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Free Vibration of a Timoshenko Beam with Arbitrary Nonuniformities, Discontinuities and Constraints

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Abstract

Natural frequencies and mode shapes of a Timoshenko beam with arbitrary non-uniformities, discontinuities, discrete spring/mass constraints and boundary conditions are computed by developing a new method based on the state transition matrix for spatially varying state equations. Algorithms to treat discontinuities in material and geometrical properties and discontinuities due to discrete spring constraints are clearly presented. Equations for natural frequencies and mode shapes are derived for following boundary conditions: clamped-free, attached springs/masses at both ends and pinned–pinned. Numerical results are presented and compared to those in the existing literature.

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Abbreviations

\(a_{1} (x)\)–\(a_{3} (x)\) :

Defined by Eqs. (12)–(14)

\(b_{1} (x)\), \(b_{2} (x)\) :

Defined by Eqs. (15) and (16)

\(A(x)\) :

Area of cross section

\(E(x)\) :

Young’s modulus of elasticity

EI(x):

E(x)I(x)

\(G(x)\) :

Shear modulus of elasticity

GA(x):

G(x)A(x)

\(I(x)\) :

Area-moment of inertia of cross section

\(J_{i}\) :

Attached mass-moment of inertia at \(x_{i}\); \(i = 1,2,\,3, \ldots ,\,n + 2\)

\(k_{i}\) :

Attached translational spring constant at \(x_{i}\); \(i = 1,2,\,3, \ldots ,\,n + 2\)

\(k_{ti}\) :

Attached rotational spring constant at \(x_{i}\); \(i = 1,2,\,3, \ldots ,\,n + 2\)

\(\ell\) :

Length of the beam

\(m_{i}\) :

Attached mass at \(x_{i}\); \(i = 1,2,\,3, \cdots ,\,n + 2\)

\(n\) :

Number of discontinuities

\(P\) :

System matrix for state equations, Eq. (19)

v 0(x):

\(\ell \phi_{0} (x)\)

\(w(x,t)\) :

Transverse displacement of beam

\(w_{0} (x)\) :

Amplitude of \(w(x,t)\)

\(x\) :

\(x_{r} /\ell\), Nondimensional spatial coordinate

\(x_{i}\) :

Locations of discontinuities, \(i = 2,\,3, \ldots ,\,n + 1\) (\(n > 0\))

\(x_{r}\) :

Dimensional spatial coordinate

\(\kappa\) :

Shear factor

\(\lambda\) :

Nondimensional frequency squared, defined by Eq. (11)

\(\nu\) :

Poisson’s ratio

\(\rho (x)\) :

Material density

ρA(x):

ρ(x)A(x)

\(\phi (x,t)\) :

Angular displacement of cross section

\(\phi_{0} (x)\) :

Amplitude of \(\phi (x,t)\)

\({{\varvec{\upchi}}}(x)\) :

\(4 \times 1\) Spatial state vector, Eq. (18)

\(\chi_{i} (x)\) :

iTh spatial state variable, Eq. (18)

\(\Psi (x)\) :

\(4 \times 4\) Spatial state transition matrix

\({{\varvec{\uppsi}}}_{j} (x)\) :

jTh column of matrix \(\Psi (x)\)

\(\psi_{ij} (x)\) :

Elements of matrix \(\Psi (x)\)

\(\omega\) :

Natural frequency

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  1. Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA, 16802, USA

    Alok Sinha

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Sinha, A. Free Vibration of a Timoshenko Beam with Arbitrary Nonuniformities, Discontinuities and Constraints. J. Vib. Eng. Technol. 11, 2099–2108 (2023). https://doi.org/10.1007/s42417-022-00690-x

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