Free Vibrations of Imperfect Cantilever Bars under Self-Weight Loading

1987 ◽  
Vol 201 (5) ◽  
pp. 345-347 ◽  
Author(s):  
L N Virgin
Keyword(s):  
Gravitational Field ◽  
Natural Frequency ◽  
Free Vibrations ◽  
Equilibrium States ◽  
Small Oscillations ◽  
Initial Imperfections ◽  
The Relationship ◽  
Weight Loading

This brief study examines the relationship between the natural frequency of small oscillations and the length of vertical cantilever struts in a gravitational field. The analysis uses a Rayleigh approach and includes post-buckled equilibrium states and emphasizes the influence of initial imperfections.

Free vibration analysis and post-critical free vibrations of nanocomposite rotating beams reinforced with graphene platelet

2021 ◽  
pp. 107754632110511
Author(s):  
Arameh Eyvazian ◽  
Chunwei Zhang ◽  
Farayi Musharavati ◽  
Afrasyab Khan ◽  
Mohammad Alkhedher
Keyword(s):  
Natural Frequency ◽  
Thermal Environment ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Weight Fraction ◽  
Free Vibrations ◽  
Graphene Platelet ◽  
The Impact

Treatment of the first natural frequency of a rotating nanocomposite beam reinforced with graphene platelet is discussed here. In regard of the Timoshenko beam theory hypothesis, the motion equations are acquired. The effective elasticity modulus of the rotating nanocomposite beam is specified resorting to the Halpin–Tsai micro mechanical model. The Ritz technique is utilized for the sake of discretization of the nonlinear equations of motion. The first natural frequency of the rotating nanocomposite beam prior to the buckling instability and the associated post-critical natural frequency is computed by means of a powerful iteration scheme in reliance on the Newton–Raphson method alongside the iteration strategy. The impact of adding the graphene platelet to a rotating isotropic beam in thermal ambient is discussed in detail. The impression of support conditions, and the weight fraction and the dispersion type of the graphene platelet on the acquired outcomes are studied. It is elucidated that when a beam has not undergone a temperature increment, by reinforcing the beam with graphene platelet, the natural frequency is enhanced. However, when the beam is in a thermal environment, at low-to-medium range of rotational velocity, adding the graphene platelet diminishes the first natural frequency of a rotating O-GPL nanocomposite beam. Depending on the temperature, the post-critical natural frequency of a rotating X-GPL nanocomposite beam may be enhanced or reduced by the growth of the graphene platelet weight fraction.

scholarly journals Mpemba Effect- the Effect of Time

2021 ◽  
Author(s):  
Jianan Wang
Keyword(s):  
Gravitational Field ◽  
Surface Area ◽  
Time Effect ◽  
Light Quantum ◽  
Unit Surface Area ◽  
Heat Flows ◽  
Unit Surface ◽  
Average Wavelength ◽  
Nature Of Time ◽  
The Relationship

This paper draws the following conclusions on the nature of time by analyzing the relationship between time and speed, the relationship between time and gravitational field, the gravitational redshift of the photon, and the black-body radiation theorem: Time on an object is proportional to the amount of energy flowing out (or in) per unit time (observer’s time) per unit surface area of the object. When an object radiates energy outward: t'=μB(T) =μσT 4=μnhν/st Where t’ is the time on the object, μ is a constant, B(T) is the radiosity，the total energy radiated from the unit surface area of the object in unit time (observer’s time), σ is the Stefan-Boltzmann constant, T is the absolute temperature, n is the number of the photons radiated, ν is the average frequency of the photons radiated, s is the surface area of the object and t is the time on the observer. When the object radiates energy outward, the higher the energy density of the space (for example the stronger the gravitational field of the space), the smaller the radiosity B(T) of the object in the space, the longer the average wavelength of the light quantum emitted by the object, the slower the time on the object, the longer the life of the system. When the object radiates energy outward, the faster the object moves relative to the ether, the higher the energy density of the local space in which the object is located, the smaller the radiosity B(T) of the object, the longer the average wavelength of the light quantum radiated by the object, the slower the time on the object, and the longer the life of the system. When the object radiates energy outward, the higher the temperature of the object, the greater the object's radiosity B(T), the shorter the average wavelength of the light quantum radiated by the object, the faster the time on the object, and the shorter the life of the system. Applying the above conclusions about the nature of time, the author analyzes the Mpemba effect and the inverse Mpemba effect, and reaches the following conclusion: the Mpemba effect is the time effect produced when heat flows from objects into space, and the "inverse" Mpemba effect is the time effect produced when heat flows from space into objects.

scholarly journals COSMOLOGICAL CONSTANT AND THE SPEED OF LIGHT

2001 ◽  
Vol 10 (01) ◽  
pp. 41-48 ◽  
Author(s):  
W. R. ESPÓSITO MIGUEL ◽  
J. G. PEREIRA
Keyword(s):  
Gravitational Field ◽  
Cosmological Constant ◽  
Electromagnetic Waves ◽  
Light Propagation ◽  
Wave Optics ◽  
Speed Of Light ◽  
Positive Cosmological Constant ◽  
Velocity Of Light ◽  
Refractive Medium ◽  
The Relationship

By exploring the relationship between the propagation of electromagnetic waves in a gravitational field and the light propagation in a refractive medium, it is shown that, in the presence of a positive cosmological constant, the velocity of light will be smaller than its special relativity value. Then, restricting again to the domain of validity of geometrical optics, the same result is obtained in the context of wave optics. It is argued that this phenomenon and the anisotropy in the velocity of light in a gravitational field are produced by the same mechanism.

Postbuckling and Free Vibrations of Composite Beams

2007 ◽  
Author(s):  
Samir A. Emam ◽  
Ali H. Nayfeh
Keyword(s):  
Natural Frequency ◽  
Axial Load ◽  
Closed Form Solution ◽  
Composite Beams ◽  
Free Vibrations ◽  
Form Solution ◽  
Mode Shapes ◽  
Axial Stiffness ◽  
Fundamental Natural Frequency ◽  
Transverse Vibrations

An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse vibrations of a composite laminated beam accounting for the midplane stretching are presented. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single equation governing the transverse vibrations. This equation is a nonlinear fourth-order partial-integral differential equation. We find that the governing equation for the postbuckling of a symmetric or antisymmetric composite beam has the same form as that of a metallic beam. A closed-form solution for the postbuckling configurations due to a given axial load beyond the critical buckling load is obtained. We followed Nayfeh, Anderson, and Kreider and exactly solved the linear vibration problem around the first buckled configuration to obtain the fundamental natural frequencies and their corresponding mode shapes using different fiber orientations. Characteristic curves showing variations of the maximum static deflection and the fundamental natural frequency of postbuckling vibrations with the applied axial load for a variety of fiber orientations are presented. We find out that the line-up orientation of the laminate strongly affects the static buckled configuration and the fundamental natural frequency. The ratio of the axial stiffness to the bending stiffness is a crucial parameter in the analysis. This parameter can be used to help design and optimize the composite beams behavior in the postbuckling domain.

Analytical Prediction for Free Response of Rotationally Ring-Shaped Periodic Structures

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Dongsheng Zhang ◽  
Shiyu Wang ◽  
Jianping Liu
Keyword(s):  
Natural Frequency ◽  
Critical Speed ◽  
Particle Number ◽  
Periodic Structures ◽  
Wave Number ◽  
Analytical Prediction ◽  
Free Response ◽  
Centrifugal Effect ◽  
Thin Ring ◽  
The Relationship

The in-plane wave motion is analytically examined to address the stationary deflection, natural frequency splitting, and mode contamination of the rotationally ring-shaped periodic structures (RRPS). The governing equation is developed by the Hamilton's principle where the structure is modeled as a thin ring with equally-spaced particles, and the centrifugal effect is included. The free responses are captured by the perturbation method and determined as closed-form expressions. The results imply that the response of stationary RRPS is characterized as standing wave, and the natural frequencies can split when the wave number n and particle number N satisfying 2n/N = int. Also the splitting behavior is determined by the relative angle between the particle and wave antinode. The coefficients of the mode contamination are also obtained. For rotating RRPS, the invariant deflections due to the centrifugal force are estimated at different rotating speeds. It is found that, for certain waves satisfying 2n/N = int, the natural frequency exceeds that of the corresponding smooth ring at the critical speed, and furthermore, the critical speed of the backward traveling wave is lower than that of the forward one. The contamination coefficients of the two kinds of waves are also obtained and they have different magnitudes. All results verify that the splitting and contamination can be determined by the relationship among the mode order, wave number, particle number, and relative position between the particle and antinode. Numerical examples and comparisons with the existing results in the literature are presented.

On the Relationship between Lenses and Inner Rings in Spiral Galaxies

1985 ◽  
Vol 6 (2) ◽  
pp. 202-205 ◽  
Author(s):  
M. P. Schwarz
Keyword(s):  
Gravitational Field ◽  
Surface Density ◽  
Gas Flow ◽  
Spiral Galaxies ◽  
Background Field ◽  
Barred Galaxies ◽  
The Relationship

AbstractIn simulations of gas flow in the gravitational field of model barred galaxies which we have described elsewhere, structures resembling inner rings have formed. The model rings encircle the bar as observed in real galaxies, but are more elongated than the average inner ring. In this paper we show that the addition of a lens-like component to the background field results in much rounder rings. Indeed the shape and position of the model rings are very sensitive to any steep gradients in the azimuthally averaged surface density near the ends of the bar.

scholarly journals Transient responses in Piezoceramic Multilayer Actuators Taking into Account External Viscoelastic Layer

2020 ◽  
pp. 255-266
Author(s):  
Ludmila Grigoryeva
Keyword(s):  
Natural Frequency ◽  
Elastic Layer ◽  
Initial Conditions ◽  
Mechanical Energy ◽  
Mechanical Damage ◽  
Electrical Energy ◽  
Free Vibrations ◽  
Viscoelastic Layer ◽  
Radial Vibrations ◽  
Cylinders And Spheres

The work develops a generalized approach to the study of thickness (radial) vibrations arising in the piezoceramic plates, cylinders, spheres under electrical loads. The state of the problem and the main approaches, used in the problems of studying the oscillations of electroelastic bodies, are described. The use of multilayer elements with electroded interface surfaces and variable direction of polarization of the layers increases the conversion efficiency of electrical energy into mechanical energy, so multilayer piezoceramic plates, cylinders, spheres with changing polarization directions with electroded interfaces are considered. Because of piezoelectric elements are often embedded in the housing and supplemented with matching layers to protect against mechanical damage, it is necessary to study their effect on the oscillations of the element. The proposed approach makes it possible to study the vibrations of plane, cylindrical and spherical bodies with layers made of various electroelastic and elastic materials. Numerical implementation is carried out using finite differences. Nonstationary oscillations of PZT-4 ceramic elements at zero initial conditions are investigated. Oscillations of multilayer plates, cylinders and spheres with and without an external elastic or viscoelastic reinforcing layer under impulse and harmonic unsteady loads are investigated and compared. There are found own frequencies for 5-layer bodies of different geometry with and without an external layer. The first natural frequency for cylinder and sphere corresponds to the radial mode of oscillations, while the second natural frequency for cylinders and spheres and the first for flat bodies are almost equal and correspond to thickness mode. The transient processes in the elements under impulse loads and the influence of the outer elastic layer (housing or matching layer) are studied, taking into account the Rayleigh attenuation. It is established that for a flat layer the outer layer increases the amplitude and the period of free vibrations after removing the load, and for cylinders and spheres it decreases. The presence of an elastic layer enhances the third and dampens the fourth natural frequency of the transducer, thereby expanding the frequency range of its operation.

scholarly journals A Rotor Looseness‒Rub-Impact‒Resonance Fault Mechanism Modeling

2020 ◽  
pp. 1-12
Author(s):  
HU Jun ◽  
◽  
HU Jun ◽  
Keyword(s):  
Analytical Solution ◽  
Natural Frequency ◽  
Rotating Machinery ◽  
Modeling Method ◽  
Vector Analysis ◽  
Engineering Practice ◽  
Coupling Faults ◽  
Fault Deformation ◽  
Fault Mechanism ◽  
The Relationship

In view of the limitation of the numerical method for coupling faults of rotating machinery, a modeling method for coupling faults of rotor, DeformationMotion Vector Analysis Method, is purposed in order to explain the fault mechanism more thoroughly and universally. Firstly, the “Deformation-Motion” is defined by fault and the relationship between fault deformation and periodic signal is established by Fourier series. Secondly, the “Vector Analysis” analyzes the deformation-motion in the dynamic coordinate system, and establishes the relationship between the fault deformation-motion and the periodic excitation force by using the dynamic and static coordinate systems. Thirdly, the analytical solution is obtained by harmonic balance method, and the coriolis acceleration term is first found in the analytical solution of rotating machinery failure. By further analysis of the analytical solution, we find that the general trend of the natural frequency of the rotor system will gradually approach the excitation source frequencies and eventually lead to the resonance: that is the law of Natural Frequency Migration Resonance Theory. Finally, the validity of the modeling method and the correctness of the analytical solution are verified by engineering practice

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