A Next Generation Electrodynamic 

Shaker for the Primary Calibration of 

Accelerometers

The NIST and AFMETCAL Super Shaker Project

Bev Payne 

Kari K. Harper 

NISTIR 7724

 
 
 

NISTIR 7724

A Next Generation Electrodynamic 

Shaker for the Primary Calibration of 

Accelerometers

The NIST and AFMETCAL Super Shaker Project

Bev Payne 

Kari K. Harper

Manufacturing Engineering Laboratory 

Manufacturing Metrology Division 

September 2010

U.S. Department of Commerce 

Gary Locke, Secretary 

National Institute of Standards and Technology 

Patrick D. Gallagher, Director
 
Table of Contents

Introduction..................................................................................................................... 1 

Design Criteria................................................................................................................ 1 

Description of the Shaker Assembly........................................................................ 2 

Moveable Magnet Assemblies.................................................................................... 6 

Moving Element and Carriage Description............................................................. 7 

Shaker Operation........................................................................................................... 7 

Shaker Performance Tests.......................................................................................... 8 

Calibration by Reciprocity......................................................................................... 10 

Calibration by Laser Fringe-Counting Interferometry........................................ 12 

Calibration by Minimum-Point Method Laser Interferometry .......................... 13 

Calibration by Sine-Approximation Method Using Quadrature Laser 

Interferometry ............................................................................................................... 15 

Summary of Uncertainties......................................................................................... 18 

Performance.................................................................................................................. 18 

Summary ........................................................................................................................ 19 

References..................................................................................................................... 20 

Super Shaker Characterization ................................................................................ 21 

 
 
A Next Generation Electrodynamic Shaker for the Primary Calibration 

of Accelerometers 

The NIST and AFMETCAL Super Shaker Project 

Bev Payne, Guest Researcher, NIST 

Kari K. Harper, Mechanical Engineer, NIST 

Introduction 

A new shaker system was designed to allow direct primary calibration of 

accelerometers using both reciprocity and interferometer techniques over the 

frequency range 40 Hz to 10,000 Hz. The system uses dual magnets and a dualcoil moving element. The design uses damped flexures to support the moving 

element. The dual-coil feature eliminates the need for an additional shaker for 

reciprocity calibrations. To accommodate improved comparison calibrations, the 

moving element is compact in design and is equipped with axially-oriented 

mounting tables at each end. Provision of easy laser interferometer beam access 

to both ends of the moving element allows interferometric accelerometer 

calibrations. 

In order to maximize the potential for design optimization, two Super Shaker 

systems were built: Super Shaker I located at the Air Force Metrology Laboratory 

(AFMETCAL) Heath, OH, and Super Shaker II located at NIST. The design also 

allows comparison calibrations to be performed over this same frequency range. 

This report gives performance data for both shakers, with the AFMETCAL shaker 

referred to in this report as SS1, and the NIST shaker as SS2. The SS2 shaker 

was the second shaker constructed and incorporates minor changes from the 

original design. The principal change was a more flexible flexure system for the 

moving element support. 

Design Criteria 

Primary calibration methods for accelerometers (reciprocity and laser 

interferometry) require high purity of the displacement signal wave form. This is 

especially true for reciprocity calibrations. For reciprocity measurements, the 

procedure does not take into account the fact that the accelerometer is 

measuring motion in only one direction, but rather the calculations involve 

measurement of current differences in the drive coil. Therefore, even a small 

amounts of distortion and cross-axis motion adversely affect the reciprocity 

calibration. A general rule is that cross-axis motion should be less than 5% and 

harmonic distortion less than 2% in order to perform reciprocity calibrations at 1 

to 2% uncertainty. For laser interferometer fringe-counting calibration, only the 

1 
on-axis displacement amplitude is measured. Provided that the laser 

beam/interferometer is properly aligned, the cross-axis motion of the shaker is 

not as significant in a fringe-counting calibration as in the reciprocity calibration. 

In designing the Super Shaker [1,2], care was taken to minimize both harmonic 

distortion and cross-axis motion. Minimization of the latter was especially 

important to our goal of improving reciprocity calibrations. Among the critical 

factors considered were: 

1. Magnet dimensions, especially the air gap size, roundness, symmetry, and 

finish. 

2. Moving element dimensions, including diameter, symmetry, and 

roundness. 

3. Position and repeatability of position of the moving element in the 

magnetic gap assembly. 

4. Symmetry of coil windings and uniformity of coil wire. 

5. Suspension system for the moving element in the magnetic gap assembly. 

6. Isolation of critical parts of the shaker to prevent cross coupling of various 

vibrating parts of the assembly. 

The dimensional tolerances for all Super Shaker assemblies were specified to be 

no greater than 30 m (0.001 inch). 

Description of the Shaker Assembly 

The Super Shaker, shown in Fig. 1, is 0.64 m (25 in) wide, 1.22 m (48 in) long, by 

about 0.5 m (20 in) high. It is mounted on soft rubber pads on a heavy flat table, 

pneumatically isolated from ground vibration. 

The frame consists of three supporting pedestals, one at each end and one in the 

middle, and four large diameter shafts running end to end, two clamped near the 

bottom of the three pedestals, and two clamped near the top of the pedestals. 

Two magnet assemblies, equipped with air bearings, are supported by the two 

horizontal rods which are the upper shafts of the frame assembly. The air 

bearings allow each magnet assembly to be moved away from the moving 

element to provide the access needed for attachment of accelerometers or 

masses. The operating axis is parallel to the upper shafts and centered in the 

plane bisecting them. 

A common cause of calibration inaccuracy is interference from frame 

resonances. The interferences are resisted by internal dampers inside the three 

pedestals and by damping inside the four shafts. The pedestals and shafts are 

2 
also filled with sand to further dampen any vibration in the frame. These 

interferences are further resisted by the large magnet mass, and small resulting 

motion, as compared to the moving element mass. The small magnet motions 

are isolated from the frame by rubber pads. 

Figure 1a. SS2 at NIST 

Figure 1b. SS1 with instrumentation and optical components 

of laser interferometer at AFMETCAL, Heath, OH

3 
4 

Figure 1c. SS1 mounted on isolation table at AFMETCAL, Heath, OH 

A carriage on the middle pedestal supports the moving element, centered on the 

operating axis. The moving element is symmetrical, with coils on each end. 

Machined from a solid aluminum block, the moving element, shown in Fig. 2, is 

symmetrically shaped with respect to a bisecting plane perpendicular to the 

operating axis. The moving element, supported by damped flexures attached to 

the pedestal in the middle, carries the accelerometer to be calibrated. 

Figure 2. Moving element of SS1 
Placement of the specimen (accelerometer) mounting tables at the ends of deep 

central cavities of the moving element allows specimens to be mounted close t

the bisecting plane and inside the envelopes of the coils located at each end. 

The outer structure between the two coils is enlarged to provide axial stiffnes

high frequency operation. The shaker is equipped with removable moun

tables (or cones), made from beryllium with a nickel coating to prevent 

scratching, Fig. 3a. The cones have threaded steel inserts for mounting 

accelerometers. The beryllium mounting cones extend the frequency range of the 

shaker to 10,000 Hz. Special wrenches were designed to enable

o 

s for 

ting 

 easy removal or 

ttachment of the cones to the moving element, Fig. 3b and 3c. 

mounting tables, shown with tables loaded on wrench 

a

Figure 3a. Nickel-coated 

beryllium mounting tables for 

moving element, shown w

Figure 3b. Wrench for attaching and 

the removing mountin

ith 

titanium threaded inserts 

g tables, shown with 

table on wrench 

Figure 3c. Wrenches for attaching and removing 

5 
Moveable Magnet Assemblies 

lement. Residual magnet motions are isolated from the frame by rubber pads. 

g sleeves, 

rovided to allow this to be done without demagnetizing the permanent magnets. 

The magnet assembly is shown in Fig. 4. Each magnet assembly has two major 

parts: the outer permanent magnet and an inner air gap assembly. The mass of 

the permanent magnets, 68kg, has been maximized within practical constraints

in order to minimize the motion of the magnet assembly relative to the moving 

element, 225g. This reduces interference to the rectilinear motion of the moving

e

Each air gap assembly is equipped with inner and outer copper shadin

a large axial port for the accelerometer cable, and four small ports for 

interferometer beam access. Each assembly is also equipped with additional 

ports for air cooling as needed. The air gap assemblies are de-mountable to 

facilitate installation of new assemblies of different air gap sizes, should future

applications require moving elements of different size. Magnetic keepers are 

Figure 4. Inner gap assembly (left) and complete magnet assembly (right) 

p

6 
Moving Element and Carriage Description 

The moving element and carriage assembly are shown in Fig. 5, attached to th

middle pedestal of the shaker. The length of the assembly is 101.6 mm (4 i

the diameter is 63.5 mm (2.5 in). The structure is aluminum and the mounting 

tables are of beryllium as described above, and are 25.4 mm (one inch) in 

diameter. The first axial resonance is approxim

e 

n) and 

ately 22 kHz. The diameter of the 

ounting table is adequate to allow the laser beam the access required for 

ultiple-reflection interferometer calibrations. 

o maximize versatility, the four laser beam ports provided in each air gap 

ed 

lexures establish the rest position of the 

oving element, guide its motion along the system axis, and resist motions 

erpendicular to the system axis. 

alternating 

c or comparison calibrations, either or both of the coils may be 

riven. Both coils are driven when large displacements and accelerations 

are desired. 

m

m

Right-side table located 

inside cylinder here. 

Left-side table located inside cylinder 

here. Accelerometer cable is visible.

Figure 5. Moving element and carriage assembly, shown with 

accelerometer attached 

T

assembly for this purpose are positioned at 90 degree intervals. 

The carriage of the Super Shaker supports the moving element on four damp

beryllium copper flexures. These f

m

p

Shaker Operation 

The vibratory drive force for the moving element is generated by an 

current in one or both of the drive coils in the air gap magnetic fields. For 

reciprocity calibrations, only one moving element coil is driven. For 

interferometri

d

7 
A low-distortion power amplifier amplifies a signal from an external signal so

to provide the alternating current. As necessary, power amplifier heating is 

reduced by connecting high-power resistors in series with the amplifier ou

The use of these resistors is important for high frequency interferometric 

operation and critical for pulse operation. The current applied to the drive coil(

Harmonic Distortion, SS1 and SS2

0.0

1.0

2.0

3.0

4.0

5.0

10 100 1000 10000

Frequency, Hz

Harmonic Distortion, %

SS1

SS2

urce 

tput. 

s) 

(typically above 20 kHz) resonances of 

e accelerometer or moving element. 

Shaker Performance Tests 

ents 

erformed on this system. 

ests and results for these factors are detailed below. 

nic 

nt and magnet structure, as 

escribed in the Design Criteria section above. 

 

must have low levels of harmonic distortion in order to minimize interference 

caused by excitation of high frequency 

th

Extensive performance tests were conducted on shakers SS1 and SS2. These 

tests examined harmonic distortion, cross-axis motion, relative motion between

the two mounting surfaces of the moving element. unwanted movement of the

optics relative to the shaker, and random effects on repeated measurem

(repeatability). The first three of these factors were observed to have a 

significant influence on the uncertainty of calibrations p

T

A Fast Fourier Transform (FFT) signal analyzer was used to measure the 

harmonic distortion components of an accelerometer mounted on the shaker. 

The tests were run at approximately 20 m/s2

 (2 gn

1

) and the maximum harmo

component at each frequency is shown in Fig. 6. Harmonic distortion can be 

minimized by careful design of the moving eleme

d

Figure 6. Total harmonic distortion for SS1 and SS2 

1 gn = 9.80665 m/s2

 

8 
The cross-axis motion was measured with a small tri-axial accelerometer having 

a mass of 5 grams. The cross-axis data is plotted in Fig. 7. These data shoul

considered as an approximate indication of the true cross-axis motion since

calibration masses could not be attached 

d be 

to the table at the same time the 

ghtweight accelerometer was attached. 

o 

 was then 

ured displacement of the two 

ounting tables are shown graphically in Fig. 8. 

li

Super Shakers 1 and 2 Cross Axis Motion

0

10

20

30

40

50

60

70

80

90

100

10 100 1000 10000

Frequency, Hz

Cross Axis Motion, %

SS1

SS2

Figure 7. Cross axis motion for SS1 and SS2 

Relative motion between the two mounting tables was measured by placing a 

polished 20 gram tungsten dummy mass on the mounting table where the test 

accelerometer is normally mounted. A single point laser vibrometer was used t

measure the displacement at three positions on the mass, approximately 120 

degrees apart. The average displacement taken at the three positions

compared to the displacement as measured by the quadrature laser 

interferometer on the other mounting table which has a small mirror attached in

the axial center. The differences (%) in the meas

m

9 
Relative Motion of SS1 and SS2

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Frequency, Hz

Relative Motion, 

%

SS2

SS1

Figure 8. Relative motion between the two mounting tables of the shaker 

Calibration by Reciprocity 

Reciprocity calibration of accelerometers has been in use at NIST for many years 

as a primary method for calibration. It is distinct in that it does not rely on optical 

instrumentation. It is not based on the wavelength of light, but upon mass for its 

primary reference. For reciprocity calibrations, a series of five small tungsten 

carbide precision masses are attached one at a time to the right mounting table, 

with the accelerometer to be calibrated on the left mounting table (see figure 5). 

The calibration is briefly describe below, and in more detail in the references 

listed at the end of this paper [3], [4], and [5]. 

Since the current in the driver coil is one of the measured quantities, it is 

necessary to keep the mean position of the moving element constant relative to 

the magnet. The Super Shaker’s moving element operates in a horizontal 

orientation (figure 5). This avoids the problem encountered in vertical oriented 

shakers of repositioning the driver coil to the exact same position in the magnetic 

field when different masses are attached. The flexures of the Super Shaker tend 

to keep the moving element in approximately the same position relative to the 

magnet, regardless of the value of the mass attached. Better repositioning of the 

moving element between mass changes is obtained by delaying the next 

measurement by a few minutes, to allow the flexures to reposition. 

For reciprocity calibration, the test accelerometer is mounted on the left, as 

shown in Fig. 5; the mirror is removed from the right, and the right mounting table 

10 
is used for attaching the calibration masses. At sufficiently low frequencies the 

magnitude of the accelerometer sensitivity (S) may be approximated [3-5] by: 

 S = [(JR )/( 2 f )]1/2 , (1) 

where: 

parameter J equals the zero intercept of a linear fit of m/(Ym-Y0) vs. m, where Ym

and Y0 are the measured transfer admittance with and without the calibration 

mass, m. 

The transfer admittance is calculated by dividing the current of the driver coil by 

the output voltage of the accelerometer while the shaker is energized by the left 

coil. In a separate measurement, R is the voltage ratio of the output voltage of 

the accelerometer to the output voltage across the left coil when used as a 

velocity sensor while the shaker is energized by the right coil. For the voltage 

ratio measurements no calibration mass is attached to the right table. The test 

frequency is represented by f.

It is critical that the coil of the reciprocal transducer remain in the same static 

position in the magnetic gap so that the magnetic field to which the coil is 

exposed will not vary as a function of mass loading. This position must remain 

stable so that changes in coil current for different masses can be measured 

accurately. Temperature should remain constant so that the physical dimensions 

of the shaker, the transduction characteristics of the reciprocal transducer, and 

the resistance used for current measurements will remain constant. For greater 

stability, the coil is driven at about 20 m/s2

 (2 gn) and allowed to remain 

energized throughout the measurements for all five masses. This results in 

measured voltages of typically 5 mV to 500 mV. The two digital voltmeters are 

placed close to the moving element so that the connecting cables can be kept 

short to reduce electrical noise. A thermocouple was attached to the 

accelerometer and the temperature was stabilized to about ±0.2 oC. 

Figure 9 shows data for reciprocity and fringe-counting calibrations on a singleended accelerometer at 100 Hz. 

The estimated combined expanded uncertainty associated with the calibration 

results obtained by reciprocity at 100 Hz is 0.26 % based on the current 

implementation of the measurement apparatus. This combined expanded 

uncertainty was calculated in accordance with methodologies described in the 

Guide to the Expression of Uncertainty in Measurement [6] using Type A and 

Type B evaluations of uncertainty components, including those contained in ISO 

documents on the calibration of vibration and shock transducers [7]. Using a 

coverage factor of 2, the estimated expanded relative uncertainty, U, at 100 Hz 

is: 0.5 %. 

11 
For a set of ten calibrations by each method, the maximum difference between 

any two data points of either of the two methods was never more than 0.3 %, and 

the averages of the results obtained with the two methods agree within 0.04 %. 

-0.40

-0.20

0.00

0.20

0.40

1 2 3 4 5 6 7 8 9 10

RECIPROCITY

FRINGE COUNTING

FRINGE COUNTING AVG

RECIPROCITY AVG

% DEVIATION FROM MEAN

RUN NUMBER

Figure 9. Data for reciprocity and fringe-counting calibrations on a singleended accelerometer at 100 Hz

Calibration by Laser Fringe-Counting Interferometry 

A flat mirror is mounted on the table located on the right side of the moving 

element opposite the table upon which the accelerometer is mounted (Fig.10). 

The shaker design allows optical access to the shaker mounting tables through a 

center hole in the magnet (figure 4). The acceleration, in meters per second 

squared, is given by 

A = 2f

 2 / 2 , ( 2 ) 

where is the wavelength of He-Ne laser light in meters,  is the number of 

fringe counts per vibration cycle, and f is frequency in hertz [2,8]. To provide 

additional displacement, both the right and left coils can be driven in a push-pull 

configuration. 

In addition to distortion and cross-axis motion which are critical to reciprocity 

calibrations, the critical elements of interferometer operation also include 

interferometer alignment and mechanical coupling between the shaker and the 

interferometer components. The shaker system was designed to minimize this 

coupling. The mass of the moving element (approximately 225 g) is very small 

12 
compared to the mass of the magnet (approximately 68 kg) in order to minimize 

magnet movement. The shaker system is installed on an air isolation table to 

minimize the effects of floor movements. The magnets are moved into place on 

air-bearing guides, and during operation the magnets are locked into place by 

turning off the air to the bearings. 

DETECTOR

LASER

MIRROR

Figure 10. Schematic for fringe-counting interferometer setup on the Super 

Shaker 

Calibration by Minimum-Point Method Laser Interferometry 

 

The minimum-point method is based on the determination of displacement 

corresponding to the zero crossings of the Bessel function of the first kind and 

first order, J1. This method utilizes the same setup as shown in Fig. 10 for 

calibrations performed in the frequency range of 1 kHz to 5 kHz. In order to 

reject spectral components present in the output of the photo-detector except 

that corresponding to the fundamental frequency of vibration, the output of the 

photo-detector is filtered using a narrow band-pass filter centered at the 

fundamental frequency of vibration. The amplitude of vibration is then adjusted 

until the null corresponding to the desired zero crossing of J1 is obtained. Peak 

displacement amplitudes of the first six zero crossings of J1 for a He-Ne laser 

(=632.8 nm) are listed in Table 1. For minimum-point calibrations, at each 

tested frequency the number of the zero crossing was selected to obtain 

displacements corresponding to accelerations of approximately 50 m/s2 or 

greater. 

13 
Table 1. Peak Displacement Amplitudes for the First Six Zero Crossings 

Zero Crossing No. Displacement 

(nm) 

0 0.00 

1 192.95 

2 353.29 

3 512.31 

4 670.95 

5 829.42 

Once the displacement, d, is determined, the acceleration is calculated using the 

following equation: 

A=(2f)

2 d. (3)

The sensitivity of the accelerometer is then obtained by dividing the 

accelerometer output voltage by the acceleration, A. 

Calibration data using fringe-counting and minimum-point methods for a singleended accelerometer is shown in Fig. 11. 

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

10 100 1000 10000

Fringe Ct.

Min. Pt.

Single Ended Accelerometer

Frequency, Hz

% Deviation Re 100 Hz

Figure 11. Calibration data for a single-ended accelerometer using 

fringe-counting and minimum-point methods 

The estimated combined uncertainty associated with the calibration results 

obtained by minimum-point interferometry at 1 kHz is 0.22 % based on the 

current implementation of the measurement apparatus. This combined 

14 
uncertainty was calculated in accordance with methodologies described in the 

Guide to the Expression of Uncertainty in Measurement [6] using Type A and 

Type B evaluations of uncertainty components, including those contained in ISO 

documents [7] on the calibration of vibration and shock transducers. Using a 

coverage factor of 2, the estimated expanded relative uncertainty, U, for the 

minimum-point method at 1 kHz is: 0.5 %, and the estimated expanded relative 

uncertainty, U, at 100 Hz is: 0.3 % using fringe-counting. Data for the two 

calibration methods at the overlapping frequency of 1 kHz agree to within less 

than 0.1 %. 

Calibration by Sine-Approximation Method Using Quadrature Laser 

Interferometry 

A piezo-electric shaker, as illustrated in figure 12, with a single mounting table 

and a driving element on the opposite end is typically used for accelerometer 

calibrations. The accelerometer is mounted at the center of the shaker table and 

mirrors are mounted around the perimeter of the accelerometer. 

LASER

Figure 12. Conventional threemirror method for calibration 

using a single coil or piezoelectric shaker for interferometric 

calibration 

Displacement must be measured at three or more positions and then averaged to 

get a good representation of the displacement of the plane on which the 

accelerometer is mounted. 

Another approach is shown in Fig. 13 using the dual-coil Super Shaker. Here the 

accelerometer is mounted on the table opposite the reflecting mirror. Using 

reflection from the center of the mirror will enable accurate measurement of the 

displacement of the accelerometer using only the center position of the moving 

mirror provided the relative motion between the surfaces of the two tables is 

small. Tests conducted at NIST have shown that significant relative motion is 

experienced between tables fabricated from stainless steel or aluminum for 

15 
frequencies higher than 5 kHz. Using a mounting table fabricated from beryllium 

with nickel plating provides for much smaller relative motion between the tables, 

and permits calibration of accelerometers up to 10,000 Hz using only one 

measuring point at the center of the mirror. 

Photodetector 2 

Photodetector 1 

Super

Shaker

Piezo-Driver 

Reference Mirror 

Polarizer

/4

Moving mirror 

Polarized laser

Beam Splitter

(non-polarizing)

Polarizing

Accelerometer Beam Splitter

Figure 13. Setup using quadrature laser interferometer for 

sine-approximation, minimum-point, and fringe-counting 

methods 

 

Using the setup shown in Fig. 13, calibration data can be obtained using the 

dual-coil shaker, for frequencies up to 1 kHz using fringe-counting, and up to 5 

kHz using the minimum-point method by monitoring the output from either 

detector 1 or detector 2. These methods and procedures are documented in 

other NIST publications [9] and in ISO 16063 - Part 11 [7]. In order to calibrate at 

frequencies higher than 5 kHz, the sine-approximation method is used. This 

method is useful at higher frequencies where calibrations at smaller 

displacements must be made due to amplitude limitations in electro-dynamic 

shakers. The current implementation of the NIST dual-coil shaker limits the 

acceleration to approximately 200 m/s2

. In actual practice the sine-approximation 

method gives good results over a wide frequency range, 100 Hz to 10,000 Hz. It 

is also much less demanding on the shaker, since it can operate at lower 

accelerations than the minimum-point method for higher frequencies. Typically, 

16 
the sine-approximation method is used at NIST for calibration of accelerometers 

at accelerations of 200 m/s2

 to 500 m/s2 from 100 Hz to 10 kHz, and fringecounting is used for frequencies of 100 Hz and below. This range of acceleration 

is used to minimize the heating effects on the shaker. The minimum-point 

method can be used as check on the other methods, but is not usually used at 

NIST on a daily basis. 

0.100

0.110

0.120

0.130

0.140

0.1 1 10 100

Fringe Ct (single pt) dual

coil shaker

Sine Approx. (single pt)

dual coil shaker

Fringe Dis.(3 pt) 

piezo shaker

Sensitivity (pC/m/s2

)

Frequency (kHz)

Data using 3pt and single pt methods

Figure 14. Calibration data on a single-ended accelerometer 

using single point (SS2 shaker) and three point (piezo-electric 

shaker) techniques 

Calibration data using fringe-counting and sine-approximation methods for a 

single-ended accelerometer on SS2 is shown in Fig. 14. This calibration uses a 

single point measurement, and thereby requires minimal adjustment to the 

interferometer. Data using fringe-disappearance on a piezo-electric shaker is 

also shown in Fig. 14. This data requires measurements at three positions on the 

shaker and thus requires realignment for each position. 

The combined relative uncertainties for both the fringe-counting and sineapproximation methods were calculated in accordance with methodologies 

described in the Guide to the Expression of Uncertainty in Measurement [6] using 

Type A and Type B evaluations of uncertainty components, including those 

contained in ISO documents [7] on the calibration of vibration and shock 

transducers. For the fringe-counting method, using a coverage factor of 2, the 

estimated expanded uncertainty, U, at 100 Hz is 0.5 %. For the sineapproximation method, again using a coverage factor of 2, the estimated 

expanded relative uncertainty, U, at 1 kHz is 0.5 %, at 5 kHz is 0.75 %, and at 10 

kHz is 1% for the NIST Super Shaker. 

17 
Summary of Uncertainties 

The overall uncertainties of the two shakers were calculated as stated above 

using the ISO 16063-11 standard [7]. Figure 15 gives the results of these 

calculations for the fringe-counting, minimum-point, and sine-approximation 

methods. The estimated expanded uncertainties, U, for frequencies 40 Hz to 

100 Hz were found to be less than 0.5 % using fringe-counting. These results are 

published in BIPM/CIPM key comparison CCAUV.V-K1 (Vibration), 2001 [10] for 

SS1. Additional details on the uncertainty calculations are given in the Appendix 

of this document. 

Uncertainty for SS1 and SS2

0.00

0.50

1.00

1.50

2.00

2.50

3.00

100 1000 10000

Frequency, Hz

Uncertainy (K=2), %

SS1

SS2

Figure 15. Calculated uncertainties for SS1 and SS2 

Performance 

The two shakers SS1 (AFMETCAL) and SS2 (NIST) were used to calibrate a 

single-ended accelerometer. The results, given in Fig. 16, show good agreement 

between the two systems and are within the uncertainty estimates. 

18 
0.0096

0.0098

0.0100

0.0102

0.0104

0.0106

100 1000 10000

Frequency, Hz

Sensitivity, V/gn

SS1

SS2

SINGLE ENDED ACCELEROMETER CALIBRATION ON SS1 AND SS2

Figure 16. Calibration results of a single-ended accelerometer on 

NIST SS2 and AFMETCAL SS1 

Summary

A dual-coil electrodynamic shaker has been developed for improved 

accelerometer calibration by both reciprocity and interferometric methods. The 

shaker uses a uniquely designed aluminum moving element with two mounting 

tables and a damped flexure support system for the moving element. The shaker 

has very low harmonic distortion and low cross motion over most of the operating 

range. Test results show that the shaker has a useful range of 30 Hz to 10 kHz 

for primary calibrations. 

The shaker can utilize all four primary methods for accelerometer calibrations: 

reciprocity, fringe-counting, minimum-point, and sine-approximation. Reciprocity 

calibration is limited to frequencies of 1 kHz or less, because of inductive 

interaction between the two coils, unless special procedures are used to cancel 

these effects. Fringe-counting is used from 30 Hz to 1 kHz. The minimum-point 

method is used from about 100 Hz to 5 kHz. The upper limit is set by the upper 

acceleration limit of the shaker, which is about 200 m/s2

 at the higher 

frequencies. Sine-approximation is used from 100 Hz to 10 kHz. 

Calibrations on NIST’s SS2 have estimated uncertainties of 0.5 % to 1.0 % over 

the frequency range 30 Hz to 10 kHz. SS1 (AFMETCAL) can obtain calibrations 

with estimated uncertainties of 0.5% to 2.4% over the frequency range of 30 Hz 

to 10 kHz. SS2 at NIST uses softer flexures for the moving element, which may 

19 
provide smaller relative motion between the two mounting tables. However the 

specific cause of the difference in uncertainties is not presently known. 

References 

1. Payne B. F., Booth G. B., “The NIST Super Shaker Project,” Proc. Metrologie 

95, 1995, 296-301. 

2. Payne, B., Evans, D., “Comparison of Results of Calibrating the Magnitude of 

the Sensitivity of Accelerometers by Laser Interferometry and Reciprocity,” 

Metrologia, 1999, 36, 391-394. 

3. S. Levy and R.R. Bouche, Calibration of Vibration Pickups by the Reciprocity 

Method, J. Research NBS 57, 227-243 (1956) RP2714. 

4. D.C. Robinson, M.R. Serbyn, B. F. Payne, A Description of NBS Calibration 

Services in Mechanical Vibration and Shock, NBS Technical Note 1232, 1987. 

5. Payne, B., Evans, D., “Primary Calibration of Accelerometers at NIST Using a 

Dual Coil Shaker,” Proceedings of the 72nd Shock and Vibration Symposium, Nov 

12-16, 2001, Destin, FL, pp. 138-142. 

6. “Guide to the Expression of Uncertainty in Measurement”, 2nd Ed., 

International Organization for Standardization, Switzerland, 1995, ISBN 92-

10188-9 

7. International Standard ISO 16063-11, Primary vibration calibrations by laser 

interferometry, International Organization for Standardization, Geneva, 1999. 

8. Payne, B., Harper, K., “Improvements in Accelerometer Calibration Using 

Fringe-Counting and Minimum-point Methods,” Proceedings of the 74th Shock 

and Vibration Symposium, Nov. 12-16, 2001, Destin, FL, pp. 138-142. 

9. Payne, B. “Single Point Optical Calibration of Accelerometers at NIST,” 

Seventh International Conference on Vibration Measurements by Laser 

Techniques: Advances and Applications, Proc. of SPIE Vol. 6345, 634504, 

(2006). 

10. BIPM/CIPM key comparison CCAUV.V-K1 (Vibration), 2001 

20 
21 

Appendix: Super Shaker Characterization 
Super Shaker I Harmonic Distortion Data August 2008 

Frequency 

(Hz)

Distortion at 2g 

(%)

Distortion at 3g 

(%)

25 1.230 0.6340 0.4250 0.3480 0.51

400 0.44160 0.41200 0.46300 0.3400 0.43500 0.44600 0.61700 0.66800 0.65900 0.4

1000 0.171500 0.682000 0.123000 0.254000 0.175000 0.196000 0.167000 0.458000 0.069000 0.08

10000 0.42

22 
Super Shaker I Plot of Harmonic Distortion Data August 2008 

0.01.02.03.04.05.0

10 100 1000 10000

Distortion at 2g (%)Distortion at 3g (%)

Frequency, 

Hz

Harmonic Distortion (%) Super Shaker I

23 
Super Shaker I Cross Axis Motion August 2008 

RSS Total % Uncert

Frequency (Hz) X axis(mV) z axis(mV) y axis(mV) x/z % y/z % Cross Axis Cross Axi

40 23.90 758.50 5.50 3.2% 0.73% 3.2% 0.0650 23.00 684.10 6.10 3.4% 0.89% 3.5% 0.0780 22.40 616.20 5.90 3.6% 0.96% 3.8% 0.08

100 46.00 1136.00 12.60 4.0% 1.11% 4.2% 0.08200 99.10 1071.00 231.70 9.3% 21.63% 23.5% 0.47300 17.10 1035.00 19.70 1.7% 1.90% 2.5% 0.05400 14.00 1041.00 11.00 1.3% 1.06% 1.7% 0.03500 6.00 570.00 1.1% 0.00% 1.1% 0.02600 0.60 705.00 16.20 0.1% 2.30% 2.3% 0.05700 16.60 706.00 9.10 2.4% 1.29% 2.7% 0.05800 1.20 702.00 14.00 0.2% 1.99% 2.0% 0.04900 5.50 970.40 32.70 0.6% 3.37% 3.4% 0.07

1000 0.18 709.00 23.00 0.0% 3.24% 3.2% 0.061500 10.20 966.00 17.60 1.1% 1.82% 2.1% 0.042000 10.70 699.50 17.80 1.5% 2.54% 3.0% 0.062500 53.00 1061.00 29.60 5.0% 2.79% 5.7% 0.113000 14.70 672.80 71.40 2.2% 10.61% 10.8% 0.223500 125.30 1003.00 144.90 12.5% 14.45% 19.1% 0.384000 54.30 650.90 25.30 8.3% 3.89% 9.2% 0.184500 53.70 967.40 134.60 5.6% 13.91% 15.0% 0.305000 23.70 643.50 54.70 3.7% 8.50% 9.3% 0.195500 39.90 964.60 66.80 4.1% 6.93% 8.1% 0.166000 51.10 641.70 10.70 8.0% 1.67% 8.1% 0.166500 82.20 931.80 35.60 8.8% 3.82% 9.6% 0.197000 27.70 606.10 77.60 4.6% 12.80% 13.6% 0.277500 266.10 966.00 2.80 27.5% 0.29% 27.5% 0.558000 267.30 793.90 31.70 33.7% 3.99% 33.9% 0.688500 224.00 814.90 44.70 27.5% 5.49% 28.0% 0.569000 46.10 700.00 266.70 6.6% 38.10% 38.7% 0.779500 238.40 725.40 67.90 32.9% 9.36% 34.2% 0.68

10000 252.00 610.80 60.00 41.3% 9.82% 42.4% 0.64

24 
Super Shaker I Plot of Cross Axis Motion August 2008 

0%

10%20%30%40%50%60%70%80%90%

100%

10 100 1000 10000

x/z %y/z %

Super Shaker 1 Cross Axis Motion Aug 2008

Frequency (Hz)

25 
Super Shaker I Relative Motion September 2009 

Freq (Hz) VIB Pos* 1 (nm) QLI Pos 1 (nm) % diff. VIB Pos 2 (nm) QLI Pos 2 (nm) % diff. VIB Pos 3 (nm) QLI 

500 5890.0 5875.0 -0.3 5912.0 5892.0 -0.3 6939.0 

1000 1254.0 1255.0 0.1 1472.0 1476.0 0.3 1758.0 2000 341.6 341.1 -0.1 364.0 363.8 -0.1 442.9 3000 175.3 175.0 -0.2 162.0 162.2 0.1 193.2 4000 102.5 102.6 0.1 88.3 89.4 0.7 106.3 5000 63.6 64.0 0.6 66.3 66.7 0.5 66.4 6000 46.4 47.0 1.1 45.2 45.7 1.0 46.4 7000 32.9 33.2 0.8 32.5 32.9 1.1 32.7 8000 26.3 26.4 0.3 26.0 26.1 0.6 26.4 9000 20.1 20.4 1.5 20.2 20.3 0.7 20.1 

10000 16.1 16.5 2.3 16.4 16.3 -0.1 16.2 

*VIB Pos = location (on mirror mounted on first mounting table) off of which the vibrometer laser was reflecting; similarly QL

mirror mounted on second mounting table) off of which the laser from the quadrature interferometer was reflecting. Each spoints about the center sequentially. These were averaged to yield an estimate of the motion of the center of each mirror. 

26 
Super Shaker I Plot of Relative Motion September 2009 

-2.0-1.0

0.01.02.0

0 1000 2000 3000 4000 5000 6000 7000 8000 

Frequency, Hz

Difference, %

Relative Motion of SS1 Mounting Tables

27 
Super Shaker I Repeatability Tests May 2009 

Repeatability Test SS1 Both coils in use; acceleration amplitude: 2-3g AF accelerometer May 28 

Freq S(run1) S(run2) S(run3) Mean Run1_Run2 % Run1_Run3 % Run2_Run3 % Prob. Error 

100 0.009963 0.009994 0.010032 0.01000 -0.3 -0.7 -0.4 1.35E-05 200 0.009973 0.010001 0.010018 0.01000 -0.3 -0.5 -0.2 8.85E-06 300 0.009972 0.009994 0.010026 0.01000 -0.2 -0.5 -0.3 1.06E-05 400 0.009983 0.009989 0.010027 0.01000 -0.1 -0.4 -0.4 9.29E-06 500 0.010013 0.010001 0.009973 0.01000 0.1 0.4 0.3 7.99E-06 600 0.010034 0.010003 0.010015 0.01002 0.3 0.2 -0.1 6.09E-06 700 0.009999 0.01001 0.009967 0.00999 -0.1 0.3 0.4 8.70E-06 800 0.010016 0.010009 0.010022 0.01002 0.1 -0.1 -0.1 2.53E-06 900 0.010022 0.01003 0.010029 0.01003 -0.1 -0.1 0.0 1.70E-06 

1000 0.009999 0.010026 0.010003 0.01001 -0.3 0.0 0.2 5.67E-06 1500 0.010028 0.010033 0.010021 0.01003 0.0 0.1 0.1 2.35E-06 2000 0.010029 0.010057 0.010014 0.01003 -0.3 0.1 0.4 8.50E-06 2500 0.010047 0.010068 0.010077 0.01006 -0.2 -0.3 -0.1 6.00E-06 3000 0.010035 0.010056 0.010024 0.01004 -0.2 0.1 0.3 6.33E-06 4000 0.010062 0.010082 0.010094 0.01008 -0.2 -0.3 -0.1 6.30E-06 4500 0.01005 0.010073 0.010046 0.01006 -0.2 0.0 0.3 5.67E-06 5000 0.010086 0.010106 0.010078 0.01009 -0.2 0.1 0.3 5.62E-06 5500 0.010098 0.01011 0.010106 0.01010 -0.1 -0.1 0.0 2.38E-06 6000 0.010101 0.010142 0.010128 0.01012 -0.4 -0.3 0.1 8.12E-06 6500 0.010053 0.010098 0.010071 0.01007 -0.4 -0.2 0.3 8.82E-06 7000 0.010064 0.010133 0.010083 0.01009 -0.7 -0.2 0.5 1.39E-05 7500 0.010179 0.010182 0.010157 0.01017 0.0 0.2 0.2 5.32E-06 8000 0.010138 0.010204 0.010205 0.01018 -0.6 -0.7 0.0 1.50E-05 8500 0.010212 0.010204 0.010164 0.01019 0.1 0.5 0.4 1.00E-05 9000 0.010196 0.010234 0.010231 0.01022 -0.4 -0.3 0.0 8.23E-06 

10000 0.010224 0.01025 0.010194 0.01022 -0.3 0.3 0.5 1.09E-05 

Probable Error of the mean calculated according to CRC "Standard Mathematical Tables"

28 
Super Shaker I Repeatability Tests May 2009 

Repeatability Test Super Shaker 1

0.0099

0.00995

0.01

0.01005

0.0101

0.01015

0.0102

0.01025

0.0103

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Frequency (Hz)

Sensitivity (pC/g)

Indicated Sensitivity, Run 1Indicated Sensitivity, Run 2Indicated Sensitivity, Run 3Mean Indicated Sensitivity

29 
 Super Shakers I and II Uncertainty Estimates August 2010 

Uncertainty Estimates for Super Shakers I and II

SS1 SS2 SS1 SS2 SS1 SS2 SS1 SS2 SS1 SS2 SS1 SS2 SS1 SS2 

Freq. (Hz) Voltage

Cross 

Axis 

Motion

Cross 

Axis 

Motion

Frequency 

stability

Frequency 

stability

Amplifier 

calibration

Amplifier 

calibration

Amplifier 

stability

Amplifier 

stability Distortion Distortion

Repeatability Repeat- ability Relative Motion Relativ

Motion 

100 0.04 0.06 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 200 0.04 0.35 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 300 0.04 0.04 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 400 0.04 0.03 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 500 0.04 0.02 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 600 0.04 0.03 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 700 0.04 0.04 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 800 0.04 0.03 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 900 0.04 0.05 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 

1000 0.04 0.05 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 1500 0.04 0.032 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 2000 0.04 0.045 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.05 2500 0.04 0.086 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.1 0.05 3000 0.04 0.163 0.07 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.2 0.05 4000 0.04 0.138 0.17 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.5 0.06 4500 0.04 0.225 0.2 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.4 0.30 5000 0.04 0.139 0.11 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.07 0.07 0.40 0.00 6000 0.04 0.122 0.15 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.10 0.10 0.8 0.38 7000 0.04 0.204 0.21 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.10 0.10 0.9 0.40 8000 0.04 0.509 0.13 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.10 0.10 0.2 0.30 9000 0.04 0.580 0.15 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.10 0.10 0.8 0.30 

10000 0.04 0.636 0.3 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.05 0.10 0.10 1.00 0.27 

30 
Super Shakers I and II Plot of Uncertainty Estimates August 2010 

Uncertainty Estimates for SS1 and SS2

0.000.501.001.502.002.503.00

100 1000 1000

Frequency, Hz

Uncertainy (K=2), %

Total K=2Total K=2

31 
Comparison of Calibration Results for Same Accelerometer Tested on Both Super Shaker

Freq. (Hz) RUN1 RUN2 RUN3 RUN4 RUN5

AVG SS1 

SEPT 09

 SS

SEPT 0

100 0.009976 0.009978 0.009926 0.009965 0.01001 0.00997 0.01001200 0.009976 0.009987 0.010002 0.010001 0.00999 0.01004300 0.009988 0.009983 0.009995 0.009983 0.00999 0.01006400 0.009988 0.009979 0.009982 0.009981 0.00998 0.00998 0.01003500 0.009973 0.009979 0.009985 0.00998 0.009998 0.00998 0.01002600 0.009971 0.009981 0.00997 0.009969 0.00997 0.01005700 0.009979 0.009968 0.009981 0.009981 0.00998 0.01001800 0.00998 0.009973 0.009981 0.009977 0.009969 0.00998 0.01001900 0.010008 0.010002 0.010024 0.009995 0.00999 0.01000 0.01007

1000 0.010000 0.009993 0.00999 0.00998 0.009998 0.00999 0.010041500 0.010017 0.010009 0.010017 0.010005 0.01001 0.010042000 0.010018 0.010022 0.010028 0.010025 0.010019 0.01002 0.010033000 0.010037 0.010036 0.010038 0.010023 0.010039 0.01003 0.010044000 0.010011 0.010052 0.010041 0.010104 0.01004 0.01005 0.010055000 0.010093 0.010069 0.010024 0.01009 0.010032 0.01006 0.010036000 0.010032 0.01001 0.010043 0.010028 0.009953 0.01001 0.010047000 0.010024 0.010019 0.01002 0.01009 0.010091 0.01005 0.010088000 0.010005 0.010046 0.010034 0.010021 0.010014 0.01002 0.010019000 0.009994 0.009988 0.009987 0.009998 0.009957 0.00998 0.01001

10000 0.009966 0.010045 0.009987 0.009984 0.010008 0.01000 0.01000

32 
33 

Plot of Comparison of Calibration Results for Same Accelerometer Tested on Both Super Sha

0.00960.00980.01000.01020.01040.0106

100 1000 1000

AVG SS1 SEPT 09 SS2 SEPT 09

KISTLER 8002K SN2039714 SS1 (W) AND SS2 SEPT 2009

SS1 IS AVG OF 5 RUNS