A variational formulation for vibration analysis of curved beams with arbitrary eccentric concentrated elements

Abstract

In this paper, a modified variational method is developed to study the free and forced vibration of curved beams subjected to different boundary conditions. An arbitrary number of eccentric concentrated elements (ECEs) attached to the beams are taken into account. A modified variational principle and least-square weighted residual method are employed to impose the continuity constraints on the internal interfaces and the boundaries of the curved beam. The shear and inertial (or radial–tangential–rotational coupling) effects are incorporated into the system kinetic and potential functional using the generalized shell theory. To test the efficiency and accuracy of the present method, both free and force vibrations of the curved beams are examined under various boundary conditions including non-classical boundary conditions. Good agreement is observed between the results obtained by the present method and those from finite element program ANSYS. Corresponding curved beams with non-eccentric concentrated elements are also developed to investigate the effects of the ECEs on the vibration behaviors of the curved beams.

Appendices

Appendix A. Disjoint generalized mass and stiffness matrices of a curved beam

The generalized mass and stiffness matrices of a curved beam in Eq. (16) are, respectively, organized as

$$\begin{aligned} \mathbf{M}= & {} \hbox {diag}\left[ {\mathbf{M}_{\mathrm{1}} ,\mathbf{M}_2 ,\ldots ,\mathbf{M}_{N_\mathrm{b} } } \right] \end{aligned}$$

(A.1)

$$\begin{aligned} \mathbf{K}= & {} \hbox {diag}\left[ {\mathbf{K}_{\mathrm{1}} ,\mathbf{K}_2 ,\ldots ,\mathbf{K}_{N_\mathrm{b} } } \right] \end{aligned}$$

(A.2)

where M$_{i}$ and K$_{i}$ are the mass and stiffness matrices of the ith beam segment, respectively, and can be given by

$$\begin{aligned} \mathbf{M}_i= & {} \int _{\theta _i } {\left[ {{\begin{array}{ccc} {\mathbf{M}_{uu}^i }&{}\quad \mathbf{0}&{}\quad \mathbf{0} \\ \mathbf{0}&{}\quad {\mathbf{M}_{ww}^i }&{}\quad \mathbf{0} \\ \mathbf{0}&{}\quad \mathbf{0}&{}\quad {\mathbf{M}_{\varphi \varphi }^i } \\ \end{array} }} \right] R\mathrm{d}\theta } \end{aligned}$$

(A.3)

$$\begin{aligned} \mathbf{K}_i= & {} \int _{\theta _i } {\left[ {{\begin{array}{ccc} {\mathbf{K}_{uu}^i }&{}\quad {\mathbf{K}_{uw}^i }&{}\quad {\mathbf{K}_{u\varphi }^i } \\ {\mathbf{K}_{uw}^{i,T} }&{}\quad {\mathbf{K}_{ww}^i }&{}\quad {\mathbf{K}_{w\varphi }^i } \\ {\mathbf{K}_{u\varphi }^{i,T} }&{}\quad {\mathbf{K}_{w\varphi }^{i,T} }&{}\quad {\mathbf{K}_{\varphi \varphi }^i } \\ \end{array} }} \right] \mathrm{d}\theta } \end{aligned}$$

(A.4)

The elements of the mass matrices are:

$$\begin{aligned} \mathbf{M}_{uu}^i= & {} \rho A\mathbf{U}_i^\mathrm{T} \mathbf{U}_i , \end{aligned}$$

(A.5a)

$$\begin{aligned} \mathbf{M}_{ww}^i= & {} \rho A\mathbf{W}_i^\mathrm{T} \mathbf{W}_i , \end{aligned}$$

(A.5b)

$$\begin{aligned} \mathbf{M}_{\varphi \varphi }^i= & {} \rho I\varvec{\Phi }_i^\mathrm{T} \varvec{\Phi }_i \end{aligned}$$

(A.5c)

The elements of the stiffness matrices are:

$$\begin{aligned} \mathbf{K}_{uu}^i= & {} \frac{EA}{R}\frac{\partial \mathbf{U}_i^\mathrm{T} }{\partial \theta }\frac{\partial \mathbf{U}_i }{\partial \theta }+\frac{\kappa GA}{R}{} \mathbf{U}_i^\mathrm{T} \mathbf{U}_i \end{aligned}$$

(A.6a)

$$\begin{aligned} \mathbf{K}_{uw}^i= & {} \frac{\kappa GA}{R}{} \mathbf{U}_i^\mathrm{T} \frac{\partial \mathbf{W}_i }{\partial \theta }-\frac{EA}{R}\frac{\partial \mathbf{U}_i^\mathrm{T} }{\partial \theta }\mathbf{W}_i \end{aligned}$$

(A.6b)

$$\begin{aligned} \mathbf{K}_{u\varphi }^i= & {} -\kappa GA\mathbf{U}_i^\mathrm{T} \varvec{\Phi }_i \end{aligned}$$

(A.6c)

$$\begin{aligned} \mathbf{K}_{ww}^i= & {} \frac{\kappa GA}{R}\frac{\partial \mathbf{W}_i^\mathrm{T} }{\partial \theta }\frac{\partial \mathbf{W}_i }{\partial \theta }+\frac{EA}{R}{} \mathbf{W}_i^\mathrm{T} \mathbf{W}_i \end{aligned}$$

(A.6d)

$$\begin{aligned} \mathbf{K}_{w\varphi }^i= & {} -\kappa GA\frac{\partial \mathbf{W}_i^\mathrm{T} }{\partial \theta }\varvec{\Phi }_i \end{aligned}$$

(A.6e)

$$\begin{aligned} \mathbf{K}_{\varphi \varphi }^i= & {} \frac{EI}{R}\frac{\partial \varvec{\Phi }_i^\mathrm{T} }{\partial \theta }\frac{\partial \varvec{\Phi }_i }{\partial \theta }+\kappa GAR\varvec{\Phi }_i^\mathrm{T} \varvec{\Phi }_i \end{aligned}$$

(A.6f)

Appendix B. Generalized interface stiffness matrices $\hbox {K}_\lambda$ and $\hbox {K}_\kappa$

According to the organization of the elements in the generalized displacement vector, assembling all interface matrices leads to the generalized interface matrices $\mathbf{K}_\lambda$ and $\mathbf{K}_\kappa$. The interface matrix $\mathbf{K}_\lambda ^{i}$ of the curved beam at the interface located at $\theta$=$\theta$$_{i}$ is given by

$$\begin{aligned} \mathbf{K}_\lambda ^i =\left[ {{\begin{array}{llllll} {\mathbf{K}_{u_i u_i }} &{}\quad {\mathbf{K}_{u_i w_i }} &{}\quad {\mathbf{K}_{u_i \varphi _i }} &{}\quad {\mathbf{K}_{u_i u_{i+1} }} &{}\quad {\mathbf{K}_{u_i w_{i+1} }} &{}\quad {\mathbf{K}_{u_i \varphi _{i+1} } } \\ {\mathbf{K}_{u_i w_i }^\mathrm{T} } &{}\quad {\mathbf{K}_{w_i w_i }} &{}\quad {\mathbf{K}_{w_i \varphi _i }} &{}\quad {\mathbf{K}_{w_i u_{i+1} }} &{}\quad {\mathbf{K}_{w_i w_{i+1} }} &{}\quad {\mathbf{K}_{w_i \varphi _{i+1} } } \\ {\mathbf{K}_{u_i \varphi _i }^\mathrm{T} } &{}\quad {\mathbf{K}_{w_i \varphi _i }^\mathrm{T} } &{}\quad {\mathbf{K}_{\varphi _i \varphi _i }} &{}\quad {\mathbf{K}_{\varphi _i u_{i+1} }} &{}\quad {\mathbf{K}_{\varphi _i w_{i+1} }} &{}\quad {\mathbf{K}_{\varphi _i \varphi _{i+1} }} \\ {\mathbf{K}_{u_i u_{i+1} }^\mathrm{T} } &{}\quad {\mathbf{K}_{u_i w_{i+1} }^\mathrm{T} } &{}\quad {\mathbf{K}_{u_i \varphi _{i+1} }^\mathrm{T} } &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad \mathbf{0} \\ {\mathbf{K}_{w_i u_{i+1} }^\mathrm{T} } &{}\quad {\mathbf{K}_{w_i w_{i+1} }^\mathrm{T} } &{}\quad {\mathbf{K}_{w_i \varphi _{i+1} }^\mathrm{T} } &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad \mathbf{0} \\ {\mathbf{K}_{\varphi _i u_{i+1} }^\mathrm{T} } &{}\quad {\mathbf{K}_{\varphi _i w_{i+1} }^\mathrm{T} } &{}\quad {\mathbf{K}_{\varphi _i \varphi _{i+1} }^\mathrm{T} } &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad \mathbf{0} \\ \end{array} }} \right] \end{aligned}$$

(B.1)

The detailed elements of the interface matrix $\mathbf{K}_\lambda ^i$ are:

$$\begin{aligned} \mathbf{K}_{u_i u_i }= & {} \varsigma _u \frac{EA}{R}\left( {\frac{\partial \mathbf{U}_i^\mathrm{T} }{\partial \theta }{} \mathbf{U}_i +\mathbf{U}_i^\mathrm{T} \frac{\partial \mathbf{U}_i }{\partial \theta }} \right) \end{aligned}$$

(B.2a)

$$\begin{aligned} \mathbf{K}_{u_i w_i }= & {} \left( {\frac{\varsigma _w \kappa GA}{R}-\frac{\varsigma _u EA}{R}} \right) \mathbf{U}_i^\mathrm{T} \mathbf{W}_i \end{aligned}$$

(B.2b)

$$\begin{aligned} \mathbf{K}_{u_i \varphi _i }= & {} \mathbf{0} \end{aligned}$$

(B.2c)

$$\begin{aligned} \mathbf{K}_{u_i u_{i+1} }= & {} -\frac{\varsigma _u EA}{R}\frac{\partial \mathbf{U}_i^\mathrm{T} }{\partial \theta }{} \mathbf{U}_{i+1} \end{aligned}$$

(B.2d)

$$\begin{aligned} \mathbf{K}_{u_i w_{i+1} }= & {} -\frac{\varsigma _w \kappa GA}{R}{} \mathbf{U}_i^\mathrm{T} \mathbf{W}_{i+1} \end{aligned}$$

(B.2e)

$$\begin{aligned} \mathbf{K}_{u_i \varphi _{i+1} }= & {} \mathbf{0} \end{aligned}$$

(B.2f)

$$\begin{aligned} \mathbf{K}_{w_i w_i }= & {} \frac{\varsigma _w \kappa GA}{R}\left( {\frac{\partial \mathbf{W}_i^\mathrm{T} }{\partial \theta }{} \mathbf{W}_i +\mathbf{W}_i^\mathrm{T} \frac{\partial \mathbf{W}_i }{\partial \theta }} \right) \end{aligned}$$

(B.2g)

$$\begin{aligned} \mathbf{K}_{w_i \varphi _i }= & {} -\varsigma _w \kappa GA\mathbf{W}_i^\mathrm{T} \varvec{\Phi }_i \end{aligned}$$

(B.2h)

$$\begin{aligned} \mathbf{K}_{w_i u_{i+1} }= & {} -\frac{\varsigma _u EA}{R}\mathbf{W}_i^\mathrm{T} \mathbf{U}_{i+1} \end{aligned}$$

(B.2i)

$$\begin{aligned} \mathbf{K}_{w_i w_{i+1} }= & {} -\varsigma _w \frac{\kappa GA}{R}\frac{\partial \mathbf{W}_i^\mathrm{T} }{\partial \theta }{} \mathbf{W}_{i+1} \end{aligned}$$

(B.2j)

$$\begin{aligned} \mathbf{K}_{w_i \varphi _{i+1} }= & {} \mathbf{0} \end{aligned}$$

(B.2k)

$$\begin{aligned} \mathbf{K}_{\varphi _i \varphi _i }= & {} \frac{\varsigma _\varphi EI}{R}\left( {\frac{\partial \varvec{\Phi }_i^\mathrm{T} }{\partial \theta }\varvec{\Phi }_i +\varvec{\Phi }_i^\mathrm{T} \frac{\partial \varvec{\Phi }_i }{\partial \theta }} \right) \end{aligned}$$

(B.2l)

$$\begin{aligned} \mathbf{K}_{\varphi _i u_{i+1} }= & {} \mathbf{0} \end{aligned}$$

(B.2m)

$$\begin{aligned} \mathbf{K}_{\varphi _i w_{i+1} }= & {} \varsigma _w \kappa GA\varvec{\Phi }_i^\mathrm{T} \mathbf{W}_{i+1} \end{aligned}$$

(B.2n)

$$\begin{aligned} \mathbf{K}_{\varphi _i \varphi _{i+1} }= & {} -\frac{\varsigma _\varphi EI}{R}\frac{\partial \varvec{\Phi }_i^\mathrm{T} }{\partial \theta }\varvec{\Phi }_{i+1} \end{aligned}$$

(B.2o)

The generalized matrix $\mathbf{K}_\kappa ^i$ due to the least-squares weighted terms at the interface located at $\theta =\theta _{i}$ is given below

$$\begin{aligned} \mathbf{K}_\kappa ^i =\left[ {{\begin{array}{cccccc} {{\bar{\mathbf{K}}}_{u_i u_i }} &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{u_i u_{i+1} }} &{}\quad \mathbf{0} &{}\quad \mathbf{0} \\ \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{w_i w_i }} &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{w_i w_{i+1} }} &{}\quad \mathbf{0} \\ \mathbf{0} &{}\quad \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{\varphi _i \varphi _i }} &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{\varphi _i \varphi _{i+1} }} \\ {{\bar{\mathbf{K}}}_{u_i u_{i+1} }^\mathrm{T} } &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{u_{i+1} u_{i+1} }} &{}\quad \mathbf{0} &{}\quad \mathbf{0} \\ \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{w_i w_{i+1} }^\mathrm{T} } &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{w_{i+1} w_{i+1} }} &{}\quad \mathbf{0} \\ \mathbf{0} &{}\quad \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{\varphi _i \varphi _{i+1} }^\mathrm{T} } &{}\quad \mathbf{0} &{}\quad \mathbf{0} &{}\quad {{\bar{\mathbf{K}}}_{\varphi _{i+1} \varphi _{i+1} }} \\ \end{array} }} \right] \end{aligned}$$

(B.3)

where the detailed elements are obtained by

$$\begin{aligned} {\bar{\mathbf{K}}}_{u_i u_i }= & {} \varsigma _u \kappa _u \mathbf{U}_i^\mathrm{T} \mathbf{U}_i , \end{aligned}$$

(B.4a)

$$\begin{aligned} {\bar{\mathbf{K}}}_{u_i u_{i+1} }= & {} -\varsigma _u \kappa _u \mathbf{U}_i^\mathrm{T} \mathbf{U}_{i+1} \end{aligned}$$

(B.4b)

$$\begin{aligned} {\bar{\mathbf{K}}}_{w_i w_i }= & {} \varsigma _w \kappa _w \mathbf{W}_i^\mathrm{T} \mathbf{W}_i , \end{aligned}$$

(B.4c)

$$\begin{aligned} {\bar{\mathbf{K}}}_{w_i w_{i+1} }= & {} -\varsigma _w \kappa _w \mathbf{W}_i^\mathrm{T} \mathbf{W}_{i+1} \end{aligned}$$

(B.4d)

$$\begin{aligned} {\bar{\mathbf{K}}}_{\varphi _i \varphi _i }= & {} \varsigma _\varphi \kappa _\varphi \varvec{\Phi }_i^\mathrm{T} \varvec{\Phi }_i , \end{aligned}$$

(B.4e)

$$\begin{aligned} \mathbf{K}_{\varphi _i \varphi _{i+1} }= & {} -\varsigma _\varphi \kappa _\varphi \varvec{\Phi }_i^\mathrm{T} \varvec{\Phi }_{i+1} \end{aligned}$$

(B.4f)

Appendix C. Generalized mass and stiffness matrices of ECEs

Assuming that there are $N_{r}$ ECEs on the rth beam segment, and then the generalized mass matrix$\mathbf{M}_c^r$ introduced by the ECEs for the rth beam segment can be written as

$$\begin{aligned} \mathbf{M}_c^r =\left[ {{\begin{array}{ccc} {{\bar{\mathbf{M}}}_{uu}^r }&{}\quad \mathbf{0}&{}\quad {{\bar{\mathbf{M}}}_{u\varphi }^r } \\ \mathbf{0}&{}\quad {{\bar{\mathbf{M}}}_{ww}^r }&{}\quad \mathbf{0} \\ {{\bar{\mathbf{M}}}_{u\varphi }^{r,T} }&{}\quad \mathbf{0}&{}\quad {{\bar{\mathbf{M}}}_{\varphi \varphi }^r } \\ \end{array} }} \right] \end{aligned}$$

(C.1)

The elements in the mass matrix are:

$$\begin{aligned} {\bar{\mathbf{M}}}_{uu}^r= & {} \sum _{m=1}^{N_r } {\left( {m_r^m \mathbf{U}_r^\mathrm{T} (\theta _r^m )\mathbf{U}_r (\theta _r^m )} \right) } , \end{aligned}$$

(C.2a)

$$\begin{aligned} {\bar{\mathbf{M}}}_{u\varphi }^r= & {} \sum _{m=1}^{N_r } {\left( {e_r^m m_r^m \mathbf{U}_r^\mathrm{T} (\theta _r^m )\varvec{\Phi }_r (\theta _r^m )} \right) } \end{aligned}$$

(C.2b)

$$\begin{aligned} {\bar{\mathbf{M}}}_{ww}^r= & {} \sum _{m=1}^{N_r } {\left( {m_r^m \mathbf{W}_r^\mathrm{T} (\theta _r^m )\mathbf{W}_r (\theta _r^m )} \right) } , \end{aligned}$$

(C.2c)

$$\begin{aligned} {\bar{\mathbf{M}}}_{\varphi \varphi }^r= & {} \sum _{m=1}^{N_r } {\left( {\left( {J_r^m +m_r^m (e_r^m )^{2}} \right) \varvec{\Phi }_r^\mathrm{T} (\theta _r^m )\varvec{\Phi }_r (\theta _r^m )} \right) } \end{aligned}$$

(C.2d)

In the same way, the stiffness matrix $\mathbf{K}_c^r$ introduced by the ECEs for the rth beam segment can be written as

$$\begin{aligned} \mathbf{K}_c^r =\left[ {{\begin{array}{ccc} {{\bar{\mathbf{K}}}_{uu}^r }&{}\quad \mathbf{0}&{}\quad {{\bar{\mathbf{K}}}_{u\varphi }^r } \\ \mathbf{0}&{}\quad {{\bar{\mathbf{K}}}_{ww}^r }&{}\quad \mathbf{0} \\ {{\bar{\mathbf{K}}}_{u\varphi }^{r,T} }&{}\quad \mathbf{0}&{}\quad {{\bar{\mathbf{K}}}_{\varphi \varphi }^r } \\ \end{array} }} \right] \end{aligned}$$

(C.3)

The elements in the stiffness matrix are given below

$$\begin{aligned} {\bar{\mathbf{K}}}_{uu}^r= & {} \sum _{m=1}^{N_r } {\left( {k_{u,r}^m \mathbf{U}_r^\mathrm{T} (\theta _r^m )\mathbf{U}_r (\theta _r^m )} \right) } , \end{aligned}$$

(C.4a)

$$\begin{aligned} {\bar{\mathbf{K}}}_{u\varphi }^r= & {} \sum _{m=1}^{N_r } {\left( {e_r^m k_{u,r}^m \mathbf{U}_r^\mathrm{T} (\theta _r^m )\varvec{\Phi }_r (\theta _r^m )} \right) } \end{aligned}$$

(C.4b)

$$\begin{aligned} {\bar{\mathbf{K}}}_{ww}^r= & {} \sum _{m=1}^{N_r } {\left( {k_{w,r}^m \mathbf{W}_r^\mathrm{T} (\theta _r^m )\mathbf{W}_r (\theta _r^m )} \right) } , \end{aligned}$$

(C.4c)

$$\begin{aligned} {\bar{\mathbf{K}}}_{\varphi \varphi }^r= & {} \sum _{m=1}^{N_r } {\left( {\left( {k_{\varphi ,r}^m +k_{u,r}^m (e_r^m )^{2}} \right) \varvec{\Phi }_r^\mathrm{T} (\theta _r^m )\varvec{\Phi }_r (\theta _r^m )} \right) } \end{aligned}$$

(C.4d)

whereby, the generalized mass and stiffness matrices deduced by the ECEs in Eq. (16) can be obtained, respectively, as:

$$\begin{aligned} \mathbf{M}_c= & {} \hbox {diag}\left[ {\mathbf{M}_c^1 ,\mathbf{M}_c^2 ,\ldots ,\mathbf{M}_c^{N_\mathrm{b} } } \right] , \end{aligned}$$

(C.5a)

$$\begin{aligned} \mathbf{K}_c= & {} \hbox {diag}\left[ {\mathbf{K}_c^1 ,\mathbf{K}_c^2 ,\ldots ,\mathbf{K}_c^{N_\mathrm{b} } } \right] \end{aligned}$$

(C.5b)