The Elements by a System of Transfer equations (EST) method offers exact solutions to various vibration problems of trusses, beams, and frames. The method can be regarded as an improved or modified transfer matrix method. Using the EST method, the roundoff errors generated by multiplying transfer arrays are avoided. It is assumed that the bars of trusses are connected by frictionless joints. Longitudinal vibration of a truss bar is described by a differential equation. In a direction perpendicular to the longitudinal axis, no bending can occur. In a transverse direction the rigid bar displacements vary linearly. The rigid bar rotational moment of inertia is taken into account. The transfer equations for the truss bar are presented. The transverse displacements at the joint (node) of an elastic and a rigid bar are equal. The essential boundary conditions at joints for the differential equation are the compatibility conditions of the displacements of truss elements. The natural boundary conditions at joints are the equilibrium equations of longitudinal elastic forces and transverse inertial forces of rigid bars.
1. Pestel, E. C. and Leckie, F. A. Matrix Method in Elastomechanics. McGraw-Hill, New York, 1963.
https://doi.org/10.1115/1.3629714
2. He, B., Rui, X., and Zhang, H. Transfer matrix method for natural vibration analysis of tree system. Math. Probl. Eng., 2012, Article ID 393204.
https://doi.org/10.1155/2012/393204
3. Pilkey,W. D. and Wunderlich, W. Mechanics of Structures: Variational and Computational Methods. CRC Press, Boca Raton, 1994.
4. Lahe, A. The transfer matrix and the boundary element method. Proc. Estonian Acad. Sci. Engng., 1997, 3, 3–12.
5. Lahe, A. The EST Method. Structural Analysis. Tallinn University of Technology Press, Tallinn, 2014.
6. Lahe, A. Varrassüsteemide võnkumine. EST-meetod. Tallinna Tehnikaülikooli kirjastus, Tallinn, 2018 (in Estonian).
7. Ramsay, A. NAFEM benchmark challange No 5: Dynamic characteristics of a truss structure. NAFEMS Benchmark Magazine, 2016, 5, 1–4.
8. Arndt, M. O método dos elementos finitos generalizados aplicado à análise devibrações l’ivres de estruturas reticuladas. PhD thesis. Curitiba, Universiade Federal do Paraná, 2009.
9. Braun, K. A., Dietrich, G., Frik, G., Johnsen, Th. L., Straub, K., and Vallianos, G. Linear Dynamic Analysis. Lecture Notes and Example Problems. ASKA UM 212 Part II, Revision A, Stuttgart, 1982; and ISD-Report No. 155, Stuttgart, 1974.
10. Argyris, J. H. and Mlejnek, H-P. Dynamics of Structures. Elsevier Science Publishers B.V., North-Holland, 1991.
11. Arndt, M., Machado, R. D., and Scremin, A. The Generalized Finite Element Method Applied to Free Vibration of Framed Structures. In Advances in Vibration Analysis Research (Farzad Ebrahimi, ed.). IntechOpen, Rijeka, Croatia, 2011, 187–212.
https://doi.org/10.5772/15545
12. Felippa, C. A., Guo, Q., and Park K. C. Mass matrix templates: general description and 1D examples. Arch. Comput. Meth. Eng., 2015, 22(1), 1–65.
https://doi.org/10.1007/s11831-014-9108-x
13. Guo, Q. Developing Optimal Mass Matrices for Membrane Triangles with Corner Drilling Freedoms. MSci thesis. Department of Aerospace Engineering Sciences, University of Colorado at Boulder, 2012.
14. Lahe, A. Ehitusmehaanika. Tallinna Tehnikaülikooli kirjastus, Tallinn, 2012 (in Estonian).
15. Ceauşu, C., Craifaleanu, A., and Dragomirescu, Cr. Transfer matrix method for forced vibrations of bars. U.P.B. Sci. Bull., Ser. D, 2010, 72, (2), 35–42.
16. Muñoz Romero, J. J. Finite-element Analysis of Flexible Mechanisms Using the Master-Slave Approach with Emphasis on the Modelling of Joints. PhD thesis. Imperial College London, University of London, 2004. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.412409 (accessed 2019-15-02).
17. Zeng, P. Composite element method for vibration analysis of structures, part I: principle and C0 element (Bar). J. Sound Vib., 1998, 218(4), 619–658.
https://doi.org/10.1006/jsvi.1998.1853
