Patent Publication Number: US-2006009930-A1

Title: Monitoring and correcting for non-translational motion in a resonance measurement apparatus

Description:
BACKGROUND OF THE INVENTION  
      1. Field of the Invention  
      The present invention relates generally to the field of resonance compensation, and more specifically to compensating for hard disk drive head suspension mechanical resonance effects, such as torsion and sway.  
      2. Description of the Related Art  
      Disk drives are magnetic recording devices used for the storage of digital information. Digital information is recorded on substantially-concentric tracks on either surface of one or more magnetic recording disks. Disks are rotatably mounted on a spindle motor, and read/write heads mounted to actuator arms access information on the disks, with the heads rotated by a voice coil motor (VCM). The voice coil motor rotates the pivoting arms, or suspensions, and moves the heads radially over the surface of the disk or disks. The read/write heads must generally be accurately positioned on the disk to ensure proper reading and writing of information that will define the data storage tracks. After the system writes the servo patterns on the disks, each disk can be added to a hard drive assembly.  
      The information contained in servo and data tracks on the disk surface must be written in a precise manner. In a typical configuration, one or more data heads are employed and are connected to a rotatable base via a suspension, or arm, that has particular mechanical characteristics. The resonance behavior of the arm is highly determinative of the accuracy of data written and read by the head, as significant resonance in the arm or suspension can significantly disturb head positions  
      Testing of disk drive suspension mechanical resonance characteristics requires devices such as a Head Resonance Tester, or HRT, to excite various resonance modes on the combined suspension and head assembly by shaking the assembly. The HRT, or shaker, may excite resonance modes within certain ranges, including but not limited to 1 KHz to 40 KHz. If the head is mounted onto the suspension and centered along a suspension axis, the HRT may shake the suspension and head arrangement in a line perpendicular to the suspension axis, which may be called the Z axis. In essence, the base of the suspension is fixed and shaken in a direction perpendicular thereto, such as the arrangement shown in  FIG. 6 . Loaded HRT  601  includes accelerometer  401 , shaker block  402 , suspension  404 , and head  403 . The amplitude of the HRT excitation in the arrangement shown may be measured by the accelerometer  401 .  
      For the arrangement shown in  FIG. 5 , when the direction of acceleration does not remain well aligned in the Z direction, the accuracy of the resonance curve generated by the accelerometer may be compromised. Such misalignment can result from resonance modes in the HRT, which can be very difficult to prevent over large frequency ranges. Further, the interaction between the suspension and the HRT, specifically the action of the suspension resonance on the shaker block, can also cause the suspension to torque and move in directions other than the Z direction in the arrangement shown. The additional movement is not measured by the accelerometer but is a component describing the rotational motion of the shaker block  402 . These spurious motions, sometimes referred to as shaker modes, provide an additional non-characterized excitation to the suspension and head and thereby limit the accuracy and usefulness of resonance measurements.  
      It would be beneficial to have a system that overcomes the shortcomings of previous designs and provides a more accurate assessment and correction of resonance curves for a mechanical arrangement, such as a drive head and suspension combination.  
     SUMMARY OF THE INVENTION  
      According to the present invention, there is provided a system and method for assessing and potentially correcting for non-translational motion in a resonance measurement apparatus. The design measures the response of an article, such as an HGA assembly including a read/write head, as well as a more complete characterization of the excitation of a shaking device, such as a head resonance tester, and computes a correction factor using either two or three point measurement. The correction factor generally geometrically relates the measurement at the shaking device and translates it into an excitation measurement at the article, such as the head of the HGA. The correction factor may be evaluated by subjecting the arrangement to further vibration at varying frequencies and assessing the response. Measurement of the shaking device may be accomplished using an accelerometer or by optical measurement using a light beam. Measurement of the article is performed using a light beam, which may be a light beam generated in the same manner as that used to measure the response of the shaking device.  
      According to one aspect of the present invention, there is provided a method for providing an enhanced response representation for an article mounted to a shaker device. The method comprises measuring an article response quantity at the apparatus, determining a shaker excitation quantity at the shaker device, computing a correction factor geometrically relating the shaker excitation quantity to the article response quantity, and determining the response representation based on applying the correction factor to raw shaker and article data.  
      According to a second aspect of the present invention, there is provided a system for determining a excitation representation of a shaking device having an article attached thereto. The system comprises an article response measurement device for measuring the article response when shaken at at least one shaking frequency. The system also comprises a shaking device excitation measurement apparatus for measuring the shaking device excitation when shaken at at least one shaking frequency. The system further comprises a computing device for computing a correction factor based on the shaking device excitation relative to the article response and applying the correction factor to received article and shaking device excitation.  
      According to a third aspect of the invention, there is provided a method for determining resonance behavior of a component affixed to a resonance inducing device. The method comprises identifying component resonance behavior for a location on the component for at least one frequency, determining resonance inducing device excitation behavior for at least one location on the resonance inducing device for at least one frequency, computing a correction factor based on the component resonance behavior and the resonance inducing device excitation behavior at varying frequencies, said correction factor geometrically relating the component and the resonance inducing device, and applying the correction factor to data received for the component and the resonance inducing device.  
      According to a further aspect of the present invention, there is provided a method for compensating for non-translational motion of a component. The method comprises determining component frequency responses of the component at varying frequencies, determining motion device frequency excitation of a motion device employed to shake the component. The determining occurs at varying frequencies. The method further comprises computing a correction factor based on the component frequency response and the motion device frequency excitation and applying motion to the component and evaluating the quality of the correction factor by applying the correction factor to data from the component and the motion device.  
      These and other objects and advantages of the present invention will become apparent to those skilled in the art from the following detailed description of the invention and the accompanying drawings. 
    
    
     DESCRIPTION OF THE DRAWINGS  
       FIG. 1  illustrates a suspension and head, also called an HGA, that may be employed with the current invention;  
       FIG. 2  illustrates a shaker device used with a disk and spindle arrangement in accordance with the current invention;  
       FIG. 3  presents a multiple HGA arrangement that may be utilized in accordance with one aspect of the current invention;  
       FIG. 4  illustrates a shaker device, accelerometer, and an article such as a suspension and head (HGA) used in one aspect of the current invention encountering ideal shaker motion;  
       FIG. 5  is a shaker device, accelerometer, and an article such as a suspension and head (HGA) used in one aspect of the current invention encountering non-ideal shaker motion;  
       FIG. 6  is a top view of the interferometer arrangement used in accordance with one aspect of the current invention and measuring a point on the head;  
       FIG. 7  shows a top view of the interferometer arrangement used in accordance with one aspect of the invention and measuring a point on the shaker block;  
       FIG. 8  illustrates a perspective view of one interferometer arrangement that may be employed with the invention disclosed herein;  
       FIG. 9  is a view of three point correction geometry;  
       FIG. 10  illustrates two point correction geometry; and  
       FIG. 11  presents possible configurations used for three point correction.  
    
    
     DETAILED DESCRIPTION OF THE INVENTION  
      The present system addresses resonance effects encountered when testing a suspension arrangement, which may comprise resonance effects associated with a single head on a disk drive suspension or multiple heads on a multiple-arm positioner depending on the application. The system and inventive aspects discussed herein may be employed in other shaking devices but are described with specific reference to head/suspension arm aspects.  
      The present disclosure will briefly describe the design wherein the head/suspension arrangement may be ultimately employed, followed by a detailed description of the resonance effect correction design.  
      The novel design presented herein may be used in association with a media reading/writing device, such as a HDD. The method and apparatus disclosed may be used in association with other devices including a drive head and support configuration while still in the scope of the present invention.  
      One such design with which the present method and apparatus may be used includes a single head and associated suspension arrangement as presented in  FIG. 1 , or alternately in a multiple head and multi-arm positioner arrangement as presented in  FIG. 3 .  FIG. 2  illustrates a disk arrangement wherein a rotatable spindle  201  maintains the disk and is typically rotated by a motor (not shown). The rotating spindle may maintain a plurality of media disks, or may maintain a single disk or virtually any number of disks. In the configuration shown, one disk is employed and is secured by locking cap  202 .  
       FIG. 3  presents a positioner  301  used to maintain a series of heads  303 ( a ) through (n), typically individual read/write heads that perform both reading and writing functions, where each head flies over the surface of media, which are typically hard disks.  
       FIG. 4  illustrates a suspension and head arrangement subjected to shaking motion, with the arrows indicating an even movement of the shaker block.  FIG. 5  illustrates a suspension and head arrangement subjected to uneven movement of the shaker block in the Z direction. In this view, the lower portion of the shaker block moves in a distance greater than the upper portion, inducing torque in the suspension and providing inexact resonance response recorded by accelerometer  401 .  
      In the arrangement shown in  FIGS. 4 and 5 , shaker excitation amplitude is measured with the accelerometer  401 , providing the acceleration of the shaker, suspension, and head measured at the base of the suspension as a function of time (a(t)). At the same time, the system has the ability to measure the response of the head at the remote end of the suspension using a laser interferometer (not shown). If the velocity of the head is v(t), the acceleration measured at the base of the suspension and the velocity of the head may be converted into Fourier representations a(ω) and v(ω), where ω is the frequency of the sinusoidal excitation and a(ω) v(ω) are the complex acceleration and velocity values at the frequency ω. The resonance gain of the suspension is:  
               G   ⁡     (   ω   )       =     20   ⁢   Log   ⁢            ω   ⁢           ⁢     V   ⁡     (   ω   )           A   ⁡     (   ω   )                        (   1   )             
 
      This resonance gain is nonzero when the suspension flexes from resonance modes such as torsion and sway.  
       FIG. 6  illustrates a top view of the overall system  601  including the laser interferometer  606  used to measure velocity and/or acceleration at the head  403 . The system illustrated also includes the motor  604  and a prism  605  used for vertical beam alignment.  FIG. 7  illustrates a top view of the overall system where the system and the laser beam  603  have been repositioned to enable measurement of the shaker block  402  rather than the head  403 . The interferometer  606  positions a laser beam to a point, such as the tip of the head  403 , which is reflected back to the system, and the returning light wave and delay thereof provides the amount of movement of the head over a fixed time period. Accelerometer  401 , shaker block  402 , and head  403  are as shown in  FIGS. 6 and 7 . As shown in  FIG. 2 , disk  602  may be located proximate the head. As shown in  FIG. 8 , interferometer device  800  includes laser  801 , motor  802 , and prism  803  used for vertical alignment. The arrangement is mounted on translation stage  804 . When the measurement of the shaker block or shaker device  402  is desired, the laser beam  603  is translated horizontally to the position shown in  FIG. 7 .  
      The shaker block  402  has typically been assumed to be a rigid body. In assuming that the shaker block  402  is a rigid body, it is further assumed that movement of a point on the shaker block  402  defines a local translational movement of the shaker block. Further, movement of multiple points on the face of the shaker block  402  provides greater knowledge of the movement of any point on the block. Movement of the block in the arrangement shown tends to induce non-translational motion in the HGA and/or head  403  attached to the shaker block  402  that is unrelated to the naturally occurring resonance modes inherent in the HGA. These aberrant non-translational motions have been observed to consist of mainly two rotations. From  FIG. 9 , theoretical shaker block  402  is centered at known point C, while head  403  is located at a distance therefrom. The distance from the center of the shaker block to the head is Ly in the y direction and L x  in the x direction. Off axis motions have generally been observed in two rotational aspects, namely an up-and-down rotation having a rotational velocity w x  and a left-and-right rotation (having a rotational velocity w y . The positive orientation of x and y and the rotational velocities w x  and w y  are as shown in  FIG. 9 .  
      A device such as a shaker has six degrees of freedom. The present design addresses non-translational motion in part by essentially ignoring translation effects in X and Y directions, and rotation about the Z axis, as measurable resonance gain from excitation in these directions is essentially zero in the arrangement presented. The remaining three degrees of freedom, which are translation in the Z direction and rotation about the X axis (w x ) and Y axis (w y ), dominate coupling to the main resonance modes of the suspension in the arrangement illustrated.  
      Data in the foregoing design, such as that shown in  FIGS. 4 and 5 , may be measured and received in various forms. One form in which data may be received is in terms of a transfer function, representing the ratio of the response to the excitation, or the ratio of the head motion to the motion of the base):  
               Transfer   ⁢           ⁢   function     =       V   H       V   T               (   2   )             
 
 where V H  represents the velocity at the head and V T  is the velocity at the base. If these quantities are decomposed into their Fourier frequency components, the transfer function can be written as:  
               Transfer   ⁢           ⁢   function     =         V   H     ⁡     (   ω   )           V   T     ⁡     (   ω   )                 (   3   )             
 
 The velocity V T  of the base can be deduced from the acceleration A C  of the base, i.e.  
               V   T     =         A   c     ⁡     (   ω   )       jω             (   4   )             
 
 where j=√{square root over (−1)}. Using this, the transfer function can be written as:  
               Transfer   ⁢           ⁢   function     =       jω   ⁢           ⁢       V   H     ⁡     (   ω   )             A   c     ⁡     (   ω   )                 (   5   )             
 
 The transfer function may be determined by two measurements, namely the velocity of the head measured by a laser interferometer and the acceleration of the base measured by an accelerometer. 
 
      The transfer function is defined in terms of frequency, and is measured after subjecting the hardware to vibration to various frequencies and taking measurements therefrom. For example, the arrangement of  FIG. 4  may be exposed to a low frequency vibration. The acceleration at the accelerometer  401  and velocity of the head  403  is measured and saved. Vibration may be halted and subsequently continued, or may not be halted and simply continued. The continuation of vibration may be at a second frequency, wherein measurements are again taken. This frequency changing and measurement continues through the range of available frequencies. As noted, the problem with these measurements is the inability to know a more complete description of the excitation of the base, in particular its rotational motion.  
      Correction of the head response transfer function based on knowledge of input motion is made in the present system in one of two general manners, namely three point correction or two point correction. Three point correction is employed in the situation where two rotations are considered added to the main translation. Two point correction considers only one rotation to the main translation. The type of processing available and the types of modes observed during shaking, as well as the construction of the shaking device and the suspension/head are factors considered in determining whether to apply two or three point correction.  
      Three point correction is illustrated in  FIG. 9 . Head  403  is adjoined to suspension  902 , with shaker block  402  having a center point  904 . The distance from the head to the center of the shaker device is L x  and L y . Various points on the shaker block may be chosen for measurement purposes, but the present arrangement employs measurement points  905 ,  906 , and  907 , with the distance between measurement points  905  and  906 / 907  equal to L 2  and the distance between points  905 / 906  and  907  equal to L 3 . The velocity in the Z direction (i.e. perpendicular to the plane of  FIG. 9 ) at each of these points with respect to center point  904  are as follows: 
 
 V   1   =V   T   −w   y ( L     2   / 2 )−w x ( L     3   / 2 ) 
 
 V   2   =V   T   −w   y ( L     2   / 2 )+w x ( L     3   / 2 ) 
 
 V   3   =V   T   +w   y ( L     2   / 2 )+w x ( L     3   / 2 )  (6) 
          where V T  is the translational velocity in the Z direction. V 1  is the velocity at point  905 , V 2  is the velocity at  906 , and V 3  is the velocity at  907 . Solving for the translational velocity and the two rotational velocities,  
                 V   T     =         V   1     +     V   3       2       ⁢     
     ⁢       w   x     =         V   3     -     V   2         L   3         ⁢     
     ⁢       w   y     =         V   2     -     V   1         L   2                 (   7   )             
    Velocity at the head  403  is: 
 
 V   H   =V   T   −w   y ( Lx )− w   x ( L   y )  (8) 
       

      Replacing w x  and w y  by the associated velocity differences divided by lengths, and writing the resultant equation in terms of transfer functions provides:  
               V   H     =       V   T     ⁡     (     1   +       (     2   ⁢     Lx     L   2         )     ⁢           jω   ⁢           ⁢     V   1       Ac     -       jω   ⁢           ⁢     V   2       Ac             jω   ⁢           ⁢     V   1       Ac     +       jω   ⁢           ⁢     V   3       Ac           +       (     2   ⁢       L   Y       L   3         )     ⁢           jω   ⁢           ⁢     V   2       Ac     -       jω   ⁢           ⁢     V   3       Ac             jω   ⁢           ⁢     V   1       Ac     +       jω   ⁢           ⁢     V   3       Ac             )               (   9   )             
 
 where A C  is the acceleration measured by the accelerometer  401 . For any i in the set 1, 2, 3, . . . , H, the i th  transfer function can be denoted by:  
             Ci   =       jω   ⁢           ⁢   Vi     Ac             (   10   )             
 
 Using Equation 10, Equation 9 may be re-written as:  
               V   H     =       V   T     ⁡     (     1   +       (     2   ⁢       L   x       L   2         )     ⁢         C   1     -     C   2           C   1     +     C   3           +       (     2   ⁢       L   y       L   3         )     ⁢         C   2     -     C   3           C   1     +     C   3             )               (   11   )             
 
 For a V Hm  representing the velocity measured by the laser and A C  the acceleration measured at the accelerometer, setting  
           A   H     =     jω   ⁢           ⁢     V   H         ,       jω   ⁢           ⁢     V   T       =   Ac     ,         jω   ⁢           ⁢     V   Hm       Ac     =     C   H       ,       
 
 the resultant transfer function relating the velocity measured by the laser at the head to the geometrically implied acceleration at the head is:  
                 jω   ⁢           ⁢     V   Hm         A   H       =       C   H     ⁢     1     (     1   +       (     2   ⁢       L   X       L   2         )     ⁢         C   1     -     C   2           C   1     +     C   3           +       (     2   ⁢       L   Y       L   3         )     ⁢         C   2     -     C   3           C   1     +     C   3             )                 (   12   )             
 
 The value on the left side of Equation 12 is referred to as the corrected transfer function. 
 
      If the top middle point of the shaker block is used for the third measurement as shown in  FIG. 11 , the corrected transfer function becomes:  
                 jω   ⁢           ⁢     V   Hm         A   H       =       C   H     ⁢     1     (     1   +       (     4   ⁢       L   X       L   2         )     ⁢         C   1     -     C   2           2   ⁢     C   3       +     C   1     +     C   2           +       (     2   ⁢       L   Y       L   3         )     ⁢         C   1     +     C   2     -     2   ⁢     C   3             2   ⁢     C   3       +     C   1     +     C   2             )                 (   13   )             
 
      From Equation (12) or (13), the measured transfer function called C H  in Equation (12) and (13), is multiplied by a correction factor to determine the corrected transfer function. Two point correction is illustrated in  FIG. 10 . Measurement points  1001  and  1002  are collinear with the head  403  and center point  904 . w x  in this configuration has no effect at the head  403  since the distance from the head  403  to the X axis is zero. The relevant rotational velocity is thus designated by w. Translational velocity at points  1001  and  1002  are: 
 
 V   1   =V   T   +w ×( L / 2 ) 
 
 V   2   =V   T   −w ×( L   2 )  (14) 
 
 where V 1  is the velocity at point  1001 , V 2  is the velocity at  1002 , and V T  is the translational velocity in the Z direction (i.e. perpendicular to the plane of  FIG. 10 ). 
 
      Solving and using the previous calculations, the corrected transfer function relating velocity measured by the laser at the head to the geometrically implied acceleration at the head is:  
                 jω   ⁢           ⁢     V   Hm         A   H       =       C   H     ⁢     1     (     1   +       (     2   ⁢       L   H     L       )     ⁢         C   1     -     C   2           C   1     +     C   2             )                 (   15   )             
 
      This is denoted as the corrected transfer function for two point correction. Again, C H  represents the measured transfer function relating V Hm , the velocity of the head measured by the laser, and Ac, the acceleration measured at the accelerometer, equal to  
           jω   ⁢           ⁢     V   Hm       Ac     .       
 
 The value to the right of C H  in the equation above represents the correction factor which when multiplied by C H  yields the corrected transfer function. 
 
      The foregoing two and three point correction values relate to the gain G(ω) in the following manner. Corrected acceleration is described by the foregoing correction factors. For example, in two point correction, the term  
       (     1   +       (     2   ⁢       L   H     L       )     ⁢         C   1     -     C   2           C   1     +     C   2             )       
 
 may be multiplied by the measured acceleration at the accelerometer to produce the geometrically implied acceleration at the head,  
                 A   H     ⁡     (   ω   )       =         A   C     ⁡     (   ω   )       ⁢     (     1   +       (     2   ⁢       L   H     L       )     ⁢         C   1     -     C   2           C   1     +     C   2             )               (   16   )             
 
      The aforementioned term describes the extra acceleration contributed by a combination of the lever arm  905  to the head  403  and the twisting of the shaker plane. The term is equal to one if no tilting exists.  
      If this term is reasonably constant on repetition of the measurement, it may be stored and used to compute a corrected suspension gain G c (ω):  
                 G   C     ⁡     (   ω   )       =     20   ⁢   Log   ⁢            ω   ⁢           ⁢       V   Hm     ⁡     (   ω   )             A   H     ⁡     (   ω   )                        (   17   )             
 
      Hence corrected gain G c  is a function of the frequency, velocity at the head, and the geometrically implied acceleration at the head, and either the two point or three point correction factor or geometric values outlined above.  
      In operation, the arrangement of  FIGS. 6 and 7  is subjected to vibration across various frequencies. Both the accelerometer  401  and the laser take initial readings for acceleration at the accelerometer  401  and velocity at the head  403 , respectively. Based on these initial readings, the system determines the foregoing two point and/or three point correction factors. These correction factors may then be evaluated for adequate performance, such as subjecting the head and suspension combination to further vibrations and applying the correction factors, and determining the actual gain of the determined signal. If the gain varies greatly from the expected signal gain, particularly if the gain and phase exceed that of the expected signal, a re-measurement and improved correction factor may be warranted. Gains and performance can vary over different frequencies. Certain frequencies where the correction factor is applied may produce worse responses than other frequencies, so tuning the correction factors may be advantageous. The tuning of correction factors, and specifically the determination of gain and frequency responses at different frequencies is typically within the realm of those skilled in the art.  
      The result of the foregoing description is a more accurate characterization of the frequency response of the head in different vibration conditions. This information may be used in a variety of ways, including but not limited to changing the overall design, changing individual component designs, minimizing the effects of the resonances encountered, and so forth. Thus the head and suspension response in a fully implemented HDD can be more accurately measured using the foregoing arrangement and methodology.  
      An alternate aspect of the present design is employing an acceleration value derived not from the accelerometer but from velocity measurements on the base of the shaking device. In this aspect, the resonance amplitude gain of the HGA structure is the ratio of the amplitude of the responding measured head displacement D Hm  versus the amplitude of an applied displacement D H  of a known element, such as the block or base to which the HGA is attached. As in the aspect presented above, the ratio of these amplitudes is given for each frequency of a Fourier decomposed spectrum of frequencies. Thus resonance gain may be defined as a transfer function, or alternately on the logarithmic scale as Gain, according to the following equations: 
 
Transfer function= D   Hm   /D   H   (18a) 
 
Gain=20*log( D   Hm   /D   H )  (18b) 
 
      Acceleration is the derivative of velocity, and velocity is the derivative of displacement. Thus these quantities are related by a multiplication by jω, such that head velocity V H  equals head displacement DH multiplied by jω, and head acceleration AH equals head velocity V H  multiplied by jω. Further, measured head velocity V Hm  equals measured head displacement D Hm  multiplied by jω, and measured head acceleration A hm  equals measured head velocity V Hm  multiplied by jω. Thus resonance gain can be described as ratios of displacements, velocities, and/or accelerations, such as:  
                     Transfer   ⁢           ⁢   function     =       D   Hm     /     D   H                   =       V   Hm     /     V   H                   =       A   Hm     /     A   H                   =     jω   ⁢           ⁢       V   Hm     /     A   H                       (   19   )                     Gain   =     20   *     log   ⁡     (       D   Hm     /     D   H       )                     =     20   *     log   ⁡     (       V   Hm     /     V   H       )                     =     20   *     log   ⁡     (       A   Hm     /     A   H       )                     =     20   *     log   ⁡     (     jω   ⁢           ⁢       V   Hm     /     A   H         )                       (   20   )             
 
      Again, the applied acceleration at the head may differ from the acceleration measured at the accelerometer  401 . Acceleration and the associated velocity have the same tilting components. Thus, the relationship between the implied velocity at the head V H  and translation velocity V T  at the center of the block to which the HGA is attached is:  
               V   H     =       V   T     ⁡     (     1   +       (     2   ⁢       L   X       L   2         )     ⁢         C   1     -     C   2           C   1     +     C   3           +       (     2   ⁢       L   Y       L   3         )     ⁢         C   2     -     C   3           C   1     +     C   3             )               (   21   )             
          while the relationship between the implied acceleration at the head and acceleration at the accelerometer, which is at the center of the block to which the HGA is attached, is similar:  
               A   H     =     Ac   ⁡     (     1   +       (     2   ⁢       L   X       L   2         )     ⁢         C   1     -     C   2           C   1     +     C   3           +       (     2   ⁢       L   Y       L   3         )     ⁢         C   2     -     C   3           C   1     +     C   3             )               (   22   )             
       

      The terms in parentheses in the foregoing equations represent the geometrical equations accounting for angular rotation components. Stated another way,  
                 jω   ⁢           ⁢     V   Hm         A   H       =         jω   ⁢           ⁢     V   Hm         A   C       ⁢     1     (     1   +       (     2   ⁢       L   X       L   2         )     ⁢         C   1     -     C   2           C   1     +     C   3           +       (     2   ⁢       L   Y       L   3         )     ⁢         C   2     -     C   3           C   1     +     C   3             )                 (   23   )             
 
      From this, the alternate aspect of the present design uses a different measurement to extract the same resonance gain values. In place of using the accelerometer measured acceleration, the system measures the translational velocity on the block V T  using the interferometer and subsequently deriving the gain. The translational velocity V T  is derived from the average of velocities V 1  and V 3  measured at the aforementioned locations, so the two different measurement methodologies provide the same gain value by utilizing the displacement/velocity/acceleration frequency relationships presented above. In other words, use of the relationship: 
 
 jωV   T   =A   C   (24) 
          yields the resonance gain derived based on the velocity measurements without use of the accelerometer measurement according to the following equation:  
                         jω   ⁢           ⁢     V   Hm         A   H       =         jω   ⁢           ⁢     V   Hm         jω   ⁢           ⁢     V   T         ⁢     1     (     1   +       (     2   ⁢       L   X       L   2         )     ⁢         C   1     -     C   2           C   1     +     C   3           +       (     2   ⁢       L   Y       L   3         )     ⁢         C   2     -     C   3           C   1     +     C   3             )                     =         jω   ⁢           ⁢     V   Hm         jω   ⁢           ⁢     V   T         ⁢     1     (     1   +       (     2   ⁢       L   X       L   2         )     ⁢         V   1     -     V   2           V   1     +     V   3           +       (     2   ⁢       L   Y       L   3         )     ⁢         V   2     -     V   3           V   1     +     V   3             )                 ⁢     
     ⁢       where   ⁢           ⁢     V   T       =         V   1     +     V   3       2               (   25   )             
       

      The foregoing equation (25) removes reliance on C i  from the above recited two point and three point correction equations, as the C i  values, equal to jω V i /A C , cancel out the A C  accelerometer measurements. The A C  values drop out of the C i  equations above when divided in the fashion presented. Thus the correction terms are based on the V i  values representing the velocities at the points on the block. This obviates the need for accelerometer  401 , and provides for the same level of correction with the use of the laser arrangement of  FIGS. 6 and 7 , used to measure head and block displacement, velocity, and acceleration.  
      In operation of this alternate aspect, the system first measures the head when exposed to vibrations at varying frequencies, and may measure head displacement, head velocity, and/or head acceleration. Subsequent to this measurement, the system is altered to evaluate the response of the block, or any other rigid device employed to shake the article, by measuring the displacement, velocity, or acceleration of the block at one or more points when exposed to different frequencies. These values, representing the excitations at varying frequencies, are employed to determine a correction factor according to the foregoing calculations. The result is a correction factor that may be employed to determine performance of the HGA without the need for an accelerometer attached to the shaking device.  
      While the invention has been described in connection with specific embodiments thereof, it will be understood that the invention is capable of further modifications. This application is intended to cover any variations, uses or adaptations of the invention following, in general, the principles of the invention, and including such departures from the present disclosure as come within known and customary practice within the art to which the invention pertains.