Vibrating cantilever beams

Fig. 7. Sketch of the apparatus for measuring inverse piezoelectric effect in polymer films. B vibrating cantilever beam, E electromagnetic exciter, P electromagnetic pick-up, S specimen film, M weight, C oscillator, Vt, V2 and V3 voltmeters, Sw switch, V d.c. bias source for measuring electrostriction effect. Drawn after Kawai (1) (1969) by permission of the Japan Society of Applied Physics...

In the ranges of 0.0081 < t/L < 0.0121 and 1 < n < 5 the corresponding equations of vibrating cantilever beams ("clamped-free" end conditions) have been applied successfully to a unidirectional graphite fibre reinforced epoxy composite revealing an E i to G12 ratio of 40 [5]. In the case of unidirectional on-axis specimens, however, 0 2 is the "longitudinal transverse shear modulus" and the terms "in-plane shear" and "interlaminar shear" are not defined. 

Equation 28 states, that co is proportional to t. The effect of the thickness of the gel film on the frequency of the first resonance mode has been investigated. When the buoyancy is taken into account, the experimental results have quantitatively followed Eq. 28. It has been found that the buoyancy plays an important role in the occurrence of the electric field-associated vibration of gel film. The vibration of the gel film in an electric field has thus roughly analyzed as a mechanical bending vibration of a uniform cantilever beam. 

Two beams of the same material are clamped at one end to form a cantilever beam. One beam has a square cross section of thickness D, and the other beam has a circular cross section of diameter D. The beams are set in vibration by a tap near the free end. If the length of the beams is the same, which beam will have the higher frequency of vibration What is the ratio of their frequencies ... 

A horizontal cantilever beam is made of an idealized material that has only two retardation times 10 and 1000 s. The beam is bent downward for 100 s. Then it is bent upward for 1 s and released without any vibrations taking place. Describe the motion of the beam for the next 10,000 s. 

Forced sinusoidal vibration of thin cantilever beam specimen by electrodynamic shaker... 

Several methods have been described for measuring the bending modulus of hbers. The balanced hber method of Scott and Robbins [73,74] appears to be the easiest to handle experimentally (except for very curly hair) and provides less scatter than the other methods [74], The vibrating-reed method (oscillating hber cantilever) has also been used with human hair [10], The cantilever beam method [75], the loop deformation method [76], and the center load beam method [76] have been described for textile hbers. 

A vertical vessel is modeled as a cantilever beam whereas a horizontal vessel is modeled as a simply supported beam. A cantilever is a much more prone to vibration and deflection then a simply supported beam, therefore the POV is typically much higher. Guiding a vessel supported in a structure will greatly alter its POV because it changes the mode of vibration. 

For a column under live load, maximum deformation and internal force of nodal sections are closely related to natural vibration period and vibration mode. Therefore, natural vibration period of a column must be determined for calculating wind load and seismic load of the column (Wei, 1985). Columns used in chemical plants or refinery plants typically have one end fixed and the other free, which may be considered as cantilever beam structures. Tall columns are high-rise flexible buildings with less damping resulting in simplified free vibration (Jeary, 1997). 

Fig. 7. Schematic diagram for the single cantilever beam apparatus for measurement of the complex Young s modulus. (A forced vibration nonresonance method.)...

A simple one-dimensional model of electrically-induced dynamic deformation or vibration of a cantilever beam made with such IPMC artificial muscle strips is given by the following equations ... 

Suppose that a microfabricated cantilever beam is to be used as a sensor for detecting the presence of a certain bacterium in a hquid solution see Figure 1.31. Following its fabrication, the beam is first coated with an antibody that is specific to the particular bacterium of interest and its free vibration characteristics are determined. The beam is then dipped into the solution to be tested. If the bacteria of interest are present, they will attach themselves to the surface of the beam, attracted by the antibody. The beam is then removed from the solution and its free vibration characteristics are re-examined. Estimate the influence of an added mass due to the attached bacterium cells, say rib cells each of mass mb, on the fundamental natural frequency of the beam. (The effectiveness of this approach for detection of bacteria has been demonstrated by Ilic et al. (2000).)... 

As mentioned above rectangular AFM cantilever beams can also be forced to torsional vibrations. In this case the experimental set-up is such that an ultrasonic transducer emits shear waves into the sample causing in-plane surface vibrations. The shear wave transducer is oriented so that the surface vibrations are polarized perpendicular to the long axis of the cantilever. If low excitation amplitudes (0.1 nm) are applied and if the excitation frequency is set close to a contact resonance frequency, the amplitude and the phase of the cantilever vibration contain now information about the local lateral tip-sample stiffness. Used as imaging quantity, they yield images of shear stiffness. By increasing the lateral excitation amplitude much above... 

The tensile strength and stress of the IPMC are measured in the same manner as those of the IP. The bending stiffness of a fully hydrated IPMC sample is estimated using the free oscillation attenuation method. By bending the sample to the appropriate initial displacement, the free vibration response can be recorded. The natural frequency of the cantilever, is obtained from the fast Fourier transform of the free vibration response curve. The stiffness of the IPMC, Egg, is determined using Eq. 4, which is based on the thin cantilever beam theory of material mechanics ... 

T = fundamental period of vibration of the vessel assuming a uniformly loaded cantilever beam fixed at the base by the following ... 

T. Bailey and J. E. Hubbard. Jr., Diatiibuted piexoelectric>palymer active vibration oootrol of a cantilever beam. AlAA J. Guidance and Cantrol 8 605-611 (1985). 

Figure 1.3 The resonant gate field effect transistor, one of the first MEMS devices. A released metal cantilever beam forms the gate electrode over the diffused source-drain channel. The input signal is applied to the input force plate, which causes the cantilever beam to vibrate, modulating the current through the transistor. Maximum vibration occurs at the resonant frequency of the cantilever beam, enabling the device to act as a high-Q electromechanical filter. (Reprinted with permission from IEEE Trans. Electron Devices, The resonant gate transistor, H.C. Nathanson, W.E. Newell, R.A. Wickstrom and J.R. Davis Jr., 1967 IEEE.)...

Nonlinear Dynamic Seismic Analysis, Table 10 Maximum values of the kinematical components (/, t) (m). v(/, t) (m), w(/, i) (m), and BJ l, i) (rad) of the cantilever beam of application section Forced Vibrations or Beam of Monosynunetric Cross Section Under Eccentric Transverse Loading for load cases (i), (ii)... 

Stochastic response of Eq. 39 to a Gaussian white noise f(t) = (t) was investigated in Ibrahim and Roberts (1976). That paper considered vibrations of an autoparametric system cmisisting of a primary mass, subjected to a broadband excitation, and a vertically oriented cantilever beam. 

Yuya et al. [243] extracted the elastic modulus of single electrospun PAN nanofibre dynamically through the natural frequencies of a pair of AFM microcantilevers linked by a nanofibre segment (Fig. 4.24b). The theory of this technique is based on the dynamic relationship between the fibre stiffness (i.e. spring constant) and the resonance frequencies of cantilever vibration mode. On the other hand, Liu et al. [244] used atomic force acoustic microscopy (AFAM) based on ultrasonic frequency oscillations to excite an AFM cantilever when the tip was in contact with a sample. A different approach based on a model of the resonant frequency that is dependent on the bob s free flight was employed to measure the elastic modulus of as-spun nylon 6, 6. A ball was glued to a nanofibre and suspended from a cantilever beam that was attached to a piezoelectric-actuated base [245]. 

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