Natural frequencies of a rotating curved cantilever beam: A perturbation method-based approach:
20 Jan 2020-Vol. 234, Iss: 9, pp 1706-1719
TL;DR: In this article, the blades of propellers, fans, compressor and turbines are modeled as curved beams and the finite element method is employed to determine the modal characterist of the blades.
Abstract: The blades of propellers, fans, compressor and turbines can be modeled as curved beams. In general, for industrial application, finite element method is employed to determine the modal characterist...
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TL;DR: In this article, a layered periodic curved beam (LPCB) concept was introduced to minimize the effect of peak attenuation at a specific narrow frequency range within the attenuation band or disintegrates it into two bands.
14 citations
TL;DR: In this paper, the elastic wave propagation and dispersion characteristics of a curved tapered frame structure were investigated analytically, and it was shown that a small periodic bent angle cross-section produces a complete, viz. axial and flexural band gap in the low-frequency region, and conicity enhances the width of the band.
Abstract:
In this work, the elastic wave propagation and dispersion characteristics of a curved tapered frame structure is investigated analytically. Separately, wave propagation through uniform curved and straight tapered beam were reported in the existing literature; however, no literature reports the influence of simultaneous bent and taper on the wave propagation. In particular, the band characteristics for the curved and tapered beam with two types of cross-sections, i.e., rectangular and circular, are presented. The paper elucidates that introducing a small periodic bent angle cross-section produces a complete, viz. axial and flexural band gap in the low-frequency region, and conicity enhances the width of the band. It is also evidenced that a curved tapered frame with a solid circular cross-section induces a wider band gap than the rectangular section. A complete first normalized bandwidth of 159% is achievable for the circular cross-section and 123% in the case of the rectangular section. The complete result is presented in a non-dimensional framework for wider applicability. An analysis of a finite tapered curved frame structure also demonstrates the attenuating characteristics obtained from the band structure of the infinite structure. The partial wave mode conversion, i.e., generation of coupled axial and flexural mode from a purely axial or flexural mode in an uncoupled medium is observed. This wave conversion is perceived in reflected and transmitted waves while this curved tapered frame is inserted between the two uniform cross-section straight frames.
6 citations
TL;DR: In this article , a simulation model based on magnetocrystalline anisotropy was presented to investigate the vibrational properties of cantilever-based sensors with magnetic hardening.
Abstract: In recent investigations of magnetoelectric sensors based on microelectromechanical cantilevers made of TiN/AlN/Ni, a complex eigenfrequency behavior arising from the anisotropic ΔE effect was demonstrated. Within this work, a FEM simulation model based on this material system is presented to allow an investigation of the vibrational properties of cantilever-based sensors derived from magnetocrystalline anisotropy while avoiding other anisotropic contributions. Using the magnetocrystalline ΔE effect, a magnetic hardening of Nickel is demonstrated for the (110) as well as the (111) orientation. The sensitivity is extracted from the field-dependent eigenfrequency curves. It is found, that the transitions of the individual magnetic domain states in the magnetization process are the dominant influencing factor on the sensitivity for all crystal orientations. It is shown, that Nickel layers in the sensor aligned along the medium or hard axis yield a higher sensitivity than layers along the easy axis. The peak sensitivity was determined to 41.3 T−1 for (110) in-plane-oriented Nickel at a magnetic bias flux of 1.78 mT. The results achieved by FEM simulations are compared to the results calculated by the Euler–Bernoulli theory.
1 citations
TL;DR: In this article, a perturbation method is used to estimate the natural frequencies of rotating twisted beams, where the twist angle and the rotating speed are treated as the perturbations parameters.
Abstract: Structures such as turbomachinery blades, industrial fans, propellers, etc. can be modeled as twisted beams. The study of dynamics of these structures is vital as operational failure of such structures can have catastrophic consequences. As the inclusion of twist and rotation complicates the problem, Finite Element (FE) method is widely used to determine the modal characteristics of rotating twisted beams. In this work, a novel formula is derived to estimate the natural frequencies of rotating twisted beams. The formula is derived using the perturbation method. The twist angle and the rotating speed are treated as the perturbation parameters. In general, the dynamics of rotating twisted beams is coupled in the two transverse planes. However, in the first part of the work the problem is assumed to be uncoupled and it is shown that this assumption is valid under certain cases. In the second part, the problem of general coupled dynamics is solved. Interesting insights based on the formula are presented. The accuracy of the derived formula is verified by comparing it with the literature and FE simulation results. It has been shown that the formula is valid over a fairly large range of twist angles and rotating speeds. In contrast to the detailed FE simulation, the derived analytical formula will be better suited for design iterations in industrial practice.
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Book•
01 Jan 1965
TL;DR: In this paper, a new chapter on computational methods that present the basic principles on which most modern computer programs are developed is presented, which introduces an example on rotor balancing and expands on the section on shock spectrum and isolation.
Abstract: This edition features a new chapter on computational methods that presents the basic principles on which most modern computer programs are developed It introduces an example on rotor balancing and expands on the section on shock spectrum and isolation
2,196 citations
Book•
01 Sep 1981
TL;DR: In this article, the authors discuss the development of Vibration Analysis of Continuous Structural Elements (SSA) and their application in the field of deep shell physics, including the following:
Abstract: Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Historical Development of Vibration Analysis of Continuous Structural Elements References Deep Shell Equations Shell Coordinates and Infinitesimal Distances in Shell Layers Stress-Strain Relationships Strain-Displacement Relationships Love Simplifications Membrane Forces and Bending Moments Energy Expressions Love's Equations by Way of Hamilton's Principle Boundary Conditions Hamilton's Principle Other Deep Shell Theories Shells of Nonuniform Thickness References Radii of Curvature References Equations of Motion for Commonly Occurring Geometries Shells of Revolution Circular Conical Shell Circular Cylindrical Shell Spherical Shell Other Geometries References Nonshell Structures Arch Beam and Rod Circular Ring Plate Torsional Vibration of Circular Cylindrical Shell and Reduction to a Torsion Bar References Natural Frequencies and Modes General Approach Transversely Vibrating Beams Circular Ring Rectangular Plates That are Simply Supported Along Two Opposing Edges Circular Cylindrical Shell Simply Supported Circular Plates Vibrating Transversely Examples: Plate Clamped at Boundary Orthogonality Property of Natural Modes Superposition Modes Orthogonal Modes from Nonorthogonal Superposition Modes Distortion of Experimental Modes Because of Damping Separating Time Formally Uncoupling of Equations of Motion In-Plane Vibrations of Rectangular Plates In-Plane Vibration of Circular Plates Deep Circular Cylindrical Panel Simply Supported at All Edges Natural Mode Solutions by Power Series On Regularities Concerning Nodelines References Simplified Shell Equations Membrane Approximations Axisymmetric Eigenvalues of a Spherical Shell Bending Approximation Circular Cylindrical Shell Zero In-Plane Deflection Approximation Example: Curved Fan Blade Donnell-Mushtari-Vlasov Equations Natural Frequencies and Modes Circular Cylindrical Shell Circular Duct Clamped at Both Ends Vibrations of a Freestanding Smokestack Special Cases of the Simply Supported Closed Shell and Curved Panel Barrel-Shaped Shell Spherical Cap Inextensional Approximation: Ring Toroidal Shell The Barrel-Shaped Shell Using Modified Love Equations Doubly Curved Rectangular Plate References Approximate Solution Techniques Approximate Solutions by Way of the Variational Integral Use of Beam Functions Galerkin's Method Applied to Shell Equations Rayleigh-Ritz Method Southwell's Principle Dunkerley's Principle Strain Energy Expressions References Forced Vibrations of Shells by Modal Expansion Model Participation Factor Initial Conditions Solution of the Modal Participation Factor Equation Reduced Systems Steady-State Harmonic Response Step and Impulse Response Influence of Load Distribution Point Loads Line Loads Point Impact Impulsive Forces and Point Forces Described by Dirac Delta Functions Definitions and Integration Property of the Dirac Delta Function Selection of Mode Phase Angles for Shells of Revolution Steady-State Circular Cylindrical Shell Response to Harmonic Point Load with All Mode Components Considered Initial Velocity Excitation of a Simply Supported Cylindrical Shell Static Deflections Rectangular Plate Response to Initial Displacement Caused by Static Sag The Concept of Modal Mass, Stiffness Damping, and Forcing Steady State Response of Shells to Periodic Forcing Plate Response to a Periodic Square Wave Forcing Beating Response to Steady State Harmonic Forcing References Dynamic Influence (Green's) Function Formulation of the Influence Function Solution to General Forcing Using the Dynamic Influence Function Reduced Systems Dynamic Influence Function for the Simply Supported Shell Dynamic Influence Function for the Closed Circular Ring Traveling Point Load on a Simply Supported Cylindrical Shell Point Load Traveling Around a Closed Circular Cylindrical Shell in Circumferential Direction Steady-State Harmonic Green's Function Rectangular Plate Examples Floating Ring Impacted by a Point Mass References Moment Loading Formulation of Shell Equations That Include Moment Loading Modal Expansion Solution Rotating Point Moment on a Plate Rotating Point Moment on a Shell Rectangular Plate Excited by a Line Moment Response of a Ring on an Elastic Foundation to a Harmonic Point Moment Moment Green's Function References Vibration of Shells and Membranes Under the Influence of Initial Stresses Strain-Displacement Relationships Equations of Motion Pure Membranes Example: The Circular Membrane Spinning Saw Blade Donnell-Mushtari-Vlasov Equations Extended to Include Initial Stresses References Shell Equations with Shear Deformation and Rotary Inertia Equations of Motion Beams with Shear Deflection and Rotary Inertia Plates with Transverse Shear Deflection and Rotary Inertia Circular Cylindrical Shells with Transverse Shear Deflection and Rotary Inertia References Combinations of Structures Receptance Method Mass Attached to Cylindrical Panel Spring Attached to Shallow Cylindrical Panel Harmonic Response of a System in Terms of Its Component Receptances Dynamic Absorber Harmonic Force Applied Through a Spring Steady-State Response to Harmonic Displacement Excitation Complex Receptances Stiffening of Shells Two Systems Joined by Two or More Displacement Suspension of an Instrument Package in a Shell Subtracting Structural Subsystems Three and More Systems Connected Examples of Three Systems Connected to Each Other References Hysteresis Damping Equivalent Viscous Damping Coefficient Hysteresis Damping Direct Utilization of Hysteresis Model in Analysis Hysteretically Damped Plate Excited by Shaker Steady State Response to Periodic Forcing References Shells Made of Composite Material Nature of Composites Lamina-Constitutive Relationship Laminated Composite Equation of Motion Orthotropic Plate Circular Cylindrical Shell Orthotropic Nets or Textiles Under Tension Hanging Net or Curtain Shells Made of Homogeneous and Isotropic Lamina Simply Supported Sandwich Plates and Beams Composed of Three Homogeneous and Isotropic Lamina References Rotating Structures String Parallel to Axis of Rotation Beam Parallel to Axis of Rotation Rotating Ring Rotating Ring Using Inextensional Approximation Cylindrical Shell Rotating with Constant Spin About Its Axis General Rotations of Elastic Systems Shells of Revolution with Constant Spin About Their Axes of Rotation Spinning Disk References Thermal Effects Stress Resultants Equations of Motion Plate Arch, Ring, Beam, and Rod Limitations Elastic Foundations Equations of Motion for Shells on Elastic Foundations Natural Frequencies and Modes Plates on Elastic Foundations Ring on Elastic Foundation Donnell-Mushtari-Vlasov Equations with Transverse Elastic Foundation Forces Transmitted Into the Base of the Elastic Foundation Vertical Force Transmission Through the Elastic Foundation of a Ring on a Rigid Wheel Response of a Shell on an Elastic Foundation to Base Excitation Plate Examples of Base Excitation and Force Transmission Natural Frequencies and Modes of a Ring on an Elastic Foundation in Ground Contact at a Point Response of a Ring on an Elastic Foundation to a Harmonic Point Displacement References Similitude General Similitude Derivation of Exact Similitude Relationships for Natural Frequencies of Thin Shells Plates Shallow Spherical Panels of Arbitrary Contours (Influence of Curvature) Forced Response Approximate Scaling of Shells Controlled by Membrane Stiffness Approximate Scaling of Shells Controlled by Bending Stiffness References Interactions with Liquids and Gases Fundamental Form in Three-Dimensional Curvilinear Coordinates Stress-Strain-Displacement Relationships Energy Expressions Equations of Motion of Vibroelasticity with Shear Example: Cylindrical Coordinates Example: Cartesian Coordinates One-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Inviscid Compressible Liquids and Gases (Acoustics) Interface Boundary Conditions Example: Acoustic Radiation Incompressible Liquids Example: Liquid on a Plate Orthogonality of Natural Modes for Three-Dimensional Solids, Liquids, and Gases References Discretizing Approaches Finite Differences Finite Elements Free and Forced Vibration Solutions References Index
1,166 citations
"Natural frequencies of a rotating c..." refers background or methods in this paper
...E1 and E2 denote the two extreme positions: (a) first mode shape, (b) second mode shape, (c) third mode shape and (4) fourth mode shape....
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...From equation (4), we note that is a function of the dimensional frequency ~ !....
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...Thus, evaluating the left- and right-hand sides of equation (4) at ~ ! 1⁄4 ~ !j and squaring both sides, we get...
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...At each ð ~ != Þj, the values of and are obtained by using equation (4)....
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Book•
07 Aug 1985
TL;DR: Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free oscillations of Systems with Quadratic Nonlinearity General Systems with Odd nonlinearities Nonlinear Systems Subject to Harmonic Excitations Multifrequency Excitations Parametric Excitations Boundary-Layer Problems Linear Equation with Variable Coefficients Differential Equations and a Large Parameter Solvability Conditions Index
Abstract: Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free Oscillations of Systems with Quadratic Nonlinearities General Systems with Odd Nonlinearities Nonlinear Systems Subject to Harmonic Excitations Multifrequency Excitations Parametric Excitations Boundary-Layer Problems Linear Equations with Variable Coefficients Differential Equations with a Large Parameter Solvability Conditions Index
429 citations
TL;DR: In this paper, a new dynamic modeling method was proposed to derive the equation of motion of a rotating cantilever beam, which can be directly used for the vibration analysis including the coupling effect, which could not be considered in the conventional modelling method.
320 citations
"Natural frequencies of a rotating c..." refers background in this paper
...The derivation of the centrifugal effect in the potential energy expression (third term in equation (23)) and the kinetic energy expression as given above, is presented in detail in the Appendix....
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...The Lagrangian is defined as L 1⁄4 T V:Here, V is given by equation (23) and T is given by the simplified from of equation (25) (given above)....
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217 citations
"Natural frequencies of a rotating c..." refers background or methods in this paper
...1=R ð Þ ! 0, the governing equation (1) as well as the natural boundary conditions (2b) reduce to their straight Euler–Bernoulli beam counterparts....
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...Figure 3 shows the comparison of the mode shape for a curved cantilever beam with opening angle 40 obtained using (1) exact method (equation (8)) and (2) approximate method (equation (15))....
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...Thus, there are two choices of shape function available for transverse displacement, namely (1) the exact mode shape (as given in equation (8)) and (2) the simplified mode shape (as given in equation (15))....
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