}\gamma \sin \gamma \\ \overset{2}{B}_{12}=\dfrac{1}{4}\sin \gamma -\dfrac{1}{4}\gamma \cos \gamma \\ \overset{3}{B}_{12}=\dfrac{1}{4}\sin \gamma \\ \overset{3}{B}_{22}=0\\ \overset{4}{B}_{12}=\dfrac{1}{4}\cos \gamma \\ \overset{4}{B}_{21}=\dfrac{1}{2}. \end{array} \end{aligned}$$

(40)

1.2 Solutions for the matrices ${{\mathbf {A}}}_i$

Let us introduce the following brief notations:

$$\begin{aligned} {{\mathfrak {a}}} =\overset{1}{B}_{1i},\quad {{\mathfrak {b}}}=\overset{2}{B}_{1i},\quad {{\mathfrak {c}}}=\overset{3}{B} _{1i},\quad {{\mathfrak {d}}}=\overset{3}{B}_{2i},\quad {{\mathfrak {e}}}=\overset{4}{B}_{1i},\quad {{\mathfrak {f}}}=\overset{4}{B}_{2i}. \end{aligned}$$

Utilizing now boundary condition (12) valid for rotationally restrained beams we get the following linear equations from Property 4—from Eq. (27)—to calculate the unknown nonzero elements in the matrices ${{\mathbf {A}}}_i$ ($i=1,2$):

$$\begin{aligned}&\left[ \begin{array}{cccccc} \cos \vartheta &{} \sin \vartheta &{} \sin \vartheta -\vartheta \cos \vartheta &{} -({{\mathcal {M}}}+1)\vartheta &{} -\cos \vartheta -\vartheta \sin \vartheta &{} 1\\ \cos \vartheta &{} -\sin \vartheta &{} -\sin \vartheta +\vartheta \cos \vartheta &{} ({{\mathcal {M}}}+1)\vartheta &{} -\cos \vartheta -\vartheta \sin \vartheta &{} 1\\ -\sin \vartheta &{} \cos \vartheta &{} \vartheta \sin \vartheta &{} -{{\mathcal {M}}} &{} -\vartheta \cos \vartheta &{} 0\\ \sin \vartheta &{} \cos \vartheta &{} \vartheta \sin \vartheta &{} -{{\mathcal {M}}} &{} \vartheta \cos \vartheta &{} 0\\ \sin \vartheta -{{\mathcal {K}}}\cos \vartheta &{} -\cos \vartheta -{{\mathcal {K}}}\sin \vartheta &{} \{2\cos \vartheta -\vartheta \sin \vartheta - &{} 0 &{} \{2\sin \vartheta +\vartheta \cos \vartheta - &{} 0 \\ &{} &{} -{{\mathcal {K}}}\left( -\sin \vartheta -\vartheta \cos \vartheta \right) \} &{} -{{\mathcal {K}}}\left( \cos \vartheta -\vartheta \sin \vartheta \right) \} &{} \\ -\sin \vartheta +{{\mathcal {K}}}\cos \vartheta &{} -\cos \vartheta -{{\mathcal {K}}}\sin \vartheta &{} \{2\cos \vartheta -\vartheta \sin \vartheta + &{} 0 &{} \{-2\sin \vartheta -\vartheta \cos \vartheta &{} 0 \\ &{} &{} +{{\mathcal {K}}}\left( \sin \vartheta +\vartheta \cos \vartheta \right) \} &{} &{} +{{\mathcal {K}}}\left( \cos \vartheta -\vartheta \sin \vartheta \right) \} &{} \\ \end{array}\right] \nonumber \\&\quad \times \left[ \begin{array}{c} \overset{1}{A}_{1i}\\ \overset{2}{A}_{1i}\\ \overset{3}{A}_{1i}\\ \overset{3}{A}_{2i}\\ \overset{4}{A}_{1i}\\ \overset{4}{A}_{2i} \end{array}\right] =\left[ \begin{array}{c} -{{\mathfrak {a}}}\cos \vartheta -{{\mathfrak {b}}}\sin \vartheta -{{\mathfrak {c}}}\left( \sin \vartheta -\vartheta \cos \vartheta \right) +{{\mathfrak {d}}}({{\mathcal {M}}}+1)\vartheta +{{\mathfrak {e}}}\left( \cos \vartheta +\vartheta \sin \vartheta \right) -{{\mathfrak {f}}}\\ {{\mathfrak {a}}}\cos \vartheta -{{\mathfrak {b}}}\sin \vartheta +{{\mathfrak {c}}}\left( -\sin \vartheta +\vartheta \cos \vartheta \right) +{{\mathfrak {d}}}({{\mathcal {M}}} +1)\vartheta -{{\mathfrak {e}}}\left( \cos \vartheta +\vartheta \sin \vartheta \right) +{{\mathfrak {f}}}\\ {{\mathfrak {a}}}\sin \vartheta -{{\mathfrak {b}}}\cos \vartheta -{{\mathfrak {c}}} \vartheta \sin \vartheta +{{\mathfrak {d}}}{{\mathcal {M}}}+{{\mathfrak {e}}} \vartheta \cos \vartheta \\ {{\mathfrak {a}}}\sin \vartheta +{{\mathfrak {b}}}\cos \vartheta +{{\mathfrak {c}}} \vartheta \sin \vartheta -{{\mathfrak {d}}}{{\mathcal {M}}}+{{\mathfrak {e}}} \vartheta \cos \vartheta \\ {{\mathfrak {a}}}\left( {{\mathcal {K}}}\cos \vartheta -\sin \vartheta \right) + {{\mathfrak {b}}}\left( \cos \vartheta +{{\mathcal {K}}}\sin \vartheta \right) + {{\mathfrak {c}}}\left[ 2\cos \vartheta -\vartheta \sin \vartheta - {{\mathcal {K}}}\left( \sin \vartheta +\vartheta \cos \vartheta \right) \right] \\ +{{\mathfrak {e}}} \left[ 2\sin \vartheta +\vartheta \cos \vartheta + {{\mathcal {K}}}\left( \cos \vartheta -\vartheta \sin \vartheta \right) \right] \\ {{\mathfrak {a}}}\left( {{\mathcal {K}}}\cos \vartheta -\sin \vartheta \right) +{{\mathfrak {b}}}\left( -\cos \vartheta -{{\mathcal {K}}}\sin \vartheta \right) +{{\mathfrak {c}}}\left[ 2\cos \vartheta -\vartheta \sin \vartheta +{{\mathcal {K}}}\left( \sin \vartheta +\vartheta \cos \vartheta \right) \right] \\ -{{\mathfrak {e}}}\left[ 2\sin \vartheta +\vartheta \cos \vartheta -{{\mathcal {K}}} \left( \cos \vartheta -\vartheta \sin \vartheta \right) \right] \\ \end{array}\right] \end{aligned}$$

With

$$\begin{aligned} D_{1}= & {} 2\sin ^{2}\vartheta +{{\mathcal {K}}}\left( \vartheta -\sin \vartheta \cos \vartheta \right) \nonumber \\ D_{2}= & {} {{\mathcal {M}}}\vartheta +2\left( 1+{{\mathcal {M}}}\right) \vartheta \cos ^{2}\vartheta -3{{\mathcal {M}}}\sin \vartheta \cos \vartheta \nonumber \\&+{{\mathcal {K}}}\left[ {{\mathcal {M}}}\left( \vartheta \cos \vartheta -2\sin \vartheta \right) \sin \vartheta +\left( 1+{{\mathcal {M}}}\right) \vartheta ^{2}+\vartheta \sin \vartheta \cos \vartheta \right] \end{aligned}$$

(41)

the solutions sought are

$$\begin{aligned} \overset{1}{A}_{1i}= & {} \frac{2{{\mathfrak {b}}}\sin \vartheta \cos \vartheta +2{{\mathfrak {c}}}\vartheta -2{{\mathfrak {d}}}{{\mathcal {M}}} \left( \sin \vartheta -\vartheta \cos \vartheta \right) +{{\mathcal {K}}}\left[ {{\mathfrak {c}}}\vartheta ^{2} -{{\mathfrak {b}}} \cos ^{2}\vartheta +{{\mathfrak {d}}}{{\mathcal {M}}}\left( \cos \vartheta -\vartheta \sin \vartheta \right) \right] }{D_{1}}, \end{aligned}$$

(42a)

$$\begin{aligned} \overset{2}{A}_{1i}= & {} \frac{1}{D_{2}}\left\{ {{\mathfrak {a}}} \left[ 2(1+{{\mathcal {M}}})\vartheta \sin \vartheta \cos \vartheta +{{\mathcal {M}}}\left( 2\cos ^{2}\vartheta -\sin ^{2}\vartheta \right) \right] \right. \nonumber \\&+{{\mathfrak {e}}}\left( 3{{\mathcal {M}}}\vartheta ^{2}+2\vartheta ^{2} -2{{\mathcal {M}}}\right) -{{\mathfrak {f}}} {{\mathcal {M}}} \left( \vartheta \sin \vartheta -2\cos \vartheta \right) \nonumber \\&+\left. {{\mathcal {K}}}\left[ {{\mathfrak {a}}}\left( \left[ 1+{{\mathcal {M}}}\right] \vartheta \sin ^{2}\vartheta +2{{\mathcal {M}}}\sin \vartheta \cos \vartheta \right) +{{\mathfrak {e}}} \vartheta \left( -2{{\mathcal {M}}}+\left[ 1+{{\mathcal {M}}}\right] \vartheta ^{2}\right) \right. \right. \nonumber \\&\left. \left. +{{\mathfrak {f}}}{{\mathcal {M}}}\left( \vartheta \cos \vartheta +\sin \vartheta \right) \right] \right\} , \end{aligned}$$

(42b)

$$\begin{aligned} \overset{3}{A}_{1i}= & {} \frac{1}{D_{2}}\left\{ {{\mathfrak {a}}}{{\mathcal {M}}} -{{\mathfrak {e}}}\left[ {{\mathcal {M}}}\cos ^{2}\vartheta -2{{\mathcal {M}}} \sin ^{2}\vartheta +2\vartheta (1+{{\mathcal {M}}})\sin \vartheta \cos \vartheta \right] \right. \nonumber \\&\left. +{{\mathfrak {f}}}{{\mathcal {M}}}\cos \vartheta +{{\mathcal {K}}} \left[ {{\mathfrak {a}}}\left( 1+{{\mathcal {M}}}\right) \vartheta +{{\mathfrak {e}}}\left( 1+{{\mathcal {M}}}\right) \vartheta \cos ^{2}\vartheta -2{{\mathfrak {e}}}{{\mathcal {M}}}\sin \vartheta \cos \vartheta +{{\mathfrak {f}}}{{\mathcal {M}}} \sin \vartheta \right] \right\} , \end{aligned}$$

(42c)

$$\begin{aligned} \overset{3}{A}_{2i}= & {} \frac{1}{D_{2}}\left\{ 2{{\mathfrak {a}}}\cos \vartheta +2{{\mathfrak {e}}}\left( \vartheta \sin \vartheta -\cos \vartheta \right) \right. \nonumber \\&\left. +2{{\mathfrak {f}}}\cos ^{2}\vartheta +{{\mathcal {K}}}\left[ 2{{\mathfrak {a}}} \sin \vartheta -2{{\mathfrak {e}}}\vartheta \cos \vartheta +{{\mathfrak {f}}}\left( \vartheta +\sin \vartheta \cos \vartheta \right) \right] \right\} , \end{aligned}$$

(42d)

$$\begin{aligned} \overset{4}{A}_{1i}= & {} \frac{-2{{\mathfrak {c}}}\cos \vartheta \sin \vartheta +{{\mathfrak {d}}}{{\mathcal {M}}}\sin \vartheta +{{\mathcal {K}}}\left( {{\mathfrak {b}}} -{{\mathfrak {c}}}\sin ^{2}\vartheta -{{\mathfrak {d}}}{{\mathcal {M}}} \cos \vartheta \right) }{D_{1}}, \end{aligned}$$

(42e)

$$\begin{aligned} \overset{4}{A}_{2i}= & {} \frac{1}{D_{1}}\left\{ -2{{\mathfrak {b}}}\sin \vartheta -2{{\mathfrak {c}}}\left( \sin \vartheta +\vartheta \cos \vartheta \right) \right. \nonumber \\&+{{\mathfrak {d}}}\left[ {{\mathcal {M}}}\vartheta \cos ^{2}\vartheta +3{{\mathcal {M}}}\sin \vartheta \left( \cos \vartheta +\vartheta \sin \vartheta \right) +2\vartheta \sin ^{2}\vartheta \right] \nonumber \\&\left. +{{\mathcal {K}}}\left[ 2{{\mathfrak {b}}}\cos \vartheta -2{{\mathfrak {c}}} \vartheta \sin \vartheta +{{\mathfrak {d}}}\vartheta ^{2} \left( 1+{{\mathcal {M}}}\right) -{{\mathfrak {d}}}\vartheta \left( 1+{{\mathcal {M}}}\right) \sin \vartheta \cos \vartheta -2{{\mathfrak {d}}}{{\mathcal {M}}}\cos ^{2}\vartheta \right] \right\} . \end{aligned}$$

(42f)

It should be mentioned that if (${{\mathcal {K}}}=0$) [${{\mathcal {K\rightarrow \infty }}}$] in ${{\mathbf {A}}}_i$, these solutions coincide with those valid for (pinned-pinned) [fixed-fixed] beams [47].