Cantilever beam resonator

The two 10 X ISOtim connecting beams would have a mass of 

If the shuttle is made up of two plates that are 50 x 100 pm and are connected by a plate that is 50 x 150 pm, the mass of the shuttle would be 

The total mass would be the sum of the beams, connectors, and shuttles, or 10.6 x 10 kg. The resonant frequency can then be approximated by 

A fixed-free cantilever beam M-Test structure fabricated in Polyl of the PolyMUMPS process can also be used as a resonator by exciting it into resonance with an AC drive signal applied to a PolyO counter-electrode defined below the end of the beam. To estimate the resonant frequency we use the spring constant for a fixed-free beam that is 20 pm wide and 150 pm long subjected to a point load at its free end  

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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. 

BENDING OF A BEAM. The complex dynamic Young s modulus can be determined from the forced, non-resonant oscillations of a single or double cantilever beam. The apparatus considered in this paper is the Dynamic Mechanical Thermal Analyzer (DMTA) (6), manufactured by Polymer Laboratories, Inc. Figure 3 shows the experimental setup for the single cantilever measurement. A thin sample is clamped at both ends. One end is attached to a calibrated shaker through a drive shaft. 

The second ORNL microcantilever project, funded by the ATF and technically co-directed hy TSA, involves polymer-coated micro cantilevers. In this case, the explosive molecule would be adsorbed on the surface of the polymer, and the swelling or change in physical properties would be measured indicating the presence of an explosive. Several polymers are currently being developed to provide the necessary selectivity for the different types of explosives signatures. The sensitivity is similar to the non-coated cantilever estimated limit of detection of low parts-per-billion (with proper collection and preconcentration system, LOD should approach low parts-per-trillion). Figure 6 illustrates two modes of detection for the coated cantilevers beam stress response and beam resonant frequency response. 

Figure 6. Illustration of the two modes of detection for the coated cantilevers beam stress response and beam resonant frequency response...

Shorter and thinner CNT are associated with higher change in resonant frequency resulting from axial CNT strain and temperature and pressure increment. Apart from CNT, the resonant frequency shift technique can also be applied on cantilever beam due to change in mass as a result of molecular adsorption [15]. 

The resonance and antiresonance frequencies are associated with the frequencies of minimum and maximum impedance for a particular resonance phenomenon, respectively. A continuous structure, like a cantilever beam, may have many resonance and antiresonance frequencies alternating in an ordered fashion with respect to frequency. 

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... 

Other transduction methods include mass-sensitive methods like the cantilever-based sensors [30]. Cantilevers are usually made of small silicon plate (length 100 pm) with a low force constant. They are fixed at one end and can vibrate at the other end. Silicon can easily be functionalized with bioelements such as antibodies or aptamers. Upon binding of the analyte to the bioelement at the surface of the cantilever, the resonance frequency of the cantilever is changing, which can be detected with help of a laser beam shining the cantilever surface [31]. 

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.)...

Micromechanical resonator has emerged as a promising component for rapidly developing telecommunication systems. These resonators also have a wide range of sensing applications. High-frequency resonators can use capacitive or piezoelectric transduction. To understand the importance of miniaturization, let us consider a cantilever beam. The natural frequency of the beam is given by the relation ... 

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]. 

Many different methods have been developed for detecting the minute deflection of the cantilever (Sarid, 1991). In this. section, we present several important ones, including vacuum tunneling (Binnig, Quate, and Gerber, 1986), mechanical resonance (Diirig, Gimzewski, and Pohl 1986), optical interferometry (Martin et al., 1988 Erlandson et al., 1988), and optical beam deflection (Meyer and Amer, 1988). 

Fig. 15.5. Force-gradient detection via mechanical resonance. Experimental and calculated resonance frequencies of a cantilever for various positions of the tunneling tip. Upper family of curves, for the copper, and lower, for the steel beam. (Reproduced from Diirig et al., 1986, with permission.)...

Scientists can fabricate microelectromechanical devices such as the cantilever above, which is a beam of silicon anchored at one end. The beam has a resonant vibrational frequency near 13 X 106 hertz (13 MHz) when stimulated with a piezoelectric vibrator. (A piezoelectric crystal, such as quartz, is one whose dimensions change in response to an electric field.) When 93 attograms (93 X 10, 8g) of an organic compound bind to the gold dot near the end of the cantilever, the vibrational frequency decreases by 3.5 kHz because of the extra mass on the beam. The minimum mass that can be detected is estimated as 0.4 attogram. 

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