Experimental Observations of Nonlinear Nonplanar Oscillations of a Cantilever Beam

2007 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh ◽  
Bashar K. Hammad
Keyword(s):  
Cantilever Beam ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Low Frequency ◽  
Harmonic Load ◽  
Bending Vibrations ◽  
Out Of Plane ◽  
Frequency Components ◽  
Plane Bending ◽  
The Difference

The dynamics of a thin cantilever beam undergoing combined torsion and bending vibrations are examined experimentally. The beam’s fundamental natural frequencies in the two orthogonal bending motions and in torsion are fv1 = 5.719 Hz, fw1 = 189.730 Hz, and fφ1 = 138.938 Hz, respectively. A base-excitation shaker imparts a harmonic load that acts parallel to the width of the beam. First, the response of the beam is examined when the excitation frequency is equal to the fundamental torsion natural frequency (i.e., f = 138.9 Hz). For low levels of excitation, the motion consists mainly of hardly noticeable twisting vibrations. For high levels of excitation, the energy of the first torsion mode excites the first out-of-plane bending mode. In this case, the beam responses exhibit modulated vibrations containing both high-frequency and low-frequency components. Second, the beam is excited at the frequency f = 132.0 Hz, which is in the neighborhood the difference of these two natural frequencies. For large excitation levels, the beam vibrates with large-amplitude out-of-plane bending motions that exhibit chaotically intermittent behaviors.

Investigation of Spectral Sensitization V. A Study of the Visible and Infrared Spectra of Some Very Pure 2-bis-Benzoxazolyl, 2-bis-Indolinyl, and 2-bis-Quinolyl Cyanine Iodides

1967 ◽  
Vol 21 (2) ◽  
pp. 71-80 ◽  
Author(s):  
A. Leifer ◽  
D. Bonis ◽  
M. Boedner ◽  
P. Dougherty ◽  
A. J. Fusco ◽  
...  
Keyword(s):  
Infrared Spectra ◽  
Absorption Maximum ◽  
Low Frequency ◽  
Cyanine Dyes ◽  
Visible Range ◽  
Bending Vibrations ◽  
Out Of Plane ◽  
Phenyl Rings ◽  
Plane Bending ◽  
Bending Modes

A detailed study of the visible spectra in solution and the infrared spectra in the solid state has been made for the following vinylogous series of cyanine dyes: [2-bis(3-ethylbenzoxazolyl)] cyanine iodides, [2-bis-(1-ethyl-3,3-dimethylindolinyl)] cyanine iodides, and [2-bis-(1-ethylquinolyl)] cyanine iodides. Each dye, to be acceptable for study, had to be chromatographically pure, give a correct microchemical elemental analysis, and be free of electron-spin-resonance (free-radical) signals. The characteristic red shift of the principal absorption maxima was observed for these dyes in the visible range as the number of methine linkages increased. A careful comparison of the visible spectral data of the [2-bis(3-ethylbenzoxazolyl)] cyanine iodides with those of the corresponding [2-bis(3-ethylbenzothiazolyl)] and [2-bis(3-ethylbenzoselenazolyl)] cyanine iodides, [Leifer et al., Appl. Spectry. 20, (1966)] indicates that the electronegativity of the atom S, Se, or O in the heterocyclic rings probably affects the wavelength of the principal absorption maximum. As the electronegativity of the Group VI.A atom increases, the principal absorption maximum shifts slightly toward the blue. Assignments of vibrational modes to separate absorption regions have been made for these vinylogous series of dyes. Each vinylog has a characteristic pattern of resonant-conjugated stretching modes in the region 1600–1400 cm−1. Some of these modes exhibit a low frequency shift as the resonant-conjugated chain increases. Comparison of the benzoxazolyl modes with those of the corresponding benzothiazolyl and benzoselenazolyl modes indicates that they are probably a function of the electronegativity of S, Se, or O in the heterocyclic rings. As the electronegativity of the VI.A atom increases, the resonant-conjugated stretching modes shift to higher frequencies. There are also characteristic bands in the 1600–1400 cm−1 region which are present in all the vinylogs of each series of these dyes. These bands have been assigned to the stretching modes of the fused phenyl rings present in these dyes. It appears that the fused phenyl stretching modes are a function of the groupings C(CH3)2, O, S, Se present in the indolinyl, benzoxazolyl, benzothiazolyl, and benzoselenazolyl heterocyclic rings, respectively. The sulfur and selenium atoms affect these modes the least while the oxygen atom affects these modes the most. Assignments have been made for the aromatic CH out-of-plane bending modes in the region 800–700 cm−1 for these vinylogous series of cyanine dyes. In the spectra of the 2-bis-indolinyl and 2-bis-quinolyl cyanine iodides, there is a band appearing in the region 1000–900 cm−1 which changes systematically with an increase in the number of hydrogens on the bridge. This band has been assigned to the out-of-plane bending vibrations of the hydrogens in a trans configuration on the bridge. No evidence of a cis isomer was observed in the spectra.

Investigation of Spectral Sensitization IV. A Study of Visible and Infrared Spectra of Some Very Pure Bis—Benzothiazolyl and Bis—Benzoselenazolyl Cyanine Iodides

1966 ◽  
Vol 20 (5) ◽  
pp. 289-297 ◽  
Author(s):  
A. Leifer ◽  
D. Bonis ◽  
M. Boedner ◽  
P. Dougherty ◽  
M. Koral ◽  
...  
Keyword(s):  
Infrared Spectra ◽  
Low Frequency ◽  
Bending Vibrations ◽  
Crystal Field Effect ◽  
Spin Resonance ◽  
Visible Spectra ◽  
Out Of Plane ◽  
Phenyl Rings ◽  
Plane Bending ◽  
Cis Isomer

A detailed study of the visible spectra in solution and the infrared spectra in the solid state have been made for the following vinylogous series of dyes: [2-bis(3-ethylbenzothiazolyl)] cyanine iodides and [2-bis(3-ethylbenzoselenazolyl)] cyanine iodides. Each dye, to be acceptable for study, had to be chromatographically pure, give a correct microchemical elemental analysis, and be free of electron-spin resonance (free radical) signals. The characteristic red shift of the principal absorption maxima was observed for these dyes in the visible as the number of methine linkages increased. Assignments of vibrational modes to separate absorption regions have been made for these vinylogous series of dyes. Each vinylog has a characteristic pattern of resonant-conjugated stretching modes in the region 1600–1400 cm−1. These modes exhibit a low-frequency shift as the resonant-conjugated chain length increases. There are bands absorbing in the regions 1594–1572 cm−1 and 1470–1453 cm−1 which are present in all the vinylogs and which have been assigned to the aromatic stretching vibrations of the fused phenyl rings present in all these dyes. There is a band appearing in the region 1000–900 cm−1 which changes systematically with an increase in the number of hydrogens on the bridge and with substitution on the bridge. This band has been assigned to the out-of-plane bending vibrations of the hydrogens in a trans configuration on the bridge. No evidence of a cis isomer was observed in the spectra. There is a band near 760 cm−1 which is split into a doublet and has been assigned to the aromatic CH out-of-plane bending vibrations of the four adjacent hydrogens on the fused phenyl rings. This splitting has been attributed to a crystal field effect which gives rise to in-phase and out-of-phase vibrations of the same groups in two different molecules.

In-Plane Vibration Modes of Arbitrarily Thick Disks

1997 ◽  
Author(s):  
Kevin I. Tzou ◽  
Jonathan A. Wickert ◽  
Adnan Akay
Keyword(s):  
Natural Frequencies ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Three Dimensional Model ◽  
Bending Vibration ◽  
Annular Disk ◽  
Out Of Plane ◽  
Plane Bending

Abstract The three-dimensional vibration of an arbitrarily thick annular disk is investigated for two classes of boundary conditions: all surfaces traction-free, and all free except for the clamped inner radius. These two models represent limiting cases of such common engineering components as automotive and aircraft disk brakes, for which existing models focus on out-of-plane bending vibration. For a disk of significant thickness, vibration modes in which motion occurs within the disk’s equilibrium plane can play a substantial role in setting its dynamic response. Laboratory experiments demonstrate that in-plane modes exist at frequencies comparable to those of out-of-plane bending even for thickness-to-diameter ratios as small as 10−1. The equations for three-dimensional motion are discretized through the Ritz technique, yielding natural frequencies and mode shapes for coupled axial, radial, and circumferential deformations. This treatment is applicable to “disks” of arbitrary dimension, and encompasses classical models for plates, bars, cylinders, rings, and shells. The solutions so obtained converge in the limiting cases to the values expected from the classical theories, and to ones that account for shear deformation and rotary inertia. The three-dimensional model demonstrates that for geometries within the technologically-important range, the natural frequencies of certain in- and out-of-plane modes can be close to one another, or even identically repeated.

Nonlinear Combination Resonances in Cantilever Composite Plates

1995 ◽  
Author(s):  
Kyoyul Oh ◽  
Ali H. Nayfeh
Keyword(s):  
Natural Frequencies ◽  
Excitation Frequency ◽  
Power Spectra ◽  
Low Frequency ◽  
Composite Plates ◽  
High Amplitude ◽  
Mode Shapes ◽  
Laser Vibrometer ◽  
Combination Resonance ◽  
Response Curves

Abstract We experimentally investigated nonlinear combination resonances in a graphite-epoxy cantilever plate having the configuration (–75/75/75/ – 75/75/ – 75)s. As a first step, we compared the natural frequencies and mode shapes obtained from the finite-element and experimental modal analyses. The largest difference in the obtained frequencies was 2.6%. Then, we transversely excited the plate and obtained force-response and frequency-response curves, which were used to characterize the plate dynamics. We acquired time-domain data for specific input conditions using an A/D card and used them to generate time traces, power spectra, pseudo-state portraits, and Poincaré maps. The data were obtained with an accelerometer monitoring the excitation and a laser vibrometer monitoring the plate response. We observed the external combination resonance Ω≈12(ω2+ω5) and the internal combination resonance Ω≈ω8≈12(ω2+ω13), where the ωi are the natural frequencies of the plate and Ω is the excitation frequency. The results show that a low-amplitude high-frequency excitation can produce a high-amplitude low-frequency motion.

Comparison of Failure Modes of Piping Systems With Wall Thinning Subjected to In-Plane, Out-of-Plane and Mixed Mode Bending Under Seismic Load: An Experimental Approach

2007 ◽  
Author(s):  
Izumi Nakamura ◽  
Akihito Otani ◽  
Masaki Shiratori
Keyword(s):  
Failure Modes ◽  
Seismic Load ◽  
Piping System ◽  
Shake Table ◽  
Shake Table Tests ◽  
Wall Thinning ◽  
Piping Systems ◽  
Out Of Plane ◽  
Plane Bending ◽  
The Difference

In order to investigate the influence of degradation on the dynamic behavior and failure modes of piping systems with local wall thinning, shake table tests using 3-D piping system models were conducted. About 50% full circumferential wall thinning at elbows was considered in the test. Three types of models were used in the shake table tests. The difference of the models was the applied bending direction to the thinned wall elbow. The bending direction considered in the tests was either of the in-plane bending, out-of-plane bending, or mixed bending of the in-plane and out-of-plane. These models were excited under the same input acceleration until failure occurred. Through these tests, the vibration characteristic and failure modes of piping models with wall thinning under seismic load were obtained. The test results showed that the out-of-plane bending is not significant for a sound elbow, but should be considered for a thinned wall elbow, because the life of piping models with wall thinning subjected to out-of-plane bending may reduce significantly.

Infrared spectra and hydrogen bonding in some methyl α-glycopyranosides

1975 ◽  
Vol 28 (2) ◽  
pp. 335 ◽  
Author(s):  
AJ Michell
Keyword(s):  
Crystal Structure ◽  
Hydrogen Bonding ◽  
Infrared Spectra ◽  
Bending Vibrations ◽  
Ambient Temperatures ◽  
Out Of Plane ◽  
Stretching Vibrations ◽  
Plane Bending

Spectra in the OH stretching and out-of-plane bending regions of four methyl α-glycopyranosides of known crystal structure have been obtained at ambient and sub-ambient temperatures. Partial deuteration has been used to uncouple the stretching vibrations and assist in assignments of the out-of-plane bending vibrations.

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