Second Mode Of Vibration Research Articles

134 固定辺で面内弾性拘束を受ける長方形板のカオス振動実験

Experimental results are presented on chaotic vibrations of a rectangular plate with an in-plane elastic constraint. Opposite edges of the plate are clamped and the other edges are simply supported. One side of the clamped edges is connected with elastic springs and is movable to an in-plane direction. Under the selected in-plane compressive force, the plate has the conditions of internal resonances. By exciting the plate with periodic lateral acceleration, chaotic responses are obtained in specific frequency regions of excitation. The chaotic responses are examined with the Fourier spectra, the principal components analysis and the maximum Lyapunov exponents. It is found that the chaotic responses are generated with the condition of the internal resonances of multiple modes. In the dominant chaotic responses, the ultra-sub-harmonic resonance of 2/3 order with the first mode of vibration and the ultra-sub-harmonic resonance of 4/3 order with the second mode of vibration contribute to the chaotic response. By changing exiting amplitude, instability boundaries of the chaotic responses are also clarified.

Numerical study of self-adaptive vibration suppression for flexible structure using interior inlay viscous fluid unit method

This paper investigates the usage of an interior inlay viscous fluid unit as a new vibration suppression method for flexible structures via numerical simulations. The first and second modes of vibration for a beam have been calculated using the commercial computational fluid dynamic package Fluent6.1, together with the liquid surface distribution and the fluid force. The calculated results show that the inlay fluid unit has suppressive effects on flexible structures. The liquid converges self-adaptively to locations of larger vibrations. The fluid force varies with the beam vibration at a phase difference of more than 180°. Thus the fluid force suppresses the beam vibration at most of the time.

Experiments on Chaotic Vibrations of a Cantilevered Beam under Vibroimpact

Experimental results are presented on chaotic vibrations of a cantilevered beam under vibroimpact. A rigid bar that is located close to the free end of the beam limits the amplitude of the beam under lateral periodic acceleration, and then asymmetric vibroimpacts are induced. In the frequency region of the impact vibration near the fundamental resonance of the lowest mode, two regions of chaotic responses are observed. There is one impact in an excitation cycle in the higher frequency region of the chaos, while occurrence of impacts in the lower frequency region is less frequent than the former one. The maximum Lyapunov exponent of the chaotic response in the higher frequency region takes higher value than that of the chaos in the lower frequency. Mode contributions to the chaos are inspected by the principal component analysis. The lowest mode of vibration prevails in the vibroimpacting response. In the regions of the chaos, contribution of the second mode of vibration to the chaotic responses increases up to 5%. As the exciting frequency is increased, the contribution of the second vibration mode to the chaos becomes larger owing to the increase in the amplitude of impact vibration. Furthermore, contribution of the second vibration mode drastically increases as the super-harmonic resonance of the second mode is generated. The maximum Lyapunov exponent increases as the contribution of the second mode increases, which implies the close relation between the complexity of the chaotic responses and the participation of the higher modes of vibration.

Predictive Active Control of Building Frames with a Solo Neurocontroller

We present an efficient generalized control scheme using a solo neural network controller (neuro-controller) for the predictive single mode response control of building frames subjected to earthquake excitation. The control scheme is developed by considering the time delay that exists between the measurement of response and the actual application of the control force. Response measurements are taken from only one point of the structure. The neurocontroller is directly trained to provide the control force signal for a predetermined response reduction, called the target reduction. The neurocontroller is trained such that it can provide control force for target reduction for any building frame ranging between 5 and 15 stories. As an illustrative example, a 10-story shear-building frame is considered whose period of first mode of vibration is about three times that of the second mode of vibration. The efficiency of the control scheme is demonstrated by controlling the first mode response of the frame subjected to El Centro and Treasure Island earthquake records. It is shown that the control scheme provides response reductions close to the target reductions for the unknown problem used for testing the efficiency of the control scheme.

크랙을 가진 회전 외팔보의 동특성 해석

In this paper, we studied about the dynamic behavior of a cracked rotating cantilever beam. The influences of a rotating angular velocity, the crack depth and the crack position on the dynamic behavior of a cracked cantilever beam have been studied by the numerical method. The equation of motion is derived by using the Lagrange's equation. The cracked cantilever beam is modeled by the Euler-Bernoulli beam theory. The crack is assumed to be in the first mode of fracture and to be always opened during the vibrations. The lateral tip-displacement and the axial tip-deflection of a rotating cantilever beam is more sensitive to the rotating angular velocity than the depth and position of crack. Totally, as the crack depth is increased, the natural frequency of a rotating cantilever beam is decreased in the first and second mode of vibration. When the crack depth is constant, the natural frequencies of a rotating cantilever beam are proportional to the rotating angular velocity in the each direction.

Application of a Genetic Algorithm for Optimal Damper Distribution within the Nonlinear Seismic Benchmark Building

This paper presents a systematic method for identifying the optimal damper distribution to control the seismic response of a 20-story benchmark building. A genetic algorithm with integer representation was used to determine the damper locations. Both H2- and H∞-norms of the linear system transfer function were utilized as the objective functions. Moreover, frequency weighting was incorporated into the objective functions so that the genetic algorithm emphasized minimization of the response in the second mode of vibration instead of the dominant first mode. The results from numerical simulations of the nonlinear benchmark building show that, depending on the objective function used, the optimal damper locations can vary significantly. However, most of dampers tend to be concentrated in the lowermost and uppermost stories. In general, the damper configurations evaluated herein performed well in terms of reducing the seismic response of the benchmark building in comparison to the uncontrolled building.

The influence of the design methodology in the response of transmission towers to wind loading

From a theoretical approach, the design procedure for the establishment of wind loading on transmission towers was reviewed and current procedures, such as Davenport's gust response factor (GRF), were compared with the statistical method using influence lines (SIL), which is considered more realistic. This latter approach can account for unbalanced loading effects, shear and axial loads and the effects of higher modes of vibration in the calculation of the response factors. Several responses due to certain assumed transverse wind characteristics were calculated for some typical transmission towers. The main findings were: (a) Peak loads calculated using SIL were larger than peak loads given by the GRF. (b) The dynamic response of transmission structures is strongly dependent on the turbulence intensity level and its spectrum. (c) For members in which there is reversal in the forces on the load position, the resonant response in the second mode of vibration was bigger, even by four to five times, than the corresponding one in the first mode. Although this effect is not as severe in terms of resulting stresses when all the components are computed in the peak responses, it can lead to fatigue problems. From the current results it can be concluded that the incorporation of the dynamic properties of transmission structures in the design methodologies is needed and that the statistical method using influence lines is a more correct approach since it allows for the inclusion of a larger number of factors in the design methodology.

118 Static and dynamic measurements on an aerospace composite beam by embedded fiber optic sensors

Two rectangular plates, assimilable to beams, one with embedded fiber optic sensors and one with the sensor glued on its surface have been manufactured using a prepreg carbon fiber fabric. The sensors, specifically two Fiber Bragg Gratings (FBGs), have been embedded in the same in-plane position but at different depths so as to be in opposite position with respect to the neutral axis. A second plate with lower natural frequency has been tested to prove the feasibility of a modal analysis by the use of a fiber Bragg Grating. The results from the static tests have shown excellent agreement with the theory. From the dynamic tests performed with the Fiber Bragg Grating the second mode of vibration was retrieved. The comparison with the results obtained with the accelerometer was very good.

A broadband Terfenol-D driven flextensional projector

Since 1994 Lockheed Martin has developed a number of Terfenol-D driven Class VII inverse flextensional underwater projectors, known as ‘‘Terfenol Dogbone Flextensional Projectors.’’ The underlying motivation for these projectors has been to match the high-strain, high-energy density, and relatively high-compliance characteristics of magnetostrictive Terfenol-D alloys with the high-compliance characteristics of the dogbone flextensional shell. The resulting projector is reliable, rugged, and has a smaller size, lower-frequency and a broader bandwidth relative to conventional ceramic driven flextensional projectors. The Class VII shell, by virtue of its shape, is designed to naturally accumulate axial compressive stress on the drive assembly with depth. Hence, the dogbone shaped shell is dynamically more compliant to better match the lower modulus of the Terfenol driver. Additionally, both the fundamental bending mode and the second mode of vibration provide dominantly in-phase radiation and well behaved beam patterns. Recently, a small permanent magnet biased projector demonstrated a mechanical quality factor of unity and high source level in its fundamental mode of vibration. A 3-dB band from 2.9 to 6.4 kHz (fundamental mode) and 10 to 18 kHz (second mode) where achieved. The merits, challenge, and future outlook for this technology are discussed. [Work supported by ONR Code 321.]

COFFEE HARVESTING I: DETERMINATION OF THE NATURAL FREQUENCIES OF THE FRUIT STEM SYSTEM IN COFFEE TREES

As part of a study of mechanized harvesting of coffee by vibration, an investigation of the natural frequencies ofthe fruit stem in coffee trees was undertaken. Theoretical models of one and two degrees of freedom were used to estimatethe natural frequencies. Results obtained with these models indicate that frequencies of resonance for ripe cherries (redcherries) and halfripe ( pinton ) are lower than the value obtained for green cherries. For ripe cherries in the first vibrationmode, the frequency of resonance ranges from 1450 to 1500 c.p.m while in the second mode of vibration the frequency ofresonance is above 28,000 c.p.m.

Vibration Control of Beams Using Electro-Magnetic Compressional Damping Treatment

A new class of structural damping treatments is introduced. This class is the electro-magnetic damping treatment (EMDT) which relies in its operation on a viscoelastic damping layer sandwiched between two magnetic layers. Interaction between the magnets generates magnetic forces that enhance the compressional damping mechanism of the viscoelastic layer. With proper tuning of the magnetic forces, in response to the structural vibration, undesirable resonances and catastrophic failures can be avoided. The fundamentals and the underlying phenomena associated with the EMDT are investigated theoretically and experimentally. A finite element model is developed to describe the interaction between the dynamics of flexible beams, the viscoelastic damping layer and the magnetic layers. The validity of the developed finite element model is checked experimentally using aluminum beams treated with EMDT patches. The beam/EMDT system is subjected to sinusoidal excitations and its multi-mode response is monitored when the magnetic layers are activated or not. Several control strategies are considered to activate the magnetic layers including simple PD controllers. The performance of the uncontrolled and controlled system is determined at various operating conditions. Attenuation of 49.4 percent is obtained for the amplitude of first mode of vibration (5.2 Hz) with control voltage of 0.2 volts. The attenuation increases to 72.56 percent for the second mode of vibration (28.6 Hz) with a control voltage of 1.68 volts. Comparisons with conventional Passive Constrained Layer Damping (PCLD) treatments emphasize the potential of the EMDT treatment as an effective means for controlling structural vibrations. [S0739-3717(00)00603-6]

Linerr motor based on a piezo-bimorph-element

Abstract A piezoelectric linear motor was developed using a bimorph PZT element. The piezoelectric linear motor can move forward and backward, using the resonance of only a piece of the piezoelectric element. The linear motor consists of a piezoelectric reed, two tremblers and two guide rails. The vibration of the reed and tremblers causes the linear motor to move. If the linear motor moves forward at the first vibration mode of the vibrating reed, then the linear motor will move backward at the second mode of vibration. The piezoelectric linear motor moves, using the resonance of the piezo-electric reed.

Results of recent development to measure and analyze the dynamic properties of materials

Although the vibrating beam technique has been widely used and demonstrated to be an effective technique for measuring the dynamic properties of materials, it has its own limitations. This is because the present analysis uses the fourth‐order beam equations which do not adequately handle the boundary conditions of the test specimens especially for the fundamental mode of vibration. Because of that, measurements on the first and sometimes the second modes of vibration have been omitted which limits the frequency range where measurements could be made. To overcome such limitations, a new approach using finite element analysis has been developed to compute the dynamic properties of materials for several modes of vibration, including the lower ones. This analysis is suited for either extensional or shear deformation, depending on the geometry of the specimen. Results of using this analysis to compute the dynamic properties of the various proposed materials for the S2WG79 standard will be presented and discussed.

Experiment on Chaotic Vibrations of a Cantilevered Beam Deformed by a Stretched Cable.

This paper presents the experimental results on chaotic vibrations of a cantilevered beam deformed by a stretched cable. The beam is deformed to a postbuckled configuration by a cable that is extended from the top to bottom of the beam. We call the deformed beam a 'tendon beam'. As a lateral load is applied to the beam statically, the beam changes to another equilibrium position accompanied with a snap through deformation. When the beam is subjected to periodic excitation, a large-ampltiude response is generated by resonance. Chaotic motion appears near the fundamental resonance region as well as the region in the higher mode of vibration. Both chaotic motions are confirmed as the chaos due to maximum Lyapunov exponent with a positive number. By recording the Poincare section of the chaos behavior, fractal attractors are clearly focused in the fundamental resonance region. The more complicated attractor is obtained in the resonance region of the second mode of vibration. The correlation dimension of the chaos in the second mode is higher than that of the fundamental mode. It is generated by the two-to-thirtees internal resonance between the fundamental mode and second mode.

Natural frequencies of arch bridges with slab decks by the finite element method

A study of the free-vibration natural frequencies of an arch bridge is presented. The bridge consists of two structural components namely a circular arch and a flat deck, monolithically connected together at the crown of the arch. A two-dimensional mathematical model is used in the analysis; the curved component is modelled by curved-beam finite elements and the deck by straight-beam finite elements. Both symmetrical and anti-symmetrical modes of vibration are studied for such bridges when the ends of the arches and decks are either hinged or fixed to rigid abutments. The effect of varying the relative stiffnesses of the two components on the fundamental and second modes of vibration is also presented and compared with the corresponding frequencies of the arch component vibrating on its own.

Spatially weighted, grating-based, two-mode, elliptical-core optical fiber vibration sensors

Photoinduced refractive-index changes in two-mode, elliptical-core optical fibers affect the beat length and the sensor sensitivity. Chirped gratings are written by attaching such fibers to cantilever beams positioned in a strained state. We show that fibers with in-line chirped gratings, with the chirp being shaped in the form of a vibration-mode shape, can be used as spatially weighted fiber sensors for vibration analysis. We demonstrate enhanced detection of the first and second modes of vibration of a cantilever beam using this process; vibration mode suppression of the order of 10 dB is obtained.

Longitudinal Motion of a Thick Transversely Isotropic Hollow Cylinder

The propagation of a longitudinal harmonic wave in a transversely isotropic shell has been investigated in the development of ultrasonic techniques for thick hollow composite cylinders. The characteristic equation for satisfying the stress-fee inner and outer cylindrical boundaries has been obtained in an exact form in terms of the wavelength, the cylinder radii and the material constants. The phase velocity of the fundamental mode is calculated for a wide range of the wavelength for various cylinder radii for some typical sample materials. The shell wave speeds for the second mode of vibration are also presented. Comparisons are made between shell wave speeds and plate wave speeds. The spread of the wave speeds for the composite shells is shown to be much wider than that for an isotropic shell.

The apparent mass of the seated human body in the fore-and-aft and lateral directions

Apparent mass frequency response functions of the seated human body have been measured in the fore-and-aft and lateral directions using random vibration in the frequency range 0.25–20 Hz. The measurements were made with a rigid seat so as to measure the horizontal force transmitted to the body. Eight subjects were used to investigate the apparent mass of the seated body with and without backrest contact for a range of three vibration magnitudes. The body appeared to have two heavily damped modes of vibration when there was no backrest contact. The first mode had a resonance frequency at about 0·7 Hz for both the fore-and-aft and the lateral directions. The second mode was less pronounced than the first, particularly for the lateral direction, and had a resonance frequency in the region of 1.5–3 Hz. The resonance frequency of the second mode of vibration, but not the first, decreased with increasing magnitude of vibration. There appeared to be only one mode of vibration when the motion of the upper body was restrained with a backrest, with a resonance frequency in the region of 3·5 Hz for the fore-and-aft direction and at about 1·5 Hz for the lateral direction. The effect of the backrest was particularly pronounced for the fore-and-aft direction.

Experimental and theoretical investigations of the effect of root flexibility on the vibration characteristics of cantilever beams

The combined effect of rotary inertia, shear deformation and root flexibility on the vibration characteristics of cantilever beams is investigated experimentally. The experimental results are compared with theoretical results obtained by using two different methods of analysis, the finite element method and the numerical integration method. Excellent agreement is obtained. Novel linear relationships are deduced for the reduction of frequencies due to the combined effect of rotary inertia, shear deformation and root flexibilities for the first and second modes of vibration.

A Study on Damping Capacity of a Jointed Cantilever Beam : 2nd Report, Comparison between Theoretical and Experimental Values

The large-sized machine tools are often made of welded steel and to increase their damping capacity a jointed beam or a damping joint is often used. The behaviour of the damping capacity relating to the first, second and third modes of vibration of the jointed beam had experimentally been studied in the previous report, however, the method for the theoretical calculation of damping capacity of it has not been established yet. In this paper, the relationships between the slip ratio and the damping capacity have been theoretically investigated considering the first and second modes of vibration of the jointed cantilever beams used in the previous study. Through comparison between the experimental values obtained from the previous study and the theoretical values, it has been confirmed that the damping capacity of a jointed cantilever beam can be well estimated by using a method proposed here.