Free Vibrational Characteristics of Pretensioned Cable Roofs

1972 ◽  
Vol 94 (1) ◽  
pp. 31-37 ◽  
Author(s):  
G. N. Bathish
Keyword(s):  
Equilibrium Position ◽  
Equations Of Motion ◽  
Static Equilibrium ◽  
Transverse Load ◽  
Mode Shapes ◽  
Shear Rigidity ◽  
Membrane Theory ◽  
Special Cases ◽  
Vibrational Characteristics ◽  
Nonlinear Membrane

Cable roofs are analyzed using nonlinear membrane theory. The cable network is simulated by an equivalent thin elastic prestressed membrane without shear rigidity. The equations of motion for the vibrating membrane are formulated by considering the static equilibrium position of the membrane as the position of the membrane after it undergoes nonlinear deformation due to uniform transverse load over the projected area of the membrane. The equations of motion are then solved for natural frequencies and associated mode shapes by restricting the analysis to small amplitude vibrations about the static equilibrium position. Two special cases are considered: a flat rectangular membrane, and a membrane that has the shape of a hyperbolic paraboloid surface that is rectangular in plan. Comparisons between linear and nonlinear membrane theory solutions are presented.

Effects of Warping and Pretwist on Torsional Vibration of Rotating Beams

1984 ◽  
Vol 51 (4) ◽  
pp. 913-920 ◽  
Author(s):  
K. R. V. Kaza ◽  
R. E. Kielb
Keyword(s):  
Aspect Ratio ◽  
Torsional Vibration ◽  
Equations Of Motion ◽  
Turbine Blades ◽  
Mode Shapes ◽  
Compressor Blades ◽  
Rotating Beams ◽  
Special Cases ◽  
Torsional Frequency ◽  
Compressor And Turbine Blades

The effect of pretwist and warping on the torsional vibration of short-aspect-ratio rotating beams is examined for application to the modeling of turbofan, turboprop, and compressor blades. The equations of motion and the associated boundary conditions by using both Wagner’s hypothesis and Washizu’s theory are derived and a few minor limitations of the Wagner’s hypothesis, as applied to thick blades, are pointed out and discussed. The equations for several special cases are solved in a closed form. Results are presented indicating the effect of warping, pretwist, and rotation on torsional vibration of beams as aspect ratio is varied. The results show that the structural warping and pretwist terms have a significant effect on torsional frequency and mode shapes of short-aspect-ratio blades whereas the inertial warping terms have negligible effect. Since the torsional frequencies and mode shapes are very important in aeroelastic analyses by using modal methods, the structural warping terms should be included in modeling turbofan, turboprop, compressor, and turbine blades.

Dynamic Characteristics of Resonant Beams With Passive Thermal Stress Regulators

2018 ◽  
Author(s):  
Tyler Kellar ◽  
Pezhman Hassanpour
Keyword(s):  
Boundary Condition ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Mode Shapes ◽  
Elastic Boundary ◽  
Special Cases ◽  
Angular Displacements

This paper addresses the dynamic characteristics of a beam with a particular elastic boundary condition. In this elastic boundary condition, the lateral and angular displacements of the beam are coupled through the elastic constraints. The dynamic characteristic, namely natural frequencies and mode shapes of vibrations are frequently encountered in the design and modeling of resonant micro-structures. The governing equations of motion of the beam is derived using Euler-Bernoulli beam theory considering the elastic coupling between the transverse and rotational displacements of the beam’s end. The characteristic equation for the natural frequencies and mode shapes of vibration is derived by applying the method of separation of variables to the governing partial differential equation of motion. The natural frequencies and mode shapes of the system are derived for various combinations of compliance values of the elastic support and are compared with those of several special cases, namely clamped-free, clamped-guided, clamped-pinned and clamped-clamped beams.

Symmetrical Solutions for Edge-Loaded Annular Elastic Membranes

2009 ◽  
Vol 76 (3) ◽  
Author(s):  
Weiwei Yu ◽  
Dale G. Karr
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Excellent Agreement ◽  
Radial Stress ◽  
Axisymmetric Deformation ◽  
Membrane Theory ◽  
General Solutions ◽  
Special Cases ◽  
Elastic Membranes ◽  
Nonlinear Membrane

The Föppl–Hencky nonlinear membrane theory is employed to study the axisymmetric deformation of annular elastic membranes. The general solutions for displacements and stresses are established for arbitrary edge boundary conditions. New exact solution results are developed for central loading and edge forcing conditions. Both positive and negative radial stress solutions are found. Comparisons are made for special cases to previously known solutions with excellent agreement.

Free Vibration of a Thin-Film Inflated Torus

2003 ◽  
Author(s):  
L. Moxey ◽  
H. Hamidzadeh
Keyword(s):  
Thin Film ◽  
Material Properties ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Inner Pressure ◽  
Special Cases ◽  
The Mathematical Model ◽  
Uniform Membrane

In this paper an overview of the mathematical model for an inflated thin-film toroidal structure is provided. In particular, attention has been confined to determine the free vibrations of inflated circular toroidal members. The provided solution is based on an improved set of equations of motion for uniform membrane. From which the dependence of natural frequencies on material properties, dimension, mode of vibration, and the inner pressure can be investigated. The validity of the developed models was verified by comparing some of the computed results with those available for special cases. Numerical results for natural frequencies and mode shapes are provided for a specific thin-film inflated torus.

scholarly journals Development of a Five-Degree-of-Freedom Seated Human Model and Parametric Studies for Its Vibrational Characteristics

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Jong-Jin Bae ◽  
Namcheol Kang
Keyword(s):  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Human Model ◽  
Whole Body ◽  
Mode Shapes ◽  
Vibrational Characteristics ◽  
Parametric Studies ◽  
Linearized Model ◽  
Nonlinear Equations Of Motion ◽  
Whole Body Vibrations

This study focuses on the biodynamic responses of a seated human model to whole-body vibrations in a vehicle. Five-degree-of-freedom nonlinear equations of motion for a human model were derived, and human parameters such as spring constants and damping coefficients were extracted using a three-step optimization processes that applied the experimental data to the mathematical human model. The natural frequencies and mode shapes of the linearized model were also calculated. In order to examine the effects of the human parameters, parametric studies involving initial segment angles and stiffness values were performed. Interestingly, mode veering was observed between the fourth and fifth human modes when combining two different spring stiffness values. Finally, through the frequency responses of the human model, nonlinear characteristics such as frequency shift and jump phenomena were clearly observed.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

A new dynamic mesh algorithm for studying the 3D transient flow field of tilting pad journal bearings

2016 ◽  
Vol 230 (12) ◽  
pp. 1470-1482 ◽  
Author(s):  
Mengxuan Li ◽  
Chaohua Gu ◽  
Xiaohong Pan ◽  
Shuiying Zheng ◽  
Qiang Li
Keyword(s):  
Flow Field ◽  
Equilibrium Position ◽  
Transient Flow ◽  
Journal Bearings ◽  
Dynamic Mesh ◽  
Static Equilibrium ◽  
Time Step ◽  
Tilting Pad ◽  
Grid Nodes ◽  
Tilting Pad Journal Bearings

A new dynamic mesh algorithm is developed in this paper to realize the three-dimensional (3D) computational fluid dynamics (CFD) method for studying the small clearance transient flow field of tilting pad journal bearings (TPJBs). It is based on a structured grid, ensuring that the total number and the topology relationship of the grid nodes remain unchanged during the dynamic mesh updating process. The displacements of the grid nodes can be precisely recalculated at every time step. The updated mesh maintains high quality and is suitable for transient calculation of large journal displacement in FLUENT. The calculation results, such as the static equilibrium position and the dynamic characteristic coefficients, are consistent with the two-dimensional (2D) solution of the Reynolds equation. Furthermore, in the process of transient analysis, under conditions in which the journal is away from the static equilibrium position, evident differences appear between linearized and transient oil film forces, indicating that the nonlinear transient calculation is more suitable for studying the rotor-bearing system.

Observations on the Steady-State Solution of an Extremely Flexible Spinning Disk With a Transverse Load

1983 ◽  
Vol 50 (3) ◽  
pp. 525-530 ◽  
Author(s):  
R. C. Benson
Keyword(s):  
Steady State ◽  
Hybrid System ◽  
Fourth Order ◽  
Steady State Solution ◽  
Transverse Load ◽  
System Of Differential Equations ◽  
Membrane Theory ◽  
Small Disk ◽  
Spinning Disk ◽  
Flexible Disk

The steady deflection of a transversely loaded, extremely flexible, spinning disk is studied. Membrane theory is used to predict the shapes and locations of waves that dominate the response. It is found that waves in disconnected regions are possible. Some results are presented to show how disk stiffness moderates the membrane waves, the most important result being an upper bound on the highest ordered wave of significant amplitude. A hybrid system of differential equations and boundary conditions is developed to replace the pure membrane formulation that is singular, and the full fourth-order plate formulation that is numerically sensitive. The hybrid formulation retains the salient features of the flexible disk response and facilitates calculations for very small disk stiffnesses.

A Method for Use of Cyclic Symmetry Properties in Analysis of Nonlinear Multiharmonic Vibrations of Bladed Disks

2004 ◽  
Vol 126 (1) ◽  
pp. 175-183 ◽  
Author(s):  
E. P. Petrov
Keyword(s):  
High Efficiency ◽  
Equations Of Motion ◽  
Linear Part ◽  
Forced Response ◽  
Mode Shapes ◽  
Solution Technique ◽  
Disk Model ◽  
Sector Model ◽  
Bladed Disk ◽  
Symmetry Properties

An effective method for analysis of periodic forced response of nonlinear cyclically symmetric structures has been developed. The method allows multiharmonic forced response to be calculated for a whole bladed disk using a periodic sector model without any loss of accuracy in calculations and modeling. A rigorous proof of the validity of the reduction of the whole nonlinear structure to a sector is provided. Types of bladed disk forcing for which the method may be applied are formulated. A multiharmonic formulation and a solution technique for equations of motion have been derived for two cases of description for a linear part of the bladed disk model: (i) using sector finite element matrices and (ii) using sector mode shapes and frequencies. Calculations validating the developed method and a numerical investigation of a realistic high-pressure turbine bladed disk with shrouds have demonstrated the high efficiency of the method.

scholarly journals Relativistic equations for elementary particles

1948 ◽  
Vol 192 (1029) ◽  
pp. 195-218 ◽  
Keyword(s):  
Elementary Particle ◽  
Wave Equation ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Wave Form ◽  
Total Charge ◽  
Field Equations ◽  
Relativistic Approximation ◽  
Special Cases ◽  
Linear Differential

From the general principles of quantum mechanics it is deduced that the wave equation of a particle can always be written as a linear differential equation of the first order with matrix coefficients. The principle of relativity and the elementary nature of the particle then impose certain restrictions on these coefficient matrices. A general theory for an elementary particle is set up under certain assumptions regarding these matrices. Besides, two physical assumptions concerning the particle are made, namely, (i) that it satisfies the usual second-order wave equation with a fixed value of the rest mass, and (ii) either the total charge or the total energy for the particle-field is positive definite. It is shown that in consequence of (ii) the theory can be quantized in the interaction free case. On introducing electromagnetic interaction it is found that the particle exhibits a pure magnetic moment in the non-relativistic approximation. The well-known equations for the electron and the meson are included as special cases in the present scheme. As a further illustration of the theory the coefficient matrices corresponding to a new elementary particle are constructed. This particle is shown to have states of spin both 3/2 and 1/2. In a certain sense it exhibits an inner structure in addition to the spin. In the non-relativistic approximation the behaviour of this particle in an electromagnetic field is the same as that of the Dirac electron. Finally, the transition from the particle to the wave form of the equations of motion is effected and the field equations are given in terms of tensors and spinors.

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