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

A double-ended tuning fork (DETF) sensor, comprising:

• a first and a second DETF, each of said DETFs having a first end and a second end;
• a proof mass;
• a support frame,
• at least one hinge rotatably connecting said proof mass to said support frame;
• said first ends of said DETFs spaced apart and connected to said proof mass and said second ends of said DETFs connected to said support frame;
• wherein said first DETF and said second DETF are constructed having two tines;
• first mass balances projecting from each of said tines of said first DETF, said first mass balances sized and positioned to form a first second-order non-linearity term associated with said first DETF;
• and
• second mass balances projecting from each of said tines of said second DETF, said second mass balances sized and positioned to form a second second-order non-linearity term associated with said second DETF such that said second second-order non-linearity term is substantially equal in sign and magnitude to said first second-order non-linearity term.

A double-ended tuning fork (DETF) sensor, comprising:

• a generally planar silicon substrate formed with an epitaxial layer on one surface thereof;
• first and second DETFs formed in said epitaxial layer, each of said DETFs having a first and second tine joined at a first end and a second end;
• a support frame formed in said substrate,
• a proof mass formed in said substrate, said proof mass rotatably suspended from said support frame;
• said first ends of said DETFs spaced apart and connected to said proof mass and said second ends of said DETFs connected to said support frame;
• first trim balancing tabs projecting outwardly from each said tine of said first DETF, said first trim balancing tabs sized and positioned to form a first second-order non-linearity term associated with said first DETF;
• second trim balancing tabs projecting outwardly from each said tine of said second DETF, said second trim balancing tabs sized and positioned to form a second second-order non-linearity term associated with said second DETF, and
• wherein said size and said position of each of said first trim balancing tabs and said second trim balancing tabs are determined such that said first and said second second-order non-linearity terms are substantially equalized in sign and magnitude.

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 DETF using finite-element methods, said design including nominal no-load frequencies and scale factors, 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 DETF,
• said second DETF having at least one mass balance formed thereon,
• a movable structure,
• a fixed support structure,
• at least one hinge member rotatably connecting said movable structure to said fixed structure, and
• said first and second DETFs spaced apart and connected between said movable structure and said fixed structure;
• c. within said finite-element model, applying forces to said movable structure to cause said movable structure to rotate about said hinge member 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 a plurality of 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 mass balance configurations;
• h. creating a model of second order non-linearity as a function of said mass balance configurations;
• and
• i. selecting said mass balance configurations such that said difference frequency is a minimum.

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.