Self-compensating resonantly vibrating accelerometer driven in multiple vibrational modes

An apparatus comprising:

• (a) substantially identical first and second vibrating sensors, each vibrating sensor exhibiting a corresponding fundamental vibrational mode and a set of higher-order vibrational modes corresponding to that fundamental vibrational mode, each fundamental and higher-order vibrational mode of each vibrating sensor being characterized by a corresponding fundamental or higher-order resonant mode frequency;
• (b) a first excitation-and-detection circuit structured and connected so as to (i) drive the first vibrating sensor at a first selected resonant mode frequency f1 of a first one of the corresponding fundamental or higher-order modes thereof, and (ii) drive the second vibrating sensor at a first selected resonant mode frequency f2 of a first one of the corresponding fundamental or higher-order modes thereof that corresponds to the first selected resonant mode frequency of the first vibrating sensor;
• (c) a second excitation-and-detection circuit structured and connected so as to (i) drive the first vibrating sensor at a second selected resonant mode frequency F1 of a second one of the corresponding fundamental or higher-order modes thereof, with F1≠f1, and (ii) drive the second vibrating sensor at a second selected resonant mode frequency F2 of a second one of the corresponding fundamental or higher-order modes thereof that corresponds to the second selected resonant mode frequency of the first vibrating sensor, with F2≠f2;
• (d) a third excitation-and-detection circuit structured and connected so as to (i) drive the first vibrating sensor at a third selected resonant mode frequency F′1 of a third one of the corresponding fundamental or higher-order modes thereof, with F′1≠f1 and F′1≠F1, and (ii) drive the second vibrating sensor at a third selected resonant mode frequency F′2 of a third one of the corresponding fundamental or higher-order modes thereof that corresponds to the third selected resonant mode frequency of the first vibrating sensor, with F′2≠f2 and F′2≠F2;
• (e) a proof mass connected to the first and second vibrating sensors so that (i) acceleration of the apparatus in a first direction along a sensing axis causes the proof mass to apply tensile and compressive loads to the first and second vibrating sensors, respectively, (ii) acceleration of the apparatus in a second direction along the sensing axis, opposite the first direction, causes the proof mass to apply compressive and tensile loads to the first and second vibrating sensors, respectively, and (iii) the corresponding tensile or compressive loads applied by the proof mass to the first and second vibrating sensors cause each difference frequency Δf=f1−f2, ΔF=F1−F2, and ΔF′=F′1−F′2 to vary monotonically with acceleration of the apparatus along the sensing axis;
• and
• (f) a processing module that is structured, connected, or programmed so as to receive one or more output signals from one or more or all of the first, second, or third excitation-and-detection circuits indicative of one or more or all of f1, f2, F1, F2, F′1, or F′2, and to generate from the one or more output signals a measurement of the acceleration of the apparatus along the sensing axis, the measurement being based at least in part on a linear or nonlinear function of one or more or all of f1, f2, F1, F2, F′1, or F′2.