Abstract
This work treats the lateral harmonic forcing, with spatial dependencies, of a two-segment beam. The segments are compact so Timoshenko theory is employed. Initially the external transverse load is assumed to be spatially constant. The goal is the determination of frequency response functions. A novel approach is used, in which material and geometric discontinuities are modeled by continuously varying functions. Here logistic functions are used so potential problems with slope discontinuities are avoided. The approach results in a single set of ordinary differential equations with variable coefficients, which is solved numerically, for specific parameter values, using MAPLEĀ®. Accuracy of the approach is assessed using analytic and assumed mode Rayleigh-Ritz type solutions. Free-fixed and fixed-fixed boundary conditions are treated and good agreement is found. Finally, a spatially varying load is examined. Analytic solutions may not be readily available for these cases thus the new method is used in the investigation.
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Abbreviations
- A i ,:
-
Beam segment cross section area
- E i ,:
-
Beam segment material Youngās modulus
- G i ,:
-
Beam segment material shear modulus
- k i ,:
-
Beam segment material shear coefficient
- M,:
-
Bending moment
- q,:
-
External force per unit length acting on the beam
- f 1, f 2, f 3, f 4, f 5,:
-
Non-dimensional functions for material/geometrical properties
- I i ,:
-
Beam segment cross section area moment of inertia
- L,:
-
Beam length
- P,:
-
Compressive axial force acting on beam
- t,:
-
Time
- V,:
-
Shear force
- w,:
-
Beam displacement in the y direction
- Y,:
-
Non-dimensional beam displacement in the vertical direction
- xyz,:
-
Inertial reference system (coordinates x, y, z)
- x,:
-
Beam axial coordinate
- Ī³,:
-
Shear strain
- Īø,:
-
Rotational angle of the beam cross section
- Ī½,:
-
Non-dimensional frequency (Ī©/Ī©0)
- Ī¾,:
-
Non-dimensional spatial coordinate
- Ļ i ,:
-
Beam segment mass density
- Ļ,:
-
Non-dimensional time
- Ļ i ,:
-
Beam segment material Poissonās ratio
- Ī©,:
-
Frequency
- Ī©0,:
-
Reference frequency
References
Mazzei, A.J., Scott, R.A.: Harmonic forcing of a two-segment Euler-Bernoulli beam. In: Dervilis, N. (ed.) Special Topics in Structural Dynamics, Volume 6: Proceedings of the 35th IMAC, a Conference and Exposition on Structural Dynamics 2017, pp. 1ā15. Springer International Publishing, Cham (2017)
Eisenberger, M.: Dynamic stiffness matrix for variable cross-section Timoshenko beams. Commun. Numer. Methods Eng. 11(6), 507ā513 (1995)
Mazanoglu, K.: Natural frequency analyses of segmented TimoshenkoāEuler beams using the RayleighāRitz method. J. Vib. Control. 23(13), 2135ā2154 (2015)
Mazzei, A.J., Scott, R.A.: Resonances of compact tapered inhomogeneous axially loaded shafts. In: Allemang, R., De Clerck, J., Niezrecki, C., Wicks, A. (eds.) Special Topics in Structural Dynamics, Volume 6, pp. 535ā542. Springer, New York (2013)
Bishop, R.E.D., Price, W.G.: The vibration characteristics of a beam with an axial force. J. Sound Vib. 59(2), 237ā244 (1978)
Esmailzadeh, E., Ohadi, A.R.: Vibration and stability analysis of non-uniform Timoshenko beams under axial and distributed tangential loads. J. Sound Vib. 236(3), 443ā456 (2000)
Saito, H., Otomi, K.: Vibration and stability of elastically supported beams carrying an attached mass under axial and tangential loads. J. Sound Vib. 62(2), 257ā266 (1979)
Irie, T., Yamada, G., Takahashi, I.: Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force. J. Sound Vib. 70(4), 503ā512 (1980)
Kounadis, A.N., Katsikadelis, J.T.: Coupling effects on a cantilever subjected to a follower force. J. Sound Vib. 62(1), 131ā139 (1979)
Cowper, G.R.: The shear coefficient in Timoshenkoās beam theory. J. Appl. Mech. 33(2), 335ā340 (1966)
Chiu, T.C., Erdogan, F.: One-dimensional wave propagation in a functionally graded elastic medium. J. Sound Vib. 222(3), 453ā487 (1999)
Craig, R.R., Kurdila, A., Craig, R.R.: Fundamentals of Structural Dynamics. Wiley, Hoboken (2006)
Kelly, S.G.: Advanced Vibration Analysis. CRC/Taylor & Francis, Boca Raton (2007)
Eslami, M.R.: Finite Elements Methods in Mechanics. Springer, Cham (2014)
Chihara, T.S.: Introduction to Orthogonal Polynomials. Gordon and Breach, London (1978)
Bhat, R.B.: Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. J. Sound Vib. 105(2), 199ā210 (1986)
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Mazzei, A.J., Scott, R.A. (2019). Harmonic Forcing of a Two-Segment Timoshenko Beam. In: Dervilis, N. (eds) Special Topics in Structural Dynamics, Volume 5. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-75390-4_1
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DOI: https://doi.org/10.1007/978-3-319-75390-4_1
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