Abstract:
A pendulum arm flexure which supports a pendulum bob for oscillation has predictable and reproducible characteristics. Holders retain a specific predetermined length of uniform diameter elongated fiber at ends of the fiber and permit flexing only along a defined length of the fiber between the holders during oscillation. Energy conserving material of the fiber temporarily stores and releases energy when flexing.

Description:
CROSS REFERENCE TO RELATED INVENTION 
     This invention is related to an invention for a Double Pendulum Gravimeter and Method of Measuring Gravity Using the Same, described in U.S application Ser. No. 14/182,091, filed concurrently herewith and assigned to the assignee hereof, now U.S. Pat. No. 9,291,742. The subject matter of this application is incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     This invention relates to a pendulum, and more particularly, to a new and improved pendulum arm in the form of a flexure which is made of energy-conserving material, such as quartz, and which has a structure that is capable of reproduction in multiple substantially identical units, all of which exhibit substantially identical length, flex, and resonant operating characteristics. Further still, the present invention relates to a new and improved method of construction of such a pendulum arm flexure. 
     A pendulum is formed by a mass or “bob” that is connected to one end of a pendulum arm. The other end of the pendulum arm is pivotally connected to a stationary structure at a point of suspension or a center of motion. Energy imparted to the bob causes it to swing back and forth in an arc of oscillation at the point of suspension. Gravity sustains the oscillation of the bob until friction dissipates the oscillation energy of the swinging bob. 
     The time required for the pendulum bob to swing from one maximum amplitude end point in the arc of oscillation back to that same point is the period (T) of the swing. The period (T) of the swing, the gravity (g) and the length of the pendulum arm (L) are related to one another in an ideal pendulum by the following equation (1):
 
 T= 2 π[L/g]   1/2   (1)
 
Knowing or measuring two of the three variables length (L), gravity (g) or period (T) permits the other variable to be calculated. In this manner, a pendulum may be used as a measurement device for determining gravity (g), or precise time intervals (T), or the frequency (f) of the oscillation of the pendulum. The period (T) and the frequency (f) are inversely related to one another by the following well known equation (2):
 
 f= 1/ T   (2)
 
     It is desirable to minimize the oscillation energy loss associated with the swinging pendulum. Oscillation energy losses have the effect of changing the period (T) and/or increasing the frequency (f). A changing period (T) or frequency (f) makes it very difficult to calculate with precision the quantity which is to be measured with the pendulum. Adding energy to replace that energy lost to friction is very difficult in a pendulum, because the added energy may create aberrations in the swing of the pendulum which in turn affect the ability to precisely measure the desired variable. While energy loss in a pendulum cannot be avoided altogether, minimizing the energy loss has the effect of enhancing the accuracy of measurement. 
     One significant source of energy loss in a pendulum is the friction at the point of suspension where the pendulum arm connects to the stationary structure. The friction from the movement of the pendulum arm relative to the stationary structure dissipates energy. Even a knife-edge point of suspension creates enough friction to adversely affect the period (T) and frequency (f) in a precision pendulum. 
     One known technique of diminishing energy loss at the point of suspension is to prevent the pendulum arm from moving relative to the stationary structure. To do so, the pendulum arm must be formed as a resilient flexure which is rigidly connected to the stationary structure at the point of suspension. The other end of the flexure is rigidly connected to the pendulum bob. The rigidly connected ends of the flexure do not move relative to the objects to which they are connected, so there is no frictional loss associated with relative movement at these points. Instead, the flexure bends back and forth as the bob swings in its arc of oscillation. 
     One known pendulum flexure is formed from a resilient, energy conserving material, such as quartz (fused silica) or other similar amorphous material. Flexing the material in one direction temporarily stores energy as intermolecular or van der Waals forces within the resilient material of the flexure. When the flexure flexes in the opposite direction, the stored energy is released. In this manner, a significant quantity of the oscillation energy is preserved, minimizing the loss of oscillation compared to the frictional losses from relative mechanical movement. 
     The known pendulum arm flexure is formed of quartz or other energy-conserving material. Examples are described in two theses: A Pendulum Gravimeter for Measurement of Periodic Annual Variations in the Gravitational Constant, by William F. Hoffman, Princeton University, Jan. 1962; and A Pendulum Gravimeter for Precision Detection of Scalar Gravitational Radiation, by David R. Curott, Princeton University, May 1965. The quartz pendulum arm flexures described in these theses are formed by heating the center section of a solid quartz rod until it achieves a viscous and flowable state, and then stretching the viscous center section to draw it out to a long, small diameter fiber extending between the larger unchanged ends of the rod. The rod transitions or necks down from the full diameter ends to the small diameter center fiber. The transitions occur in an unpredictable manner according to the uniformity of heat distribution in the center section of the quartz rod, the amount of heat energy in the center section prior to stretching, the rate at which the solid rod is stretched, and the viscosity of the heated center portion from which the fiber is formed, among other variables. The fiber itself is not of a uniform diameter, because the stretching occurs in an uncontrolled manner. The necked down transition portions between the full diameter ends of the rod and the center fiber are also variable in characteristics, due to the transitions occurring in an uncontrolled manner. 
     As a consequence of these uncontrolled variables, the length (L) of the pendulum arm is not predictable, and the flex characteristics of the flexure are also unpredictable. The necked down transition portions do not precisely demarcate points which establish the length (L) of the fiber which forms the pendulum arm. The thinnest portions of the necked down transition portions adjacent to the fiber may flex slightly along with the fiber, thereby varying the length (L) of the pendulum arm. Furthermore, the nonuniform diameter or thickness of the fiber itself will have different flexure characteristics. 
     These idiosyncratic aspects of known prior art quartz pendulum arm flexures are not of principal concern in those pendulum devices which utilize only a single pendulum supported by a single flexure. The operating characteristics of the pendulum device are adapted to the unique characteristics of the single flexure. However, in pendulum devices which require two flexures to support a single bob, or in pendulum devices which use two separate pendulums operating at the same oscillation frequency, it is important that multiple pendulum arm flexures have substantially the same length, flex and resonant operating characteristics. Pendulum arm flexures having substantially the same length, flex and resonant operating characteristics achieve predictable oscillatory behavior. Using pendulum arm flexures which have significantly different length, flex and resonant operating characteristics result in undesirable modes of movement of a single pendulum supported by two flexures. The undesirable modes of movement consume additional energy and adversely affect the desired operation of the pendulum. In addition, in double or multiple pendulum devices, significantly different length, flex and resonant operating characteristics of multiple pendulum arm flexures create substantial difficulties in attempting to coordinate and synchronize the motions of multiple pendulums, or may make synchronized operation achievable only when accompanied by substantial and undesirable energy loss. 
     SUMMARY OF THE INVENTION 
     The pendulum arm flexure of the present invention is made from quartz or other energy-conserving material, and has a definite length (L) and a substantially uniform diameter fiber extending between opposite ends to which holders are attached. As a consequence of these characteristics, the length, flex and resonant operating characteristics of the flexure are predictable and therefore reproducible in multiple substantially identical ones of the pendulum arm flexures, each of which has substantially identical operating characteristics. Two of these substantially identical pendulum arm flexures may be used effectively to suspend a single bob in single pendulum device. Multiple ones of these substantially identical pendulum arm flexures may be used to suspend the bobs of multiple pendulums in a multiple pendulum device. The definite and determinable operating characteristics of the pendulum arm flexure of the present invention minimize or eliminate undesirable modes of motion which consume additional oscillating energy of the pendulum. The definite and determinable operating characteristics of the pendulum arm flexure reduce the need, and components required, to add energy to an oscillating pendulum, thereby simplifying the operation of the pendulum. The present invention also involves a method of constructing such a pendulum arm flexure having these desirable characteristics. 
     In accordance with these considerations, one principal aspect of the invention is a pendulum arm flexure for supporting a pendulum bob from a support structure. The flexure comprises an elongated fiber having opposite ends, and a holder connected at each opposite end of the elongated fiber. One holder is adapted to rigidly connect the pendulum arm flexure to either the pendulum bob or the support structure, and the other holder is adapted to rigidly connect the pendulum arm flexure to the other one of the pendulum bob or the support structure. Each holder includes an inner end adjacent to the fiber, and the fiber extends continuously between the inner ends of the holders. The inner end of each holder has a larger cross-sectional size than the cross-sectional size of the adjacent fiber. The inner end of each holder transitions abruptly in cross-sectional size relative to the cross-sectional size of the connected fiber. The fiber has a precise length measured between the abrupt transitions at the inner ends of the opposite holders. The fiber has resiliency characteristics which permit flexing along the length of the fiber between the inner ends of the holders during oscillation of the pendulum. Each holder has rigidity characteristics which prevent flexing of the holder at its inner end during oscillation of the pendulum. The fiber is formed of energy conserving material which temporarily stores energy expended in flexing the fiber in one direction as intermolecular force and then releases the stored energy when the fiber flexes in the opposite direction. 
     Other aspects of the pendulum arm flexure include some or all the following described features. The fiber has a substantially uniform cross-sectional size between the inner ends of the opposite holders. The resiliency characteristic of the fiber is substantially uniform along the length of the fiber between the inner ends of the holders. An electrically conductive coating covers the fiber and each holder. Each holder is integrally connected to the fiber, such as by integral fusion. The fiber and both holders are separately formed before each holder and the fiber are integrally fused together. The fiber and both holders are formed of the same material, which is preferably capable of viscously flowing upon the application of sufficient heat, such as a glass or quartz material. 
     Another principal aspect of the invention is a method of constructing a pendulum arm flexure which supports a pendulum bob from a support structure, in which the pendulum arm flexure comprises an elongated fiber having opposite ends and a holder located at each opposite end of the elongated fiber, with each holder adapted to connect the pendulum arm flexure to one of the pendulum bob or the support structure. The method comprises forming first and second holders separately from one another and from an elongated fiber, connecting the first holder to one end of the fiber and connecting the second holder to the other end of the fiber at a predetermined distance from the first holder to establish the length of the pendulum arm flexure which will undergo oscillation. 
     Other subsidiary aspects of the construction method include some or all of the following described features. Each holder is integrally connected to the ends of the separate fiber, by for example, fusing each holder and the fiber. The holders and the fiber are formed of the same material, such as quartz, which is capable of fusion upon the application of sufficient heat. Each holder is formed to include an opening within which to receive the one end of the fiber, the end of the fiber is inserted into the opening of each holder, and the holder and the end of the fiber inserted the opening are fused to integrally connect each holder to each end of the fiber. The holder and the end of the fiber are fused by the application of heat sufficient to melt the holder and the fiber while directing a stream of cover gas over the fiber adjacent to the inner end of each holder to cool the fiber and prevent melting of the fiber adjacent to the holder. The fiber is formed to have a substantially uniform cross-sectional size along its length between the holders. A center section of a rod of material from which the fiber is formed is heated sufficiently to make the center section of the rod viscous and flowable, and opposite ends of the rod are moved away from one another at a substantially constant rate to draw the viscous center section of the rod into an elongated and substantially uniform and reduced cross-sectional length of material, from which the fiber is obtained. Opposite ends of the rod are moved away from one another at a substantially constant rate by suspending the rod vertically above a hollow tube of electrically conductive material, attaching a magnet to the lower end of the vertically suspended rod, and moving the end of the heated rod into the tube at a substantially constant rate established by eddy currents induced in the electrically conductive tube which create a magnetic force that counteracts force from the magnet and causes the magnet to move downward at a substantially constant rate. The exterior of the flexure may be coated with an electrically conductive material. 
     A more complete appreciation of the present invention and its scope may be obtained from the accompanying drawings, which are briefly summarized below, from the following detailed description of presently preferred embodiments of the invention, and from the appended claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a perspective block and generalized component illustration of a double pendulum device which incorporates multiple ones of the pendulum arm flexures of the present invention. 
         FIG. 2  is a perspective view of one pendulum arm flexure shown in  FIG. 1  and which incorporates the present invention. 
         FIG. 3  is an enlarged axial section view of one end of the pendulum arm flexure shown in  FIG. 2 . 
         FIG. 4  is an enlarged axial section view of the end of the pendulum arm flexure shown in  FIG. 3 , prior to fusing a fiber in a retainer portion of a holder to form the integral flexure shown in  FIGS. 2 and 3 . 
         FIG. 5  is a partial enlarged axial section view similar to  FIG. 4 , showing a holder and an end of a fiber prior to inserting the end of the fiber into the holder and prior to integrally fusing those components together to create the flexure shown in  FIGS. 2 and 3 . 
         FIG. 6  is a perspective view of a prior art pendulum arm flexure with respect to which the present invention is an improvement. 
         FIG. 7  is a perspective view of a rod from which the prior art pendulum shown in  FIG. 6  is constructed. 
         FIG. 8  is an enlarged axial section view of one end of the prior art pendulum arm flexure shown in  FIG. 6 . 
         FIG. 9  is a generalized perspective view of a glass lathe holding a tube and a rod upon which actions are performed to construct the pendulum arm flexure shown in  FIGS. 2-5 . 
         FIGS. 10A-10J  are perspective, axial section, enlarged and partial sequential views which illustrate actions performed on the tube and the rod shown in  FIG. 9  to construct a holder shown in  FIGS. 4 and 5  of the pendulum arm flexure shown in  FIGS. 2 and 3 . 
         FIGS. 11A-11C  are perspective, axial section, enlarged and partial sequential views which illustrate actions performed to construct a fiber of the pendulum arm flexure shown in  FIGS. 2-5 . 
         FIGS. 12A-12C  are perspective, axial section, enlarged and partial sequential views which illustrate actions performed on the holder illustrated in  FIGS. 10A-10J  and the fiber illustrated in  FIGS. 11A-11C , to connect the holder and the fiber as shown in  FIGS. 2 and 3 . 
         FIG. 13A  is a partial view similar to  FIG. 3 , showing an electrically conductive coating applied to the exterior of the pendulum arm flexure shown in  FIGS. 2 and 3 .  FIGS. 13B and 13C  are generalized illustrations of actions taken to apply the electrically conductive coating shown in  FIG. 13A . 
     
    
    
     DETAILED DESCRIPTION 
     Four pendulum arm flexures  20 , each of which incorporates the present invention, are used in a double pendulum device  22  shown in  FIG. 1 . The double pendulum device  22  may be a gravimeter used to measure gravity (g), or a clock used to measure intervals of time (T) or to establish a frequency (f). An example of a double pendulum device  22  used as a gravimeter is described in the above cross-referenced U.S patent application. 
     The double pendulum device  22  includes a first pendulum  24  and a second pendulum  26 . The first pendulum  24  comprises a pendulum bob  28  and two pendulum arm flexures  20  which suspend the pendulum bob  28  from a pendulum suspension structure  30 . The second pendulum  26  comprises a pendulum bob  32  and two pendulum arm flexures  20  which suspend the pendulum bob  32  from a pendulum suspension structure  34 . One end of each pendulum arm flexure  20 , the upper end  36  as shown in  FIG. 1 , is connected to one of the pendulum suspension structures  30  or  34 . The other end of each pendulum arm flexure  20 , the lower end  38  as shown in  FIG. 1 , is connected to one of the bobs  28  or  32 . In this manner, two pendulum arm flexures  20  support each pendulum bob  28  and  32  from each pendulum suspension structure  30  and  34 . Both pendulum suspension structures  30  and  34  are connected to a support post  40  which extends from and forms part of a rigid base  42  of the device  22 . 
     The upper ends  36  of the two flexures  20  associated with each pendulum  24  and  26  are rigidly connected to the suspension structures  30  and  34 . The lower ends  38  of the two flexures  20  associated with each pendulum  24  and  26  are rigidly connected to respectively opposite ends of the bobs  28  and  32 . The points of connection of the upper ends  36  of the flexures  20  to the pendulum suspension structures  30  and  34 , and the points of connection of the lower ends  38  of the flexures  20  to the bobs  28  and  32 , cause the pendulums  24  and  26  to swing or oscillate in a common plane of oscillation. Preferably, the pendulums  24  and  26  oscillate 180° out of phase with one another, meaning that when the pendulum  24  reaches its maximum amplitude point in its arc of oscillation on the left (as shown), the pendulum  26  reaches its maximum amplitude in its arc of oscillation on the right (as shown), and vice versa. The maximum amplitude points of the pendulum bobs  28  and  32  in their arcs of oscillation are sensed by amplitude sensors  44  and  46 , respectively, both of which are attached to the base  42 . Although the bobs  28  and  32  are shown supported below the suspension structures  30  and  34 , the flexures  20  could also be suspend the bobs above suspension structures in appropriate circumstances. 
     The length, flexure and resonant operating characteristics of the pendulum arm flexures  20  are substantially identical in accordance with the present invention, as discussed in greater detail below. The weight and center of mass distribution of the pendulum bobs  28  and  32  are also substantially identical. Consequently, the pendulums  24  and  26  experience substantially identical natural or resonant oscillation characteristics. The substantially identical natural resonant oscillation characteristics of each pendulum  24  and  26  causes one pendulum  24  or  26  to oscillate at a frequency (f) or period (T) which is substantially identical to the frequency or period of the other pendulum  26  or  24 . 
     When oscillating at their natural resonant frequencies, the pendulums  24  and  26  conserve the maximum amount of oscillation energy. Stated alternatively, the pendulums  24  and  26  minimize the loss of oscillation energy when operating at their natural resonant frequencies. The natural resonant frequency energy storage and loss characteristic of any resonant system is defined by a term referred to as “Q”. When operating at a high Q, a resonant system conserves the maximum amount of the resonant energy and minimizes the loss of oscillating energy. 
     The pendulums  24  and  26  preferably have substantially identical high Q&#39;s and natural resonant frequencies. The pendulums  24  and  26  swing in substantially identical arcs of oscillation, maintain substantially identical maximum amplitude points, and do so while losing a minimum amount of oscillation energy from the unavoidable frictional energy loss associated with any moving mechanical system. These desirable characteristics result in major part from the consistent, predictable and reproducible characteristics of each pendulum arm flexure  20 . 
     More details concerning each pendulum arm flexure  20  are shown in  FIGS. 2-5 . The pendulum arm flexure  20  comprises a holder  50  at each end  36  and  38  of the flexure  20 . The holders  50  are adapted to connect the ends  36  and  38  to a pendulum bob (e.g,  28  or  32 ,  FIG. 1 ) and to a suspension structure (e.g.,  30  or  34 ,  FIG. 1 ). A fiber  52  extends between the opposite ends  36  and  38  of the holders  50 . The fiber  52  is formed from material, such as quartz (fused silica), which provides a high degree of energy conservation due to the storage and release of intermolecular forces when the material is mechanically flexed or bent and then released to resume its initial non-flexed position. Preferably, the holders  50  are also formed of the same type of material as the fiber  52 . A further desirable characteristic of this type of energy-conserving material is a capability to become viscous, flow and melt upon the application of sufficient heat, as discussed below. 
     The fiber  52  has a substantially uniform diameter and substantially uniform material characteristics along its length between the holders  50 . The fiber  52  flexes when the pendulum swings in its arc of oscillation. The holders  50  do not flex to any significant degree when the pendulum swings in its arc of oscillation, because the holders  50  are themselves rigid and rigidly connected to the pendulum suspension structures  30  and  34  and to one of the pendulum bobs  28  and  32 . 
     Each holder  50  has a uniform diameter tubular portion  54  located at its outer end. A middle portion  56  of each holder  50  is formed as a hollow frustoconical-shaped or necked-down transition which extends inward from the tubular portion  54  toward an inner tubular retainer portion  58  of each holder  50 . The retainer portion  58  extends inward from the transitional portion  56  and connects to the fiber  52 . Initially, before the fiber  52  is connected to the tubular retainer portion  58 , a small axial opening  60  extends from an inner end  62  of the retainer portion  58  into the transitional portion  56  ( FIGS. 4 and 5 ). An outer end  64  of the fiber  52  is inserted into the axial opening  60  ( FIG. 4 ). The retainer portion  58  is heated until it and the outer end  64  of the fiber  50  melt and fuse together into an integral solid mass  66  ( FIG. 3 ). The integral fusion of the end  64  of the fiber  52  and the retainer portion  58  makes the end  64  of the fiber  52  integrally and rigidly a part of the holder portion  50 . In this manner, the fiber  52  is rigidly and integrally joined to each holder  50 , as shown in  FIG. 3 . 
     Due to its integral connection to the retainer portion  58  of the holder  50 , the outer end  64  of the fiber  52  is not able to flex relative to the retainer portion  58  or relative to the holder  50 . Flexing of the fiber  52  is only possible beginning at the point where the fiber  52  adjoins the inner end  62  of the retainer portion  58  and along the length of the fiber  52  to the point adjoining the inner end  62  of the retainer portion  58  of the holder  50  at the opposite end of the flexure  20 . The fiber  52  flexes only between the inner ends  62  of the retainer portions  58  of the opposite holders  50  ( FIG. 2 ), due to the rigid connection of the holders  50  to the pendulum bobs and to the suspension structures. 
     The specific positions of the inner ends  62  of the holders along the length of the fiber  52  precisely define the effective oscillation length (L) of the pendulum arms along which flexure occurs. The length of the pendulum arm (L) is precisely and definitely established by a center portion  68  of the fiber  52  extending between the distinct inner ends  62  of the retainer portions  58  of the oppositely positioned holders  50 . The length of the center portion  68  of the fiber  52  between the opposite ends  62  of the retainer portions  58  is precisely set before fusing the end  64  of the fiber  52  and the retainer portion  58  of the second holder  50  of the flexure  20 . The length (L) of the pendulum is controlled by the extent to which the center portion  68  is exposed after inserting and fusing the end  64  of the fiber  52  in the axial opening  60 . Controlling the length (L) of the flexure  20  in this manner allows multiple ones of the pendulum arm flexures  20  to be constructed having substantially identical lengths (L). 
     In contrast to the precise and controllable length (L) of the fiber  52  of the flexure  20  ( FIG. 2 ), a known prior art flexure  70 , shown in  FIGS. 6 and 8 , exhibits an effective oscillation length (L) which is significantly indeterminable. The prior art flexure  70  is formed from a single integral rod  72  of quartz material, shown in  FIG. 7 . A center portion  74  of the rod  72  is heated until it becomes viscous and flowable, which allows cylindrical ends  76  of the rod  72  to be separated and pulled in opposite directions, thereby drawing the viscous center portion  74  into two opposite frustoconically shaped transitional portions  78  between which a considerably smaller diameter center fiber  80  extends. The center fiber  80  continues in a portion  82  which diminishes further in diameter from the transitional portion  78 , until ultimately a center portion  84  of the fiber  80  reaches a somewhat consistent diameter over some indeterminate length. The entire prior art flexure  70  is formed simultaneously in this manner. 
     The two transitional portions  78  are variable and nonuniform in their thickness and length characteristics. The diminishing-diameter portions  82  of the center fiber  80  are also variable and nonuniform in their thickness and length. The variability in thickness arises from the lack of precise control in drawing the viscous center portion  74  of the rod  72  into the transitional portions  78 , the portion  82  and the center fiber  84 . Most importantly, however, the variable transitional portions  78  and the diminishing-diameter portions  82  do not precisely establish the beginning and ending points at which the fiber  80  flexes. Flexure may occur in some indeterminate location within the transitional portions  78  and/or in the diminishing-diameter portions  82 . Without such a specific point at which the fiber  80  is allowed to flex, it is impossible to determine with precision the effective length (L) of the flexure  70  during oscillation, as shown graphically in  FIG. 6 . 
     Another category of problems associated with the prior art flexure  70  is that its resonant oscillatory characteristics are substantially indeterminable. The flexible transitional portions  78  and the diminishing-diameter portions  82  of the fiber  80  vary in thickness or diameter, and that variability introduces different flexure characteristics in those portions  78  and  80  compared to the more uniform flexing characteristics of the center portion  84  of the fiber  80 . The variability in thickness of the flexing portions  78  and  82  of the flexure  70  create different mechanical flex characteristics, which leads to variability in the natural resonant frequency characteristics of the flexure  70 . These variable characteristics make it very difficult or impossible to predict the oscillating characteristics of the prior art flexure  70 . 
     A prior art pendulum arm flexure  70  of the type shown in  FIG. 6  will perform satisfactorily in a pendulum which utilizes only a single such flexure  70  to support a pendulum bob. In those circumstances, the flexure and natural resonant frequency operating characteristics of the flexure  70  are simply measured, and then the remaining aspects of the pendulum are adapted to the measured characteristics of the flexure  70 . In other words, the operating characteristics of the pendulum device are adapted to the unique characteristics of the pendulum arm flexure. In the case of a single flexure-single bob pendulum device, consistency in the characteristics of the pendulum arm flexure is not necessarily essential. 
     On the other hand, a prior art pendulum arm flexure  70  is not satisfactory for use where multiple pendulum arm flexures are used to support a single pendulum bob, or where multiple pendulums must oscillate in synchronization with one another, or where the pendulum device requires or depends upon predictable length, flex and natural resonant frequency operating characteristics of the pendulum arm flexure. To obtain optimal performance in such situations, each pendulum arm flexure should have substantially identical and predictable length (L) and natural resonant frequency operating characteristics. Without such substantially predictable characteristics, the oscillation of a single pendulum bob supported by two flexures will not oscillate in the desired manner with minimum loss of oscillation energy, and/or the two pendulums will not oscillate in synchronization with one another with minimum loss of oscillation energy. Excessive energy loss becomes a substantial and significant problem in the use of these prior pendulum devices. 
     The pendulum arm flexure  20  of the present invention solves these problems by having a substantially predictable effective length (L) and predictable flexure and natural resonant frequency characteristics. As a consequence, the present invention permits the construction of multiple substantially identical pendulum arm flexures  20  on a repeatable, predictable and consistent basis, thereby assuring that the pendulum devices in which multiple ones of those flexures  20  are utilized will operate as desired with minimal loss of oscillating energy. 
     A method of constructing each pendulum arm flexure  20  to yield consistent and predictable characteristics entails separately constructing two holders  50  and a single fiber  52  ( FIGS. 4 and 5 ) and then joining them together to form the flexure  20  ( FIG. 2 ).  FIGS. 10A-10J  illustrate the construction of one holder  50 . The other holder  50  of the flexure  20  is formed in the same manner.  FIGS. 11A-11C  illustrate the construction of a single fiber  52 .  FIGS. 12A-12C  describe joining the two holders  50  to the fiber  52  to form the flexure  20  ( FIG. 2 ).  FIGS. 13A-13C  describe placing an electrically conductive coating on the exterior of the flexure  20  ( FIG. 2 ). 
     Construction of the holder  50  commences, as shown in  FIG. 9 , by fusing an end of a quartz tube  90  to a solid quartz rod  92 . The fusion preferably occurs while the tube  90  and the rod  92  are held in chucks  94  and  96  of spindles  98  and  100 , respectively, of a conventional glass lathe  102 . The spindles  98  and  100  are sometimes referred to as the headstock and tailstock of the lathe, respectively. The spindles  98  and  100  rotate coaxially about a single working axis of the lathe  102 , and the chucks  94  and  96  hold one or two workpieces and rotate them about that working axis. As shown in  FIG. 9 , the tube  90  and rod  92  constitute the workpieces. One of the spindles is movable longitudinally along the working axis, to move the workpiece held by that spindle axially relative to the workpiece held by the other spindle. The flexure  24  is preferably constructed by actions performed by using the glass lathe  102 , as described in connection with  FIGS. 10A-10J and 12A-12C . 
     The ends of the tube  90  and the rod  92  are brought into contact with one another, by movement of the spindle  100  toward the spindle  98 . Heat from a heat source such as an methane-oxygen flame or a laser is directed onto the contacting ends and adjacent portions of the tube  90  and the rod  92 . Sufficient heat is applied to melt and fuse together the contacting ends of the tube  90  and the rod  92 , causing the tube  90  and the rod  92  to be integrally connected to one another. The heat for fusing the tube  90  and the rod  92  together is applied while the tube  90  and the rod  92  are rotated by the spindles  98  and  100 , thereby uniformly distributing the heat and uniformly fusing together the ends of the tube  90  and the rod  92 . The fused-together tube and rod are thereafter allowed to cool to room temperature. 
     Next, as shown in  FIG. 10A , heat from a methane-oxygen flame or laser is applied to heat a center portion  104  of the fused-together ends of the tube  90  and rod  92 . The heat is applied while the fused-together tube  90  and rod  92  are rotating in the glass lathe  102  ( FIG. 9 ), thereby evenly distributing the heat throughout the center portion  104 . Sufficient heat is applied to make the center portion  104  viscous and flowable. 
     Thereafter as shown in  FIG. 10B , the opposite ends of the fused together tube  90  and rod  92  are moved axially away from one another by separating the spindles  98  and  100  from one another ( FIG. 9 ), thereby extending the length of the viscous center portion  104  of the tube  90  and simultaneously drawing it radially inward into a frustroconically shaped necked down tube portion  106  and a frustroconically shaped necked down solid portion  108 . The necked down tube portion  106  is hollow to the location where the tube  90  was fused to the rod  92  ( FIGS. 9 and 10 ), and the necked down solid portion  108  is complete integral material since it was formed from the rod  92 . The necked down portions  106  and  108  are thereafter allowed to cool to room temperature. 
     The necked down tube portion  106  is then heat worked to thicken a shoulder area  110  of the necked down portion  106  and to reduce the internal diameter of an axial opening  112  through a neck area  114  of the necked down portion  106 , as shown in  FIG. 100 , while the tube  90  and rod  92  rotate in the glass lathe  102  ( FIG. 9 ). Applying heat to the necked down tube portion  106  while rotating it in the glass lathe causes the viscous glass to accumulate in the shoulder area  110  and in the neck area  114 , due to surface tension of the viscous material. Consequently, the amount of material in a shoulder area  110  and in the neck area  114  increases. The increased material in the neck area  114  reduces the diameter of the axial opening  112 . The configuration shown in  FIG. 100  is allowed to cool to room temperature. 
     The necked down tube portion  106  is then cut away from the necked down solid portion  108 , as shown in  FIG. 10D . Cutting is accomplished by scoring the neck area  114  with a diamond cutter  116  at a location  118  on the neck area  114  adjacent to the end of the necked down solid portion  108  while the tube  90  and rod  92  rotate in the glass lathe  102  ( FIG. 9 ), and then applying axial separation force from the spindles  98  and  100  ( FIG. 9 ) to separate the neck area  114  from the necked down solid portion  108  at the scored location  118 . The location for scoring the neck area  114  and separating the necked down portions  106  and  108  should be measured from the shoulder area  110  to extend about 30-50% more than the desired final axial length of the retainer portion  58  ( FIGS. 4 and 5 ). The separated necked down tube portion  106  is thereafter used to create one holder  50  ( FIGS. 4 and 5 ) as further described below. 
     Next, as shown in  FIG. 10E , heat is applied to the neck area  114  of the necked down tube portion  106 , until the neck area  114  becomes viscous. The viscous material in the heated neck area  114  accumulates and reduces the length of the neck area  114  and increases the thickness of the walls of the neck area  114  to reduce the inside diameter of the axial opening  112  through the neck area  114 , as shown in  FIG. 10F . The heat is applied to accumulate the viscous material in the neck area  114  until the diameter of the axial opening  112  is reduced to the desired diameter of the final size of the axial opening  60  in the retainer portion  58  ( FIGS. 4 and 5 ). In a preferred embodiment described herein, the desired diameter of the axial opening  112  is approximately 50 microns (μ). Preferably a video camera with visual enlargement and measurement capabilities is used to visualize the effects and gauge the diameter of the axial opening  112  as the heat is applied. 
     Thereafter, as shown in  FIG. 10G , the necked down tube portion  106  with its shortened and reduced internal diameter neck area  114  ( FIG. 10F ) is reattached by heat fusion to the necked down rod portion  108  from which it was previously separated ( FIG. 10D ). Attachment in this manner allows the necked down tube portion  106  with its shortened and reduced internal diameter neck area  114  to be cut at the desired length of the retention portion  58  ( FIGS. 4 and 5 ). 
     Cutting the shortened and reduced internal diameter neck area  114  of the necked down tube portion  106  to the desired length of the retention portion  58  ( FIGS. 4 and 5 ) is illustrated in  FIG. 10H . The desired length is measured at location  120 , and the rotating neck area  114  is scored lightly with very light contact from the diamond cutter  116 , while the attached necked down tube portion  106  and the necked down rod portion  108  rotate in the glass lathe. A small amount of liquid, preferably water, is applied at the scored location  120 , and the spindle  100  of the glass lathe  102  ( FIG. 9 ) is moved slightly axially relative to the spindle  98  to separate the necked down tube portion  106  from the necked down rod portion  108  at the scored location  120 . 
     The end  122  of the separated neck area  114  is thereafter heat or flame polished, as shown in  FIG. 10I . The heat from the polish gathers any slight projections or irregularities of material resulting from mechanically separating the necked down tube portion  106  from the necked down rod portion  108  ( FIG. 10H ). Any slight projections or irregularities surrounding the reduced internal diameter axial opening  112  are thereby removed, to prevent those slight projections from inhibiting the insertion of the fiber  52  into the axial opening  60  when the flexure  20  is constructed ( FIGS. 4 and 5 ). As a result of the actions described in conjunction with  FIG. 10I , the neck area  114  of the necked down tube portion  106  assumes the final configuration of the retainer portion  58  of the holder  50  ( FIGS. 3-5 ). The thickened shoulder area  110  of the necked down tube portion  106  has previously assumed the final configuration ( FIG. 10C ) of the transitional portion  56  of the holder  50  ( FIGS. 2-5 ). 
     The holder  50  is completed by cutting the tube  90  with a wet saw  124  at a position  126  spaced along the cylindrical tube  90  from the transitional portion  56  or the shoulder area  110 , as shown in  FIG. 10J . The cut end of the cylindrical tube at position  126  is heat or flame polished to eliminate any slight projections resulting from wet sawing the tube  90 . Eliminating any such slight projections in this manner has eliminates stress concentration points which might cause the holder  50  to break. 
     Construction of the fiber  52  of the flexure  20  ( FIGS. 2-5 ) commences with use of a small diameter solid quartz rod  130 , shown in  FIG. 11A . The rod  130  has a diameter of about 1 mm and is of a manageable length of approximately 100 mm, for example. The rod  130  has been ultra sonically cleaned for approximately 10 minutes in a 2% micro-90 solution, and then rinsed in tap water. Next, the rod is rinsed in de-ionized water and then hot air dried. The rod is then etched for three minutes in a 25% hydrofluoric acid solution, followed by a tap water rinse, a de-ionized water rinse and then air dried. Once prepared in this manner, the rod  130  is subjected to the actions which form part of it into the fiber  52 . 
     An upper end  132  (as shown) of the rod  130  is connected by a conventional clamp  134  to a stationary structure  136 , or otherwise held in a stationary position, while the remainder of the rod  130  hangs vertically downward from the upper stationary-supported end  132 . Another conventional clamp  138  is attached to a lower end  140  (as shown) of the rod  130 . A relatively small magnet  142  which produces substantial magnetic flux, such as a conventional rare earth magnet, is connected to the clamp  138 . The lower end  140  of the vertically suspended rod  130 , the lower clamp  138  and the magnet  142  are located vertically above a center opening  144  of a vertically oriented tube  146 . The electrically conductive tube  146  is electrically conductive and is formed from relatively low electrical resistance material such as copper or aluminum. 
     Heat from a methane-oxygen flame or from a laser is applied along a middle section  150  of the rod  130  between its ends  132  and  140 , as shown in  FIG. 11A . The heat uniformly heats the middle section  150  of the rod  130  until it becomes viscous and flowable. Gravity acts on the viscous middle section  150  and the lower end  140  of the rod  130 , on the clamp  138  and on the magnet  142 , causing the viscous middle section  150  of the rod  130  to stretch as shown in  FIG. 11B . The lower end  140  of the rod  130 , the clamp  138  and the magnet  142  move downward into the center opening  144  of the tube  146  under the influence of gravity. 
     The magnetic flux from downward moving magnet  142  induces eddy currents in the conductive tube  144 . The eddy currents flow circumferentially around the conductive tube  144 , and create a magnetic flux and upward oriented magnetic force within the tube which opposes the magnetic flux of the magnet  142 , thereby creating an oppositional force to resist the downward movement of the magnet  142  under the influence of gravity. The magnitude of eddy currents induced in the tube  144  is related to the speed at which the magnet  142  descends within the tube. The amount of oppositional magnetic force created by the eddy currents in the tube  144  increases with the speed of descent of the magnet  142 . At a sufficient rate of descent, the oppositional force from the eddy currents interact with the magnetic flux from the magnet  142  to counterbalance the gravitational force on the viscous middle section  150  and the lower end  140  of the rod  130 , on the clamp  138  and on the magnet  142 , causing the speed of the descent of the magnet  142 , the clamp  138  and the lower end  140  of the rod  130  to stabilize at a constant downward velocity. 
     The viscous middle section  150  of the rod  130  stretches at a constant rate once the constant downward velocity of the magnet  142 , the clamp  138  and the lower end  140  of the rod  130  are stabilized in their downward descent rate. At some point in the constant downward descent after the viscous middle section  150  has stretched considerably, the middle section  150  cools and its viscosity decreases enough to increase the mechanical resistance to further downward descent, causing the lower end  140  of the rod  130  and the clamp  138  and the magnet  142  to slow and ultimately gently terminate further descent within the tube  144 . 
     The constant rate of descent of the lower end  140  of the rod  130 , the clamp  138  and the magnet  142  within the conductive tube  144  stretches the middle section  150  of the rod  130  at a constant rate while the middle section  150  remains viscous. The constant rate stretching of the middle section  150  of the rod  130  has the effect of drawing down the viscous middle section  150  to a substantially constant diameter along its length. Transitional portions of the middle section  150  adjacent to the ends  132  and  140  of the rod  130  experience a reduction in diameter, but those transitional portions are not part of the constant diameter section  150  and are not used to form the fiber  52  of the flexure  20  ( FIGS. 1-5 ). The constant diameter middle section  150  is drawn down to approximately 30μ in diameter for use as the fiber  52 , in the preferred example of the flexure  20  described herein. 
     By starting with similar rods  130  ( FIG. 11A ), which have the same initial diameter and which have been subjected to the same preconditioning described above, multiple middle sections  150  having substantially the same diameter are created from each of the similar rods  130 . The middle sections  150  each have substantially the same diameter as a result of using the same magnet  142  moving downward within the same tube  144  to achieve a constant rate of descent and constant rate of stretching of the middle section  150 . In this manner, multiple similar fibers  52  are created for each of multiple similar flexures  20  ( FIGS. 1-3 ). 
     Next, as shown in  FIG. 11C , the constant diameter middle section  150  is cut at a position  152  near one end of the constant diameter middle section  150 , preferably with a scissors. A video camera with microscopic expansion and measuring capabilities is used to select the position 152  at which the constant diameter middle section  150  is cut. The position  152  is selected to avoid including any portion of the larger diameter transitional portions between the ends  132  and  140  and the constant dia meter center section  150 . A piece  154  of the constant diameter middle section  150  remains connected to one of the ends  132  or  140  (end  132  is shown in  FIG. 11C ). As is explained below, leaving the piece  154  connected to the end  132  facilitates construction of the flexure  20  ( FIGS. 2-5 ). A part of the piece  154  becomes a fiber  52  for one or more flexures  20  ( FIGS. 2-5 ). 
     Construction of the flexure  20  commences by chucking a holder  50  ( FIG. 10J ) into one of the spindles of the glass lathe ( FIG. 9 ). The end  132  of the rod  130  to which the piece  154  remains connected ( FIG. 11C ) is chucked into the other one of the spindles of the glass lathe ( FIG. 9 ). The cut end of the uniform diameter piece  154  ( FIG. 11C ) constitutes the end  64  of the fiber  52  ( FIGS. 4 and 5 ). 
     The end  64  of the constant diameter piece  154  is inserted into the axial opening  60  of the holder  50 , as shown in  FIG. 12A . The end  64  of the piece  154  is extended substantially completely through the axial opening  60  in the retainer portion  58  of the holder  50 . Heat from a hydrogen-oxygen flame or a laser, for example, is applied to the exterior of the retainer portion  58  to fuse the retainer portion  58  and the end  64  into the single integral mass  66  ( FIG. 3 ). 
     Simultaneously with the application of the heat, a stream  156  of cover gas, such as argon or helium, is directed from a nozzle  158  onto the piece  154  at a position directly adjoining the inner end  62  of the retainer portion  58  of the holder  50 . The stream  156  of cover gas cools the piece  154  adjacent to the end  62  to prevent the piece  154  from becoming sufficiently viscous so that the larger mass  66  of the molten retainer portion  58  and end  64  do not draw material from the piece  154  outside of the end  62 . In this manner, the stream  156  of cover gas ensures that the diameter of the piece  154  adjacent to the end  62  of the retainer portion  58  remains constant in diameter and is not diminished in diameter when the retainer portion  58  and the inner end  64  of the piece  154  are integrally fused together. Consequently, the diameter of the piece  154  fiber  52  immediately adjacent to the inner end  62  of the holder  50  remains the same diameter as the fiber  52  at other locations along the length of the fiber  52 . 
     Next, as shown in  FIG. 12B , a portion  160  is cut out of the piece  154 , thereby separating the end  132  and transitional portion ( FIG. 12A ) from the piece  154 . The portion  160  has sufficient length to form the fiber  52  of the flexure  20  ( FIGS. 2-5 ) and to form the end  64  of the fiber  52  ( FIG. 4 ). The remaining portion of the piece  154  and the connected end  132  are removed from the chuck of the spindle of the glass lathe, for later use in fabricating another fiber  52  for another flexure  20 , if the length of the remaining piece  154  is sufficient for that purpose. 
     Thereafter, as shown in  FIG. 12C , another holder  50  is inserted in the chuck of the spindle of the glass lathe. The end of the portion  160  is inserted into the axial opening  60  of the second holder  50 . The extent of insertion precisely establishes the length of the fiber  52  between the inner ends  62  of the holders  50  ( FIG. 2 ). A video camera with microscopic expansion and measuring capabilities is used to establish the precise length of the fiber  52 . The actions described above in connection with  FIG. 12A , including the application of the stream  156  of cover gas from the nozzle  158 , are repeated to fuse the end  64  of the portion  160  and the retainer portion  58  of the second holder  50  into the integral mass  66  ( FIG. 3 ), thereby completing the formation of the flexure  20  ( FIGS. 1-3 ). 
     Construction of the flexure  20  is completed by applying a thin conductive layer  162  of electrically conductive material, such as gold palladium, to the exterior of the holder  50  and the fiber  52  of the flexure  20 , as shown in  FIG. 13A . The conductive layer  162  electrically connects the flexure  20  to one of the pendulum suspension structures  30  or  34 , which are connected through the post  40  to the base  42  ( FIG. 1 ). In essence, the flexures  20  are electrically connected to the same common reference potential as the surrounding components, thereby draining any electrostatic charge that might otherwise accumulate on the flexures  20  during use. An accumulation of static charge on the flexure will electrostatically attract and repel the flexure with respect to adjoining structures and thereby adversely influence the oscillation characteristics of the pendulum. Adverse influences on the oscillation of the pendulum create inaccuracies in the quantity being measured by the pendulum device. 
     The conductive layer  162  is applied as discussed in connection with  FIGS. 13B and 13C . To begin, as shown in  FIG. 13B , the flexure  20  is placed inside a chamber  164 . An oxygen plasma is directed onto the flexure  20  within the chamber  164 . The oxygen plasma oxidizes any hydrocarbon impurities on the surface of the flexure  20 , turning those impurities into carbon dioxide and thereby leaving the flexure  20  clean. A clean surface of the flexure is essential to achieving good adherence of the layer  162  ( FIG. 13A ). 
     Next, as shown in  FIG. 13C , the clean flexure is placed in a sputter coating chamber  166 . The conductive layer  162  is sputter coated or deposited in the conventional manner onto the quartz or other energy-conserving material of the clean flexure  20 . The electrically conductive layer  162  is very thin in depth and uniform in thickness, for example a few microns, a few hundred angstroms, or a few molecules in depth. Despite the relative thinness of the layer  162 , its thickness is sufficient to prevent any electrical charge from accumulating on the flexure. 
     The electrically conductive layer  162  is sufficiently thin and flexible to avoid adversely influencing the flex characteristics of the flexure  20 . As a result, the flex and oscillating characteristics of the flexure  20  are established principally by the flex characteristics of the quartz or other energy-conserving material which forms the fiber  52  ( FIG. 2 ). The electrically conductive layer  162  also remains adherent and sufficiently flexible to avoid cracking or separating during oscillation of the pendulum. The electrically conductive layer  162  does not diminish the strength or integrity of the quartz or other energy-conserving material from which the flexure  20  is formed. The metallic conductive layer  162  should not be highly stressed, create excessive tension on the exterior of the fiber  52  and/or create nucleations on the underlying quartz or other energy-conserving material, because such effects weaken the fiber  52  and makes it prone to break after a time of oscillation. Any stresses from the conductive layer  162  should be compatible with and comparable to the stresses occurring within the quartz or other energy conserving material of the fiber  52  during oscillation. Preferably, the electrically conductive coating is gold palladium. A conductive layer which is not satisfactory for long-term oscillation is tin oxide. 
     Forming the flexure  20  with the structure described and in the manner described results in a substantial improved flexure compared to the known prior art flexures used in pendulums. 
     The length of the fiber  52  between the inner ends  62  of the two oppositely positioned retainer portions  58  of the holders  50  is precisely established by use of the microscopic expansion and measuring capabilities of the video camera, when the second holder  50  is fused as described in connection with  FIG. 12C . In this manner, the precise oscillation length (L) of the fiber  52  is established. 
     The use of the stream  156  of cover gas ( FIG. 12A ) maintains the constant diameter of the fiber  52  between the inner ends  62  of the retainer portions  58  of the holders  50  ( FIG. 2 ). Consequently, the fiber  52  is not weakened at the point where it is fused to the holder  50 . The fiber  52  is not more prone to fail from the mechanical stress of vibration at the location where it is fused to the holder  52 , since the diameter of the fiber  52  remains undiminished at this position. 
     Similarly, since the diameter of the fiber  52  remains constant at the position adjoining the inner ends  62  of the retainer portions  58 , the uniform diameter of the fiber  52  along its entire length establishes substantially similar flex and natural resonant frequency operating characteristics. Furthermore, these natural resonant frequency operating characteristics are similar among multiple flexures  20  constructed in the manner discussed above, due to the substantially constant and uniform diameter fibers  52  obtained from substantially uniform diameter middle sections  150  ( FIGS. 11B and 11C ) of multiple rods  130  which have been processed in a substantially similar manner as described above. 
     The retainer portions  58  of each of the holders  50  are of sufficient mass and rigidity to prohibit any flexure. Consequently, only the fiber  52  flexes between the inner ends  62  of the retainer portions  58  of the holders  50  at opposite ends of the flexure  20  ( FIG. 2 ). The natural resonant frequency operating characteristics of the pendulum results substantially only from the characteristics of the fiber  50 , allowing the natural resonant frequency and length characteristics to be predetermined and made uniform among multiple ones of the flexures  20 . 
     The flexure of the present invention prevents the accumulation of electrostatic charges. Aberrations in the oscillation of the pendulum due to the accumulation of static charge are avoided, and as a consequence, the quantity (e.g., gravity) measured by the pendulum is more accurate. 
     The method of constructing the pendulum armed flexure  20  as discussed above involves uniform, precise and repeatable actions. As a consequence, multiple pendulum arm flexures having substantially identical length, flex and natural resonant frequency operating conditions can be produced on a controllable, precise and repeatable basis. The substantially identical characteristics allow two pendulum arm flexures to be used effectively to suspend a single bob in a single pendulum device, and/or allow multiple similar pendulums to be used effectively in multiple pendulum devices. Undesirable modes of motion are avoided by using multiple pendulum arm flexures having substantially identical length, flex and natural resonant frequency operating characteristics. The loss of oscillation energy is avoided by using multiple pendulum arm flexures according to the present invention. 
     The significance of these and other improvements and advantages will become apparent upon gaining a full appreciation of the present invention. Preferred embodiments of the invention and many of its improvements have been described with a degree of particularly. The detail in describing the preferred examples is not necessarily intended to limit the scope of the invention. The scope of the invention is defined by the following claims.