Trim balancing of second-order non-linearity in double ended tuning fork resonators

A method for designing double-ended tuning fork (DETF) sensors such that the second-order non-linearity of a first DETF and of a second DETF substantially cancel, the method comprising the steps of:

• a. designing said first and said second DETFs using finite-element methods, said design including nominal no-load frequencies and scale factors and including subjecting said DETFs to deformations with rotation and transverse displacement experienced in actual use, and said DETFs having substantially equal second-order non-linearity terms;
• b. creating a finite-element model, said finite-element model including at least
• said first and said second DETFs, each said DETF having one or more mass balance projecting therefrom,
• a movable proof mass,
• a fixed support frame,
• at least one hinge rotatably connecting said movable proof mass to said fixed support frame, and
• said first and second DETFs spaced apart and connected between said movable proof mass and said fixed support frame;
• c. within said finite-element model, applying forces to said movable proof mass causing said movable proof mass to rotate about said hinge and applying full-scale force loadings directly to said DETFs;
• d. within said finite-element model, said DETFs being in a deformed, pre-stressed state, using non-linear analysis to find a vibration frequency F1 of said first DETF and a vibration frequency F2 of said second DETF;
• e. repeating step d for at least four different forces, said forces being between a negative full-scale force and a positive full-scale force;
• f. using curve-fitting techniques to find the non-linearity in the difference frequency, where said difference frequency equals F1−F2;
• g. repeating steps c, d, e and f for a plurality of different mass balance configurations for each said first DETF and said second DETF;
• h. creating a model of second order non-linearity as a function of said mass balance configurations relative to each said first DETF and said second DETF;
• i. selecting said mass balance configurations where said difference frequency is a minimum;
• and
• j. verifying said mass balance configurations by creating a complete design embodying said selected mass balance configurations.