VIBRATIONS OF POWER LINES IN A STEADY WIND: III. THE FREE VIBRATIONS OF A HEAVY STRING

1936 ◽  
Vol 14a (1) ◽  
pp. 16-24
Author(s):  
R. Ruedy
Keyword(s):  
Natural Frequency ◽  
Finite Length ◽  
Standing Waves ◽  
Free Vibrations ◽  
Power Lines ◽  
Middle Portion ◽  
Zero Values ◽  
Hermitian Polynomials ◽  
The Waves

When a long heavy wire is fastened to two supports under such a tension that it is nearly straight, the span behaves in the same manner as a stretched string, but when there is appreciable sag, standing waves form mainly in the middle portion; apart from damping, their shape is described by Hermitian polynomials, which in the case of high overtones resemble sine waves. The change in the tension along the line causes attenuation of the waves. By keeping the ratio [Formula: see text] less than about unity the span is rendered vibration-proof, at least near the ends. The natural frequency of strings of finite length depends on the zero values of the Hermitian polynomials.

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.

VIBRATIONS ON POWER LINES IN A STEADY WIND: V. RESONANCE OF STRINGS WITH STRENGTHENED ENDS

1938 ◽  
Vol 16a (12) ◽  
pp. 215-225
Author(s):  
R. Ruedy
Keyword(s):  
Total Mass ◽  
Linear Density ◽  
Unit Length ◽  
Standing Waves ◽  
Power Lines ◽  
Resonance Frequencies ◽  
The Law

The resonance frequencies, and in particular all the overtones of a string along which the linear density varies according to the law ρ(1 + λx/L)m, are slightly higher than the frequencies of a uniform string of the same total mass when the ratio between the mass of an element at the end and a corresponding element at the centre is varied between 1 and 25. In order to bring a string with strengthened ends into resonance it is necessary not only that the force acting on unit length of the string be of the same frequency as one of the resonance frequencies, and that its strength varies along the string in proportion to the amplitudes of the corresponding standing waves, but it must also be proportional to the mass of each element. It is therefore more difficult to produce true resonance in a string with strengthened ends than in a uniform string.

VIBRATIONS ON POWER LINES IN A STEADY WIND: VI. FORCED VIBRATIONS OF UNIFORM STRINGS AND OF STRINGS WITH STRENGTHENED ENDS UNDER THE ACTION OF UNIFORM PERIODIC FORCES

1939 ◽  
Vol 17a (1) ◽  
pp. 1-13
Author(s):  
R. Ruedy
Keyword(s):  
Plane Wave ◽  
Standing Waves ◽  
Wave Pattern ◽  
Power Lines ◽  
Practical Calculation ◽  
Square Root ◽  
Damping Force ◽  
Force And Motion ◽  
At Resonance ◽  
Point To Point

A study of the complete equation expressing the action of a driving force, periodic in time but constant throughout the length of the string and opposed by a damping force proportional to the velocity, leads to formulae suitable for the practical calculation of the shape of standing waves that are produced by a plane wave of sound or by a steady wind. At resonance the amplitude at the midpoint of a uniform string set into a plane wave of sound is proportional to the diameter, to the square root of the intensity of the wave (G erg per sq. cm. per sec.), and inversely proportional to the order of the overtone and to the square root of the frequency. Damping causes the lag between force and motion to differ from point to point, particularly near the nodes, so that even at resonance the wave pattern is not rigorously stationary. On the average, the lag increases from the value zero, obtained when the ratio v/v0 between applied frequency and fundamental frequency is zero, to ±π/2 when v/v0 = 1, increases again from −π/2 through 0 at v/v0 = 2, to π/2 at v/v0 = 3, and so on.

Broadband Random Phase Diffuser for Ultrasonic Imaging

1979 ◽  
Vol 1 (4) ◽  
pp. 325-332
Author(s):  
Gerard A. Alphonse ◽  
David Vilkomerso
Keyword(s):  
Solid Angle ◽  
Random Phase ◽  
Ultrasonic Imaging ◽  
Standing Waves ◽  
Acoustic Imaging ◽  
Point Sources ◽  
Phase Plate ◽  
Reflected Waves ◽  
Large Aperture ◽  
The Waves

In reflective imaging, waves must be scattered by the object over a broad solid angle so that some of the reflected waves impinge upon the collecting aperture. Surfaces such as biological specimens under study in acoustic imaging are considered smooth at the wavelengths used (e.g., 1 mm) and therefore act as specular reflectors. In order to obtain reflection over a broad spatial range, large aperture, sector or compound scanning are used. In certain types of systems, diffuse insonification is sometimes used by imaging a raster of random phase points onto the surface. However interference between the waves from these point sources produces random fringes or “speckle-like” patterns overlaying the image. In optics these fringes have been reduced by rotating the diffuser. A similar approach has been taken here. This paper describes a simple random phase plate having two levels, 0° and 180 phase that can, by rotation, change the relative phases of the diffuse insonification points so as to reduce the speckle-like effect in the image. The temporal bandwidth of the random phase plate is narrow because of standing waves in it. To reduce standing waves the diffuser is intimately coupled to a wedged transducer. This combination is used to obtain diffuse insonification with broad spatial and temporal bandwidth.

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.

scholarly journals NEGATIVE PHASE VELOCITY IN NONLINEAR OSCILLATORY SYSTEMS — MECHANISM AND PARAMETER DISTRIBUTIONS

2007 ◽  
Vol 21 (23n24) ◽  
pp. 4170-4177 ◽  
Author(s):  
ZHOUJIAN CAO ◽  
PENGFEI LI ◽  
HONG ZHANG ◽  
GANG HU
Keyword(s):  
Phase Velocity ◽  
Natural Frequency ◽  
Excitable Media ◽  
Negative Phase ◽  
High Dimensional ◽  
Wave Source ◽  
Oscillatory Systems ◽  
The Waves ◽  
Parameter Distributions ◽  
Negative Phase Velocity

Waves propagating inwardly to the wave source are called antiwaves which have negative phase velocity. In this paper the phenomenon of negative phase velocity in oscillatory systems is studied on the basis of periodically paced complex Ginzbug-Laundau equation (CGLE). We figure out a clear physical picture on the negative phase velocity of these pacing induced waves. This picture tells us that the competition between the frequency ωout of the pacing induced waves with the natural frequency ω0 of the oscillatory medium is the key point responsible for the emergence of negative phase velocity and the corresponding antiwaves. ωoutω0 > 0 and |ωout| < |ω0| are the criterions for the waves with negative phase velocity. This criterion is general for one and high dimensional CGLE and for general oscillatory models. Our understanding of antiwaves predicts that no antispirals and waves with negative phase velocity can be observed in excitable media.

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 Determination of Cable Tension Force using Accelerometer

2019 ◽  
Vol 9 (2) ◽  
pp. 224-229
Keyword(s):  
Natural Frequency ◽  
Tensile Force ◽  
Tensile Load ◽  
Free Vibrations ◽  
Tension Force ◽  
Main Element ◽  
Accelerometer Sensor ◽  
Simple Method ◽  
Long Span ◽  
Axial Tensile

Cable is the main element in a long span structure and is often used for special structures such as long span bridges, roofs and other structures that require a long span. The stiffness of the cable is determined by the amount of axial tensile force acting on the cable, and hence, the magnitude of the actual tensile force on the cable is an important factor to be determined and monitored. One simple method for determining the actual tensile force on a cable is to calculate the tensile force from the first natural frequency of the cable. However, it is important to ensure that the formulas used to calculate the tensile force are accurate. This research aims to determine the level of accuracy and the factors that influence the accuracy of the formula to determine the tension force of the cable from the natural frequency value of the cable. The methodology used in this research project was by applying free vibrations to the cable with given axial tensile load and measuring the acceleration that occurred with an accelerometer sensor. By using Fast Fourier Transform (FFT), the natural frequency value of the cable can be calculated and the actual tensile strength in the cable can be determined. From the experiment conducted, it was found that the length of the cable affects the accuracy of the measurement of the natural frequency and the magnitude of tensile force of the cable. The strain that occurs on the cable plays a very important role to the accuracy of the formulas used.

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