The Effect of Lumped Parameters on Beam Frequencies

Abstract

SummaryAn investigation has been made into the errors involved in using certain lumped parameter methods for the solution of beam frequencies. It is found that existing methods are not consistent for all boundary conditions. A new dynamic stiffness matrix has been formulated, which gives consistently good results for even a few elements. The error in the solution is always inversely proportional to the fourth power of the number of elements used.

Finite elements applied to panel flutter.

“…C ONSIDERABLE progress has been made in the past few years in applying finite-element techniques to dynamic elastic problems. A major development has been in distributing the mass over each element and appears to have been presented first by Leckie and Lindberg, 1 Archer, 2 -3 and Argyris 4 for beam vibration problems. Further applications to plate vibrations have been given by Lindberg, 5 Zienkiewicz and Cheung, 6 Guyan, 7 and Dawe.…”

“…Also included in Table 1 are theoretical values developed from the simplified buckling relation cr cr = CrjE(t/R) (1) The values of C and 17 in Eq. (1) were determined using results obtained by Gerard.…”

“…(1) were determined using results obtained by Gerard. Considering first the plasticity reduction factor rj, Gerard used linear stability theory 1 (2) (3) Values of rj using Eqs. (2) and (3) were computed from modulus data obtained from the compression stress-strain curves shown in Fig.…”

Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

“…Limiting our attention to mechanical problem and choosing, in the literature those works whose spirit is closer to ours, the application of Hencky's modeling concept can be found again in some more recent works which deal with single beams, see [], or with assemblages of beams, see []. Moreover Hencky‐type models can be used to conceive specially targeted lattice materials structured at different scales, see [].…”

The Fundamental Flexural Vibration of a Cantilever Beam of Rectangular Cross Section with Uniform Taper

“…Other methods of calculating the natural frequencies of lateral vibrations of tapered beams have been given by Scanlan and Rosenbaum <8) , namely, Rayleigh's energy method, the Rayleigh-Ritz method, Stodola's method and the Myklestad method. Matrix methods using influence coefficients have been given by Thomson" 0 , Duncan and Collar 110 ', Leckie and Lindberg (11) and Lindberg U2) . Housner and Keightley°3 ) have determined the natural frequencies and the mode shapes for the first three modes by means of a digital computer and Martin (1,1) has derived some correction factors for the effect of taper on flexural frequencies.…”