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On the whole spectrum of Timoshenko beams. Part II: further applications

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Abstract

The problem of free vibrations of the Timoshenko beam model has been addressed in the first part of this paper. A careful analysis of the governing equations has shown that the vibration spectrum consists of two parts, separated by a transition frequency, which, depending on the applied boundary conditions, might be itself part of the spectrum. Here, as an extension, the case of a doubly clamped beam is considered. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided: this allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. This case is a meaningful example of more general ones, where the wave-numbers equation cannot be written in a factorized form and hence must be solved by general root-finding methods for nonlinear transcendental equations. These theoretical results can be used as further benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. finite element models.

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Abbreviations

\({\mathbf{A}}\) :

Coefficient matrix for the homogenous system

\({\mathbf{A}_r}\) :

Coefficient matrix for the reduced homogenous system

\({\mathbf{X}}\) :

Unknown column matrix for the homogenous system

\({\mathbf{X}_r}\) :

Unknown column matrix for the reduced homogenous system

\({\mathbf{0}}\) :

Right-hand-side column matrix for homogeneous system

\({\mathbf{0}_{r}}\) :

Right-hand-side column matrix for reduced homogeneous system

A :

Cross section area

\({A_{1}}\), \({A_{2}}\), \({A_{3}}\), \({A_{4}}\) :

Integration constants for \({V}\), first part of the spectrum

\({A_{1n}}\), \({A_{2n}}\), \({A_{3n}}\), \({A_{4n}}\) :

Integration constants for the n-th eigenmode

B :

Cross section depth (and width)

\({B_{1}}\), \({B_{2}}\), \({B_{3}}\), \({B_{4}}\),:

Integration constants for \({\Phi}\), first part of the spectrum

\({C}\) :

Constant factor (see Eq. (3.17))

\({C_{1}}\), \({C_{3}}\), \({C_{4}}\) :

Integration constants for \({V}\), transition frequency

\({\tilde{C}_{1}}\), \({\tilde{C}_{3}}\), \({\tilde{C}_{4}}\), \({\tilde{D}_{1}}\) :

integration constants for the n-th eigenmode at transition frequency

\({D_{1}}\) :

Integration constant for \({\Phi}\), transition frequency

\({E}\) :

Young’s modulus

\({E_{1}}\), \({E_{2}}\), \({E_{3}}\), \({E_{4}}\) :

Integration constants for \({V}\), second part of the spectrum

\({E_{1n}}\), \({E_{2n}}\), \({E_{3n}}\), \({E_{4n}}\) :

Integration constants for the n-th eigenmode

\({G}\) :

Shear modulus

\({I}\) :

Cross section mass moment of inertia

\({L}\) :

Beam length

\({\tilde{L}}\) :

Special value of beam length

\({V}\) :

Vibration mode for transversal displacement

\({f_{\lambda}}\) :

Space frequency associated with wave-number \({\lambda}\)

\({f_{\lambda_n}}\) :

Space frequency associated with the n-th vibration mode

\({k}\), \({k_1}\), \({k_2}\) :

Integer values corresponding to wave-numbers of vibration modes

t :

Time variable

v :

Transversal displacement

x :

Space variable (beam abscissa)

z :

Dummy space variable

\({z^{\star}}\) :

Normalized dummy space variable

\({\Phi}\) :

Vibration mode for section rotation

\({\hat{\alpha}_1}\) :

Coefficient of eigenmode for generalized wave-number

\({\hat{\alpha}_{1n}}\) :

Value of \({\hat{\alpha}_1}\) for n-th vibration mode

\({\alpha_1}\), \({\alpha_2}\) :

Eigenmode coefficients for first/second wave-number

\({\alpha_{1n}}\), \({\alpha_{2n}}\) :

Values of \({\alpha_1}\), \({\alpha_2}\) for n-th vibration mode

\({\tilde{\alpha}_2}\) :

Eigenmode coefficient for second wave-number at transition frequency

\({\varepsilon}\) :

Predefined convergence tolerance

\({\kappa}\) :

Shear correction factor

\({\hat{\lambda}_{1}}\) :

Generalized wave-number (first part of the spectrum)

\({\lambda_1}\) :

First wave-number (second part of the spectrum)

\({\lambda_2}\) :

Second wave-number (first and second part of the spectrum)

\({\tilde{\lambda}_{2}}\) :

Second wave-number at transition frequency

\({\lambda_{1n}}\) :

First wave-number for n-th vibration mode

\({\lambda_{2n}}\) :

Second wave-number for n-th vibration mode

\({\lambda_{1}^{\star2}}\) :

First root (squared) of wave-numbers equations

\({\lambda_{2}^{\star2}}\) :

Second root (squared) of wave-numbers equations

\({\nu}\) :

Poisson’s ratio

\({\xi}\) :

Dimensionless space variable (dimensionless beam abscissa)

\({\rho}\) :

Beam density (mass per unit volume)

\({\sigma_n}\), \({\hat{\chi}_n}\) :

Eigenmode coefficient for first part of spectrum for n-th vibration mode

\({\tau_n}\), \({\chi_n}\) :

Eigenmode coefficient for second part of spectrum for n-th vibration mode

φ :

Section rotation

\({\hat{\chi}}\) :

Eigenmode coefficient for first part of spectrum, doubly clamped beam

\({\chi}\) :

Eigenmode coefficients for second part of spectrum, doubly clamped beam

\({\omega}\) :

Angular frequency

\({\tilde{\omega}}\) :

Angular frequency at the transition value (cut-off frequency)

\({\omega^{\star}}\) :

Limiting value (upper/lower bound) for angular frequency

\({\omega_n}\) :

Angular frequency (theoretical value) for n-th vibration mode

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Authors and Affiliations

  1. DICAAR—Department of Civil and Environmental Engineering and Architecture, University of Cagliari, 2, via Marengo, 09123, Cagliari, Italy

    Antonio Cazzani

  2. DADU—Department of Architecture, Design and Urban Planning, University of Sassari, Asilo Sella, 35, via Garibaldi, 07041, Alghero, SS, Italy

    Flavio Stochino & Emilio Turco

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  1. Antonio Cazzani
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Cazzani, A., Stochino, F. & Turco, E. On the whole spectrum of Timoshenko beams. Part II: further applications. Z. Angew. Math. Phys. 67, 25 (2016). https://doi.org/10.1007/s00033-015-0596-9

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