On the Vibration of a Rotating Disk

1972 ◽  
Vol 39 (4) ◽  
pp. 1143-1144 ◽  
Author(s):  
S. Barasch ◽  
Y. Chen
Keyword(s):  
Approximate Calculation ◽  
Initial Conditions ◽  
Rotating Disk ◽  
Equation Of Motion ◽  
Outer Radius ◽  
Order Equation ◽  
System Of Differential Equations ◽  
Fourth Order Equation ◽  
First Order ◽  
Inner Radius

The equation of motion of a rotating disk, clamped at the inner radius and free at the outer radius, is solved by reducing the fourth-order equation of motion to a set of four first-order equations subject to arbitrary initial conditions. A modified Adams’ method is used to numerically integrate the system of differential equations. Results show that Lamb-Southwell’s approximate calculation of the frequency is justified.

Displacement and Stress Fields in a Functionally Graded Fiber-Reinforced Rotating Disk With Nonuniform Thickness and Variable Angular Velocity

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Y. Zheng ◽  
H. Bahaloo ◽  
D. Mousanezhad ◽  
A. Vaziri ◽  
H. Nayeb-Hashemi
Keyword(s):  
Radial Direction ◽  
Volume Fraction ◽  
Rotating Disk ◽  
Outer Radius ◽  
Functionally Graded ◽  
Circumferential Direction ◽  
Stress Fields ◽  
Fiber Reinforced ◽  
Inner Radius ◽  
Nonuniform Thickness

Displacement and stress fields in a functionally graded (FG) fiber-reinforced rotating disk of nonuniform thickness subjected to angular deceleration are obtained. The disk has a central hole, which is assumed to be mounted on a rotating shaft. Unidirectional fibers are considered to be circumferentially distributed within the disk with a variable volume fraction along the radius. The governing equations for displacement and stress fields are derived and solved using finite difference method. The results show that for disks with fiber rich at the outer radius, the displacement field is lower in radial direction but higher in circumferential direction compared to the disk with the fiber rich at the inner radius. The circumferential stress value at the outer radius is substantially higher for disk with fiber rich at the outer radius compared to the disk with the fiber rich at the inner radius. It is also observed a considerable amount of compressive stress developed in the radial direction in a region close to the outer radius. These compressive stresses may prevent any crack growth in the circumferential direction of such disks. For disks with fiber rich at the inner radius, the presence of fibers results in minimal changes in the displacement and stress fields when compared to a homogenous disk made from the matrix material. In addition, we concluded that disk deceleration has no effect on the radial and hoop stresses. However, deceleration will affect the shear stress. Tsai–Wu failure criterion is evaluated for decelerating disks. For disks with fiber rich at the inner radius, the failure is initiated between inner and outer radii. However, for disks with fiber rich at the outer radius, the failure location depends on the fiber distribution.

scholarly journals Transient Thermoelastic Analysis of Pressurized Rotating Disks Subjected to Arbitrary Boundary and Initial Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Mohammad Zamani Nejad ◽  
Azam Afshin
Keyword(s):  
Thermal Stresses ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Separation Of Variables ◽  
Outer Radius ◽  
Rotating Disks ◽  
Arbitrary Boundary ◽  
Inner Radius ◽  
Transient Thermal

This paper focuses on exact analytical solution of transient thermoelastic behaviors of rotating pressurized disks subjected to arbitrary boundary and initial conditions. The pressure, inner radius, and outer radius are considered constant. The basic thermoelasticity theory under generalized assumptions is used to solve the thermoelastic problem. Using the method of the separation of variables, the relations of temperature and transient thermal stresses in the radial direction are obtained. In the case study, the disk is considered under heat flux. Some useful discussions and numerical examples are presented. The analytical results were compared with those of the finite element method and good agreement was found. The relations obtained in this paper can be applied to any arbitrary boundary and initial conditions.

Natural Frequencies of Thick Rotating Annular Discs Using an Integrated Method

2003 ◽  
Author(s):  
Yuan Mao Huang ◽  
C. C. Lai
Keyword(s):  
Natural Frequencies ◽  
Plate Theory ◽  
Outer Radius ◽  
Interactive Software ◽  
The Finite Element Method ◽  
First Order ◽  
Integrated Method ◽  
Subspace Iteration ◽  
Inner Radius ◽  
Existing Data

An integrated method that combines the first order plate theory, Hamilton’s principle, the finite element method and the subspace iteration method is used to calculate the natural frequencies of thick rotating annular discs. The shear deformation and the rotary inertia of elastic discs with the uniform thickness are considered. Interactive software is generated for personal computers, and the effects of the disc rotational speed, the ratio of the disc thickness to the disc outer radius and the ratio of the disc inner radius to the disc outer radius on natural frequencies are analyzed. Comparison of calculated natural frequencies of discs shows good correlation with existing data. It is expected that this method can provide more accurate results compared with existing methods.

Amplitudes Decay in Different Kinds of Nonlinear Oscillators

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Mohammad A. Al-Shudeifat
Keyword(s):  
Nonlinear Oscillator ◽  
Initial Conditions ◽  
A Priori ◽  
Nonlinear Oscillators ◽  
Transformation Method ◽  
Equation Of Motion ◽  
Time Varying ◽  
Initial Displacement ◽  
Restoring Force ◽  
First Order

A transformation is employed to obtain expressions for the decay of the displacement, the velocity, and the energy for various forms of nonlinear oscillators. The equation of motion of the nonlinear oscillator is transformed into a first-order decay term plus an energy term, where this transformed equation can be decoupled into a set of two analytically solvable equations. The decoupled equations can be solved for the decay formulas. Unlike other methods in the literature, this transformation method is directly applied to the equation of motion, and an approximate solution is not required to be known a priori. The method is first applied to a purely nonlinear oscillator with a non-negative, real-power restoring force to obtain the decay formulas. These decay formulas are found to behave similarly to those of a linear oscillator. In addition, these formulas are employed to obtain an accurate formula for the frequency decay. Based on this result, the exact frequency formula given in the literature for this oscillator is generalized by substituting the initial values of the envelopes for the actual initial conditions. By this modification, the formulas for the initial and time-varying frequencies become valid for any combination of the initial displacement and velocity. Furthermore, a generalized nonlinear oscillator for which the transformation is always valid is introduced. From this generalized oscillator, the proposed transformation is applied to analyze various types of oscillators.

Analysis of optimal superconvergence of an ultraweak-local discontinuous Galerkin method for a time dependent fourth-order equation

2020 ◽  
Vol 54 (6) ◽  
pp. 1797-1820
Author(s):  
Yong Liu ◽  
Qi Tao ◽  
Chi-Wang Shu
Keyword(s):  
Discontinuous Galerkin ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Local Discontinuous Galerkin Method ◽  
Order Equation ◽  
Fourth Order Equation ◽  
One Dimensional ◽  
First Order ◽  
Local Discontinuous Galerkin ◽  
Derivatives Of

In this paper, we study superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al. [To appear in Math. Comput. DOI: https://doi.org/10.1090/mcom/3562 (2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this superconvergence is proved to be k + min(3, k) when piecewise ℙk polynomials with k ≥ 2 are used. We also prove a 2k-th order superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove superconvergence of (k + 2)-th and (k + 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k ≥ 2. Numerical experiments verify that all theoretical findings are sharp.

Computer Analysis of Flexural Vibrations of a Rotating Disk With an Arbitrary Profile

1965 ◽  
Author(s):  
J. J. Blech
Keyword(s):  
Boundary Conditions ◽  
Turbine Blades ◽  
Rotating Disk ◽  
Outer Radius ◽  
Mode Shapes ◽  
Determinantal Equation ◽  
Arbitrary Profile ◽  
Solid Disk ◽  
Inner Radius ◽  
Flexural Vibrations

A collocation method is applied to the differential equation and boundary conditions which govern the flexural vibrations of a rotating thin solid disk with an arbitrary profile, clamped at the inner radius and having a flexible ring carrying masses (turbine blades) at the outer radius. The determinantal equation, from which approximations to the reasonance frequencies and mode shapes are extracted, is derived. The results of sample calculations are compared with theoretical and test results. The treatment of other types of boundary conditions is straightforward. A brief discussion of the convergence of this method is given.

Particle motion with air resistance

2012 ◽  
Vol 8 (1) ◽  
pp. 1-15
Author(s):  
Gy. Sitkei
Keyword(s):  
Basic Equation ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Terminal Velocity ◽  
Dimensionless Form ◽  
Wood Industry ◽  
Acceptable Error ◽  
General Validity ◽  
Practical Applications ◽  
Air Resistance

Motion of particles with air resistance (e.g. horizontal and inclined throwing) plays an important role in many technological processes in agriculture, wood industry and several other fields. Although, the basic equation of motion of this problem is well known, however, the solutions for practical applications are not sufficient. In this article working diagrams were developed for quick estimation of the throwing distance and the terminal velocity. Approximate solution procedures are presented in closed form with acceptable error. The working diagrams provide with arbitrary initial conditions in dimensionless form of general validity.

Improved Diffuser/Volute Combinations for Centrifugal Compressors

2008 ◽  
Vol 130 (1) ◽  
Author(s):  
T. Steglich ◽  
J. Kitzinger ◽  
J. R. Seume ◽  
R. A. Van den Braembussche ◽  
J. Prinsier
Keyword(s):  
Cross Sections ◽  
Pressure Ratio ◽  
Pressure Rise ◽  
Outer Radius ◽  
Inner Radius ◽  
Pressure Pipeline ◽  
The Cross ◽  
Improved Performance ◽  
High Pressure Pipeline ◽  
Original Configuration

Internal volutes have a constant outer radius, slightly larger than the diffuser exit radius, and the circumferential increase of the cross section is accommodated by a decrease of the inner radius. They allow the design of compact radial compressors and hence are very attractive for turbochargers and high-pressure pipeline compressors, where small housing diameters have a favorable impact on weight and cost. Internal volutes, however, have higher losses and lower pressure rise than external ones, in which the center of the cross sections is located at a larger radius than the diffuser exit. This paper focuses on the improvement of the internal volute performance by taking into account the interaction between the diffuser and the volute. Two alternative configurations with enhanced aerodynamic performance are presented. The first one features a novel, nonaxisymmetric diffuser̸internal volute combination. It demonstrates an increased pressure ratio and lower loss over most of the operating range at all rotational speeds compared with a symmetric diffuser̸internal volute combination. The circumferential pressure distortion at off design operation is slightly larger than in the original configuration with a concentric vaneless diffuser. Alternatively, a parallel-walled diffuser with low-solidity vanes and an internal volute allows a reduction of the unsteady load on the impeller and an improved performance, approaching that of a vaneless concentric diffuser with a large external volute.

Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects

1973 ◽  
Vol 40 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S. Atluri
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Vibrations ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
Inertia Effects ◽  
Modal Expansion ◽  
General Response

This investigation treats the large amplitude transverse vibration of a hinged beam with no axial restraints and which has arbitrary initial conditions of motion. Nonlinear elasticity terms arising from moderately large curvatures, and nonlinear inertia terms arising from longitudinal and rotary inertia of the beam are included in the nonlinear equation of motion. Using a Galerkin variational method and a modal expansion, the problem is reduced to a system of coupled nonlinear ordinary differential equations which are solved for arbitrary initial conditions, using the perturbation procedure of multiple-time scales. The general response and frequency-amplitude relations are derived theoretically. Comparison with previously published results is made.

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