Abstract:
A gyroscope includes a ring-shaped resonator mounted in a housing, and a bottom plate attached to the resonator. A plurality of openings arranged substantially circumferentially on the bottom plate, and a plurality of grooves between the openings. A plurality of piezoelectric elements are located in the grooves. The resonator is substantially cylindrical. The plurality of openings are arranged substantially symmetrically. The piezoelectric elements can be outside the resonator, or inside the resonator. A cylindrical flexible suspension connecting the bottom to the resonator to the ring shaped resonator, wherein an average radius of the cylindrical flexible suspension and the ring shaped resonator, accounting for variation thickness of wall, is the same throughout.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation in part of U.S. patent application Ser. No. 11/845,073, filed on 26 Aug. 2007, entitled CORIOLIS FORCE GYROSCOPE WITH HIGH SENSITIVITY, which is a continuation of U.S. patent application Ser. No. 11/284,922, filed on Nov. 23, 2005, entitled CORIOLIS FORCE GYROSCOPE WITH HIGH SENSITIVITY, which claims priority to Ukrainian Patent Application No. 200505177, filed May 31, 2005, which are all incorporated herein by reference in their entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention is related to gyroscopes, and more particularly, to gyroscopes having high sensitivity and high signal-to-noise ratio. 
     2. Background Art 
     Vibrational gyroscopes have many advantages over conventional gyroscopes of the spinning wheel type. Thus, a vibrational gyroscope is considerably more rugged than a conventional spinning wheel gyroscope, can be started up much more quickly, consumes much less power and has no bearings which could be susceptible to wear. 
     A wide variety of vibrating members have been employed in previously proposed vibrational gyroscopes, ranging in shape from a tuning fork to a pair of torsionally oscillating coaxial spoked wheels. However the present invention is particularly concerned with vibrational gyroscopes in which the vibrating member comprises a radially vibrating annular shell, such as a hemispherical bell or a cylinder for example. In such gyroscopes the axis of the annular shell (e.g., the z axis) is the sensitive axis and the shell, when vibrating, periodically distorts in an elliptical fashion with four nodes spaced regularly around the circumference and located on the X and Y axes. Any rotation about the z axis generates tangential periodic Coriolis forces which tend to shift the vibrational nodes around the circumference of the shell and thereby generate some radial vibration at the original nodal positions on the X′ and Y′ axes. The Coriolis force can be calculated based on the relationship F c =2VxΩ, where F c  is the Coriolis force, V is the linear velocity vector of the mass elements of the resonator (shell) due to fundamental mode vibration, “x” is the vector product, and Ω is the angular velocity vector. Consequently the output of one or more transducers located at one or more of these nodal positions gives a measure of the rotation rate (relative to an inertial frame) about the Z-axis. 
     This highly symmetrical system has a number of important advantages over arrangements in which the vibrating member is not rotationally symmetrical about the z-axis. Thus, the component of vibration rotationally induced by the Coriolis forces is precisely similar to the driving vibration. Consequently, if the frequency of the driving vibration changes (e.g. due to temperature variations) the frequency of the rotationally induced component of vibration will change by an identical amount. Thus, if the amplitude of the driving vibration is maintained constant, the amplitude of the rotationally induced component will not vary with temperature. Also the elliptical nature of the vibrational distortion ensures that the instantaneous polar moment of inertia about the z-axis is substantially constant throughout each cycle of the vibration. Consequently, any oscillating torque about the z-axis (due to externally applied rotational vibration) will not couple with the vibration of the walls of the shell. Accordingly vibration gyroscopes incorporating an annular shell as the vibrating member offer superior immunity to temperature changes and external vibration. 
     However in practice, vibrational gyroscopes generally employ piezoelectric transducers both for driving and sensing the vibration of the vibrating member. In cases where a vibrating annular shell is employed, the transducers are mounted on the curved surface of the shell, generally near its rim. Since it is difficult to form a low compliance bond between two curved surfaces, the transducers must be sufficiently small to form an essentially flat interface with the curved surface of the annular shell. The output of the vibration-sensing transducers is limited by their strain capability, so that the sensitivity of the system is limited by signal-to-noise ratio. All these problems become more acute as the dimensions of the annular shell are reduced. 
       FIG. 1  illustrates how Coriolis forces are used in gyroscopes to measure the speed of rotation. As shown in  FIG. 1 , a resonator, typically in the shape of a cylinder, designated by  104  in its un-deformed state, is rotated. The vibration modes of the cylinder  104  involve “squeezing” the cylinder along with one of its two axes, thereby forming an ellipse. One of the axes, designated by X, becomes the major axis of the ellipse, and the other one, designated by Y, becomes the minor axis of the ellipse. This is the primary vibration mode of the cylindrical resonator, with the vibration mode designated by  101  in  FIG. 1 . 
     In essence, the cylindrical resonator alternates between orthogonal states, shown by  101  in  FIG. 1 . When the resonator rotates at an angular velocity Ω, a second vibration mode starts to appear, which is designated by  102  in  FIG. 1 . This is due to Coriolis force vectors  103 , which result in a Coriolis force in a combined Coriolis vector  105 . Therefore, the added standing vibration wave  102  is oriented at 45° relative to the primary vibration modes  101 . The amplitude of the standing wave  102  is related to the angular velocity of the resonator, and is processed electronically to generate a value representative of that angular velocity. It will be appreciated from  FIG. 1  that if the rotation of the resonator  104  were counterclockwise (instead of clockwise, as shown in the figure), the orientation of the resulting Coriolis force vector would be at 90° to what is indicated by  105  in  FIG. 1 , and would be detected accordingly. 
     As discussed above, conventional Coriolis force gyroscopes typically use a machined resonator cavity, or cylindrical resonator, with a number of piezoelectric elements that are attached to the body of the cylinder. Some of the piezoelectric elements are used to drive the vibration of the cylinders, and others are used to detect the standing wave due to the rotation, indicated by  102  in  FIG. 1 . A typical arrangement involves eight such piezoelectric elements arranged, equiangularly around the circumference of the resonator  104 , such that the major and minor axes of the ellipse (X and Y in  FIG. 1 ) have four piezoelectric elements used to generate the primary vibration mode  101 , and four piezoelectric elements arranged along the axes X′ and Y′, used to detect the standing wave  102  due to the Coriolis force. 
     It is relatively straightforward, using current technology, to machine a very precise resonator  104 , to extremely high tolerance. However, the piezoelectric elements are typically glued to the outside of the resonator. The overall structure, therefore, deviates from a perfectly symmetrical structure, since it is extremely difficult to glue the piezoelectric elements with perfect repeatability. Typical dimensions of such structures are on the order of a few millimeters to perhaps a centimeter for the smaller resonators, and larger dimensions for some of the bigger ones. The fact that the perfectly vibrating cylinder of  FIG. 1  becomes an asymmetrical structure has a direct effect on the gyroscope sensitivity, and the signal-to-noise ratio, since some of the Coriolis force-driven standing wave  102  and the primary vibration mode  101  begin to overlap, rather than be at a perfect 45° angle to each other. 
     Such resonators as used in Coriolis vibrational gyroscopes have several disadvantages: 
     1. locating the piezoelectric element t near a free edge of the resonator acts to dampen the standing wave. This, in turn, leads to a reduction in the Q-factor of the gyroscope, which in turn leads to restrictions on the scaling (of the size) of the gyroscope, and generally, on the sensitivity of the gyroscope. 
     2. the use in such resonators of piezoelectric elements that are glued to the surface of the resonator leads to an uneven distribution of stiffness in the resonator, when moving in a circumferential direction. This means that the axis of the standing wave does not necessarily coincide with the axes of the piezoelectric elements. This in turn leads to a nonlinearity in the scaling coefficient of the gyroscope, as well as to a reduction in the sensitivity of the gyroscope. This effect is particularly visible at relatively low rates of rotation. 
     If the bottom of the resonator has a number of openings that are typically evenly distributed in the circumferential direction (and also symmetrical relative to the axis of the resonator), together with a number piezoelectric elements in between the openings, such a construction has the following disadvantages: 
     1. locating the piezoelectric elements in between the openings does not permit an exact coincidence between the axis of the standing wave and the axes of the piezoelectric elements (in other words, the coordinate system of the two effects do not coincide). 
     2. mechanical balancing and tuning of the resonator in this case is a fairly complex procedure, although this is generally true of many such CVGs. 
     A conventional CVG gyroscope is described in U.S. Pat. No. 4,644,793, which include a cylindrical cup as a resonator. The resonator is affixed to a flat flexible plate (a membrane), while the membrane is connected to a ceramic disk on which a set of electrodes is located. Deformation of the membrane, when an AC excitation signal is supplied to the resonator causes radial vibration. When rotating about their axis, the nodes of the radial vibration move on the circumference of the resonator cup due to the action of the Coriolis force. The movement of the nodes is transferred to the membrane on which sensors are located, for example, capacitive sensors, which pickup the movement of the nodes. 
     Such a device has a number of disadvantages: 
     1. the use of a membrane in a sensing element, which needs to be clamped down on its outer edge, and loaded with the cylindrical cup on its inner edge, leads to the undesirable effect when non-inertial influences are experienced, such as vibration or shock. In this case, the membrane can resonate at its own frequencies, which can be very similar, or identical, to the primary frequency of the resonator cup. This, in turn, limits the sensitivity of the gyroscope, and can also lead to significant errors in the measurement. 
     2. placing sensors on the membrane leads to a lack of coincidence of the axis of the standing wave with the axes of the sensors, and also leads to a reduced sensitivity of the gyroscope and a lower accuracy of measurement of the angular velocity Ω. 
     3. mechanical balancing and tuning of the sensing element is still fairly complex, similar to the other gyroscopes described above. 
     Accordingly, there is a need in the art for a gyroscope with high precision, high sensitivity and a high signal-to-noise ratio. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention relates to Coriolis force gyroscopes with high sensitivity that substantially obviates one or more of the disadvantages of the related art. More particularly, in an exemplary embodiment of the present invention, a gyroscope includes a substantially cylindrical resonator mounted in a housing. A bottom plate is attached to the ring shaped resonator using a cylindrical flexible suspension, where the cylindrical flexible suspension has substantially the same radius as the ring shaped resonator but is substantially thinner than the ring shaped resonator. A plurality of openings are arranged substantially equiangularly on the bottom plate. A plurality of grooves in the bottom plate are arranged substantially equiangularly. A plurality of piezoelectric elements are arranged in the grooves. 
     The number of openings can be anywhere between 2 and 16, with eight openings preferred, with a corresponding number of piezoelectric elements. The piezoelectric elements and the grooves can be inside or outside the resonator. 
     Accordingly, the resonator structure described herein is intended to improve the sensitivity and accuracy of a CVG by using a new sensing element construction, and by locating the piezoelectric elements differently, another objective is to make the job of balancing and tuning of the sensing element simpler. 
     One embodiment of the invention includes a cylindrical resonator element in the shape of a cylinder with a bottom, a mounting element attaches the thin walled cylinder to the base of the gyroscope, and the piezoelectric elements and the electrical pickup elements are also included in the sensing element. The cylinder has a thicker upper portion—a ring shaped resonator. The bottom of the cylinder is divided into sectors with gaps, that are generally located in a symmetrical fashion and generally oriented radially from the center to the peripheral part of the cylinder bottom. Each sector includes a groove, where several piezoelectric excitation and signal pickup elements can be located. The grooves can be on both the outer and the inner portions of the bottom of the cylinder, and furthermore, the gaps that form the sectors can be at least partly located on the body of the side portion of the cylinder. 
     The proposed construction of the sensing element permits a more precise coincidence of the axes of the standing wave and the piezoelectric elements, compared to conventional gyroscopes. This leads to a simplification of the correction circuit of the standing wave and to a reduction in the energy consumption, as well as to an increase in the sensitivity and accuracy of the gyroscope. Also, the mechanical balancing and tuning of the sensing element is simplified, due to a reduction in stiffness along the axes of the flexible suspension of the resonator. This is because the mass that needs to removed in order to balance the sensing element can be removed in the areas of the grooves, by changing the widths of the grooves (here, changing the stiffness of the sectors can compensate for a mass imbalance of the ring resonator). 
     Additional features and advantages of the invention will be set forth in the description that follows, and in part will be apparent from the description, or may be learned by practice of the invention. The advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are intended to provide further explanation of the invention as claimed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS/FIGURES 
       The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention. In the drawings: 
         FIG. 1  illustrates how Coriolis forces are used in gyroscopes to measure the speed of rotation. 
         FIG. 2  illustrates a gyroscope according to one embodiment of the invention. 
         FIG. 3  illustrates a cross-sectional view of the resonator according to one embodiment of the invention. 
         FIG. 4  illustrates one embodiment of a piezoelectrode that can be used in the present invention. 
         FIGS. 5A-5B  illustrates photographs of one embodiment of a gyroscope of the invention. 
         FIG. 6  illustrates an electrical schematic of how the gyroscope is controlled and angular velocity is sensed. 
         FIG. 7  illustrates an electrical schematic of one of the elements of the control electronics. 
         FIG. 8  illustrates a plan view of the sensing element of the gyroscope in another embodiment of the invention. 
         FIG. 9  illustrates a cross-section along the line A-A of  FIG. 1 . 
         FIG. 10  illustrates an electrical schematic for the input signals and the output signals of the resonator. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings. 
       FIG. 2  illustrates one embodiment of the invention. As shown in  FIG. 2 , a resonator of a Coriolis force gyroscope, shown in a top plan view, includes a resonator body  104  and a bottom plate  206 , which has been modified in a particular way. The bottom plate  206  includes a plurality of openings  210 , which are preferably equi-angularly distributed around the periphery of the bottom plate  206 . The bottom plate  206 , therefore, in effect, has a number of “spokes” (as in wheel spokes). In between the openings  210 , a number of piezoelectric elements  208  are placed on the bottom plate.  212  in  FIG. 2  designates a mounting hole, which is used to secure the resonator  104 . 
     The piezoelectric elements  208  act to both vibrate the resonator  104  in its primary mode, and to detect the secondary vibration mode of the resonator  104 . It should be noted that without the openings  210 , the piezoelectric elements  208  will detect mostly the vibration modes of the bottom plate  206  itself, which are generally similar to the vibration modes of a membrane, such as a surface of a drum. 
     However, the addition of the openings  210  enables the piezoelectric elements  208  to detect the secondary vibration mode of the resonator. 
     Furthermore, it should be noted that the number of openings in piezoelectric elements  208  need not be eight, as shown in  FIG. 2 . In the degenerate case, only two openings and two piezoelectric elements  208  (positioned at 90° to the openings) can be used. Other combinations are possible, such as the use of three openings arranged at 120°, and correspondingly three piezoelectric elements  208  also located at 120°, and offset from the openings by 60°. It will be understood that the processing of the signals involved is somewhat more complex where the secondary vibration mode is not perfectly aligned with some of the piezoelectric elements  208  (as would be the case when eight openings and eight piezoelectric elements  208  are used), however, with modern computational technology, this is not a difficult computational problem to solve. 
     Other variations are possible, e.g., the use of 4, 5, 6 or 7 openings and corresponding piezoelectric elements  208 . As yet another possibility more than 8 such openings can be used, e.g., 16. However, manufacturability is an issue, since as the number of such openings and piezoelectric elements  208  increases, the signal-to-noise ratio and sensitivity increases, but the manufacturing costs also increase as well. 
     The piezoelectric elements  208  can be located both inside the cylindrical resonator (in other words on the side of the bottom plate  206  that faces into the resonator  104 ), or on the side of the bottom plate  206  that faces outside. 
     Generally, it is preferred to utilize as much of the area of the bottom plate  206  as possible. In other words, whatever space is available after the openings  210  are made, should preferably be used for locating the piezoelectric elements  208 . Thus, rectangular piezoelectric elements  208 , such as shown in  FIG. 2 , do not necessarily use up all the available area. A piezoelectric element  208 , such as shown in  FIG. 4 , takes advantage of more of the available area. Furthermore, the openings  210  need not be circular, but can have other shape, e.g., oval, etc. It should be noted that while such a shape is more efficient in terms of “real estate” utilization, it is also more difficult to manufacture. Therefore, the advantages provided by such a shape (in terms of device sensitivity, signal-to-noise ratio, dynamic range, etc.) should be balanced against manufacturability issues. 
     It should also be noted that the present invention is not limited to any particular method of mounting the piezoelectric elements  208  on the bottom plate  206 . For example, gluing, epoxying, or any other method known in the art can be used. Furthermore, the air from the cylindrical resonator  104  can optionally be evacuated to achieve a vacuum. For relatively small resonators, on the order of the approximately a centimeter in height, this results in only a minor improvement in performance, on the order of a few percent. For larger resonators, vacuum inside the resonator  104  may be significantly advantageous. 
       FIG. 3  illustrates a cross-sectional side view of one exemplary embodiment of the resonator of the present invention. The cylindrical body of resonator  104 , as seen in cross-section, has a diameter 2R 0 , and two portions—a relatively stiff portion, designated by  303 , and having a length L and a thickness H, and a relatively flexible portion, designated by  304 , having a length l, and a thickness h. A fitting  321  with an opening  317  for mounting is used to mount the resonator  104  on a shaft (not shown). Note that it is important that the resonator be tightly mounted, without any “play.” The values of the parameters R 0 , H, L, h, l, r and d can vary greatly, depending on the diameter of the resonator and the field of use of the gyroscope. One exemplary embodiment can have the following values: R 0 =12.5 mm, H=1 mm, L=8 mm, h=0.3 mm, l=10 mm, r=4 mm, d=5.5 mm. 
     The gyroscope described herein works as follows. A signal generator supplies an AC signal to opposite piezoelectric electrodes  208 , which are glued on the spokes. The frequency of the supplied AC signal is close to the natural vibration frequency of the resonator  104 . Due to the bending deformation of the spokes, the resonator  104  vibrates at the fundamental frequency in the 2-nd mode, oriented along the driven piezoelectric electrodes (see  101  in  FIG. 1 ). The piezoelectric elements  208 , located at 45 degrees, are therefore used to detect the signal. The signal is proportional to the angular velocity Ω, and can be demodulated, and then used to generate a signal that compensates for the displacement of the orientation of the 2× standing wave  102 , as described below with reference to  FIG. 6 . As one option, the compensation signal can be used as the output signal that represents the angular velocity. 
       FIG. 5A  is a photograph of a gyroscope according to one embodiment of the invention, with its cover removed. The resonator itself is covered by a lid  534  (and is therefore not visible in this figure), and is mounted on a base plate  538  which has a flange  530 . Circuit boards  532  are mounted as shown and attached to a ring  536  with screws. The ring  536  is used to maintain stiffness of the overall structure.  FIG. 5B  shows a gyroscope with a cover  540 . The dimensions of the device of  FIG. 5B  are about 50 mm diameter and 45-50 mm in height, although the device can easily be miniaturized further, for example, through the use of ASICs, etc. Roughly half the volume in the device of  FIGS. 5A-5B  is taken up by the electronics, the circuit boards  532 , etc., which easily lends itself to miniaturization. 
     The resonator  104  can be manufactured from any number of materials, however, to ensure high stability of the measurements, it is typically manufactured from a material with low internal losses and a high Q factor. Generally, the smaller the resonator  104 , the greater the error in the measurements. To reduce the error, the resonator  104  can be made out of materials with high Q factors. Also, temperature stability is also important for some applications, and various precision non-magnetic alloys with known elasticity properties can be used, or titanium alloys with damping coefficients of, e.g., δ=0.03%             δ=0.022%, and a temperature coefficient of Young&#39;s modulus of e=5×10 −5  l/° C. to e=9×10 −5  l/° C. Other materials can also be used, such as various alloys, fused silica, quartz, etc.
     Since the thickness of the flexible suspension portion  304  is &lt;&lt;H, its own natural vibration frequency is shifted to lower frequencies. This is seen from the equation for the frequency of vibration of the resonator, which is given by: 
     
       
         
           
             
               
                 
                   
                     ω 
                     i 
                   
                   = 
                   
                     
                       K 
                       ⁡ 
                       
                         ( 
                         i 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         h 
                         2 
                       
                       
                         R 
                         0 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         E 
                         
                           
                             ( 
                             
                               1 
                               + 
                               v 
                             
                             ) 
                           
                           ⁢ 
                           ρ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where 
               K   ⁡     (   i   )       =       i   ⁡     (       i   2     -   1     )           (       i   2     +   1     )               
is the coefficient that depends on the mode of the vibration i, E is Young&#39;s modulus, v is Poisson coefficient, ρ is the density of the material of the resonator.
 
     This means that the resonator  104  and the base on which it is mounted are widely separated in frequency space. Therefore, the flexible suspension portion  304  of the resonator  104  functions as a damper when inertial forces act on the resonator  104  (e.g., vibrational forces, shock, impacts, etc.). Furthermore, the natural frequency of the suspension is chosen such that it is significantly different from the maximum frequency of noise, which is typically around 2-3 KHz. 
     Reducing the thickness h of the suspension portion  304  reduces its rotational moment of inertia, which in turn reduces the demands on the precision of its manufacturing, and reduces the need for perfect symmetry of the manufactured item. This can be seen from the relationship of the moments of inertia M K  of the resonator and moment of inertia of the suspension M S  as they relate to the amplitude of the vibration of the resonator: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       S 
                     
                     
                       M 
                       K 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         h 
                         H 
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Therefore, when 
               h   H     ≤     1   4           
the tolerance requirements for manufacturing of the suspension portion  304  are reduced by an order of magnitude. Only the resonator portion  303  itself needs to be precisely manufactured, not the rest of the structure, which reduces manufacturing cost substantially.
 
     The bottom plate  206 , as well as the flexible suspension portion  304 , acts as elastic suspension. Since the electrodes  208  are placed on the bottom plate  206 , which increase stiffness along the axes of their orientation, it is necessary to increase the stiffness of the structure between the axes X and Y to enable the resonator  104  to vibrate along the axes X′ and Y′ in  FIG. 1 . The eight openings therefore serve this function. The stiffness of the “spokes” (along axes X′ and Y′) is given by 
               C   x     =       Eh   3       12   ⁢       (       R   0     -     r   0       )     2               
(see  FIG. 9 ), whereas the stiffness of the bottom plate  206  along the axes at 22.5° relative to the Y axis, is given by
 
               C   y     =       Eh   3       12   ⁢       (       R   0     -   d     )     2               
where d is the diameter of the openings (for circular openings). For the resonator to vibrate along the axis X (see  FIG. 8 ), the following condition must be satisfied:
 
 Cx/Cy=r   0   2   /R   0   2 ≦1  (3)
 
     Since the electrodes  208  are placed along the axes X, and their stiffness is given by 
               C   n     =         E   n     ⁢     bh     n   ⁢           ⁢   3           12   ⁢     a   3               
where b is the width of the electrodes, α is the length of the electrodes, h n —thickness of the electrodes, E n  is Young&#39;s modulus of the electrode (e.g., piezo-ceramic Young&#39;s modulus). The spokes have the stiffness given by h n =h if α is approximately equal to
 
             a   ≈       R   0     2           
and
 
     
       
         
           
             
               C 
               Σ 
             
             = 
             
               
                 
                   Eh 
                   3 
                 
                 
                   12 
                   ⁢ 
                   
                     R 
                     0 
                     2 
                   
                 
               
               ⁢ 
               
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         8 
                         ⁢ 
                         
                           E 
                           n 
                         
                         ⁢ 
                         b 
                       
                       
                         ER 
                         0 
                       
                     
                   
                   ) 
                 
                 . 
               
             
           
         
       
     
     To satisfy this condition, C Σ /C y &lt;1 has to hold true, or 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             d 
                             
                               R 
                               o 
                             
                           
                         
                         ) 
                       
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             8 
                             ⁢ 
                             
                               E 
                               n 
                             
                             ⁢ 
                             b 
                           
                           
                             ER 
                             0 
                           
                         
                       
                       ) 
                     
                   
                   &lt; 
                   1 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     It is clear that this condition is satisfied even when d≧R 0 /2. This, in turn, demonstrates that a gyroscope with such an arrangement of electrodes will have higher sensitivity than a conventional Coriolis force gyroscope. 
       FIG. 6  illustrates the electronic circuit that can be used to control the gyroscope and measure the angular velocity. Piezoelectric electrodes  208 A and  208 E receive a driving signal in the form of a sinusoidal voltage Asin(ω o t), where ω o —is the frequency equal to (or close to) the second order vibrational mode frequency of the resonator  104 , typically with an amplitude between 1 and 10 Volts, depending on the dynamic range of the gyroscope. A standing wave is generated, with four nodal points oriented along the piezoelectrodes  208 A,  208 E and  208 G,  208 C, and the four nodal points, located along the piezoelectrodes  208 B,  208 F              208 H,  208 D. In order to automatically maintain a stable amplitude of oscillation when the gyroscope is functioning, signals proportional to the amplitude of the oscillation are received from the piezoelectrodes  208 G and  208 C, are summed, and sent to the signal generator block  624 , which provides positive feedback control, as well as automatic gain control (AGC). The output of the signal generator block  624  is fed to the piezoelectrodes  208 A and  208 E. Thus, the signal generator  624  provides for generating the vibration of the resonator  104  with autostabilization of the amplitude of the vibration using the AGC.
     In the absence of rotation, when Ω=0, the signal at the nodes of the standing wave (the signal measured by the piezoelectrodes  208 B,  208 F and  208 H,  208 D) are minimal (essentially representing the drift of the zero of the gyroscope). When the resonator  104  rotates about its axis of symmetry, the piezoelectrodes  208 B,  208 F and  208 H,  208 D measure a signal, which is shifted in phase by 90 degrees relative to the driving signal Asin(ω o t), in other words, a cosine wave A 1  cos (ω o t) is measured, whose amplitude A 1  is proportional to the angular velocity Ω. This signal is received from the sense piezoelectrodes  208 B,  208 F is summed, demodulated (see  760  in  FIG. 7 ) using proportional and integral (PI) regulator (see  762  in  FIG. 7 ), then is modulated (see  764  in  FIG. 7 ) by a signal with the same frequency as the driving frequency to form a compensation signal (the signal received by the piezoelectrodes  208 A and  208 E). These operations are performed in block  622 . The inverted signal is then supplied to the control piezoelectrodes  208 H and  208 D to compensate for a signal that is generated at the nodes. Thus, a negative feedback loop is implemented, which compensates for the signal at the nodes. In this case, the feedback signal from the output of the PI regulator is proportional to the angular velocity vector Ω. 
     To reduce the zero bias drift of the gyroscope, block  626  can be used, which provides a minimum possible signal in the nodes of the standing wave when the gyroscope is calibrated. This signal is supplied to the control piezoelectrodes  208 H and  208 D with an opposite phase to the signal present in those electrodes, and which is present primarily due to imperfections of the manufacturing of the resonator  104 . This approach permits compensating for mass imbalances caused by differences in resonator cylinder wall thickness. 
     Block  620  is a programmable gain amplifier that filters the output signal, and normalizes the amplitude of the output signal of the gyroscope. 
     The sensing element, illustrated in  FIGS. 8 and 9 , includes a thin-walled cylinder  1 , that has a ring shaped resonator portion  2 , formed as a cylindrical rim having a length L and a wall thickness H. A cylindrical flexible suspension  303  includes a thin portion having a length  1  and a wall thickness h. A bottom  804  is divided into sectors  805  by the gaps  806 , which have a width b 2 . The wall thickness h of the flexible suspension  303  is smaller than the thickness of the walls of the ring-shaped resonator  2 . The ring shaped resonator  2  is attached to the suspension  303  R o  is the radius of the cylinder  1 . Inside each sector  805  there is a groove  807  having a width b 1 . 
     Piezoelectric elements  1008  are necessary to excite the resonator and to pick up the signal from the resonator, see  FIG. 10 . A joint  9  is located on the bottom  804 , inside the cylinder  1 , and is generally oriented coaxially with the cylinder  1 . Openings  10  and  11 , which are coaxial with the resonator  1  and the joint  9 , are used for flexible attachment of the sensing element to the base (not shown in figures). 
     The opening  11  in the joint  9  has a radius r, and an axially symmetrical base surface B, which is used for the manufacture of the sensing element, and which defines the direction of its axis. The joint  9  is added in order to reduce the communication between the vibrating portions of the bottom  804  and the base. This assists in the reduction of energy selection from the ring resonator  2 , where the energy is transferred to the base. Connecting the sensing element to the base is done by using the tubular portion  9 , using a conical coupling surface. This permits the centering of the sensing element relative to the base of the gyroscope. 
     The number of sectors  805 , and therefore the number of gaps  807  and grooves  807 , where the piezoelectric elements  1008 A- 1008 H are located (see  FIG. 10 ) is at least two, although more sectors and grooves  807  can be used. For example, sixteen such sectors can be used, however, the more sectors (and therefore groove  807 ), the more piezoelectric elements  1008  need to be used, but this tends to lead to a worsening of the dynamic characteristics of the gyroscope overall. Also, with a relatively large number of such sectors and grooves, the costs of manufacturing of the resonator increases. For most practical applications, eight sectors and eight grooves is approximately the optimal configuration, where the arrangement is axially symmetrical, at 45° angles, as illustrated in  FIG. 8 . 
     The grooves  807  can be located in both sides of the bottom  804 , in other words, on the outer side, as well as on the inner side. Correspondingly, the piezoelectric elements can be located either on the outer side, or on the inner side. The gaps  806 , which form the sectors  805 , are partially located on the body (side surface) of the cylinder  1 , near the flexible suspension  303 . This permits reducing the stiffness of the flexible suspension  303 . In the proposed sensing element, any method of mounting the piezoelectric elements  1008  on the bottom  804  can be used, such as gluing, epoxying, or any of the other known methods. 
     The sensing element can be manufactured from a number of materials, however, in order to ensure high stability of the excitation, it is preferable to manufacture it from material with low internal energy losses and a high Q factor, which would provide for a high quality resonator. It is also preferable that the material of the resonator have relatively stable elastic properties in the relevant working temperature range, such as Ni-SPAN-C-alloy  902  materials, as well as other high quality non-magnetic or weakly magnetic materials. The sensing element can be manufactured from quartz, such as Suprasil, because this material has high stability elastic characterization and has high Q factor, which exceeds the Q factor of metallic materials by a factor of several X. 
     The gyroscope as described herein works as follows: 
     A generator (not shown in  FIG. 10 ) provides a control signal to opposing piezoelectric elements ( 1008 A and  1008 E) in the form of a sinusoidal voltage Asinω 0 t, where ω 0  is the angular frequency of the signal that is equal to (or is close to) to its own frequency of fundamental form oscillation the sensing element. Due to deformation of the bottom  804 , there is a bending moment, which causes elliptical deformation of the suspension  303  at the second mode of oscillation. As a result, a standing wave is generated in the sensing element, with four nodes, oriented along the piezoelectric elements  1008 A and  1008 E, and  1008 G and  1008 C, and four nodes oriented along the piezoelectric elements  1008 B,  1008 F,  1008 H and  1008 D. Signal pickup is received from piezoelectric elements  1008 G,  1008 C,  1008 H and  1008 D. The piezoelectric elements  1008 G and  1008 C are used to pick up the signal from the anti-nodes of the standing wave, and the piezoelectric elements  1008 H and  1008 D are used to pick up the signals from the nodes. Since the piezoelectric elements are arranged in a symmetrical manner, the axis of the nodes, and the axis of the anti-nodes can switch places. 
     When the gyroscope rotates with the oscillating sensing element about its central axis with a constant angular velocity, a Coriolis force F c , is generated, which displaces the nodes of the standing wave along the circumference of the resonator. The piezoelectric elements  1008 B and  1008 F located at the nodes, therefore, receive a signal which is proportional to the angular velocity Ω. The signal is then processed electronically (not shown in  FIG.10 ), to generate a signal for compensating for the inertial displacement of the standing wave. The compensating signal is the output from the electronic circuit, and is proportional to the angular velocity, which is the parameter that needs to be measured. 
     It is important to achieve a maximum possible coincidence between the axis of the standing wave and the axes of the piezoelectric elements  1008 . This can be done by using finite element analysis, and should preferably take into account the geometry of the sensing element, the loading forces acting on it, and the material properties. The process is generally as follows: 
     each node of element has a generalized coordinate λ i , and all the coordinates are represented as a transpose matrix T:
 
{λ}={λ 1 , λ 2 , . . . , λ N } T ,  (5)
 
     where N is a total number of nodal displacements. 
     Within each element of the finite element analysis model, for the components of displacement vectors of any point M, the approximation through nodal displacement u i  is given, which are the unknown quantities:
 
 u   i ( M )=Φ ik ( M )λ k , i=1, 2, 3, k=1, 2, . . . , N,  (6)
 
     Where Φ ik (M) are the functions of the elements, which represent the coupling between the nodal displacement and the displacement vectors of the point M of the body:
 
{ u}={Φ}{λ}   (7)
 
     in matrix form. 
     The equilibrium equation is written based on the possible displacements, based on which the work performed by the internal forces is equal to the work performed by external forces due to the possible displacement: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∫ 
                         V 
                       
                       ⁢ 
                       
                         
                           σ 
                           · 
                           δɛ 
                         
                         ⁢ 
                         
                           ⅆ 
                           V 
                         
                       
                     
                     = 
                     
                       
                         
                           ∫ 
                           V 
                         
                         ⁢ 
                         
                           
                             
                               q 
                               → 
                             
                             · 
                             δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             → 
                           
                           ⁢ 
                           
                             ⅆ 
                             V 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           S 
                         
                         ⁢ 
                         
                           
                             
                               p 
                               → 
                             
                             · 
                             δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             → 
                           
                           ⁢ 
                           
                             ⅆ 
                             S 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
         
         where σ is the stress tensor, δε is the deformation tensor, {right arrow over (q)} is external load distributed over the volume V of the body, δ{right arrow over (u)} is the small-scale displacement of each point of the body, permitted by the constraints, and {right arrow over (p)} is load distributed over the surface S of the body. The tensor equation for the components of the deformations through the nodal displacement, for small deformations, is given by: 
       
    
     
       
         
           
             
               
                 
                   
                     
                       ɛ 
                       ij 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               ∂ 
                               
                                 Φ 
                                 ik 
                               
                             
                             
                               ∂ 
                               
                                 x 
                                 j 
                               
                             
                           
                           + 
                           
                             
                               ∂ 
                               
                                 Φ 
                                 jk 
                               
                             
                             
                               ∂ 
                               
                                 x 
                                 i 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         λ 
                         k 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     where i, j=1, 2, 3, x 1 , x 2 , x 3  are the coordinate axes, oriented along the unitary vectors {right arrow over (e)} 1 , {right arrow over (e)} 2 , {right arrow over (e)} 3 , or, in matrix form:
 
{ε}={ B}{λ},   (10)
 
where
 
               {   B   }     =       {     ∇     Φ   →       }     =     {       1   2     ⁢     (         ∂     Φ   ik         ∂     x   j         +       ∂     Φ   jk         ∂     x   i           )       }             
is the matrix that couples the deformations to the nodal displacements. The coupling between the tensor components and the deformations, for an elastic body, is given by Hookes&#39; law:
 
σ ij   =D   ijkl ε kl ,  (11)
 
     where D ijkl  are the elastic constants of the body, i, j, k, l=1, 2, 3, or, in matrix form:
 
{σ}={ D}{ε},   (12)
 
     By substituting Equation 10 into Equation 12, the dependents of the stress tensor on the nodal displacement can be calculated. Then, the equilibrium equation for an elastic body, that contains the displacement of its points is given by: 
     
       
         
           
             
               
                 
                   
                     
                       { 
                       σ 
                       } 
                     
                     = 
                     
                       
                         { 
                         D 
                         } 
                       
                       ⁢ 
                       
                         { 
                         B 
                         } 
                       
                       ⁢ 
                       
                         { 
                         λ 
                         } 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∫ 
                         V 
                       
                       ⁢ 
                       
                         D 
                         ⁢ 
                         
                           
                             ∇ 
                             
                               u 
                               → 
                             
                           
                           · 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 ∇ 
                                 
                                   u 
                                   → 
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           V 
                         
                       
                     
                     = 
                     
                       
                         
                           ∫ 
                           V 
                         
                         ⁢ 
                         
                           
                             
                               q 
                               → 
                             
                             · 
                             δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             → 
                           
                           ⁢ 
                           
                             ⅆ 
                             V 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           S 
                         
                         ⁢ 
                         
                           
                             
                               p 
                               → 
                             
                             · 
                             δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             → 
                           
                           ⁢ 
                           
                             ⅆ 
                             S 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     where 
               ∇     u   →       =       1   2     ⁢     (         ∂     u   i         ∂     x   j         +       ∂     u   j         ∂     x   i           )     ⁢       e   →     i     ⁢       e   →     j             
is the tensor operator. Relative to the final element that has a given volume V e , with a finite surface area S e , the equation can be rewritten as:
 
                         δλ   i     ⁢     {         ∫     V   e       ⁢         ∇     Φ   i       ·   D     ⁢       ∇     Φ   j       ·     λ   j       ⁢     ⅆ   V         -       ∫     V   e       ⁢         q   →     ·       Φ   →     i       ⁢     ⅆ   V         -       ∫     S   t       ⁢         p   →     ·       Φ   →     i       ⁢     ⅆ   S           }       =   0     ,           (   15   )               
where i, j=1, 2, . . . , N. Given that δλ i  are non-zero, then, to satisfy the equation, the expressions within the curly brackets must be zero. In other words, a system of linear algebraic equations can be derived, which determine the conditions of equilibrium of the finite element:
 
 {K}{λ}={ƒ},   (16)
 
where
 
               K   ij     =       ∫     V   e       ⁢         ∇     Φ   i       ·   D     ⁢     ∇     Φ   j       ⁢     ⅆ   V               
is the stiffness matrix OK of the element. Using Equations 4 and 13, this can be rewritten as {K}={B} T {D}{B}, while
 
               f   i     =         ∫     V   e       ⁢         q   →     ·       Φ   →     i       ⁢     ⅆ   V         +       ∫     S   e       ⁢         p   →     ·       Φ   →     i       ⁢     ⅆ   S                 
is the vector of the nodal forces on the element of the finite element analysis model, where i, j=1, 2, . . . , N
 
     The set of equations in (16), for all elements of the body, and given the boundary conditions, can be represented as a set of equilibrium equations written as:
 
{   K }{  λ }={  ƒ },   (17)
 
     where {  K } is a global matrix of the body&#39;s stiffness, {  λ } and {  ƒ } are the vectors of the nodal displacement and the forces acting on the body, respectively. 
     The set of equations in (17) can be used to calculate the properties of the sensing element, and the solution defines the vector of the nodal displacement. The displacement of the points of the body can then be calculated, as well as the deformation and the tension at those nodes. 
     Based on d&#39;Alembert&#39;s principle, when volumetric inertial forces 
                 q   →     inertial     =         -   ρ     ⁢         ∂   2     ⁢     u   →         ∂     t   2           =       -   ρ     ⁢         Φ   →     j     ·       λ   ¨     j                 
are used in the integral in Equation 16, the set of equations for the displacement of the finite element can be written as:
 
{ M}{{umlaut over (λ)}}+{K}{λ}={ƒ},   (18)
 
     where 
               M   ij     =       ∫     V   e       ⁢     ρ   ⁢           ⁢         Φ   →     i     ·       Φ   →     j       ⁢     ⅆ   V               
is the matrix representing the masses of the elements, {{umlaut over (λ)}} is the second time derivative of the vector of nodal displacements, ρ is the density of the material from which the body is made. The system of equations in (18) describes the eigenmodes (or natural vibrations) of the body, in the absence of external forces, and when the nodal displacements are found in the form {λ}e iωt , where ω and t are the frequency and vibration time of the vibrating body, respectively, the equations can be rewritten as:
 
[−ω 2   {  M }+{  K }]{  λ }= 0,  (19)
 
     given that the system of equations is equal to zero, the natural vibration frequencies of the body ω 1 , ω 2  . . . etc. can be calculated. Their corresponding nodal displacement eigenvectors {  λ } i , where i=1, 2, . . . represent their own resonant frequencies, can also be found. 
     Based on the above approach, the natural frequencies and the deformations of the different portions of the resonator have been calculated. Due to the gaps  806  and grooves  807 , the maximum displacement of rim of the sensing element is located in the same plane as the gaps  806  and grooves  807 . This means that when the sensing element vibrates at the second mode of frequency, the standing wave will be clearly defined relative to the piezoelectric elements, and its axis will be coincident with the axes of the piezoelectric elements. The fundamental resonant frequency of the cylindrical resonator can be given as: 
     
       
         
           
             
               
                 
                   
                     
                       ω 
                       i 
                     
                     = 
                     
                       
                         K 
                         ⁡ 
                         
                           ( 
                           i 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           h 
                           2 
                         
                         
                           R 
                           0 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           E 
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 v 
                               
                               ) 
                             
                             ⁢ 
                             ρ 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     where 
               K   ⁡     (   i   )       =       i   ⁡     (       i   2     -   1     )           (       i   2     +   1     )               
is the coefficient that depends on the mode of oscillations, E is Young&#39;s modulus of the resonator, v is Poisson&#39;s coefficient for the material of the resonator, and ρ is the density of the material.
 
     Analyzing Equation 20, it is clear that when h is less than H, the resonant frequency of the suspension  303  will shift into a lower frequency range. Thus, the sensing element and the base can be decoupled. Thus, the suspension  303  acts as a shock absorber, or a damper, when the gyroscope is subjected to non-inertial effects, such as shock, vibration, and so on. Also, the flexible suspension  303  should have its parameters selected such that its resonant frequency should not coincide with the maximum spectral component of technical noise to minimize the random component of the output. 
     Furthermore, reducing the thickness of the wall of the flexible suspension  303  reduces its rotational moment of inertia, which in turn leads to a generally looser requirement for its manufacturer. Also, the demands on the materials from which it is made are less stringent, as discussed earlier. 
     The tuning and balancing of the resonator is primarily due to geometric imperfections in the shape of the sense in elements and of the piezoelectric element  1008 , during their manufacture. The sensing element can be tuned and balanced after initial manufacture by changing the dimensions of the grooves, which simplifies the procedure considerably. 
     Having thus described embodiments of the invention, it should be apparent to those skilled in the art that certain advantages of the described method and apparatus have been achieved. It should also be appreciated that various modifications, adaptations, and alternative embodiments thereof may be made within the scope and spirit of the present invention. The invention is further defined by the following claims.