Triangular Cantilever Research Articles

Calibration procedures for frictional measurements with a lateral force microscope

This paper outlines a procedure for the calibration of the lateral force applied to a triangular cantilever tip by means of a lateral force microscope. The force is directed perpendicular to the cantilever major axis. The lateral force results in a torsion of the cantilever. The torsion signal is monitored as a voltage on a four-sector photodiode. The conversion from a torsion signal to a lateral force depends on the optical geometry of the instrument and the cantilever shape, dimensions and material properties. The calibration procedure used requires a calculation of the elasticity of the cantilever and experimental data. In case tripping forces are ignored, the lateral force can be taken as the friction force.

Friction effects in the deflection of atomic force microscope cantilevers

The conventional deflection-mode atomic force microscope operates by optically monitoring the slope near the end of a microcantilever in contact with the sample surface. This signal is usually interpreted as a measure of height change. Lateral forces from friction, surface geometry, or inclination of the cantilever to the surface also affect the slope due to cantilever buckling. We calculate the deflection of a hollow triangular model cantilever subject to both lateral and normal forces. The measured response of the servo circuit to an inclined, loaded cantilever is then determined. This shows (1) errors are always present in height measurements of structures on inhomogeneous surfaces; (2) the sensitivity to buckling can be reduced by repositioning the laser; (3) friction measurements can be accurately made by scanning in two directions and applying the proper calibration.

Comment on "Vibration of a 45" Right Triangular Cantilever Plate by a Gridwork Method"

Comment on "Vibration of a 45" Right Triangular Cantilever Plate by a Gridwork Method"

VIBRATION OF A 45 DEG RIGHT TRIANGULAR CANTILEVER PLATE BY A GRIDWORK METHOD

A free vibration solution is given for a 45° right triangular cantilever plate. A grid work analogy for plate bending is derived and applied to obtain the first 10 natural frequencies and mode shapes. The grid work solution is shown to give a close comparison with the experimentally determined natural frequencies and nodal patterns for the first five modes. It also is demonstrated that this gridwork solution gives a better comparison with the experimental results than does a particular corresponding Rayleigh Ritz solution.

Stress Analysis of Triangular Cantilever Plates

The stress distribution in triangular cantilever plates under a uniform normal load and under a tip load is investigated theoretically. Numerical solutions are presented for plates with angles of sweep from 0 deg. to 60 deg. and with tip angles of 30 deg., 45 deg., 60 deg., 75 deg. and 90 deg.

Vibration of Triangular Cantilever Plates by the Ritz Method

Abstract Using the method published by Ritz in 1909, natural frequencies and corresponding node lines have been determined for two symmetric and two antisymmetric modes of vibration of isosceles triangular plates clamped at the base and having length-to-base ratios of 1, 2, 4, and 7 and for the two lowest modes of right triangular plates clamped along one leg and having ratios of the length of the free leg to that of the clamped one of 2, 4, and 7. A nonorthogonal co-ordinate system was used which gave constant limits of integration over the area of the triangle. The co-ordinate transformation made it necessary to modify the functions used by Ritz in approximating deflections and to consider cross products in the integration. The integration was done numerically, using tables compiled by Young and Felgar in 1949. To check the accuracy of results, a solution was obtained to the problem of a vibrating cantilever beam of uniform depth and triangular plan view. The results obtained were found to be consistent with those obtained for the plates by using an eight-term series to approximate the deflections of the symmetric plates (isosceles triangles) and a six-term series to approximate the deflections of the unsymmetric plates (right triangles).