The prediction and understanding of structural resonances are required to optimize scanning probe microscope (SPM) design. Here, Euler beam theory is applied to the beetle-style SPM to derive analytic functions for the natural frequencies of three significant modes of vibration as a general function of the microscope shape, materials, and dimensions. In the first mode, the three piezoelectric legs vibrate transversely and the scanhead moves from side to side. In the second, the legs bend circumferentially and the scanhead rotates about its center. These modes have been identified previously, but here the mechanics analysis is presented in an improved form where the inertia of the piezo legs is considered, constraints on the shape of the central supporting disk are lifted, and appropriate boundary conditions are defined and enforced. In addition, we discuss a third mode that has not been previously identified. In this lowest frequency mode, two legs pivot about the stationary third leg. The predictions are tested against experimental data obtained from an atomic force microscope (AFM) built in our laboratory. We show that the mode frequencies can be determined easily using in situ motion of the AFM cantilever itself. Predicted frequencies are in good agreement with experimental results, although unpredicted modes are also observed. The simple closed-form solutions allow the designer to make quantitative comparisons when choosing the materials and dimensions used in the SPM design. Two new design criteria emerge from the analysis for optimizing the resonant response of the "beetle" scanhead: (1) the wall thickness tau of the piezo legs should be minimized, with their mean diameters increasing as tau(-1) and (2) the distance between the legs and the center of the scanhead should be adjusted to optimize the rotation mode eigenfrequency. (c) 2006 American Institute of Physics.