An Improved Component-Mode Synthesis Method to Predict Vibration of Rotating Spindles and Its Application to Position Errors of Hard Disk Drives

2005 ◽  
Vol 128 (5) ◽  
pp. 568-575 ◽  
Author(s):  
Takehiko Eguchi ◽  
Teruhiro Nakamiya
Keyword(s):  
Mathematical Model ◽  
Natural Frequencies ◽  
Hard Disk ◽  
Equation Of Motion ◽  
Component Mode Synthesis ◽  
Mode Shapes ◽  
Air Bearings ◽  
Improved Method ◽  
Stationary Part ◽  
Improved Model

This paper describes an accurate mathematical model that can predict forced vibration of a rotating spindle system with a flexible stationary part. In particular, we demonstrate this new formulation on a hard disk drive (HDD) spindle to predict its position error signal (PES). This improved method is a nontrivial extension of the mathematical model by Shen and his fellow researchers, as the improved method allows the flexible stationary part to comprise multiple substructures. When applied to HDD vibration, the improved model consists not only a rotating hub, multiple rotating disks, a stationary base, and bearings (as in Shen’s model) but also an independent flexible carriage part. Moreover, the carriage part is connected to the stationary base with pivot bearings and to the disks with air bearings at the head sliders mounted on the far end of the carriage. To build the improved mathematical model, we use finite element analysis (FEA) to model the complicated geometry of the rotating hub, the stationary base and the flexible carriage. With the mode shapes, natural frequencies, and modal damping ratios obtained from FEA, we use the principle of virtual work and component-mode synthesis to derive an equation of motion. Naturally, the stiffness and damping matrices of the equation of motion depend on properties of the pivot and air bearings as well as the natural frequencies and mode shapes of the flexible base, the flexible carriage, the hub, and the disks. Under this formulation, we define PES resulting from spindle vibration as the product of the relative displacement between the head element and the disk surface and the error rejection transfer function. To verify the improved model, we measured the frequency response functions using impact hammer tests for a real HDD that had a fluid-dynamic bearing spindle, two disks, and three heads. The experimental results agreed very well with the simulation results not only in natural frequencies but also in gain and phase.

Modeling and Free Vibration of Flex Circuits in Hard Disk Drives

2003 ◽  
Author(s):  
Jonathan Wickert
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equilibrium Shape ◽  
Local Equilibrium ◽  
Hard Disk Drives ◽  
Laser Interferometry ◽  
Hard Disk ◽  
System Level ◽  
Mode Shapes ◽  
Free Length

A flex circuit connects the stationary electronic components in a hard disk drive to the rotating arm that carries the read/write heads and positions them above data tracks on the disk. Flex circuits are conventionally formed as a laminate of polyimide substrate, adhesive, and copper conductors. Deformation of a flex circuit is discussed in the context of the following stages: the initial unstressed shape, configurations in which stresses set and relax in response to elevated temperature, equilibrium, and small amplitude vibration. The model involves displacements of the flex circuit in the directions tangent and normal to the local equilibrium shape, and those motions couple with the arm’s dynamics. Nonlinearity associated with finite curvature, partial elastic springback, and the arm’s geometry and inertia properties are incorporated within the vibration model to predict system-level natural frequencies, mode shapes, and coupling factors between the circuit and the arm. Laboratory measurements using noncontact laser interferometry validate the model with respect to the circuit’s shape, stiffness, restoring moment, and natural frequencies. The primary degrees of freedom for optimizing flex circuit design are the thicknesses of the individual layers within the circuit, free length, and the locations and slopes of the circuit’s attachment points to the arm and electronics block. The model’s predictions and trends developed from a case study in free length are discussed with a view toward reducing coupling between the circuit and arm in certain vibration modes.

Free vibration analysis of a rectangular plate carrying multiple various concentrated elements

2003 ◽  
Vol 217 (2) ◽  
pp. 171-183 ◽  
Author(s):  
J-S Wu ◽  
H-M Chou ◽  
D-W Chen
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Normal Mode ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Equation Of Motion ◽  
Mode Shapes

The dynamic characteristic of a uniform rectangular plate with four boundary conditions and carrying three kinds of multiple concentrated element (rigidly attached point masses, linear springs and elastically mounted point masses) was investigated. Firstly, the closed-form solutions for the natural frequencies and the corresponding normal mode shapes of a rectangular ‘bare’ (or ‘unconstrained’) plate (without any attachments) with the specified boundary conditions were determined analytically. Next, by using these natural frequencies and normal mode shapes incorporated with the expansion theory, the equation of motion of the ‘constrained’ plate (carrying the three kinds of multiple concentrated element) were derived. Finally, numerical methods were used to solve this equation of motion to give the natural frequencies and mode shapes of the ‘constrained’ plate. To confirm the reliability of previous free vibration analysis results, a finite element analysis was also conducted. It was found that the results obtained from the above-mentioned two approaches were in good agreement. Compared with the conventional finite element method (FEM), the approach employed in this paper has the advantages of saving computing time and achieving better accuracy, as can be seen from the existing literature.

Free Vibration of a Spinning Stepped Timoshenko Beam

2000 ◽  
Vol 67 (4) ◽  
pp. 839-841 ◽  
Author(s):  
S. D. Yu ◽  
W. L. Cleghorn
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Linear Transformations ◽  
Stepped Beam ◽  
Uniform Cross Section ◽  
Vibration Problems

The finite element method is employed in this paper to investigate free-vibration problems of a spinning stepped Timoshenko beam consisting of a series of uniform segments. Each uniform segment is considered a substructure which may be modeled using beam finite elements of uniform cross section. Assembly of global equation of motion of the entire beam is achieved using Lagrange’s multiplier method. The natural frequencies and mode shapes are subsequently reduced with the help of linear transformations to a standard eigenvalue problem for which a set of natural frequencies and mode shapes may be easily obtained. Numerical results for an overhung stepped beam consisting of three uniform segments are obtained and presented as an illustrative example. [S00021-8936(01)00101-5]

A Numerical Simulation of the Head-Disk Assembly in Magnetic Hard Disk Files: Part I—Component Models

1990 ◽  
Vol 112 (4) ◽  
pp. 593-602 ◽  
Author(s):  
O. J. Ruiz ◽  
D. B. Bogy
Keyword(s):  
Numerical Simulation ◽  
Natural Frequencies ◽  
Hard Disk Drives ◽  
Hard Disk ◽  
Coupled System ◽  
Air Bearing ◽  
Disk Drives ◽  
Air Bearings ◽  
Magnetic Hard Disk ◽  
Component Models

In previous papers the dynamics of air bearing sliders used to carry the read/write transducers in magnetic hard disk files has been studied. These studies are useful in evaluating the steady flying and stability of sliders subjected to various disturbances. They are particularly useful in finding the natural frequencies of the air bearings. However, in hard disk drives the sliders are attached to suspensions, which are highly specialized structures that connect the sliders to the positioning actuators. These suspensions have to be relatively stiff in lateral translation, but very flexible in pitch and roll. This latter feature is accomplished by the gimbal or flexure that connects the slider to the end of the suspension. The suspension-gimbal structure has its own natural frequencies, which can be excited by disturbances such as track seeking and impacting the actuator against the crash stop. In order to study the effect of these structures on the head-disk spacing it is necessary to include them in the numerical simulator. In this two part study such a simulator is developed. In Part I the component parts and their interfaces are modeled. In Part II the numerical simulation of the coupled system is accomplished and the numerical results of several sample simulations are presented and discussed.

Closed-Form Solutions of Axisymmetrical Transverse Vibrations of Concave Parabolic Thickness Annular Plates

2009 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
Partial Differential ◽  
First Order ◽  
Annular Plates ◽  
Transverse Vibrations

This paper deals with transverse vibrations of axisymmetrical annular plates of concave parabolic thickness. A closed-form solution of the partial differential equation of motion is reported. An approach in which both method of multiple scales and method of factorization have been employed is presented. The method of multiple scales is used to reduce the partial differential equation of motion to two simpler partial differential equations that can be analytically solved. The solutions of the two differential equations are two levels of approximation of the exact solution of the problem. Using the factorization method for solving the first differential equation, which is homogeneous and includes a fourth-order spatial-dependent operator and second-order time-dependent operator, the general solution is obtained in terms of hypergeometric functions. The first diferential equation and the second differential equation (nonhomogeneous) along with the given boundary conditions give so-called zero-order and first-order approximations, respectively, of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

Analyze on a Coupled Beam Vibration System by FEM (II): Case Study

2014 ◽  
Vol 933 ◽  
pp. 285-290
Author(s):  
Xi Zhang ◽  
Li Yun Yi ◽  
Hui Liu
Keyword(s):  
Finite Element Method ◽  
Natural Frequencies ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
Vibration System ◽  
Beam Vibration ◽  
Mode Localization ◽  
Beam System ◽  
Weakly Coupled

From the article of Analyze on a coupled beam vibration system by FEM (I): theory, equation of motion of the beam system with attachments is established by conventional finite element method, and some parameters, such as natural frequencies, associated mode shapes, are obtained. The mode localization and frequency loci veering phenomena of a weakly coupled beam system with multiple spring-mass systems are investigated. Studies show that for weakly coupled beam system with attachments, the mode localization and loci veering will occur once there is a disorder in the system.

Analyze on a Coupled Beam Vibration System by FEM (I): Theory

2014 ◽  
Vol 933 ◽  
pp. 281-284
Author(s):  
Juan Juan Wen ◽  
Li Yun Yi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Natural Frequencies ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
Vibration System ◽  
Element Analysis ◽  
Generalized Eigenvalue ◽  
Beam System ◽  
Coupled Beams

An approach based on finite element analysis is presented to analyze the free vibration of coupled beams with spring-mass attachments. The equation of motion of the beam system with attachments is established by conventional finite element method. The natural frequencies and the associated mode shapes are obtained from the generalized eigenvalue problem. It is found that the results from the proposed method agree well with those from analytical methods in the literature.

Vibration of Flex Circuits in Hard Disk Drives

2003 ◽  
Vol 125 (3) ◽  
pp. 335-342 ◽  
Author(s):  
J. A. Wickert
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equilibrium Shape ◽  
Local Equilibrium ◽  
Hard Disk Drives ◽  
Laser Interferometry ◽  
Hard Disk ◽  
System Level ◽  
Mode Shapes ◽  
Free Length

A flex circuit connects the stationary electronic components in a hard disk drive to the rotating arm that carries the read/write heads and positions them above data tracks on the disk. Flex circuits are conventionally formed as a laminate of polyimide substrate, adhesive, and copper conductors. Deformation of a flex circuit is discussed in the context of the following stages: the initial unstressed shape, configurations in which stresses set and relax in response to elevated temperature, equilibrium, and small amplitude vibration. The model involves displacements of the flex circuit in the directions tangent and normal to the local equilibrium shape, and those motions couple with the arm’s dynamics. Nonlinearity associated with finite curvature, partial elastic springback, and the arm’s geometry and inertia properties are incorporated within the vibration model to predict system-level natural frequencies, mode shapes, and coupling factors between the circuit and the arm. Laboratory measurements using noncontact laser interferometry validate the model with respect to the circuit’s shape, stiffness, restoring moment, and natural frequencies. The primary degrees of freedom for optimizing flex circuit design are the thicknesses of the individual layers within the circuit, free length, and the locations and slopes of the circuit’s attachment points to the arm and electronics block. The model’s predictions and trends developed from a case study in free length are discussed with a view toward reducing coupling between the circuit and arm in certain vibration modes.

Multi-crack detection in general cross-section swept wings based on natural frequency changes

2020 ◽  
pp. 107754632096401
Author(s):  
Fatemeh Barzegar ◽  
Saeedreza Mohebpour ◽  
Hekmat Alighanbari
Keyword(s):  
Mathematical Model ◽  
Natural Frequency ◽  
Cross Section ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Detection Method ◽  
Mode Shapes ◽  
Modal Strain Energy ◽  
Torsional Mode ◽  
Frequency Changes

In this article, a multi-crack detection method, which is based on natural frequency changes and the concept of modal strain energy, is for the first time developed for the general cross-section swept tapered wings under coupled bending-torsional vibration and applied to the solid and thin-walled airfoil cross-section wings. The presented method is able to handle the problems with an unknown number of cracks and predicts the number of existent cracks, their locations and depths by optimization of an appropriate objective function. The stress intensity factors of airfoil-shaped crack surfaces are obtained using an approximation method. Inputs of the detection method are natural frequencies of uncracked and cracked wings which are calculated by using a mathematical model and finite element method software ANSYS, respectively, and validated by comparison with former research studies. In the mathematical model, the Rayleigh–Ritz method is used to calculate the coupled bending-torsional mode shapes of the uncracked wing and their corresponding natural frequencies. Results demonstrate that the proposed method has precisely predicted the number, locations and depths of cracks in all case studies.

scholarly journals Observations on the vibration of axially tensioned elastomeric pipes conveying fluid

2000 ◽  
Vol 214 (3) ◽  
pp. 423-434 ◽  
Author(s):  
Y L Zhang ◽  
D G Gorman ◽  
J M Reese ◽  
J Horacek
Keyword(s):  
Mathematical Model ◽  
Penalty Function ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Function Technique ◽  
Flowing Fluid ◽  
Linear Dynamic ◽  
Axial Tension ◽  
Pipes Conveying Fluid ◽  
Conveying Fluid

A study of the effect of axial tension on the vibration of a single-span elastomeric pipe clamped at both ends conveying fluid has been carried out both experimentally and theoretically. A new mathematical model using a penalty function technique and the method of kinematic correction and fictitious loads has been developed. The influence of flowing fluid and axial tension on natural frequencies and mode shapes of the system has been described using this model and compared with experimental observations. Linear and non-linear dynamic response of the harmonically excited pipe has also been investigated for varying flow velocities and initial axial tensions.

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