Vibrating beam force sensor

A mechanical resonator having interconnected vibrating beams and counter balances whereby transfer of energy from the vibrating beams is essentially eliminated.

BACKGROUND OF THE INVENTION
 The present invention relates generally to vibrating beams, including
 piezoelectric or silicon beams that may be piezoelectrically,
 electrostatically, electromagnetically, or thermally driven, and
 particularly to vibrating beams that are utilized as force sensors, for
 example, acceleration sensors or accelerometers. In particular, the
 present invention relates to a method and apparatus for reducing the
 forces transferred to the beam supporting structure to thereby improve the
 mechanical resonance amplification factor (Q) of the vibratory system.
 A widely used technique for force detection and measurement in various
 mechanical resonators, including acceleration, and pressure sensors,
 employs one or more vibrating beams having a frequency of vibration which
 varies as a function of the force applied. An electrostatic,
 electromagnetic, piezoelectric or thermal force is applied to the beams to
 cause them to vibrate transversely or in various other modes at a resonant
 frequency. The resonant frequency of such a beam is raised when subjected
 to tension and lowered when subjected to compression. The mechanical
 resonator is designed so that the physical quantity to be measured results
 in tension or compression of the vibrating beam or beams, whereby the
 vibration frequency of the beam or beams is a measure of the amplitude of
 the quantity being measured. In one such mechanical resonator, one or more
 elongate vibrating beams are coupled between an instrument frame and a
 proof mass suspended by a flexure to measure acceleration. Acceleration
 force applied to the proof mass along a fixed axis causes tension or
 compression of the beams, which varies the frequency of the vibrating
 beams. The force applied to the proof mass is quantified by measuring the
 change in vibration frequency of the beams.
 Recently, mechanical resonators have been fabricated from a body of
 semiconductor material, such as silicon, by micromachining techniques. For
 example, one micromachining technique involves masking a body of silicon
 in a desired pattern, and then deep etching the silicon to remove portions
 thereof. The resulting three-dimensional silicon structure functions as a
 miniature mechanical resonator device, such as an accelerometer that
 includes a proof mass suspended by a flexure. Existing techniques for
 manufacturing these miniature devices are described in U.S. Pat. No.
 5,006,487, METHOD OF MAKING AN ELECTROSTATIC SILICON ACCELEROMETER and
 U.S. Pat. No. 4,945,765, SILICON MICROMACHINED ACCELEROMETER, the complete
 disclosures of which are incorporated herein by reference.
 In electrostatically driven mechanical resonators, the elongate beam(s) are
 typically vibrated by a drive electrode(s) positioned adjacent to or near
 each beam. A drive voltage, e.g., alternating current, is applied to the
 drive electrode(s) in conjunction with a bias voltage to generate an
 electrostatic force that vibrates the beam(s) at a resonant frequency.
 Motion of the beam(s), in turn, generates a current between the electrode
 and the beam(s) to produce an electrical signal representing the vibration
 frequency of the beam. Typically, high bias voltages are considered
 desirable because the current signal from the charging capacitance is
 proportional to the bias voltage. Therefore, increasing the bias voltage
 increases the signal to noise ratio of the resonator such that less
 amplifier gain is required for the oscillator circuit.
 Another important consideration in the manufacture of miniature vibratory
 force sensing mechanical resonators is to minimize variations in the
 frequency signal from the vibrating beams, except for frequency variations
 responsive to the applied force. To that end, manufacturers of these
 devices typically strive to maximize the resonance amplification factor
 (Q) of the vibrating beams, which generally represents the sharpness of
 the resonances. The resonance amplification factor, or Q, is typically
 maximized by partially or completely evacuating the chamber surrounding
 the mechanical resonator to reduce viscous damping of the resonator beams.
 Thus, mechanical resonators ideally operate in a vacuum to increase the Q
 and thereby increase the signal-to-noise ratio of the mechanical
 resonator.
 Various transducers, including accelerometers, utilize one or more
 vibrating beams that vibrate laterally in the plane of the beams or in
 various other modes. The resonant frequency of such a beam or beams is
 raised when the beam is subject to tension and lowered when subjected to
 compression. The transducer is designed so that the physical quality to be
 measured results in application of tension or compression to the vibrating
 beam or beams so that the frequency of vibration of the beam or beams is a
 measure of the amplitude of the quantity being measured. The performance
 of a vibrating beam is also degraded if energy is transferred from the
 beam to other structures, for example, the beam supporting structure,
 through rotational and transverse forces at the ends of the beam. Such
 mechanical coupling between the beam and the supporting structure can
 lower the Q of the beam and cause undesirable frequency shifts. One prior
 art method used a double-ended tuning fork having multiple beams vibrating
 out of phase to cancel rotational and transverse forces to reduce the
 energy transfer from the beam. The double-ended tuning fork utilizes two
 or more beams located side-by-side vibrating in opposite directions to
 cancel the forces appearing at the ends of the beams. The out of phase
 vibrations of the double-ended tuning fork set up equal and opposite
 reaction forces in the supporting structure at the ends of the beams which
 cancel. Examples of multiple beam resonators used to reduce energy
 transfer to the supporting structure are disclosed in U.S. Pat. No.
 4,215,570; U.S. Pat. No. 4,372,173; U.S. Pat. No. 4,415,827 and U.S. Pat.
 No. 4,901,586, the complete disclosures of which are incorporated herein
 by reference.
 Another prior art approach used vibration isolators between the ends of the
 beams and the supporting structure to reduce the transfer of energy from
 the beam to the mounting structure. Such isolators usually have an
 isolation mass at each end of the vibrating beam and a resilient member
 between each isolation mass and the supporting structure. The resilient
 members permit the beam and the isolator masses to move relative to the
 supporting structure, whereby the amount of energy transferred from the
 vibrating beam to the supporting structure is reduced. The isolation
 systems are most effective when the isolator masses are large and the
 isolation springs are compliant. Such large isolator masses and compliance
 springs result in a low resonant frequency for the isolation system which
 is undesirable, particularly in accelerometer applications. In addition,
 isolation systems attenuate the reaction forces generated in the
 supporting structure, but cannot completely eliminate them.
 U.S. Pat. No. 5,450,762, REACTIONLESS SINGLE BEAM VIBRATING FORCE SENSOR,
 the complete disclosure of which is incorporated herein by reference,
 provides yet another approach using a counter balance structure at the
 each end of the vibrating beam to cancel rotational and transverse forces
 appearing at the ends of the beam, whereby the transfer of energy from the
 beam to the mounting structure is reduced. The counter balances move in
 directions opposite to the ends of the beam in order to cancel both
 rotational moments and transverse forces normal to the longitudinal axis
 of the beam, i.e. moment and shear forces at the ends of the beam. The
 action of the counter balance generates equal and opposite reaction forces
 within the beam that cancel the moment and shear forces internally.
 Therefore, in contrast to the double-ended tuning fork, the
 counter-balanced beam transmits no energy into the supporting structure
 and no reaction force is developed within the supporting structure which
 must be cancelled by an equal and opposite force. The counter balances are
 configured relative to the beam to completely cancel only one of either
 the rotational moments or the transverse forces at the ends of the beam.
 When one of these forces is cancelled, there remains a residual amount of
 the other. The counter balance is intended to provide an optimal balance
 between the amounts of residual transverse force and rotational moment for
 a particular application. U.S. Pat. No. 5,450,762 also discloses a flexure
 interposed between the ends of the vibrating beam and the support
 structure which is intended to reduce the amount of residual torque
 applied to the mounting structure.
 The vibrating beam in the counter-balanced vibrating beam force sensors
 preferably or necessarily has particular values of four characteristics:
 (1) beam resonant frequency, (2) longitudinal stiffness, (3) sensitivity
 to force, and (4) strength. Beam design controls the values of these four
 characteristics. Often, the manufacturing process makes thickness
 variation of the beam impractical. Therefore, two dimensions, length and
 width, are typically chosen to control these four characteristics. While
 the single counter balanced vibrating beam force sensor of the prior art,
 shown and described below in FIG. 2, was configurable such that values of
 the four characteristics of beam frequency, longitudinal stiffness, force
 sensitivity, and strength, were satisfactory for prior art sized proof
 masses, shown and described in FIG. 1, some system designs have
 requirements demanding a larger proof mass. Experience has shown that a
 larger mechanism with a larger proof mass is less susceptible to
 interference from various sources and is therefore more stable.
 The moment of inertia of the pendulous proof mass and the force from
 acceleration both increase linearly with the width of the proof mass,
 while the force from acceleration increases with the square of the length
 and the moment of inertia increases with the cube of the length.
 Increasing the moment of inertia causes an undesirable increase in the
 pendulous proof mass/flexure system resonant frequency. Therefore, the
 preferred method of increasing the proof mass is to increase its width
 without changing its length. Such increases in the dimensions of the
 pendulous proof mass require dimensional changes in the vibrating beams to
 achieve satisfactory performance. Again, processing constraints do not
 generally permit changing the thickness of the beams and the four
 characteristics of resonant frequency, sensitivity to force, strength, and
 longitudinal stiffness vary differently and independently with variation
 of the beam dimensions such that control of the four characteristics is
 often unsatisfactory by mere manipulation of the length and width
 dimensions.
 Generally, satisfactory performance of a mechanism having a proof mass of
 increased dimensions requires vibrating beams having a force sensitivity
 reduced in proportion to an increase in the moment of inertia of the proof
 mass about the bending or hinge axis of the flexure, a strength and
 longitudinal stiffness increased in proportion to an increase in the
 moment of inertia, and an unchanged resonant frequency. The resonant
 frequency of a vibrating beam is proportional to its width and inversely
 to the square of its length; force sensitivity is inversely proportional
 to the cube of its width and directly proportional to the square of its
 length; strength is proportional to its width; and longitudinal stiffness
 is proportional to its width and inversely proportional to its length. Due
 to the differently and independently varying nature of the four
 characteristics with variation of the chosen dimensions of the vibrating
 beams, no combination of the width and length dimensions result in
 satisfactory values of the four characteristics for a proof mass of size
 increased over that of the prior art.
 U.S. Pat. No. 5,367,217, the complete disclosure of which is incorporated
 herein by reference, provides a four bar double-ended tuning fork device
 wherein two outer beams vibrate out of plane with two inner beams.
 According to one embodiment, the two inner beams are joined by one or more
 bridge members extending between the two beams in order to synchronize
 their motion and eliminate undesirable modes of operation. The four bar
 resonator operates similarly to the standard double-ended tuning fork
 described above, except that the beams vibrate out-of-plane rather than
 in-plane or laterally. The structure taught cannot be adapted to an
 in-plane resonator and is therefore not applicable to the problems of
 in-plane. Nor can the four bar out-of-plane resonator be combined with
 other structures taught in the prior art to solve the problems posed by an
 instrument having an in-plane resonator coupled to a proof mass of size
 increased over that of the prior art.
 SUMMARY OF THE INVENTION
 The invention recognizes and accounts for the fact that the values of the
 four vibrating beam characteristics (1) beam frequency, (2) longitudinal
 stiffness, (3) force sensitivity, and (4) strength, vary differently and
 independently with variations of the controlling dimensions of the
 vibrating beams, and that no combination of the width and length
 dimensions result in satisfactory values of the four characteristics for a
 proof mass of size increased over that of the prior art.
 The present invention provides methods and apparatus for achieving a
 vibrating beam mechanical resonator having satisfactory characteristics
 when used with a pendulous proof mass of size significantly increased over
 that of the prior art for detecting and measuring forces. These methods
 and apparatus are useful in a variety of applications, and they are
 particularly useful for measuring acceleration.
 According to one aspect of the present invention, the present invention
 includes various embodiments which overcome the problems of the prior art
 by providing a mechanical resonator having a pair of vibrating beams
 interconnected at a point approximately half-way along the length of the
 beams with one end of each vibrating beam interconnected to a support
 structure for detecting a force applied thereto, and counter balances
 extending from each vibrating beam, whereby transfer of force at said
 interconnected end of each vibrating beam to the support structure is
 essentially eliminated for beams vibrating in the plane of the support
 structure. The present invention provides counter balance structures
 disposed at each end of the vibrating beams and a joining member extending
 between the vibrating beams for mechanically joining or coupling the beams
 such that they vibrate in unison. The counter balance and mechanical
 interconnection act in combination to internally cancel forces normally
 appearing at the ends of the beam, including both rotational moments and
 transverse forces normal to the longitudinal axis of the beam. The
 combination counter balance and mechanical interconnection completely
 cancel only one of these forces, either the rotational moments or the
 transverse, i.e. shear, forces. As in the prior art device described in
 above incorporated U.S. Pat. No. 5,450,762, when one of these forces is
 cancelled, there remains a residual amount of the other. However, the
 combination counter balance and mechanical interconnection provides an
 optimal balance between the amounts of residual transverse force and
 rotational moment while maximizing the resonance amplification factor (Q)
 of the vibrating beams for a particular application wherein the proof mass
 size is significantly increased over that of the prior art.
 According to still another aspect of the present invention, the present
 invention further provides various physical embodiments in which the
 support structure includes a frame and a proof mass pliantly suspended
 therefrom with the vibrating beams being coupled to both the proof mass
 and the frame for detecting a force applied to the proof mass.
 In a preferred embodiment, the vibrating beam mechanical resonator includes
 multiple fingers intermeshed with fingers projecting from an electrode
 which are coupled to an oscillating circuit for electrostatically
 vibrating the beams in a substantially lateral direction.
 According to another aspect of the present invention, the invention
 provides methods for eliminating the transfer of energy from a mechanical
 resonator to a supporting structure thereof, whereby rotational moment and
 transverse or shear forces at the ends of the beams are substantially
 eliminated while the resonance amplification factor (Q) of the vibrating
 beams is maximized. One such method includes mechanically coupling one end
 of a pair of vibrating beams to the supporting structure for detecting a
 force applied thereto, mechanically interconnecting the vibrating beams at
 a point about half-way along their length, and counter balancing each of
 the vibrating beams. The counter balances are configured relative to the
 interconnected beams to completely cancel only one of these forces, either
 the rotational moments or the transverse forces.
 According to still another aspect of the present invention, the invention
 provides a method for substantially eliminating the transfer of energy
 from a mechanical resonator in which counter balancing the beams includes
 outwardly extending a counter balancing member from each end of the beams.
 In a preferred embodiment, the counter balancing aspect of the method
 includes counter weighting each outwardly extending counter balancing
 member.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
 In the Figures, like numerals indicate like elements.
 FIG. 1 shows a representative force detecting system or accelerometer 10 is
 illustrated according to the prior art as illustratively described in
 essential detail in above incorporated U.S. Pat. No. 5,450,762 in
 combination with U.S. Pat. No. 5,948,981, in the name of the inventor of
 the present invention and assigned to the assignee of the present
 application, the complete disclosure of which is hereby incorporated
 herein by reference. Accelerometer 10 is a miniature structure fabricated
 from a substrate 12 of semiconductor material by conventional
 micromachining techniques. Substrate 12 is preferably formed of a
 monocrystalline silicon material. Accelerometer 10 includes one or more
 flexures 14 pliantly suspending a proof mass 16 from a frame 18 for
 movement of the proof mass 16 along an input axis 20. Flexures 14 define a
 hinge axis 22 about which proof mass 16 moves in response to an applied
 force, such as the acceleration of the vehicle, aircraft or other moving
 body having the accelerometer 10 mounted thereon. Proof mass 16 has width
 dimension W measured along hinge axis 22 and a length dimension L measured
 along a pendulous axis 26 perpendicular to hinge axis 22 and extending
 between flexure 14 and the free end of proof mass 16 opposite flexure 14.
 Accelerometer 10 includes a pair of mechanical resonators 24 coupled to
 proof mass 16 and to frame 18 for measuring forces applied to proof mass
 16 by means discussed in detail below. An oscillator circuit 30, shown and
 described in detail below and in above incorporated U.S. Pat. No.
 5,948,981, electrostatically drives mechanical resonators 24 at their
 resonance frequency. In response to an applied force, mass 16 rotates
 about hinge axis 22, causing axial forces, either compressive or tensile,
 to be applied to mechanical resonators 24. The axial forces change the
 frequency of vibration of mechanical resonators 24 and the magnitude of
 this change serves as a measure of the applied force.
 Silicon substrate 12 includes an upper silicon or active layer 32
 electrically isolated from an underlying substrate 34 by an insulating
 layer 36 applied to underlying substrate 34, as shown and described in
 detail in above incorporated U.S. Pat. No. 5,948,981. Alternatively, an
 insulating layer is applied to active layer 32, as shown and described in
 above incorporated U.S. Pat. No. 5,948,981. Insulating layer 36 is
 preferably a thin layer, e.g., about 0.1 to 10.0 micrometers, of oxide,
 such as silicon oxide. Silicon substrate 12 is usually formed by oxidizing
 active layer 32 and underlying substrate 34, and then adhering the two
 layers together. A portion of active layer 32 is removed to bring layer 32
 to the desired thickness. Silicon oxide layer 36 retains its insulating
 properties over a wide temperature range to ensure effective mechanical
 resonator performance at, for example, high operating temperatures on the
 order of above about 70 to 100 degrees Celsius. In addition, insulating
 layer 36 inhibits undesirable etching of the active layer while the
 substrate is being etched, as discussed in detail in above incorporated
 U.S. Pat. No. 5,948,981.
 Proof mass 16 is typically formed from underlying substrate 34 by etching a
 slot 38 through substrate 12 and suitably etching around flexure 14.
 Mechanical resonators 24 and appropriate electrical bonds (not shown) for
 coupling mechanical resonators 24 to an oscillator circuit 30 are formed
 on active layer 32 by suitable etching techniques, such as reactive ion
 etching, anisotropic etching or the like. Preferably, the electrical bonds
 are directly coupled to oscillator circuit 30. If desired, the remaining
 portions (not shown) of active layer 32 are removed to minimize
 disturbances to the active components.
 Flexure 14 is preferably etched near or at the center of the underlying
 substrate 34, i.e., substantially centered between upper and lower
 surfaces 40, 42. Preferably, flexure 14 is formed by anistropically
 etching the flexures in a suitable etchant, such as potassium hydroxide
 (KOH). T his arrangement ensures that center of gravity 44 of
 accelerometer 10 is placed with a high degree of precision upon the center
 axis of underlying substrate 34, whereby input axis 20 is substantially
 normal to the plane of substrate 12, which in turn ensures that input axis
 20 is essentially perpendicular to the plane of proof mass 16. Preferably,
 flexure 14 is designed to exhibit simple bending motion and to limit
 S-bending motion. Accordingly, flexure 14 is preferably formed with a
 relatively short length. Alternatively, flexure 14 is formed of two or
 more flexures (not shown) arranged to pliantly suspend proof mass 16 from
 frame 18 for rotation about hinge axis 22, as shown and described in above
 incorporated U.S. Pat. No. 5,948,981.
 According to one preferred embodiment, frame 18 is isolated from external
 stresses caused by, for example, external sources of shock and vibration
 applied to the sensor or thermal stresses caused by a mismatch of
 coefficients of thermal expansion between materials of the sensor.
 External stresses and strains may be induced by, for example, the typical
 mechanical coupling of frame 18 to a silicon cover plate (not shown),
 which, in turn, is typically connected to a ceramic or metal mounting
 plate (not shown). Since the mounting and cover plates are fabricated from
 different materials, they will usually have substantially different
 coefficients of thermal expansion when heated. This mismatch in thermal
 coefficients may cause undesirable stresses and strains at the interface
 of the inner and cover plates, causing a slight distortion of frame 18.
 Many methods of isolating accelerometer 10 from Such undesirable stresses
 and strains are known to those of ordinary skill in the relevant arts. For
 example, suspending accelerometer frame 18 from a second outer frame 46 by
 flexures 48 formed by overlapping slots 49 and 50 through substrate 12,
 whereby accelerometer frame 18 is able to move relative to the outer frame
 46, as shown and described in above incorporated U.S. patent application
 Ser. No. 08/735,299. Such isolation minimizes the distortion of
 accelerometer frame 18 and thereby decreases the effects of thermal
 mismatching on mechanical resonators 24.
 Electrostatic Comb Drive
 FIG. 2 illustrates the prior art electrostatically driven mechanical
 resonators 24. Mechanical resonator 24 is formed with one or more elongate
 beams 52 mounted to enlarged or widened end supports 54, 56. Beam 52 is
 formed from active silicon layer 32 and separated from underlying
 substrate 34 so that beam 52 may be vibrated laterally relative to fixed
 end supports 54, 56. End supports 54, 56 are suitably bonded to proof mass
 16 and frame 18, respectively, by mounting pads 58, 60.
 Mechanical resonators 24 each further include an electrostatic drive for
 laterally vibrating beam 52 at its resonance frequency. The electrostatic
 drive includes an elongate electrode 62 located on one side of beam 52.
 Electrode 62 is generally parallel to and laterally spaced from beam 52,
 as shown and described in above incorporated U.S. patent application Ser.
 No. 08/735,299. Electrode 62 is preferably etched from active layer 32 and
 doped with a suitable conductive material to create the necessary charge
 carriers and to facilitate completion of the electrical circuit.
 Alternatively, electrode 62 is formed from an electrically conductive
 material deposited by conventional manufacturing methods on the surface of
 active layer 32. The conductive material is preferably gold which is a
 relatively stable material readily deposited by conventional sputtering or
 deposition methods.
 Electrode 62 is supported by a pair of support arms 64 extending laterally
 away from beam. One support arms 64 on electrode 62 is coupled to a
 bonding pad (not shown) for electrically coupling electrode 62 to
 oscillation circuit 30. Another mounting pad (not shown) electrically
 couples beam 52 to oscillation circuit 30 to complete the electrical
 circuit with electrode 62 and beam 52. Details illustrating the electrical
 interconnections of beam 52 and electrode 62 to oscillation circuit 30 are
 shown and described in above incorporated U.S. patent application Ser. No.
 08/735,299. Underlying substrate 34 may also include a bonding pad (not
 shown) for electrically connecting substrate 34 to ground. Bonding pads
 are typically formed from a suitable conductive material, such as gold.
 As shown more clearly in FIG. 2A, beam 52 includes a plurality of fingers
 66 projecting outward from a lateral surface 68 of beam 52 toward
 corresponding electrode 62. Likewise, electrode 62 includes a plurality of
 fingers 70 projecting laterally inward so that beam fingers 66 and
 electrode fingers 70 are intermeshed with each other. Fingers 66, 70 are
 each sized so that their ends 72 will not contact beam 52 or electrode 62
 when beam 52 is laterally vibrated relative to electrode 62. Usually,
 fingers 66, 70 will have a length of about 20 to 60 microns, preferably
 about 35 to 45 microns, so that fingers 66, 70 overlap each other in the
 lateral direction by about 2 to 10 microns. Electrode fingers 70 and beam
 fingers 66 are axially spaced from each other by a suitable distance to
 provide electric capacitance therebetween. Usually, beam and electrode
 fingers 66, 70 are spaced by about 2 to 10 microns from each other and
 preferably about 4 to 8 microns. Since beam fingers 66 are axially spaced
 from electrode fingers 70, the distance between fingers 66, 70 generally
 remains constant as beam 52 vibrates in the lateral direction.
 Electrostatic force is generally proportional to the square of the charge;
 the charge is in turn proportional to the voltage and to the capacitance
 between the beam and the electrode. The capacitance is inversely
 proportional to the distance between the beam and the electrode.
 Accordingly, the electrostatic force is proportional to the square of the
 voltage and inversely proportional to the square of the distance between
 the beam and the electrode. Thus, changes in the distance between the beam
 and the electrode will typically change the electrostatic force. In fact,
 this change in the electrostatic force often acts as an electrical spring
 that works opposite to the elastic force or mechanical spring of the beam
 to lower the resonance frequency. For example, as the beam moves from its
 rest position closer to the electrode, the electrostatic force increases,
 the change in force working opposite to the elastic force of the beam.
 When the beam moves from its rest position away from the electrode, the
 electrostatic force decreases, so that the change in electrostatic force
 again works against the elastic restoring force of the beam. This lowers
 the resonance frequency of the beam by a factor related to the magnitude
 of the bias voltage. Accordingly, the resonant frequency of the beams is
 generally sensitive to changes in the bias voltage.
 As shown and described in above incorporated U.S. patent application Ser.
 No. 08/735,299, the distance between intermeshed beam and electrode
 fingers 66, 70 remains substantially constant as beam 52 vibrates relative
 to stationary electrode 62. The electrostatic force between the beam and
 the electrode is generally proportional to the change in capacitance with
 distance. Since the capacitance between the intermeshed electrode and beam
 fingers changes linearly with the motion of the beams, the electrostatic
 force remains substantially constant as the beams move toward and away
 from the electrodes. Accordingly, the electrostatic force remains
 substantially constant during vibration of beam 52 and, therefore, does
 not work against the mechanical spring of beam 52 to lower the resonance
 frequency. Thus, the sensitivity to changes in bias voltage is decreased
 by 5 to 10 times compared to a similar resonator that does not incorporate
 intermeshed fingers. Reducing the sensitivity of the resonance frequency
 to changes in bias voltage increases the accuracy of the mechanical
 resonator. As a result, the mechanical resonator operates effectively with
 higher bias voltage levels, which results in a larger signal-to-noise
 ratio and requires less amplifier gain in the oscillator circuit. Usually,
 a bias voltage of about 5 to 100 Volts is applied to electrode 62 and beam
 52, preferably at least 50 Volts will be applied to the electrodes and
 beam.
 Forces applied to proof mass 16 cause it to rotate about hinge axis 22.
 This rotation generates an axial force against mechanical resonators 24.
 The axial force applied to mechanical resonators 24 proportionally changes
 the vibration frequency of beam 52 in each mechanical resonator 24. A
 relatively high resonance amplification factor (Q) makes the vibration
 frequency of the beam less sensitive to manufacturing tolerances,
 variations in the electronic components of the oscillator circuit, and
 other inputs apart from variations in the physical quantity to be
 measured. Therefore, maximization of the Q of mechanical resonators 24 is
 generally considered beneficial. Typically, Q is maximized by partially
 evacuating accelerometer 10 to reduce damping of beam 52 because the air
 between the moving beam 52 and the electrode 62 damps the movement of beam
 52 toward electrode 62. On the other hand, gas damping of proof mass 16 is
 also desirable to minimize vibrations of proof mass 16 unrelated to an
 applied force. For example, a force applied to proof mass 16 in a vacuum
 or near vacuum causes proof mass 16 to continuously swing back and forth
 about inner flexure 14 until eventually slowed to a halt by internal
 damping forces. Undesirable resonance caused by vibrations in the
 surrounding environment, other than the applied force, cause the proof
 mass to oscillate. Gas damping of proof mass 16 minimizes these
 undesirable oscillations.
 As shown and described in above incorporated U.S. patent application Ser.
 No. 08/735,299, intermeshed beam and electrode fingers 66, 70 decrease the
 damping of beam 52 at pressures above vacuum on the order of 5 to 10
 times. In fact, mechanical resonators 24 of the prior art operate
 effectively in air having substantially higher pressure levels than
 vacuum, i.e., pressure levels on the order 1/10 to 1 atmosphere, because a
 portion of the air between beam 52 and electrode 62 is located in the
 axial gaps between beam and electrode fingers 66, 70. Since fingers 66, 70
 are not moving toward and away from each other, this portion of the air
 contributes substantially less to the damping of beam 52. Accordingly,
 mechanical resonators 24 can be operated at atmospheric pressure, which
 allows proof mass 16 to be gas damped to minimize undesirable vibrations
 in the proof mass 16.
 Internal Force Balance
 FIG. 3 illustrates a double balanced vibrating beam force sensor 100 formed
 in accordance with the present invention having two counterbalanced
 vibrating beams. The vibrating beams are attached to one another at their
 approximate midpoint, or approximately halfway along their lengths, by a
 joining portion, where by the side-by-side beams are constrained to
 vibrate together. Compared to the prior art single beam sensor shown in
 FIGS. 1-2, double beam sensor 100 of the present invention shown in FIG. 3
 has the desired values for each of the four characteristics: (1) beam
 frequency, (2) longitudinal stiffness, (3) force sensitivity, and (4)
 strength, whereby satisfactory performance is achieved when used with a
 proof mass sized significantly larger than the prior art sized proof
 masses. Classical mechanical analysis shows that, in the preferred
 embodiment, double beam sensor 100 of the present invention exhibits the
 same resonant frequency, one-half the force sensitivity, twice the
 longitudinal stiffness, and twice the strength as single beam mechanical
 resonator 24 of the prior art (shown in FIGS. 1-2), when coupled to a
 proof mass (not shown) having a width W twice as large as the prior art
 proof mass and an essentially identical length L.
 The invention provides electrostatically driven mechanical resonator 100
 formed with two or more elongate beams 52. Beams 52 are joined to end
 supports 54, 56. End supports 54, 56 are suitably bonded to proof mass 16
 and frame 18, respectively, by mounting pads 58, 60. Elongate beams 52 are
 joined together at their approximate midpoint, or approximately halfway
 along their lengths, by joining portion 102, whereby beams 52 are
 constrained to move together laterally. Elongate beams 52, thus
 constrained to vibrate transversely together, vibrate at essentially the
 same natural resonant frequency in the plane of end supports 54, 56 as
 either beam vibrating singly. The natural resonant frequency, f.sub.n, of
 a beam is proportional to its cross-sectional area according to:
EQU f.sub.n.congruent.bh.sup.3 /12, (1)
 where: b=the beam thickness; and
 h=the beam height measured in the force input axis.
 Thus, it can be shown that two elongate beams 52 interconnected by joining
 portion 102 have essentially half the characteristic sensitivity to force,
 twice the characteristic strength, and twice the characteristic
 longitudinal stiffness of any single beam configured with the same length,
 width and thickness dimensions and that the interconnected beams have
 essentially the same characteristic natural resonant frequency as that of
 the single beam. Therefore, two interconnected beams suspended from a
 large proof mass have essentially the same response to an applied force as
 an identical single beam suspended from a proof mass half the size.
 Each beam 52 are preferably formed from active silicon layer 32 and
 separated from substrate 34 so that beams 52 may be vibrated laterally
 relative to fixed end supports 54, 56. Each beam 52 includes two counter
 balances 104 disposed at opposite ends of beams 52. Each counter balance
 104 includes an outwardly extending member 106 that extends beyond the end
 of beam 52. Outwardly extending member 106 is supported from beam 52 by a
 spacer 108 that holds outwardly extending member 106 in a spaced
 relationship with beam 52. The spacing between outwardly extending member
 106 and beam 52 is sufficiently large to prevent contact between outwardly
 extending member 106 and fixed end supports 54, 56 disposed at the extreme
 ends of each beam 52 during vibration. Each counter balance 104 further
 includes an optional counter weight portion 110 sized to cancel the
 rotational moments or transverse forces appearing at the ends of beam 52.
 Alternatively, outwardly extending member 106 is sized to cancel the
 rotational or transverse forces without including optional weight portion
 110.
 Each counter balance 104 is positioned along beam 52 at a point that
 rotates as the beam vibrates. Outwardly extending members 106 of counter
 balances 104 extend in a direction such that part of the motion of each
 counter balance 104 caused by the rotation of the section of beam 52 to
 which it attaches is in a direction opposite to that of beam 52, whereby
 the forces normally generated at the ends of beam 52 are internally
 cancelled by the equal and opposite forces generated by counter balance
 104. The proportions of outwardly extending member 106, spacer 108, and
 optional weight portion 110 are selected relative to the dimensions of
 beam 52 such that one of the rotational moment and the transverse forces
 at the ends of the beam are internally cancelled, although a residual
 amount of the other force remains. Choices of beam support length and
 proportions of outwardly extending member 106, spacer 108, and optional
 weight portion 110 relative to the size of beam 52 are determined for each
 particular application. The choices depend in part upon the
 characteristics of the device in which the mechanical resonator is used.
 For a pendulous accelerometer, for example, the forces on the proof mass
 suspension from forces and moments at that ends of the beam support would
 be considered along with linear and angular stiffness of the proof mass
 suspension in various directions.
 Mechanical resonator 100 further includes an electrostatic drive for
 laterally vibrating beams 52 at the resonance frequency of mechanical
 resonator 100. In the example shown in FIG. 3, only one beam 52 is
 electrostatically driven. The electrostatic drive includes an elongate
 electrode 62 located on one side of beam 52. Electrode 62 is generally
 parallel to and laterally spaced from beam 52, as shown and described
 above and in above incorporated U.S. patent application Ser. No.
 08/735,299. Electrode 62 is preferably etched from active layer 32 and
 doped with a suitable conductive material to create the necessary charge
 carriers and to facilitate completion of the electrical circuit as
 described above. Again, electrode 62 is alternatively formed from an
 electrically conductive material, such as gold, deposited by conventional
 manufacturing methods on the surface of active layer 32. Electrode 62 is
 supported by a pair of support arms 64 extending laterally away from beam.
 Electrode 62 and beam 52 are electrically coupled to complete an
 electrical circuit with oscillation circuit 30 as described above and in
 above incorporated U.S. patent application Ser. No. 08/735,299.
 One beam 52 includes plurality of fingers 66 projecting outward from a
 lateral surface 68 of beam 52 toward corresponding electrode 62 and
 intermeshing with plurality of fingers 70 projecting laterally inward
 toward beam 52. As described above and in above incorporated U.S. patent
 application Ser. No. 08/735,299, fingers 66, 70 are each sized so that
 their ends 72 will not contact beam 52 or electrode 62 when beams 52 are
 laterally vibrated relative to electrode 62. Fingers 66, 70 are axially
 spaced from each other by a suitable distance to provide electric
 capacitance therebetween, and the distance between fingers 66, 70
 generally remains constant as beams 52 vibrate laterally. The constant
 electric capacitance between fingers 66, 70 ensures that the electrostatic
 force remains substantially constant during vibration of beams 52 and,
 therefore, does not work against the mechanical spring of beams 52 to
 lower the resonance frequency.
 Oscillation Circuit
 FIG. 4 illustrates a representative oscillation circuit 30 in which
 vibrating beam 52 of mechanical resonators 100 functions as a resonator.
 As shown and described in above incorporated U.S. patent application Ser.
 No. 08/735,299, a transimpedence amplifier 202 converts a sense current
 received from vibrating beam 52 to a voltage. This voltage is filtered by
 a bandpass filter 204, which reduces noise, and its amplitude is
 controlled by an amplitude limiter 204. The resulting signal is combined
 with the output or DC bias voltage from a DC source 208 in a summing
 junction 210. The DC bias voltage generates a force between electrode 62
 and beam 52. The signal from amplitude limiter 204 modulates this force
 causing beam 52 to vibrate laterally at its resonant frequency. This
 lateral beam motion, in turn, generates the sense current. An output
 buffer 212 isolates the oscillator from external circuitry connected to an
 output 214 of oscillation circuit 30. The gain in oscillation circuit 30
 sustains oscillation of beam 52.
 Method of Manufacture
 FIGS. 5A-5C show a method of manufacturing accelerometer 10 according to
 the prior art, as shown and described in above incorporated U.S. patent
 application Ser. No. 08/735,299. An insulating layer of silicon oxide is
 first applied to underlying substrate 34, active layer 32 or both.
 Preferably, an oxide layer 320 is epitaxially grown on substantially flat
 surfaces of silicon wafers 322, 324, as shown in FIG. 5A. Silicon wafers
 322, 324 are then placed together, as shown in FIG. 5B, preferably by
 molecular bonding at elevated temperatures, e.g., on the order of about
 300 to 500 degrees Celsius. In a preferred configuration, portions of
 silicon wafers 322, 324 are removed after they have been bonded together
 to provide underlying substrate 34 having a thickness of about 300 to 700
 micrometers, preferably about 400 to 600 micrometers, and a relatively
 thin active layer 32 of about 5 to 40 micrometers, preferably about 10 to
 30 micrometers, as shown in FIG. 5C.
 Proof mass 16 and instrument frame 18 is then etched into underlying
 substrate 34 such that proof mass 16 is pliantly suspended from frame 18
 by flexure 14 and mechanical resonators 100, including joining portion 102
 and counter balances 104, are etched into active layer 32, as shown in
 FIG. 1. Insulating layer 36 inhibits undesirable etching of mechanical
 resonators 100 while underlying substrate 34 is being etched and vice
 versa. Vibrating beams 52 are etched, preferably with reactive ion
 etching, into the active layer 32. Electrodes 62 are etched from active
 layer 32 and doped with a suitable conductive material to create charge
 carriers and to facilitate completion of the electrical circuit. After the
 accelerometer components are formed into silicon wafers 322, 324, beams 52
 are mechanically coupled to proof mass 16 and frame 18, and electrodes 62
 are capacitively coupled to oscillator circuit 30.
 Those of ordinary skill in the relevant art recognize that the present
 invention is not limited to the double balanced vibrating beam force
 sensor described above and shown in the Figures. For example,
 accelerometer 10 may incorporate a double ended tuning fork or a variety
 of other mechanical resonator arrangements. However, the double balanced
 vibrating beam arrangement is generally preferred, because the transfer of
 energy from the moving beams to the rest of the components in
 accelerometer 10 is minimized, which increases the effectiveness of the
 mechanical resonator.
 Although the foregoing invention has been described in detail for purposes
 of clarity, it will be obvious that certain modifications may be practiced
 within the scope of the appended claims. For example, although the present
 invention is particularly useful for electrostatically driven resonators,
 it may also be used with other drive means, such as piezoelectric drives,
 electromagnetic drives, thermal drives or the like.