The effect of a concentrated mass on the modal characteristics of a rotating cantilever beam H H Yoo * , S Seo and K Huh School of Mechanical Engineering, Hanyang University, Seoul, Korea Abstract: Modal characteristics of rotating cantilever beams with a concentrated mass located in an arbitrary position are investigated in this paper. Equations of motion are derived by employing hybrid deformation variables. The resulting equations are linear but capture the stiffening effect induced by the rotational motion of the beam. For modelling of the concentrated mass, use is made of the Dirac delta function, which avoids increasing the degrees of freedom of the system. The resulting equations of motion are transformed into dimensionless forms in which four dimensionless parameters are identied. The effects of the dimensionless parameters on the modal characteristics of the rotating beams are examined through numerical study. It is found that the magnitude and the location of the concentrated mass signicantly inuence the modal characteristics of the rotating beam. Keywords: rotating cantilever beams, concentrated mass, equations of motion, hybrid deformation variables NOTATION a P acceleration vector of the generic point P ^ a 1 , ^ a 2 , ^ a 3 orthogonal unit vector xed to the rigid hub A cross-sectional area of the beam E Young’s modulus I 2 , I 3 two principal second area moments of inertia of the beam cross-section L undeformed length of the beam m magnitude of the concentrated mass p position vector of point P with respect to point O q 1i , q 2i , q 3i generalized coordinates corresponding to s, u 2 and u 3 r radius of the rigid hub (distance from rotation centre to point O) s arc length stretch of the neutral axis T reference period u 1 , u 2 , u 3 measure numbers of deformation in the directions of ^ a 1 , ^ a 2 and ^ a 3 (shown in F ig. 1) U strain energy of the beam v O velocity vector of point O v P velocity vector of the generic point P x distance from O to P 0 (spatial variable) a concentrated mass divided by the beam mass b the location of concentrated mass divided by the beam length g angular speed of the beam divided by the reference angular speed d hub radius divided by the beam length constant column matrix characterizing the deection shape for synchronous motion m 1 , m 2 , m 3 numbers of generalized coordinates corresponding to q 1i , q 2i and q 3i r mass per unit length of the beam r * modied mass per unit length of the beam t dimensionless time f 1i , f 2i , f 3i spatial functions for s, u 2 and u 3 o ratio of the chordwise bending natural frequency to the reference frequency A angular velocity vector of the rigid hub O angular speed of the rigid hub partial derivative of the symbol with respect to x double differentiation of the symbol with respect to x ¢ time derivative of the symbol T he M S was received on 24 July 2001 and was accepted after revision for publication on 3 September 2001. * Corresponding author: School of M echanical Engineering, Hanyang University, Sungdong-Gu- Haengdang-Don g 17, Seoul, Korea 133-791. 151 C09501 # IMechE 2002 Proc Instn Mech Engrs Vol 216 Part C: J Mechanical Engineering Science