Mechanical Oscillator

When monitored by an oscillatory feedback method, the NSOM probe is typically driven at its resonance frequency. A probe's frequency response is dependent upon the values of the spring constant, mass, and damping coefficient. The mechanical system examined in this tutorial represents the interaction of these parameters for both the tuning fork oscillator and the bent optical probe NSOM configurations. In practice, the probe is driven through a range of frequencies to generate a frequency spectrum such as that created interactively in the tutorial. The feedback resonance frequency is then set to a value corresponding to the peak in the probe frequency response curve.

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To operate the tutorial, use the mouse cursor to adjust the three sliders. As the values of Spring Constant, Mass, and Damping Coefficient are changed, they are displayed in the windows above the sliders. The graph window displays the corresponding frequency response curve for an oscillating probe having the properties selected by the sliders. The computed Q value as well as the Frequency at Maximum Magnitude are also displayed near the center of the tutorial window.

The equation describing the amplitude of the mass oscillator as a function of frequency (Hz) is given by:

A = Fo/(M2(4π 2 f 2 - (k/M)1/2)2 + 4b 2π 2f 2)1/2

Where F(0) is the driving amplitude, M is the mass of the oscillator, f is the frequency of oscillation in hertz, k is the spring constant of the material, and b is the damping coefficient. Each of these parameters may be varied in the applet to illustrate the resulting change in resonance and the change in Q.

The parameter Q, also termed the quality factor or Q-factor, in this context, is a measure of quality of the oscillator. The quality factor is defined as the oscillator's resonance frequency divided by its resonance width. It is generally beneficial to maximize the Q of the probe oscillation to achieve sensitive tip height regulation. The lower the Q of the oscillating probe, the lower the signal-to-noise ratio, which in turn results in less representative topographic information being obtained from the oscillatory feedback mechanism. For the mechanical system represented in the tutorial, the equation for the quality factor of the system is

Q=(k/M)1/2 • M/b

The larger the Q, the greater the sensitivity of the feedback signal. If the spring constant, mass, and damping coefficient are not known, then the Q can be calculated by dividing the frequency of the maximum amplitude by the width of the peak. As evident from examining the equation above, the term that has the greatest impact on the Q is the damping coefficient, b, with greater damping in the mechanical system resulting in lower Q. A lower Q results in a mechanical oscillator with less sensitivity to a disturbance in the system, such as that resulting from the interaction of an oscillating probe with a specimen.

Contributing Authors

Jeremy R. Cummings, Matthew Parry-Hill, Thomas J. Fellers, and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.