Quantitative studies of material properties and interfaces using the atomic force microscope (AFM) have important applications in engineering, biotechnology and chemistry. Emerging studies require an estimate of the stiffness of the probe so that the forces exerted on a sample can be determined from the measured displacements. Numerous methods for determining the spring constant of AFM cantilevers have been proposed, yet none accounts for the effect of the mass of the probe tip on the calibration procedure. This work demonstrates that the probe tip does have a significant effect on the dynamic response of an AFM cantilever by experimentally measuring the first few modes of a commercial AFM probe and comparing them with those of a theoretical model for a cantilever probe that does not have a tip. The mass and inertia of an AFM probe tip are estimated from scanning electron microscope images and a simple model for the probe is derived and tuned to match the first few modes of the actual probe. Analysis suggests that both the method of Sader and the thermal tune method of Hutter and Bechhoefer give erroneous predictions of the area density or the effective mass of the probe. However, both methods do accurately predict the static stiffness of the AFM probe due to the fact that the mass terms cancel so long as the mode shape of the AFM probe does not deviate from the theoretical model. The calibration errors that would be induced due to differences between mode shapes measured in this study and the theoretical ones are estimated.
Damping in a micro-cantilever beam was measured for a very broad range of air pressures from atmosphere (10 5 Pa) down to 0.2 Pa. The beam was in open space free from squeeze films. The damping ratio, due mainly to air drag, varied by a factor of 10 4 within this pressure range. The damping due to air drag was separated from other sources of energy dissipation so that air damping could be measured at 10 -6 of critical damping factor. The linearity of the damping was confirmed over a wide range of beam vibration levels. Lastly, the measured damping was compared with several existing theories for air-drag damping for both rarified and viscous flow gas theories. The measured data indicate that, in the rarefied regime the air damping is proportional to pressure, independent of viscosity, and in the viscous regime the damping is determined by viscosity.
One powerful method for measuring the motion of microelectromechanical systems (MEMS) relies on a Laser Doppler Vibrometer (LDV) focused through an optical microscope. Recent data taken under a very simple and common condition demonstrate that the velocity signal produced by the LDV with an optical microscope may be different from the velocity signal produced by the LDV without a microscope. This is especially important if one wishes to estimate acceleration by differentiating velocity. In this study, the time derivatives of LDV signals are compared against the signal from an accelerometer when the LDV is focused through an optical microscope and without the microscope system. The signal from the LDV without the microscope is almost identical to the accelerometer signal. In contrast, the signal from the LDV with the microscope exhibits a nonlinear relationship with the accelerometer signal. Both the LDV and the accelerometer were measuring a sinusoidal velocity generated by an electromechanical shaker. The Fourier transform of the acceleration from the LDV with the microscope shows a multitude of high harmonics of the excitation frequency, which have much higher amplitudes than the harmonics present in the accelerometer signal. Without the microscope, the LDV gives a much less distorted sinusoidal signal, even after time differentiation. The distortion of the signal from the LDV is periodic, with the same period as the sinusoidal drive signal. The largest distortion occurs near points of maximum negative acceleration, corresponding to the positive displacement peak of the sinusoidal oscillation. Because the measured oscillation is out of plane, pseudo-vibrations caused by speckle noise do not explain the distortion. Instead, the distortion appears to be caused by the optics of the microscope.
The shape control of thin, flexible structures has been studied primarily for edge-supported thin-plates. For applications such as electromagnetic wave reflectors, corner-supported configurations may prove more applicable since they allow for greater flexibility and larger achievable deflections when compared to edge-supported geometries under similar actuation conditions. Models of such structures provide insight for effective, realizable designs, enable design optimization, and provide a means of active shape control. Models for small deformations of corner-supported, thin laminates actuated by integrated piezoelectric actuators have been developed. However, membrane deflections expected for nominal actuation exceed those stipulated by linear, small deflection theories. In addition, large deflection models have been developed for membranes; however these models are not formulated for shape control. This paper extends a previously-developed linear model for a corner-supported thin, rectangular laminate to a more general large deflection model for a clamped-corner laminate composed of moment actuators and an array of actuating electrodes. First, a nonlinear model determining the deflected shape of a laminate given a distribution of actuation voltages is derived. Second, a technique is employed to formulate the model as a map between input voltage and deflection alone, making it suitable for shape control. Finally, comparisons of simulated deflections with measured deflections of a fabricated active laminate are investigated.
Current published models for predicting squeeze film damping (SFD), which are based on different assumptions, give widely different results in the free-molecule regime. The work presented here provides experimental data for validating SFD models in that regime. The test device was an almost rectangular micro plate supported by beam springs. The structure was base-excited. The rigid plate oscillated vertically while staying parallel to the substrate. The velocities of the plate and of the substrate were measured with a laser Doppler vibrometer and a microscope. The damping ratio was calculated by performing modal analysis of the frequency response functions. The test structures were contained in a vacuum chamber with air pressures controlled to provide a five-order-of-magnitude range of Knudsen numbers. The damping coefficients from the measurements were compared with predictions from various published models. The results show that the continuum-base Reynolds equation predicts squeeze-film damping accurately if used with correct boundary conditions. The accuracy of molecular-based models depends heavily on the assumptions used in developing the models.
Forces generated by a static magnetic field interacting with eddy currents can provide a novel method of vibration damping. This paper discusses an experiment performed to validate modeling [3] for a case where a static magnetic field penetrates a thin sheet of conducting, non-magnetic material. When the thin sheet experiences motion, the penetrating magnetic field generates eddy currents within the sheet. These eddy currents then interact with the static field, creating magnetic forces that act on the sheet, providing damping to the sheet motion. In the presented experiment, the sheet was supported by cantilever springs attached to a frame, then excited with a vibratory shaker. The recorded motions of the sheet and the frame were used to characterize the effect of the eddy current damping.
When a micro cantilever beam is excited by base shaking, electrostatic force makes the tip displacement response nonlinear with respect to the base acceleration input. This paper derives a single-degree-of-freedom model for the deflection in a micro cantilever due to electrostatic voltage for this excitation. The tip deflection due to electrostatic force is derived first as part of the total tip deflection, and then in terms of an equivalent base excitation. The relationship between electrostatic deflection and equivalent base excitation is determined numerically, but can be represented accurately by a simple curve-fit function.
This report summarizes a survey of several new methods for obtaining mechanical and rheological properties of single biological cells, in particular: (1) The use of laser Doppler vibrometry (LDV) to measure the natural vibrations of certain cells. (2) The development of a novel micro-electro-mechanical system (MEMS) for obtaining high-resolution force-displacement curves. (3) The use of the atomic force microscope (AFM) for cell imaging. (4) The adaptation of a novel squeezing-flow technique to micro-scale measurement. The LDV technique was used to investigate the recent finding reported by others that the membranes of certain biological cells vibrate naturally, and that the vibration can be detected clearly with recent instrumentation. The LDV has been reported to detect motions of certain biological cells indirectly through the motion of a probe. In this project, trials on Saccharomyces cerevisiae tested and rejected the hypothesis that the LDV could measure vibrations of the cell membranes directly. The MEMS investigated in the second technique is a polysilicon surface-micromachined force sensor that is able to measure forces to a few pN in both air and water. The simple device consists of compliant springs with force constants as low as 0.3 milliN/m and Moire patterns for nanometer-scale optical displacement measurement. Fields from an electromagnet created forces on magnetic micro beads glued to the force sensors. These forces were measured and agreed well with finite element prediction. It was demonstrated that the force sensor was fully functional when immersed in aqueous buffer. These results show the force sensors can be useful for calibrating magnetic forces on magnetic beads and also for direct measurement of biophysical forces on-chip. The use of atomic force microscopy (AFM) for profiling the geometry of red blood cells was the third technique investigated here. An important finding was that the method commonly used for attaching the cells to a substrate actually modified the mechanical properties of the cell membrane. Thus, the use of the method for measuring the mechanical properties of the cell may not be completely appropriate without significant modifications. The latest of the studies discussed in this report is intended to overcome the drawback of the AFM as a means of measuring mechanical and rheological properties. The squeezing-flow AFM technique utilizes two parallel plates, one stationary and the other attached to an AFM probe. Instead of using static force-displacement curves, the technique takes advantage of frequency response functions from force to velocity. The technique appears to be quite promising for obtaining dynamic properties. More research is required to develop this technique.