EIGENSOLUTIONS OF ANNULAR-LIKE ELASTIC DISKS WITH INTENTIONALLY REMOVED OR ADDED MATERIAL

Abstract Many examples of elastic, isotropic, stationary annular-like disks are studied analytically and experimentally for free-free and clamped-free boundary conditions. Natural frequencies and deformation shapes of the first few flexural modes including repeated roots are examined and tabulated. Disks with large circular holes or annular holes or annular slots within the disk body with a volume or mass ratio Γ of 5 to 15% are studied with particular emphasis on mode shapes as they deviate from the regular annular plate modes. Material removal cases via incisions or minor cuts at the disk rim, hub or within the body are not considered in this investigation. Material addition cases are simulated by thickening the outer rim or inner hub regions, for Γvalues up to 60%. The final example considers a gear from a helicopter tail rotor gearbox; it has 8 holes and thick rim and hub. A bi-orthogonal polynomial-trigonometrical shape function series is proposed in the Ritz minimization scheme that employs both classical thin and Mindlin's thick plate theories. The effect of number of terms is evaluated by examining an expansion of the linearly independent basis function and by calculating an overall root mean square (rms) error associated with the prediction of a mode shape. The clamped inner edge is described by 4 alternate models and the impedance boundary condition described was found to be the most satisfactory. Predictions of the semi-analytical Ritz method closely match with measured eigensolutions and results yielded by finite element models. The Ritz method is especially attractive because of significant computational savings in addition to the ease with which it can be integrated within a component mode synthesis or multi-body dynamics framework for forced response or system design studies.

[1]  C. D. Mote,et al.  Vibration and parametric excitation in asymmetric circular plates under moving loads , 1987 .

[2]  P.J.M. van der Hoogt,et al.  Accurate calculation methods for natural frequencies of plates with special attention to the higher modes , 1995 .

[3]  K. Liew,et al.  Buckling And Vibration Of Annular Mindlin Plates With Internal Concentric Ring Supports Subject To In-Plane Radial Pressure , 1994 .

[4]  Nobuyuki Iwatsuki,et al.  The Theoretical Modal Analysis of a Circular Plate With a Solid Shaft , 1994 .

[5]  K. M. Liew,et al.  Transverse Vibration Of Trapezoidal Plates Of Variable Thickness: Unsymmetric Trapezoids , 1993 .

[6]  Nobuyuki Iwatsuki,et al.  THE ESTIMATION AND MEASUREMENT OF THE SOUND POWER RADIATED FROM A CIRCULAR PLATE WITH A SOLID SHAFT , 1994 .

[7]  K. Liew,et al.  Vibration Analysis of Corner Supported Mindlin Plates of Arbitrary Shape Using the Lagrange Multiplier Method , 1994 .

[8]  Yang Xiang,et al.  Vibration of annular sector mindlin plates with internal radial line and circumferential arc supports , 1995 .

[9]  Ming-ran Lee,et al.  Analytical formulations for annular disk sound radiation using structural modes , 1994 .

[10]  Snehashish Chakraverty,et al.  Use of Characteristic Orthogonal Polynomials in Two Dimensions for Transverse Vibration of Elliptic and Circular Plates With Variable Thickness , 1994 .

[11]  C. D. Mote,et al.  Natural Frequencies Of A Thin Disk, Clamped By Thick Collars With Friction At The Contacting Surfaces, Spinning At High Rotation Speed , 1993 .

[12]  W. Soedel,et al.  Fundamental Frequencies of Cutout Square Plates With Clamped Edges , 1971 .

[13]  Charles W. Bert,et al.  Free Vibration Analysis of Annular Plates by the DQ Method , 1993 .

[14]  Peter Hagedorn,et al.  Identification of boundary conditions as a part of model correction , 1995 .

[15]  P.A.A. Laura,et al.  Comments On "Free Vibration Analysis Of Annular Plates By The Dq Method" , 1994 .

[16]  Raymond D. Mindlin,et al.  THICKNESS-SHEAR AND FLEXURAL VIBRATIONS OF A CIRCULAR DISK , 1954 .

[17]  Transverse vibrations of rectangular plates with thickness varying in two directions and with edges elastically restrained against rotation , 1979 .

[18]  K. Liew.  Frequency solutions for circular plates with internal supports and discontinuous boundaries , 1992 .

[19]  V. Thevendran,et al.  Vibration Analysis of Annular Plates with Concentric Supports Using a Variant of Rayleigh-Ritz Method , 1993 .

[20]  Sen-Yung Lee,et al.  Free vibrations of elastically restrained non-uniform plates , 1992 .