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Journal of Mechanical Science and Technology 24 (6) (2010) 1329~1335

www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-010-0336-1

Measuring displacement signal with an accelerometer†

Sangbo Han*

School of Mechanical Engineering and Automation, Kyungnam University, Masan, 631-701, Korea

(Manuscript Received September 11, 2009; Revised March 11, 2010; Accepted March 31, 2010)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract

An effective and simple way to reconstruct displacement signal from a measured acceleration signal is proposed in this paper. To re-

construct displacement signal by means of double-integrating the time domain acceleration signal, the Nyquist frequency of the digital

sampling of the acceleration signal should be much higher than the highest frequency component of the signal. On the other hand, to

reconstruct displacement signal by taking the inverse Fourier transform, the magnitude of the significant frequency components of the

Fourier transform of the acceleration signal should be greater than the 6 dB increment line along the frequency axis. With a predeter-

mined resolution in time and frequency domain, determined by the sampling rate to measure and record the original signal, reconstruct-

ing high-frequency signals in the time domain and reconstructing low-frequency signals in the frequency domain will produce biased

errors. Furthermore, because of the DC components inevitably included in the sampling process, low-frequency components of the sig-

nals are overestimated when displacement signals are reconstructed from the Fourier transform of the acceleration signal. The proposed

method utilizes curve-fitting around the significant frequency components of the Fourier transform of the acceleration signal before it is

inverse-Fourier transformed. Curve-fitting around the dominant frequency components provides much better results than simply ignoring

the insignificant frequency components of the signal.

Keywords: Acceleration signal; Curve-fitting; Displacement signal; Inverse Fourier transform; Nyquist frequency; Sampling rate

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction

Accelerometers are the most frequently used transducers to

measure vibration responses of structures. The main objective

of signal analysis involved in a structural vibration test is ob-

taining information on amplitudes, frequencies, and phase

differences of the measured accelerations. There are many

techniques available in literature to extract the parameters of a

measured signal, ranging from classical Fourier transform

methods [1-3] to relatively sophisticated ones [4-7]. Convert-

ing measured acceleration signals in the form of velocities and

displacements is necessary in cases when direct measurements

are not possible [8]. While it is quite easy to extract informa-

tion on the frequency components and root mean square val-

ues of the amplitudes of velocities and displacements from

measured acceleration signals, it is difficult to reconstruct the

corresponding time signals of structural responses in the form

of velocities and displacements.

Generally, there are two methods in converting a measured

time history of acceleration signal into a displacement signal

[9]. One is by directly integrating the acceleration signal in the

time domain. The other is by dividing the Fourier-transformed

acceleration signal by the scale factor of

inverse Fourier transform. Both methods produced a signifi-

cant amount of errors depending on the sampling resolution to

digitize the response signals [10, 11]. To have better resolution

in the time domain, there is a need to compromise coarse reso-

lution in the frequency domain with a given number of sam-

pling points and visa versa. Therefore, with a fixed resolution

in time and frequency domain, converting high-frequency

signals in the time domain and converting low-frequency sig-

nals in the frequency domain will produce biased errors. For a

pure sinusoidal signal, ignoring the insignificant components

of the Fourier transformed acceleration signal provides a rea-

sonably accurate time history of the corresponding displace-

ment signal [4]. In this paper, an improved frequency domain

method is suggested to reconstruct displacement signals from

measured acceleration signals that include multi-frequency

components with a certain amount of damping. The technique

is based on the curve-fitting method widely used in the ex-

perimental modal analysis to extract modal parameters from

frequency response functions.

2

ω−

and taking its

2. Errors in time domain reconstruction

The first source of error introduced in the reconstruction

† This paper was recommended for publication in revised form by Associate Editor

Seokhyun Kim

*Corresponding author. Tel.: +82-55-249-2623, Fax.: +82-505-999-2160

E-mail address: sbhan@kyungnam.ac.kr

© KSME & Springer 2010

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S. Han / Journal of Mechanical Science and Technology 24 (6) (2010) 1329~1335

process is time resolution of the digitized acceleration signal.

The numerical quadrature error involved in the two-point

trapezoidal rule to integrate acceleration into velocity is given

as follows [12].

3

()

( )

12

For a pure sinusoidal signal, the relationship between the

second derivative of the acceleration and the velocity is given

as

3

0

( )(2 ) ( )

a tf v t

π

=−

??

where

quency of the measurement is determined by the sampling

resolution

t

∆ with the following relationship.

1

2

t

∆

Therefore, relative error in evaluating the velocity from the

acceleration of a pure sinusoidal signal is given as

3

3

0

()

12

Ny

f

In addition, the relationship between the frequency of the

signal to be analyzed within a certain amount of error and the

Nyquist frequency of the measurement is determined as fol-

lows.

12

0.7287

Nyv

fff

ε

π

For example, if we want to reconstruct velocity signal by di-

rectly integrating a sinusoidal acceleration signal within 5%

error, the highest frequency component to be reconstructed is

0.2685

Ny

f

and within 1% of error, the highest frequency

component of the signal should be less than 0.1570

frequency component will be much lower if the integration is

performed again to reconstruct the displacement signal.

The time domain reconstruction scheme was examined with

numerically generated acceleration signals of the free vibra-

tion response of 1 degree-of–freedom(DOF) system. Fig. 1

and 2 represent the displacement signals reconstructed from

acceleration signals whose natural frequencies are 20.3 and

400 Hz, respectively, with a damping ratio of 1% by double

integration using the trapezoidal rule in the time domain. It is

assumed that acceleration signals are measured with a digital

signal analyzer having 2,048 sampling points. Since record

time is fixed at 1 second, time and frequency domain resolu-

tion of the digitized signals are 1/2,047 second and 1 Hz, re-

spectively, and the Nyquist frequency is 1023.5 Hz. Figs. 1

2

t

E a t

??

∆

=

0

tt

∆

≺ ≺

(1)

(2)

0f is the frequency of the signal. The Nyquist fre-

Ny

f

=

(3)

v

f

π

ε =

(4)

3

3

0

3

v

Ny

ε

==

(5)

Ny

f . The

and 2 show that the higher the frequency of the signals, the

worse the reconstructed displacement signals. Another aspect,

as shown by the result in Fig. 2, is that the digitized theoretical

displacement signal itself is not a good representative of the

original pure sinusoidal signal because of the poor sampling

resolution. In this example, total sampling points is 2,048, and

there are only 5 (2,048/400) points available to represent one

period of the signal. Therefore, no coarse sampling resolution

is recommended if the aim is to reconstruct the time domain

displacement signal from a signal measured with accelerome-

ter.

The second source of error is a result of unavailable infor-

mation regarding the initial conditions involved in each inte-

gration scheme. The uncertain value of the initial velocity will

produce a DC component during the successive integration of

the conversion process (Fig. 3). One methods to eliminate the

error due to the uncertain initial value of the signal is by filter-

ing out the DC component in every integration scheme or

extrapolating the acceleration signal to determine the appro-

priate initial velocity. The result of Fig. 3 can be improved

using suitable initial velocity and displacement, which can be

derived from the extrapolating scheme and the successive

elimination of the DC component (Fig. 1).

0 0.10.2 0.30.4 0.5 0.6 0.70.80.91

-1

-0.5

0

0.5

1

(a) Theoretical Displacement

Disp. in mm

0 0.1 0.20.3 0.4

Time in Second

0.50.60.7 0.8 0.91

-2

-1

0

1

Disp. in mm

(b) Displacement reconstructed by double integration

Fig. 1. Free vibration response of 1 DOF system with A natural fre-

quency of 20.3 Hz and 1% damping.

(a) Theoretical Displacement

0 0.010.020.030.040.050.06

-1

-0.5

0

0.5

1

Disp. in mm

0 0.01 0.020.030.04 0.05 0.06

-1

-0.5

0

0.5

1

Time in Second

Disp. in mm

(b) Displacement reconstructed by double integration

Fig. 2. Segment of free vibration response of 1 DOF system with a

natural frequency of 400 Hz and 1% damping.

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S. Han / Journal of Mechanical Science and Technology 24 (6) (2010) 1329~1335

1331

3. Errors in frequency domain reconstruction

From the properties of Fourier transform of the integrals,

the discrete Fourier transform of the velocity and the dis-

placement signals is given as follows:

1

2

jk

π

1

(2)

k

π

The time histories of the velocities and the accelerations are

obtained by taking the inverse Fourier transform of the coeffi-

cients in Eqs. (6) and (7)

1

(2/)

j kr N

rk

k

=

1

(2/)j kr N

rk

k

=

The error involved in the transformation of velocities and

displacements comes from the scale factor of 1/ 2 j

2

1/(2) k

π

−

in Eqs. (6) and (7). This error can be explained by

the results of the signal given in Fig. 1. Theoretically, the Fou-

rier transform of a single frequency signal is the delta function

of

0

()ff

δ

−

, where

Therefore, all the other Fourier coefficients are zero except at

the corresponding frequency value. However, due to the dig-

itization error, each Fourier coefficient has a small value,

when the Fourier coefficient of the frequency component of

the signal is divided by the scale factor, there is a distortion in

the Fourier coefficients along the frequency axis. The amount

of distortion is severer when there is DC component, leakage,

or damping in the measured signal. In this case, the difference

between the maximum and minimum value of the Fourier

coefficients is relatively small such that the scale factor plays

significant role in the conversion process.

To convert acceleration into displacement in the frequency

domain, each component of the Fourier coefficients of the

acceleration signal should be divided by

kk

VA

=

0,1, 2,, ( 1)

kN

=−

?

(6)

2

kk

DA

=−

0,1, 2,, ( 1)

kN

=−

?

(7)

0

N

v V e

π

−

=∑

0,1, 2, , (1)

rN

=−

?

(8)

0

N

d D e

π

−

=∑

0,1, 2,, (1)rN

=−

?

(9)

k

π and

0f is the frequency of the signal.

2

(2) f

π

−

, indicating

that the value of the Fourier coefficients of the acceleration

decreases inversely proportional to the square of the frequency

value after being converted into displacement. When ex-

pressed in dB scale, this decrease is equivalent to 6 dB decre-

ment. Therefore, any frequency component whose magnitude

appears below the 6 dB line in the spectrum of the original

acceleration signal would be much less than the insignificant

noise level when converted into displacement. Thus, the sig-

nificant frequency components included in the raw data signal,

as well as the possibility of the distortion in the resulted con-

verted displacement signal can be easily identified.

Suppose the measured signal has a very high frequency

component, then the discrete Fourier coefficients of the accel-

eration will be divided by the scale factor of

high value of f before it is converted into displacement.

This will cause the significant frequency component of the

signal to become less than the DC and low-frequency noise

components. Fig. 4(a) represents the magnitudes of the dis-

crete Fourier transform of the acceleration response signal

corresponding to a single degree of freedom system whose

natural frequency is 20.3 Hz and damping ratio is 3%. A cer-

tain level of leakage is expected because the frequency resolu-

tion of the measurement is 1 Hz. The magnitude of the 20.3

Hz component clearly appears below the 6 dB line [Fig. 4(a)],

2

(2) f

π

−

with

00.010.020.030.040.050.06

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Tim in Second

Disp. in mm

Theoretical Displacementdata1

Reconstructed Displacement

Fig. 3. Time history of the reconstructed displacement with unknown

initial conditions from an acceleration signal in Fig. 1.

0 20 406080 100

10

0

10

10

10

20

Frequency (Hz)

Abs(A(f))

Measured acceleration

6 dB Line

0 2040 6080 100

10

-10

10

0

10

10

Frequency (Hz)

Abs(X(f))

Theoretical displacement

Reconstructed displacement

Fig. 4. Fourier transform of the free vibration response of 1 DOF sys-

tem with a natural frequency of 20.3 Hz and 3% damping.

1

(a) Theoretical displacement

00.20.4

Time (sec.)

0.60.81

-1

0

mm

0 0.2 0.4

Time (sec.)

0.60.81

-140

-130

-120

-110

mm

(b) Reconstructed displacement

Fig. 5. Reconstructed displacement signal using the Fourier transform

shown in Fig. 4(b).

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S. Han / Journal of Mechanical Science and Technology 24 (6) (2010) 1329~1335

therefore, the magnitude of this frequency component be-

comes much less than the magnitude of the low-frequency

components [Fig. 4(b)]. Taking the inverse Fourier transform

of these Fourier coefficients will distort the time history of the

displacement (Fig. 5).

4. Reconstruction of multi-frequency component signal

As stated above, both frequency domain and time domain

methods provide a certain amount of errors depending on the

frequency components of the signal. The frequency domain

method works well with the acceleration signal measured

without leakage, in which case the magnitude of the signifi-

cant frequency component is much bigger than the non-

contributing frequency components. On the other hand, the

time domain method works well with the acceleration signal

whose frequency is well below the Nyquist frequency. How-

ever, these conditions are seldom satisfied in real situations. In

practice, structural response consists of both high-frequency

and low-frequency component signals, and leakage and DC

component always occur in the measurement.

When a damped signal is measured, the magnitude of the

significant frequency components becomes less than that of

the noise components after the Fourier coefficients are divided

by the scale factor of

ω

−

. This results in a completely dis-

torted time history of the displacement signal due to the ab-

normally exaggerated low-frequency components, as shown in

Fig. 5.

To measure the structural responses with an accelerometer

and to reconstruct the displacement signal after the measure-

ment, the time domain method is preferred. However, in this

case, time sampling should be fine enough to have a much

higher Nyquist frequency than the highest frequency compo-

nent expected in the signal to be measured.

To reconstruct a multi-frequency component signal with

minimum error in the frequency domain, there must be some

procedure to minimize the effects of the DC component and

leakage of the measurement. The effect of the DC component

and leakage on the conversion error comes from the fact that

these components can become significant after the Fourier

coefficients are divided by the scale factor of

The zero-padding method involves neglecting all unwanted

frequency components of the signal. Assuming that every

other signal component is noise except for the significant ones

appearing in the Fourier transform of the acceleration, zero

padding the noise components could reduce the effect of the

conversion factor. Digital band pass filter can be used on the

measured and recorded signal to eliminate non-significant

components of the Fourier transform of the acceleration signal.

However, adjusting the band width of the filter is much more

difficult than simply zero-padding the signal. A detailed ex-

planation of the zero-padding method is included in the refer-

ence [11].

A 3 DOF system response was examined to determine the

effectiveness of zero-padding the signal in the frequency do-

2

2

ω

−

.

main method. The measured acceleration response was as-

sumed to have natural frequencies of 80.3, 400.5 and 600.7 Hz

and 1, 0.3 and 0.1% of modal damping for the corresponding

modes, respectively. The Fourier transform of this accelera-

tion signal is scaled with

ω

−

set to zero except for the coefficients in the vicinity of the

natural frequencies. The resulted Fourier transform is com-

pared with the theoretical Fourier transform of the displace-

ment signal in Fig. 6. Reconstructed time history of the dis-

placement from this zero-padded Fourier transform is given in

Fig. 7. As expected, the overall feature of the signal can be

reconstructed, but not the exact time history, especially at the

starting part of the signal. This is because the zero-padded

Fourier transform cannot provide the phase information of the

original signal.

2

, and all other coefficients are

5. Curve-fitting the response signal

To improve the incompleteness of the reconstruction proc-

ess in the frequency domain, a better procedure is proposed.

Except for some special cases, most measured structural ac-

celerations are either steady-state responses or decaying-

transient responses. Steady-state responses consist of single-

frequency sinusoidal signals, in which the zero-padding

method can provide satisfactory results.

0 200 400

Frequency (Hz)

600800 1000

10

0

10

5

Abs(X(f))

(a) Theoretical Fourier Coef.

0 200400

Frequency (Hz)

6008001000

10

0

10

5

Abs(X(f))

(b) Zero-padded Fourier Coef.

Fig. 6. Magnitude of the Fourier transform of theoretical displacement

and reconstructed displacement with zero-padding method for a signal

with three frequency components.

0 0.0050.01

Time (sec.)

0.0150.02 0.025

-2

0

2

mm

(a) Theoretical displacement

0 0.0050.01

Time (sec.)

0.0150.02 0.025

-2

0

2

mm

(b) Reconstructed displacement

Fig. 7. Time histories of the theoretical and the reconstructed dis-

placement using the zero-padded Fourier transform given in Fig. 6.

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S. Han / Journal of Mechanical Science and Technology 24 (6) (2010) 1329~1335

1333

The decaying-transient responses of the structure, such as

those of the impact hammer test can be considered as the lin-

ear combination of 1 DOF system responses. From the exam-

ple of single-frequency sinusoidal signal, the reconstruction

error in the frequency domain methods comes from the fact

that low-frequency noise components become bigger than the

actual frequency components of the signal after the Fourier

transform of the acceleration signal is divided by the scale

factor of

ω

−

. Since the acceleration response of the struc-

ture consists of the contributing modal components of the

structure, any other frequency components appearing in the

scaled Fourier transform can be considered as irrelevant noise

components. Therefore, by extracting only the significant

frequency components of the scaled Fourier transform, a rea-

sonably accurate time history of the displacement signal can

be reconstructed.

The method of extracting the significant frequency compo-

nents of the signal is the same as the curve-fitting method used

in the experimental modal analysis. In the experimental modal

analysis, the curve-fitting method is utilized to find coeffi-

cients in a theoretical expression for the frequency response

function which most closely matches the measured data. This

task is most readily tackled by the series form for the fre-

quency response function. There are a number of curve-fitting

methods applicable in both frequency and time domains, and

detailed procedures are given in [13]. However, the basic as-

sumption for the various curve-fitting methods is that in the

vicinity of a resonance, total response is dominated by the

contribution of the mode with the closest natural frequency. In

this study, the concept stating that response signal is a linear

combination of sinusoidal signals whose frequency compo-

nents significantly appear in the Fourier transform is utilized.

By assuming that the signal is composed of individual 1 DOF

system response, each single-frequency component appearing

on the Fourier transformed signal can be reconstructed. In

addition, the linear combination of the single-frequency com-

ponents in the frequency domain is considered as the actual

Fourier transform of the displacement signal to be recon-

structed.

For 1DOF system, the Fourier transform of the response is

given as follows.

δ

ω

ζ

−+

where

() X jω is the Fourier transform of the signal;

the static deflection; r is the frequency ratio; and ζ is the

damping ratio of the signal. The curve-fitting procedure is

applied as follows.

First, individual resonance peaks are detected on the scaled

Fourier transform of the acceleration signal, and the frequency

of the maximum response is taken as the significant frequency

component of the signal.

Second, using non-linear curve-fitting algorithm, the

and ζ in Eq. (10) most closely fitting the scaled Fourier

2

2

()

12

st

X j

rjr

=

(10)

st δ is

st δ

transform of the signal along several frequency lines (in this

study, there are nine spectral lines) around the frequency of

the maximum response are determined.

Third, the theoretical expression of the Fourier transform of

the response corresponding to the one in Eq. (10) is replaced

with the scaled Fourier transform in order to eliminate the

distorted region introduced by the scaling with

ward, the final Fourier transform is inverse Fourier trans-

formed to reconstruct the time domain signal.

To evaluate the effectiveness of this method, the single

DOF system response shown in Figs. 4 and 5 is tested again.

As stated earlier, since the magnitude of the relevant fre-

quency component of the acceleration response appears below

the 6 dB line, simply scaling this Fourier transformed signal

gives a totally incorrect time history of the displacement (Fig.

5).

The Fourier transform corresponding to the displacement

signal, obtained by dividing the Fourier transform of the ac-

celeration signal by the scale factor of

taking nine spectral lines around the peak. Fig. 8 compares the

curve-fitted scaled Fourier transform of the acceleration signal

and the theoretical one. The contribution of the discrepancy

between the curve-fitted and the theoretical Fourier transform

for the frequency components far away from the peak are

insignificant because of their small values. The curve-fitted

Fourier transform zoomed around the peak is given in Fig. 9,

and the corresponding Nyquist plot is given in Fig. 10, which

demonstrates the effectiveness of this method. The results of

these figures show that recovering the significant frequency

components around the peak is the key factor in the recon-

struction process of the displacement signal. Therefore, the

effectiveness of this method depends on the accuracy of the

curve-fit, which is exactly the same as the extraction of modal

parameters in the experimental modal analysis.

The result of the reconstructed displacement time signal is

compared with the theoretical one in Fig. 11.

In using the curve-fit method, the curve-fit algorithm may

estimate a negative damping value because of the unrealisti-

cally high values of low-frequency components of the scaled

Fourier transform corresponding to the displacement signal

(Fig. 4). In such a case, the reconstructed time signal of the

displacement appears to be diverging instead of decaying, as

shown in Fig. 12. This kind of error can be easily determined

by examining the Nyquist plot of the scaled Fourier transform

the displacement signal, which always traces along the half

plane of the negative imaginary axis (Fig. 13).

The curve-fitting method is compared with the zero-

padding method by reconstructing the same 3 DOF system

response given in Fig. 6. The curve-fitted Fourier transform of

the displacement is given in Fig. 14. The curve-fitting method

accurately provides both the overall shape and the detailed

traces of the time signal as shown in Figs. 15 and 16, respec-

tively.

Comparing both the reconstructed displacement signals us-

ing the zero-padding and curve-fitting methods in Figs. 7 and

2

ω−

. After-

2

ω−

is curve-fitted