Structural stability of a light rotating beam under combined loads

Abstract

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms Abstract This paper investigates the structural stability of long boring or milling tools. The tool is modelled by a rotating cantilever beam that is subject to compression and torsion, manifested by semi-tangential torque. The three dimensional mathematical model is based on Euler-Bernoulli beam theory considering a linear three-dimensional problem. We obtain a dimensionless relationship between the relative importance of rotation, compression, and torsion that reveals the stability boundaries of the system.Consider a long boring or a milling tool (see Fig. 1(a)) modelled by a straight vertical cantilever beam. The beam rotates about its vertical axis as well as being subjected to torsion M t and compression mg. Due to the presence of torsion, we are not able to analyse the system in two-dimensions [1]. The compression can be modelled by a lumped mass m attached to the free end of the beam (see Fig. 2(a)) that is much larger than the mass of the beam. Thus, the mass of the beam might be neglected. The beam is considered to be prismatic, homogeneous, linearly elastic and inextensible. It is either in compression or in tension depending on whether it stands upward or downward, respectively. The described system might become unstable depending upon the speed of rotation, the compression, the torsion, or a combination of all three [1].The arrangement of the model and the corresponding notation can be seen in Fig. 2(a) where the gravitational acceleration is denoted by g, the angular velocity is ω, the centrifugal force is mω 2 d 1 , the compression is mg and the torsional moment vector is M t . Note that the twisting moment is assumed to be semi-tangential [3,4] depicted in Fig. 1(b), that is, the forces F acting on the beam generate an axial torque M t that is able to tilt about both the y and z axes. By taking into account only small displacement r = col v w and angles ψ , θ during buckling, the linearised form of the torque is M t = M t col 1 δ v /2 δ w /2 where M t = 4F a, and the bending components of M t come from its resolution with respect to the principal system (ξ, η, ζ) and by using the definition of the semitangential torque in the sense of Ziegler [3] (see Fig. 1(b) and (c)). In case of the principal system,

Effect of axial force on the stability of milling: Local bifurcations around stable islands

“…It is well known that the lateral stiffness of bearings exposes a hardening/softening character with respect to the axial force that is either a compressive or a tensile (see Abele et al (2010); Harris and Kotzalas (2006)). This way, a constant axial force can increase or decrease the natural frequency of the machine tool (see Beri et al (2017); Beri and Stepan (2018)). In case of the axial component of the regenerative cutting force, the dynamic behaviour of the system is increasingly intricate (see Beri and Stepan (2018)).…”