Stability of a rod with the elastic joint exposed to distributed non-conservative load

Numerous features of non-conservative mechanical systems identified in studying their stability determine significant theoretical and practical interest to analyze various options of the calculation schemes. The paper considers stability of a rectilinear rod connected at one end to a joint and loaded with follower and potential loads uniformly distributed along the rod length. The joint was rigid with respect to rotation of the rod end. In order to apply the method of expanding solution to the perturbed motion equation into a series in terms of eigenmodes, the problem of determining the system eigenfrequencies and modes was solved. The cases of separate and combined action of the follower and potential loads were considered. A study was made on the influence of rigidity of the rod fastening and damping in the system on the loads critical values and on the position of the stability region boundaries on the loading parameters plane.

References

[1] Koiter W.T. Unrealistic follower forces. J. Sound Vib., 1996, vol. 194, no. 4, pp. 636–638, doi: https://doi.org/10.1006/jsvi.1996.0383

[2] Sugiyama Y., Langthjem M.A., Ryu B.J. Realistic follower forces. J. Sound Vib., 1999, vol. 225, no. 4, pp. 779–782, doi: https://doi.org/10.1006/jsvi.1998.2290

[3] Elishakoff I. Controversy associated with the so-called «follower forces»: critical overview. Appl. Mech. Rev., 2005, vol. 58, no. 2, pp. 117–142, doi: https://doi.org/10.1115/1.1849170

[4] Beck M. Die Knicklast des einseitig eingespannten, tangential gedr?ckten Stabes. ZAMP, 1952, vol. 3, no. 3, pp. 225–228, doi: https://doi.org/10.1007/BF02008828

[5] Bolotin V.V. Nekonservativnye zadachi teorii uprugoy ustoychivosti [Nonconservative problems of elastic stability theory]. Moscow, Fizmatgiz Publ., 1961. 340 p. (In Russ.).

[6] Pfluger A. Stabilitätsprobleme der Elastostatik. Springer, 1964. 473 p.

[7] Nikolai E.L. On the stability of the rectilinear equilibrium form of a compressed and twisted rod. Izv. Leningr. politekhn. in-ta, 1928, no. 31, pp. 1–26. (In Russ.).

[8] Bigoni D.N., Kirillov O.N., Misseroni D. et al. Flutter and divergence instability in the Pfl?ger column: experimental evidence of the Ziegler destabilization paradox. J. Mech. Phys. Solids, 2018, vol. 116, pp. 99–116, doi: https://doi.org/10.1016/j.jmps.2018.03.024

[9] Elishakoff I. Resolution of the 20th century conundrum in elastic stability. World Scientific, 2014. 352 p.

[10] Sotoudeh Z., Hosking N.S. Stability analysis of columns with imperfection. AIAA J., 2017, vol. 55, no. 4, pp. 1417–1424, doi: https://doi.org/10.2514/1.J055136

[11] Rastgoo M., Fazelzadeh S.A., Eftekhari M. et al. Flow-induced flutter instability of functionally graded cantilever pipe. Int. J. Acoust. Vib., 2017, vol. 22, no. 3, pp. 320–325, doi: https://doi.org/10.20855/ijav.2017.22.3477

[12] Radin V.P., Samogin Yu.N., Chirkov V.P. et al. Reshenie nekonservativnykh zadach teorii ustoychivosti [Solving nonconservative problems of stability theory]. Moscow, Fizmatlit Publ., 2017. 236 p. (In Russ.).

[13] Radin V.P., Chirkov V.P., Novikova O.V. et al. The damping effect on critical values of nonconservative loads. Problemy mashinostroeniya i nadezhnosti mashin, 2020, no. 2, pp. 46–53, doi: https://doi.org/10.31857/S0235711920020121 (in Russ.). (Eng. version: J. Mach. Manuf. Reliab., 2020, vol. 49, no. 2, pp. 122–128, doi: https://doi.org/10.3103/S1052618820020120)

[14] Fazelzadeh S.A., Tashakorian M., Ghavanloo E. Nonconservative stability analysis of columns with various loads and boundary conditions. AIAA J., 2019, vol. 57, no. 10, doi: https://doi.org/10.2514/1.J057501

[15] Radin V.P., Chirkov V.P., Shchugorev A.V. et al. Stability of bar with flexure hinge under nonconservative loading. Spravochnik. Inzhenernyy zhurnal [Handbook. An Engineering Journal], 2017, no. 10, pp. 36–41. (In Russ.).

[16] Panovko Ya.G. Vnutrennee trenie pri kolebaniyakh uprugikh system [Internal friction at oscillations of elastic systems]. Moscow, Fizmatgiz Publ., 1960. 194 p. (In Russ.).

[17] Ziegler H. Principles of structural stability. Blaisdell, 1968. 150 p. (Russ. ed.: Osnovy teorii ustoychivosti konstruktsiy. Moscow, Mir Publ., 1971. 192 p.)

[18] BolotinV.V., Zhinzher N.I. Effects of damping on stability of elastic systems subjected to nonconservative forces. Int. J. Solids Struct., 1969, vol. 5, no. 9, pp. 965–989, doi: https://doi.org/10.1016/0020-7683(69)90082-1

[19] Volmir A.S. Ustoychivost deformiruemykh system [Stability of deformated systems]. Moscow, Nauka Publ., 1967. 984 p. (In Russ.).