Acta Mechanica

, Volume 230, Issue 1, pp 137–156 | Cite as

On the eigenfrequencies of preloaded rotationally restrained extensible circular beams by Green’s functions

  • L. P. KissEmail author
  • G. Szeidl
Original Paper


The article is devoted to the vibrations of heterogeneous curved beams with centerline extensibility accounted for. The end supports are rotationally restrained pins, and the effect of a central, either compressive or tensile, force (preload) is incorporated into the model. The coupled differential equations that govern the problem are derived from the principle of virtual work. These are replaced by a system of homogeneous Fredholm integral equations. The kernel of the integral equation system is the Green’s function matrix, which is given in closed form. The eigenvalue problem determined by the homogeneous Fredholm integral equation system is solved numerically, and the results obtained are presented graphically. If the spring stiffness tends to (zero) [infinity], the beam behaves as if it were (pinned–pinned) [fixed–fixed]. When the external force tends to zero, the model returns the eigenfrequencies of the free vibrations.

List of symbols


E-weighted area

\(E(\eta ,\zeta )\)

Young’s modulus


Components of the distributed load

\({\mathbf {G}}(\varphi ,\gamma )\)

The Green’s function matrix

\(I_{e\eta }\)

E-weighted moment of inertia to the axis \(\eta \)


Volute spring stiffness

\({\mathcal {K}}=k_{\mathrm{v}}R/I_{e\eta }\)

Dimensionless spring stiffness


Geometry–heterogeneity parameter


Bending moment


Axial force

\(P_\zeta \)

External load

\(P_{\zeta ~\mathrm {crit}}\)

Critical (buckling) load

\({\mathcal {P}}=P_{\zeta }R^2\vartheta /(2I_{e\eta })\)

Dimensionless external load

\(Q_{e\eta }\)

E-weighted 1-st moment to the axis \(\eta \)


Radius of the E-weighted centerline

\(u_o,\, w_o\)

Tangential and normal displacement components

\(U_o,\, W_o\)

Dimensionless displacement components

\({\hat{U}}_{ob},\, {\hat{W}}_{ob}\)

Dimensionless displacement amplitudes

\(\alpha _{i\;\mathrm {str.}}; \alpha _{i\;\mathrm {free}}; \alpha _{i\;\mathrm {ld.}}\)

i-th eigenfrequency of the straight-, curved unloaded-, and curved loaded beam

\(\varepsilon _{o\xi }\)

Axial strain on the E-weighted centerline

\(\varepsilon _{o\xi \;\text {crit}}\)

Critical axial strain

\(\vartheta \)

Semi-vertex angle; \(\vartheta =0.5{\bar{\vartheta }}\)

\(\lambda \)


\(\varphi ;\gamma \)

Angle coordinates

\(\rho , \rho _{\mathrm{a}}\)

Density, average density over the cross section

\(\chi ^{2}\)

Parameter, related to the strain and geometry

\(\psi _{o\eta }\)

Angle of rotation on the E-weighted centerline


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© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied MechanicsUniversity of MiskolcMiskolc-EgyetemvárosHungary

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