A NEW NUMERICAL METHOD FOR ROTATING SYSTEMS IN ENGINEERING ANALYSIS AND DESIGN SZE PAN PAN DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 A NEW NUMERICAL METHOD FOR ROTATING SYSTEMS IN ENGINEERING ANALYSIS AND DESIGN SZE PAN PAN (B.Eng. (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 Acknowledgements I wish to acknowledge the Singapore Millennium Foundation for its M.Eng scholarship grant that allows me to pursue this research. I am very grateful for the guidance and support by Prof Koh C.G. throughout this project. Thanks to my colleagues and friends, who have helped me in one way or another during the course of my study. i Table of Contents ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY iv NOMENCLATURE v LIST OF FIGURES vii LIST OF TABLES ix 1. INTRODUCTION 1.1 General 1.2 Literature Review 1 1.2.1 1.2.2 1.2.3 1.2.4 Analytical Methods Numerical Methods Moving Element Methods Disk Model 1.3 Objectives and Scope 1.4 Layout of Thesis 2. METHODOLOGY AND FORMULATION 2.1 Formulation of Plate in the Polar Coordinates 2.2 Formulation of Moving Element Method 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 Shape Functions for Degenerated Plate Elements Principle of Virtual Work Bending Stiffness Transverse Shear Stiffness Geometric Stiffness Equation of Motion 2.3 Time Domain Solution to General Rotating Dynamics 3. RESULTS 3.1 Convergence Study 3.1.1 3.1.2 Element Size Convergence Study Time-step Size Convergence Study 3.2 Case 1: Stationary Disk Subjected to Rotating Load (SD-RTL) 3.3 Case 2: Rotating Disk Subjected to Stationary Load (RD-STL) 3.4 Dynamic Response of Disks 3.4.1 3.4.2 3.4.3 In-plane Response Varying Speed Varying Load 8 13 14 16 16 21 21 24 25 27 33 36 40 45 45 46 47 49 52 55 55 56 57 ii 4. PARAMETRIC STUDIES AND APPLICATIONS 4.1 Effect of Membrane Stresses 4.2 Effect of Damping 4.3 Aerodynamic Effect 4.4 Effect of Modulus and Poisson Ratios 73 73 79 81 82 5. CONCLUSIONS AND RECOMMENDATIONS 5.1 Conclusions 5.2 Recommendations for Future studies 97 97 99 REFERENCES 100 APPENDIX 103 iii Summary Moving Element Method (MEM), a new method that incorporates moving co-ordinates into the well-known Finite Element Method (FEM), is a powerful tool in solving dynamics problems of moving loads. It has been shown earlier to be an elegant method for one-dimensional train-track problems and subsequently developed for in-plane dynamic problems of rotating disks. The method is herein further advanced into more important out-of-plane dynamic problems of rotating disks in this thesis. The study involves mainly numerical study to examine the efficiency and accuracy of the results. Dynamic response including displacements, stresses and strains are studied numerically. Two types of moving load problems, namely, stationary disk subjected to rotating transverse load (SD-RTL) and rotating disk subjected to stationary transverse load (RD-STL) are compared. The effects of various parameters are investigated in the study. They are effects of membrane stresses, damping, aerodynamic, modulus and Poisson ratios. The advantages of the MEM as a creative numerical tool in solving rotational dynamics problems are demonstrated. iv Nomenclature Coordinates ( r ,θ , z ) Material coordinates ( r ,η , z ) Material E υ ρ ri ro h D0 D k General κ σ m U T ∇2 t r v a ur , uθ , uz u, v, w ε r , ε θ , ε rθ Space coordinates Modulus of elasticity of the disk material Poisson ratio of the disk material Density of the disk material Inner radius of an annular disk Outer radius of an annular disk Thickness of the disk Eh Elasticity rigidity, −υ Eh3 Flexural rigidity, 12(1 − υ ) Shear correction factor Curvature change of deflection middle plane Membrane stresses Internal moments per unit length of the middle plane Strain energy Kinetic energy Laplace operator in polar coordinates Time Position vector Velocity vector Acceleration vector Displacements in r − , θ − and z − directions Displacements in r − , θ − and z − directions in the middle plane Radial, tangential and shear strains ε r0 , ε θ0 , ε r0θ σ r , σ θ , σ rθ MEM N N w , N r , Nη Radial, tangential and shear strains in the middle plane Nγ rz Shape functions for evaluation of shear in r − z plane Nγη z Shape functions for evaluation of shear in η − z plane W Radial, tangential and shear stresses Shape functions for displacements and slopes Shape functions for w , βη and β r respectively Displacement vector {w1 ϕ r1 ϕη w2 ϕ r ϕη L ϕη } T v D0 Db Ds Kb Ks KG M C K ζ σb εb Bb , B s , B G Pnor , P tan & Ω, Ω ⎡ ⎤ ⎢1 υ ⎥ ⎢ ⎥ Elasticity rigidity, ⎢υ ⎥ ⎢ 1−υ ⎥ ⎢0 ⎥ ⎣ ⎦ ⎡ ⎤ ⎢1 υ ⎥ ⎥ Eh3 ⎢ Bending rigidity, υ ⎥ ⎢ 12(1 − υ ) ⎢ −υ ⎥ ⎢0 ⎥ ⎣ ⎦ kEh ⎡1 ⎤ Shear rigidity, 2(1 + υ ) ⎢⎣0 ⎥⎦ Bending stiffness matrix Transverse shear stiffness matrix Geometric stiffness matrix Equivalent mass matrix Equivalent damping matrix Stiffness matrix (include K b , K s and K G ) Damping coefficient Stress due to bending of plate Stain due to bending of plate B matrices for bending, shear and geometric stiffness Normal and tangential load acting in the in-plane direction Speed and acceleration of rotation ρ ro4 h Ω Ω Dimensionless rotating speed Ω∗ Dimensionless rotating speed (in-plane) Ωro D ρ E vi List of Figures Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 3.13 Fig. 3.14 Fig. 3.15 Fig. 3.16 Fig. 3.17 Fig. 3.18 Model of annular disk subjected to load Mesh of moving elements degree-of-freedom at each node A typical element Sampling points for transverse shear Percentage difference with converged value for a stationary annular disk with various radius ratios subjected to point transverse load at the outer boundary Time history of displacement for disks of different radius ratios subjected to uniform transverse pressure Time history of displacement under point load on a disk of radius ratio 0.5 Deflection profile at the outer radius at different times Displacement for a stationary disk subjected to rotating transverse point load Displacement along outer radius for stationary disk with radius ratio 0.5 subjected to transverse point load rotating at various dimensionless speeds Dimensionless critical speeds at various modes for stationary disks with different radius ratios subjected to rotating transverse point load Deflection profile for disk with radius ratio 0.5 at various critical speeds Displacement for a rotating disk subjected to stationary transverse point load Comparison of displacement under load for the case of RD-STL and RD-RTL Comparison of displacement profiles in cases: SD-STL, RDRTL, SD-RTL and RD-STL at Ω =5.24 Dimensionless critical speeds for rotating disks with different radius ratios subjected to stationary transverse point load at various modes Comparison of results with Adams (1987) Comparison of displacement under load for RD-STL with different inner boundary conditions: Clamped and Simply Supported Rotating speed profile for the disk Radial and circumferential displacement profile for the free rotating disk Transverse displacement profile for rotating disk subjected to rotating uniform transverse pressure Rotating speed profile for the disk 42 43 43 43 44 58 58 60 60 61 63 63 64 65 67 67 68 69 69 70 70 71 71 vii Fig. 3.19 Fig. 3.20 Fig. 3.21 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 4.12 Fig. 4.13 Transverse displacement profile for rotating disk subjected to point transverse load Transverse displacement for disk subjected to harmonic load of frequency 60 rad/s Transverse displacement for disk subjected to harmonic load of frequency 98 rad/s Model of a stationary disk, which is loaded at 30° ≤ η ≤ 30° in the in-plane direction, is subjected to rotating transverse uniform pressure The radial stress , circumferential stress and shear stress distributions due to a patch normal compression load Pnor =7.5E+06Nm-1 acting on −30° ≤ η ≤ 30° at the outer radius Transverse deflection profile at the outer radius when the disk is subjected to four different magnitudes of in-plane compression patch load: Nm-1, 2.5E+06 Nm-1, 7.55E+06 Nm-1and 12.5E+06 Nm-1 Transverse deflection profile at the outer radius when the individual stresses are ignored: all stresses , no radial stress , no circumferential stress and no shear stress Displacement for disk subjected to in-plane patch compression force of three different magnitudes The radial stress, circumferential stress and shear stress distributions due to a patch normal tensile load Pnor =7.5E+06Nm-1 acting on −30° ≤ η ≤ 30° at the outer radius Transverse deflection profile at the outer radius when the disk is subjected to five different magnitudes of in-plane tensile patch load: Nm-1 , 2.5E+06 Nm-1, 7.55E+06 Nm-1, 12.5E+06 Nm1 and 17. 5E+06 Nm-1 Displacement for disk subjected to in-plane patch tensile force of four different magnitudes The radial stress, circumferential stress and shear stress distributions due to a patch normal tensile load P tan=7.5E+06Nm-1 acting on −30° ≤ η ≤ 30° at the outer radius Displacement for disk subjected to in-plane patch tangential force of four different magnitudes Dimensionless critical speeds for stationary disk subjected to uniform in-plane normal loads of various magnitudes at the outer boundary and rotating transverse point load Dimensionless critical speeds for stationary disk subjected to uniform in-plane tangential loads of various magnitudes at the outer boundary and rotating transverse point load Dimensionless critical speeds for rotating disk subjected to uniform in-plane normal loads of various magnitudes at the outer boundary and stationary transverse point load 71 72 72 85 85 86 86 87 88 88 89 90 91 92 92 93 viii w w Ω=0,ζ =0 10 -1 -2 ζ -3 0.02 -4 0.05 -5 Ω Fig. 4.15 Magnification of displacment under point load for disk with different damping coefficient 12 10 Ω 0 200 K 400 600 Fig. 4.16 Dimensionless critical speeds for rotating disk with elastic foundation of different stiffnss 94 12 10 Ω 0 Eθ Er Fig. 4.17 Dimensionless critical speeds for stationary disks with different modulus ratios subjected to rotating transverse point load at various modes 14 12 10 Ω 0 Eθ Er Fig. 4.18 Dimensionless critical speeds for rotating disks with different modulus ratios subjected to stationary transverse point load at various modes 95 20 18 16 14 12 Ω 10 -0.3 -0.2 -0.1 0.1 0.2 0.3 υr Fig. 4.19 Dimensionless critical speeds for rotating disks with modulus ratio 4.0 and different Poisson ratios subjected to stationary transverse point load 96 Conclusions and Recommendations In this thesis, the MEM has been extended to the study of disk in the transverse direction. The findings arising from the numerical study are concluded below, and recommendations for future studies are made. 5.1 Conclusions Formulation is derived based on MEM for dynamic analysis of disk subjected to relative motion with respect to the imposed transverse load. Two important relative motion cases, SD-RTL and RD-STL, are studied. The relative motion is capable of inducing the disk to instability at certain speeds. Different modes are displayed at these speeds. These two cases differ from one another in terms of the additional geometric stiffness in RD-STL. The tensile membrane stress from disk rotation can effectively delay the occurrence of instability. It also shows that any in-plane stresses would affect both the in-plane and transverse behaviors of the disks. Besides the effects of disk rotation and rotating speed, disks with larger radius ratios are found to be stiffer and have higher critical speeds. The study identifies cases when instability cannot be induced. These are SD-STL, RDRTL and a stationary disk subjected to rotating uniform transverse pressure. The first two cases not involve relative motion while the disk equivalent stiffness remains constant throughout the time in the third case. However, when membrane stresses are not axisymmetrical in the third case, instability is induced. It is concluded that that as long as there is periodic change in the disk equivalent stiffness either from the relative motion or 97 the non-axisymmetrical membrane stresses, critical conditions would be reached at certain speeds. Dynamic response of disks is then examined. It is found that when the disk rotates at varying speed, circumferential displacement and shear stress are present as a result of the acceleration (or deceleration) of the disk. These additional effects to the transverse response should not be ignored. Various responses that involve changing rotating speeds and load speeds are investigated. A number of parameters are studied. The relationships between the critical speeds and membrane stresses due to various types of in-plane loading are found. In general, tensile normal forces effectively raise the critical speeds while compressive normal forces reduce these speeds. Tangential loads not affect the behavior greatly and only reduces the critical speeds slightly as a result of the radial stresses induced due to Poisson effect. It is shown that, of all stress compoenents, radial stress exerts the greatest effect. Also, material damping is capable of reducing the magnitude of displacement greatly near critical speed. It also causes asymmetry in the displacement profile. The simplified aerodynamic effect is studied by subjecting the disk to elastic foundation to represent the cushioning effect from the air below the disk. It is found that as the elastic stiffness beneath increases, critical speeds are effectively increased. Finally, the effect of modulus ratio is illustrated. High modulus ratio does not effectively delay the occurrence of 98 instability for disk with radius ratio 0.5. For high modulus ratio of 4.0, negative Poisson values would enhance its behavior. The advantages of MEM are demonstrated in this thesis. For any membrane stress, its effect can be readily added into the formulation. Other effects like aerodynamic and modulus ratio can be incorporated with ease. It has also demonstrated sufficient accuracy when compared with alternative analytical methods. When compared to traditional FEM, MEM does not require the cumbersome task of updating the load position in handling moving load problems as the moving elements ‘move’ and is always stationary relative to the load point. Significant computational cost is saved. In steady state case that involves constant load and constant speed, the dynamic problem can be reduced to an equivalent static problem requiring only one-step solution. It is able to reflect correctly an asymmetrical displacement in the presence of material damping in steady state solution. 5.2 Recommendations for Future Studies This thesis has rather extensively looked into different possibilities of use for MEM and demonstrated its adaptability in different problems. It would further enhance the value of MEM if individual applications can be studied in details with the complicating effects such as temperature, aerodynamic forces and frictions. Experimental study can be conducted to compare with the MEM results for a realistic model in industrial applications (e.g. circular saw blades, computer memory drive and disc brakes in automotives). 99 References Adams, G.G., 1987. Critical Speeds for a Flexible Spinning Disk. International Journal of Mechanical Sciences, 29(8), pp. 525-531. Benson, R.C. and Bogy, D.B., 1978. Deflection of a Very Flexible Spinning Disk Due to a Stationary Transverse Load. Journal of Applied Mechanics, 45, pp. 636-642. Chen, D.Y. and Ren, B.S., 1998. Finite Element Analysis of the Lateral Vibration of Thin Annular and Circular Plates with Variable Thickness. Journal of Vibration and Acoustics, 120, pp.747-752. Chen, J.S., and Jhu, J., 1996. On the In-plane Vibration and Stability of a Spinning Annular Disk. Journal of Sound and Vibration, 195(4), pp.585-593. Chorng, F.L., Ju, F.L. and Ying, T.L., 2000. Axisymmetric Vibration Analysis of Rotating Annular Plates by a 3D Finite Element. International Journal of Solids and Structures, 37, pp.5813-5827. Chung, J. Oh, J.E. and Yoo, H.H., 2000. Non-linear Vibration of a Flexible Spinning Disc with Angular Acceleration. Journal of Sound and Vibration, 231(2), pp.375-391. Deng, T.T., 2002. Computational Modelling for Rotating Disks and Wheels. B.Eng Thesis, National University of Singapore. Huang, H-C., 1989. Static and Dynamic Analyses of Plates and Shells: Theory, Software and Applications. Great Britain: Springer-Verlag Berlin Heidelberg. Kim, B.C., Raman, A. and Mote, C.D., 2000. Prediction of Aeroelastic Flutter in a Hard Disk Drive. Journal of Sound and Vibration, 238(2), pp.309-325. Koh, C.G., Ong, J.S.Y., Chua, D.K.H., and Feng, J., 2003. Moving Element Method for Train-Track Dynamics. International Journal of Numerical Methods of Engineering, 56, pp. 1549-1567. Liang, D.-S., Wang, H.-J. and Chen, L.-W., 2002. Vibration and Stability of Rotating Polar Orthotropic Annular Disks Subjected to a Stationary Concentrated Transverse Load. Journal of Sound and Vibration, 250(5), pp.795-811. Liu C.F., Lee J.F. and Lee Y.T., 2000. Axisymmetric vibration analysis of rotating annular plates by a 3D finite element. International Journal of Solids and Structures, 37, pp. 5813-5827. Moreshwar, D. and Mote, C.D., 2003. In-plane Vibrations of a Thin Rotating Disk. Transactions of the ASME, 125, pp.68-72. 100 Mote, C.D., 1965. Free Vibration of Initially Stressed Circular Disks. Journal of Engineering for Industry, May, pp.258-263. Mote, CD., 1970. Stability of Circular Plates Subjected to Moving Loads. Journal of the Franklin Institute, 290(4), pp. 329-344. Olsson, M., 1991. On the Fundamental Moving Load Problem. Journal of Sound and Vibration, 145, pp. 299-307. Ono, K., Chen, J.S. and Bogy, D.B., 1991. Stability Analysis for the Head-Disk Interface in a Flexible Disk Drive. Journal of Applied Mechanics, 58, pp.1005-1014. Renshaw, A.A., 1998. Critical Speed for Floppy Disks. Journal of Applied Mechanics, 65, pp. 116-120. Renshaw, A.A., Angelo, C.D. and Mote, C.D., 1994. Aerodynamically Excited Vibration of a Rotating Disk. Journal of Sound and Vibration, 177(5), pp.577-590. Renshaw, A.A. and Mote, C.D., 1992. Absence of One Nodal Diameter Critical Speed Modes in an Axisymmetric Rotating Disk. Journal of Applied Mechanics, 59, pp. 687688. Shen, I.Y. and Song, Y., 1996. Stability and Vibration of a Rotating Circular Plate Subjected to Stationary In-plane Edge Loads. Journal of Applied Mechanics, 63, pp.121127. Son, H., Kikuchi, N., Ulsoy, G. and Yigit A.S., 2000. Dynamics of Prestressed Rotating Anisotropic Plates Subject to Transverse Loads and Heat Sources, Part I: Modelling and Solution Method. Part II: Application to a Specially Orthotropic Disk. Journal of Sound and Vibration, 236(3), pp.457-504. Srinivasan, V., and Ramamurti, V., 1980. Dynamic Response of an Annular Disk to a Moving Concentrated, In-plane Edge Load. Journal of Sound and Vibration, 72, pp. 251262. Sze, P.P., 2003. Computational Study for Rotating Disk. B.Eng Thesis, National University of Singapore. Weisensel, G.N., and Schlack, A.L., 1988. Forced Response of a Rotating Thin Annular Plate to a Moving Concentrated Transverse Load. Proceedings of the 6th International Modal Analysis Conference, pp. 1643-1647. Weisensel, G.N., and Schlack, A.L., 1993. Response of Annular Plate to Circumferentially and Radially Moving Loads. Journal of Applied Mechanics, 60, pp. 649-661. 101 Zienkiewicz, O.C. and Taylor, R.L., 1991. The Finite Element Method, v2 (4th ed.). 102 Appendix – Formulation of MEM for in-plane The annular disk is discretized in the same way as in Fig. 2.2. The same element in Fig. 2.4 is used by with only two degrees of freedom (DOFs) i.e. ur and uθ at each node. U = ( ur1 uθ1 u r uθ L ur uθ ) T (A1) Displacement vector u = (u r u θ ) T at any point in the element is expressed in terms of nodal displacements by means of shape functions, as follows. u = NU (A2) where N is a matrix containing the shape functions used: ⎧N N=⎨ ⎩0 N2 L N6 N1 N2 L 0⎫ ⎬ N6 ⎭ (A3) Note that the shape functions are necessarily defined in the (r, η ) coordinate system instead of the (r, θ ) co-ordinate system. The (r, η ) co-ordinate system rotates relative to the disk and may be called the rotating co-ordinate system. Assuming polynomial shapes of the lowest order possible, an example of the shape functions used is N1 = (r − R2 )(η − η )(η − η3 ) ΔRΔη (A4) where R1 and R2 are radii of the outer arc 1-2-3 and inner arc 4-5-6, respectively, η1, η2 and η3 are the polar angles (in radian) of radial lines 1-6, 2-5 and 3-4, respectively. The band width and sectorial angle of the element are, respectively, ΔR = R1 − R2 and Δη = η − η . Accordingly, the displacement within the element is assumed to be linear along radial lines and quadratic along arc lines. 103 The strain vector ε = ( ε rr ε θθ γ rθ ) T can be expressed in terms of nodal displacements as follows. ε = BU (A5) where ⎡ ∂N1 ⎢ ⎢ ∂r N B=⎢ ⎢ r ⎢ ∂N ⎢ r ∂ η ⎣⎢ ∂N1 r ∂η ∂N1 N1 − r ∂r ∂N ∂r N2 r ∂N r ∂η ∂N r ∂η ∂N N − r ∂r ∂N ∂r N6 L r ∂N L r ∂η L ⎤ ⎥ ⎥ ∂N ⎥ r ∂η ⎥ ∂N N ⎥ − ⎥ r ⎦⎥ ∂r (A6) Note again that the above is formulated in the rotating co-ordinate (r, η ) system. The stress vector σ is related to the strain vector through the elasticity matrix D: σ = Dε (A7) For plane stress state, ⎡ ⎤ v ⎥ ⎢ E ⎢v D= ⎥ 1− v2 ⎢ 1− v ⎥ ⎢0 ⎥ ⎦ ⎣ (A8) Substituting Equation (A5) into Equation (A7), we have the following. σ = DBU (A9) The virtual displacement method is applied. Adopting a consistent formulation, the same set of shape functions is used in the virtual system (denoted by over bar). Thus, u = NU (A10) ε = BU (A11) The internal virtual work is given by the following integral over the element considered. 104 WI = ∫ T ε σ dV (A12) Element Using Equations (A9) and (A11) and expressing in the ( r , θ ) coordinate system, it can be shown that WI = U T {∫∫ B DB h r dr dθ}U T (A13) The external virtual work involves nodal forces and inertial forces. The former is represented by nodal force vector F, whereas the latter requires acceleration at every material point in the element. Accordingly, the external virtual work is given by T WE = U F − U T ∫∫ N T ρ a h r dr dθ (A14) where a = (ar aθ ) T is the acceleration vector defined in the fixed co-ordinate system ( r , θ ) . Considering the following position vector for a material point r = (r + u r )e r + u θ e θ (A15) the velocity vector (v) and acceleration vector (a) can be obtained as follows v= dr = (r& + u& r − u θ θ& )e r + (rθ& + u r θ& + u& θ )e θ dt a= dv = (&r& + u&&r − 2u&θ θ& − u θ&θ& − ( r + u r )θ& )e r + (2r&θ& + 2u& r θ& − u θ θ& + r&θ& + u r &θ& + u&&θ )e θ dt (A16) (A17) where r& = &r& = , θ& = Ω , &θ& = at steady-state. Hence ⎡u ⎤ ⎡− u& ⎤ ⎡− 1⎤ && + 2Ω ⎢ θ ⎥ + rΩ ⎢ ⎥ − Ω ⎢ r ⎥ a=u ⎣0⎦ ⎣ u& r ⎦ ⎣u θ ⎦ (A18) 105 The acceleration with reference to the fixed frame would involve acceleration relative to the rotating frame (first term), the Coriolis acceleration (second term) and centrifugal forces (third and fourth terms). By applying the coordinate transformation η = θ + Ωt (A19) the following equations are derived ∂u(r , η) ∂u(r , η) du(r , θ) =Ω + ∂η dt ∂t (A20a) d u(r , θ) ∂ u(r , η) ∂ u(r , η) ∂ u ( r , η) = + 2Ω +Ω dt ∂t ∂t∂η ∂η2 (A20b) Making use of the shape functions as defined in Equations (A4), the above equations can be written as du(r , θ) & = Ωu, η +u& = ΩN, η U + NU dt (A21a) d u(r , θ) && + 2ΩN, U & + Ω N, U && + 2Ωu& , η +Ω u, ηη = NU =u η ηη dt (A21b) For the Coriolis acceleration term in Equation (A18), it is convenient to introduce the following shape function matrix for ease of formulation. ˆ =⎡ N ⎢N ⎣ − N1 − N2 L N2 L N6 − N6 ⎤ ⎥⎦ (A22) Thus, ⎡− u θ ⎤ ˆ ⎢ u ⎥ = NU ⎣ r ⎦ (A23a) ⎡− u&θ ⎤ ˆ ˆ & ⎢ u& ⎥ = Ω N ,η U + N U ⎣ r ⎦ (A23b) 106 Equation (A18) can now be written as && + 2Ω N, U & + Ω N, U + 2Ω N & + rΩ ⎡− 1⎤ − Ω NU ˆ U + 2Ω N ˆU a = NU ,η η ηη ⎢0⎥ ⎣ ⎦ (A24) Equating WE and WI , using the above equation and expressing the integral in the (r, η ) co-ordinate system yield the following equations of motion. M A + C V + K U = P (t ) (A25) where the equivalent mass, damping and stiffness matrices for the element are, respectively, M = ∫∫ N T ρ N h r dr dη C = 2Ω ∫∫ N T ρ N, η h r dr dη + 2Ω (A26a) ∫∫ N T ˆ h r dr dη ρN (A26b) ˆ , h r dr dη K = ∫∫ B T DB h r dr dθ + Ω ∫∫ N T ρ N, ηη h r dr dη + Ω ∫∫ N T ρ N η − Ω ∫∫ N T ρNhrdrdη (A26c) and the equivalent load vector is ⎡− 1⎤ P(t ) = F(t ) − ∫∫ N T ρ r Ω ⎢ ⎥ h r dr dη ⎣0⎦ (A27) where F is the external applied load vector to the rotating disk. By the direct stiffness method, the element matrices and the load vector are assembled to form the corresponding structure matrices and vector, leading to M s A s + C s Vs + K s U s = Ps (t ) (A28) where subscript s denotes the structure (disk). 107 In the present study, the applied load is constant and thus Ps is not a function of time. At steady state with the disk rotating at a constant angular speed Ω , Equation (27) in fact reduces to an equivalent static system K s U s = Ps (A29) For cases when the disk is rotating at non-uniform speed, the acceleration in Equation (A18) becomes ⎡ u ⎤ & ⎡ −uθ ⎤ ⎡ −u& ⎤ ⎡ −1⎤ & ⎡0⎤ && + 2Ω ⎢ θ ⎥ + rΩ ⎢ ⎥ − Ω ⎢ r ⎥ + Ω a=u + rΩ ⎢ ⎥ ⎢1 ⎥ ⎣0⎦ ⎣ ⎦ ⎣ u&r ⎦ ⎣ ur ⎦ ⎣uθ ⎦ (A30) Also, the result of transformation in Equation (A20b) becomes d 2u(r , θ ) & ∂u ∂ 2u(r ,η ) ∂ 2u(r ,η ) ∂ u( r ,η ) = Ω + + Ω + Ω dt ∂η ∂t ∂t ∂η ∂η (A31) As a result the stiffness matrix in Equation (A26) and load vector in Equation (A27) are changed to & NT ρ N K = ∫∫ BT DBhrdrdη − Ω ∫∫ NT ρ Nhrdrdη + Ω ∫∫ ˆ hrdrdη & NT ρ N, hrdrdη ˆ , hrdrdη + Ω +Ω ∫∫ NT ρ N,ηη hrdrdη + 2Ω ∫∫ NT ρ N η η ∫∫ (A32) and ⎡ −1⎤ & ⎡0 ⎤ hrdrdη P (t ) = F (t ) − ∫∫ NT ρ rΩ ⎢ ⎥ hrdrdη − ∫∫ NT ρ rΩ ⎢1 ⎥ ⎣0⎦ ⎣ ⎦ (A33) With the formulation for the RD-SL demonstrated above, the formulation for SD-RL can also be studied as a special case of the above formulation with some changes. In contrast to Equation (A18), since the disk is stationary, the acceleration (of material particle) is simply given by 108 a= d 2u(r , θ) dt (A34) Applying the coordinate transformation leads to && − 2ΩN, U & + Ω N, U && + 2ΩN, η u& + Ω 2u, ηη = NU a=u η ηη (A35) In view of the above, it can be shown that the equivalent mass matrix remains the same as in Equation (A26a). The equivalent damping and stiffness matrices are C = 2Ω ∫∫ N T ρ N, η h r dr dη (A36a) K = ∫∫ B T DB h r dr dθ + Ω ∫∫ N T ρ N, ηη h r dr dη (A36b) Hence, the formulation for the SD-RL case can be treated as a special case in the RD-SL program by simply suppressing the terms associated with the Coriolis effects and centrifugal forces. 109 [...]... lie mainly in finding ways to increase the critical speeds and expand the stability region For instance, a higher rotating speed in circular saw blades would imply a higher production in wood-cutting industry and in disk storage a shorter data access time In the classical plate theory as adopted by most analytical solutions, in- plane membrane forces of rotation are assumed to be unaffected by transverse... introduced at the end of this chapter 1.1 General Rotating disks can be found readily in vehicle wheels, automotive disk brakes, circular saw blades and computer memory disks There is demand on increasing rotating speeds Trains are to travel faster, faster circular saws imply higher productivity and higher rotating speed allow higher data access rates in disk drives The dynamic behavior of the rotating systems. .. membrane stresses on transverse behavior The advantages of MEM in solving problem, that involves relative motion, over traditional FEM are demonstrated It also offers more flexibility than analytical method as various considerations can be added in the formulation with ease With these advantages, the proposed MEM has tremendous application potential to the engineering design and analysis of rotating. .. response of stationary annular disk subjected to a moving concentrated, in- plane edge load Chen and Jhu (1996) applied Bessel function in studying the case of rotating disk subjected to stationary in- plane load Coordinate transformation is applied in both analyses While Chen has only used linear strain relationship in predicting disk response and the existence of divergence instabilities and critical speeds,... these, steady state problem of dynamic responses for a Stationary Disk subjected to Rotating Transverse Load (SD-RTL) and a Rotating Disk subjected to Stationary Transverse Load (RD-STL) are looked into The complete formulation is derived Finally, the time-domain formulations are derived in order to handle problems that involve varying speeds or loads 2.1 Formulation of Plate in the Polar Coordinates Consider... system and damping by air are represented Their effects on the stability of the disk are studied Material Damping Kim et al (2000) highlighted that the aluminum substrate in the hard disk has significant material damping and has to be modeled A damping term assumed to be proportional to the rate of bending strain is added to Equation (1.1) 7 1.2.2 Numerical Method In the last decade, several papers... industrial applications – data storage The first parameter studied is the membrane stresses due to in- plane loading i.e normal/tangential patch/uniform loads on the outer boundary Interesting observations are made for rotating disk subjected to point load located at different positions on the disk Material damping is added to the formulation and its effect to the disk behavior studied A simplified aerodynamic... to stationary load In circular saw blade, as the blade rotates to cut an object, in- plane and transverse forces is exerted to the blade at the contact For a floppy-disk drive, read/write head exerts transverse load to the disk In the case of hard-disk drive, the aerodynamic interaction between the read/write heads and disks gives rise to transverse force on the hard disk On braking, a patch transverse... to rotating load and rotating disk subjected to stationary load In both cases, relative motion between the disk and the load is involved The latter case is different from the former not only in the addition of centrifugal force but also Coriolis effect Therefore, it is only a good approximation for former case of disk rotating at very low speeds Srinivasan and Ramamurti (1980) studied the dynamic response... transverse load is applied at top and bottom of the rotating disc brake In addition to loading, there are different concerns for these applications For circular saw blades, the initial stresses due to rolling and the temperature distribution and the loading at the edge of the disk would be the main concerns For data storage disk, the membrane stresses are significant due to the high rotating speed The . friction force and aerodynamic, for the rotating disk studied. The general interests lie mainly in finding ways to increase the critical speeds and expand the stability region. For instance, a higher. A NEW NUMERICAL METHOD FOR ROTATING SYSTEMS IN ENGINEERING ANALYSIS AND DESIGN SZE PAN PAN DEPARTMENT OF CIVIL ENGINEERING NATIONAL. material damping and the load parameters on the rotating disk have found to play important roles and have been observed. Aerodynamic Adams (1987) obtained the critical speeds at which a spinning