Precise correction in laser doppler forced vibrology of soft products

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99 (2008) 156 – 160

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journal homepage: www.elsevier.com/locate/issn/15375110

Research Note: PH—Postharvest Technology

Precise correction in laser doppler forced vibrology of soft products Jiri Blahoveca,, Shinichiro Kurokib, Naoki Sakuraic a

Czech University of Life Sciences, 16521 Prague 6—Suchdol, Czech Republic Hiroshima University, Collaborative Research Center, 2-313 Kagamiyama, Higashi-Hiroshima 739-8527, Japan c Integrated Arts and Sciences, Hiroshima University, 1-7-1 Kagamiyama, Higashi-Hiroshima 739-8521, Japan b

ar t ic l e i n f o

The modulus of elasticity of spheroids can be estimated by the analysis of their proper free vibrations, and can be used for the evaluation of fruit maturity and/or ripening. The lowest

Article history:

frequency resonant peak (mode M0) was previously proposed as a basis for corrections of

Received 9 February 2007

the whole amplitude–frequency plot in relation to the forcing deformation level. This paper

Received in revised form

describes a computing method that makes it possible to incorporate M0 peak self-changes

5 September 2007

into the correction. The exact solution is given for parabolic and/or parabolic-like peaks. It

Accepted 17 September 2007

is shown that improving the correction process is important mainly for wide peaks. The

Available online 9 November 2007

methods of making precise corrections of amplitude–frequency plots in case of general, not parabolic, M0 peaks are also given. & 2007 IAgrE. Published by Elsevier Ltd. All rights reserved.

1.

Introduction

The modulus of elasticity of spheroids can be estimated by the analysis of their proper free vibrations. This method stems from fruit research in the 1960s (Abbott et al., 1968; Cooke, 1970). It was modified repeatedly using an accelerometer and sound sensing, which led to the commercial instruments (De Baerdemaeker et al., 2002) used for evaluation of fruit maturity and/or ripening. The laser Doppler vibrometer (LDV) began to be used in fruit-vibration studies in the 1990s (Muramatsu et al., 1997). The obtained results correlated well with data obtained by accelerometer sensing. The LDV as a non-contact method detected fruit vibration more accurately than was possible with an accelerometer (Terasaki et al., 2001a). The modulus of elasticity (Young’s modulus) E is calculated from the resonant frequencies f2 corresponding to the spherical mode M1, denoted usually as 0S2, by the following formula (Terasaki et al., 2001a, 2001b; Yamamoto & Haginuma, 1984;

Sato & Usami, 1962): E¼

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 3 36p4 ð1 þ nÞ 3 m2 rf 22 , 2 w

(1)

where m is the mass of the spherical product, r its density, w dimensionless natural frequency, and n Poisson’s ratio. Use of external vibrations, i.e. use of the forcing vibration systems, is the simplest way to excite the proper vibration modes of the tested product. A simple scheme of LDV applied to such a system is given in Fig. 1. The main part of the optical system is a laser to measure the product surface changes: the actual velocity of the surface radial motion is detected as a modulation of the optical signal after interference of the initial and the Doppler-modified reflected laser waves. A vibration generator with a source power amplifier and a signal generator controlled by a computer is used for forcing of the product analysis. The vibration generator works usually with an amplitude of 1 mm and the frequency response in sound range is monitored by an accelerometer connected to

Corresponding author.

E-mail address: [email protected] (J. Blahovec). 1537-5110/$ - see front matter & 2007 IAgrE. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.biosystemseng.2007.09.008

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LDV M0

Nomenclature

A AFP ALD ARV ARV0 Acor RV Acorr RV A1 A2 a E FFT f f2 H

99 (2008) 156– 160

amplitude of the vibration generator, mm

M1

amplitude–frequency plot amplitude of product vibrations registered by the laser Doppler vibrometer, mm relative amplitude of product vibrations registered by the laser Doppler vibrometer: ALD/A (dimensionless)—see Eq. (2) parameter of parabolic peak (dimensionless)—see Eq. (7) ARV corrected by precision of the forcing amplitude; based on o0—see Eqs. (3a) and (3b) ARV corrected by precision of the forcing amplitude; based on o0c—see Eqs. (3a) and (3b) amplitude of product motion, mm amplitude of product forcing deformation, mm parameter of parabolic peak, s2 rad2—see Eq. (7)

M2 m S y0 V w w n r

Modulus of elasticity, Pa fast Fourier transformation frequency, Hz peak frequency of the mode M2, Hz—see Eq. (1) horizontal laser position

the vibration generator’s table. The signals coming from the optical part are interfaced and then evaluated by the LDV software with fast Fourier transformation (FFT) giving the plot of the real vibration amplitudes versus frequencies: the amplitude–frequency plot (AFP) with the relative amplitude ARV is calculated by the following formula: ALD , (2) ARV ¼ A

Laser Doppler Vibrometer

V

o or o0 o0c

first peak in forced induced AFP of spheroidal objects second peak in forced induced AFP of spheroidal objects third the first peak in forced induced AFP of spheroidal objects mass of the tested spheroidal product, kg—see Eq. (1) parameter of Lorentz’s peak, rad s1; it represents the peak area—see Eq. (9) parameter of Lorentz’s peak, dimensionless; it has meaning of constant value—see Eq. (9) vertical laser position parameter of Lorentz’s peak, rad s1; it has meaning of the peak width—see Eq. (9) dimensionless parameter in Eq. (1) Poisson’s Ratio density of the tested spheroidal product, kg.m3 see Eq. (1) angular frequency, rad s1; o ¼ 2pf relative angular frequency: o/o0, dimensionless angular frequency at the M0 peak corrected o0

ALD  A1 2 ¼ ARV ð1 þ o2 r Þ  or , A2

(3a)

for the vertical laser position and

Optical Part

Acor RV ¼

Personal Computer

H

laser Doppler vibrometer

where ALD denotes the amplitude of the product surface motion given by the LDV system and A is the amplitude of the vibration generator’s table. In a previous paper (Blahovec et al., 2007) it was shown that A in Eq. (2) represents the superposition of two amplitudes: amplitude of the product motion A1 and amplitude of the product deformation A2 and the proportion of both the components is different for different frequencies. The level of the forcing is then determined by A2 only, so that relative frequency ARV from Eq. (2) should be corrected into the form Acor RV ¼ ALD =A2 ; consequently. Acor RV ¼

Electronic Part

157

Signal Generator Power Amplifier

Vibration Generator Fig. 1 – Laser Doppler vibrometer (LDV) applied to forced vibrating object. LDV has optical and electronic parts. The optical part can be arranged into two basic positions: vertical (V) and horizontal (H). The vibration generator induces vertical forced vibrations in the spheroidal product lying freely on its table.

ALD ¼ ARV ð1 þ o2 r Þ, A2

(3b)

for the horizontal laser position, where or in Eqs. (3a) and (3b) denotes the ratio of the actual angular frequency o to the angular frequency o0 corresponding to M0 peak—see the example in Fig. 2. The magnitude of change in AFPs due to corrections is important mainly at low frequencies and decreases with increasing frequencies. The corrections also cause a change of M0 peak position; it is shifted to lower frequencies. This paper describes the method used to determine the new angular frequency o0c that, when applied instead of o0 in Eqs. (3a) and (3b), the resulting Acor RV M0 maximum lies directly at this frequency; i.e. the frequency used for AFP correction is consistent with the corrected M0 peak.

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Relative Amplitude, dimensionless

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99 (2008) 156 – 160

20

Measured

Corrected

15 Peak shift

10

M0

5 M1

0 0

200

100

300

Frequency, Hz Fig. 2 – Example of AFP (amplitude–frequency plot) obtained for melon during ripening (vertical laser position, frequency step 1.5 Hz, Kuroki 2006, unpublished). The marks M0 and M1 denote modes of vibration (Blahovec et al., 2007). M0 peak at 54.55 Hz was used for corrections using Eq. (3a). M0 peak was shifted due to correction to 54.31 Hz. The peak locations were obtained by polynomial approximation of top parts of the peaks. The relation between frequency (f) and the angular frequency (x) is x ¼ 2pf.

2.

quadratic equation was obtained with the following solution for the vertical laser position: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ARV0  1 o0c ¼ o20  . (8a) a

Basic theory

Eq. (3a) can be rewritten into the following formula: Acor RV ¼ ARV þ

o20c ðARV  1Þ. o2

Its derivative in the M0 peak has to be zero:   o2 o2 dAcor dARV RV ¼ 1 þ 0c ¼ 0.  2ðARV  1Þ 0c 2 do do o o2

(4)

(5)

Moreover, at this point the identity o ¼ o0c has to be fulfilled, and Eq. (5) is rewritten to the following differential equation: dARV ARV  1 . ¼ o do

(6a)

Similarly for the horizontal laser position (Eq. (3b)) we obtained dARV ARV ¼ . do o

(6b)

Eqs. (6a) and (6b) determine the AFP point (ARV ,o0c) on the M0 peak (before correction) with a tangent that goes also through another AFP point (ARV ,o), where (ARV ,o) is (1, 0) and (0, 0) for the vertical laser position and the horizontal laser position, respectively.

3.

Applications

3.1.

Parabolic peaks

Parabolic peaks can be used to approximate the top parts of some real M0 peaks. They can be expressed in the following form: ARV ¼ ARV0  aðo  o0 Þ2 ,

(7)

where o0 is the peak angular frequency and parameter a describes the peak width. The higher the a, the narrower the peak. After expressing Eq. (6a) in terms of Eq. (7), the

Eq. (8a) shows that the corrected peak frequency is lower than the initial one; this difference increases with increasing ARV0 and decreasing a. The role of parameter a in formula (8a) is demonstrated in Fig. 3. For the laser horizontal position we similarly obtained: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ARV0 . (8b) o0c ¼ o20  a Eq. (8b) has properties similar to as Eq. (8a).

3.2.

Lorentz’s peaks

The vibration peaks can be described by Lorentz’s peak expression (ORIGIN Getting Started Manual, 2002): ARV ¼ y0 þ

2Sw 1 , p 4ðo  o0 Þ2 þ w2

(9)

where y0, S and w are parameters. S plays the role of the peak area and w is the peak half-width. The exact solution of our task for this equation is very difficult, but for small corrections we can approximate Eq. (9) by its Taylor set, where the first three terms are identical with parabolic Eq. (7), where ARV0 ¼ y0+2S/(pw) and a ¼ 8S/(pw3). The corrected M0 peak frequencies can then be calculated from Eqs. (8a) and (8b).

3.3.

Real peaks

In some cases, the peak top part can be approximated successfully by a polynomial of the second order and the problem could then be solved by the same method as in part 3.1. Only in a limited number of cases must the differential Eqs. (6a) and (6b) be solved numerically. Approximation should be used in all cases with a reduced number of data

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1

Angle Ratio

0.8 0.6 0.0001 0.0003

0.4 0.2 0

0.0006 0.0012

100 50 Initial Peak Position, Hz

0

150

Fig. 3 – Ratio of the parabolic peak frequencies (corrected divided by initial) plotted against initial peak frequency for the vertical laser position—see Eq. (8a) with initial peak height ARV0 ¼ 5 and different values a (values given in the figure).

Table 1 – Peak positions before (B) and after (A) correction in rad s1

20 18

Method/positions

16

Parabolic approximation of data Numerical analysis of smoothed data

Relative amplitude

14 12

B

A

308.38 308.29

307.53 307.74

B-values are given by approximation of the experimental data, Avalues are given by approximated data using Eq. (6a).

Measured Corrected

10 8 6 4 2 0 35

40

45 50 Frequency, Hz

55

60

Fig. 4 – M0 peak of melon from Fig. 2. The measured data were corrected using Eq. (3a); the corrected peak positions were determined by Eq. (8a). Six points from the top part of the measured peak were used to the parabolic approximation. The results are given in Table 1.

points, when the frequency step is higher or comparable to the desired precision of the calculated o0c. Both approximate and numerical solutions were applied to the real M0 peak given in Fig. 4. Numerical solution of Eq. (6a) was carried out in the following manner. The measured peak was first smoothed by the polynomial method using 5 points and then the first derivative was calculated for every point as a basis for determining (ORIGIN Getting Started Manual, 2002) the value of the expression: d(ARV)/do(ARV1)/o. The zero point of the expression was determined using its approximation by the polynomial of the second order through 6 points at the top of the ARV peak. This zero point corresponds to the new peak

frequency o0c. The initial peak position was determined by a similar method to that for finding the zero point of the expression d(ARV)/do. Results of both methods are given in Table 1. This table illustrates very low differences between results obtained by both the methods. We can conclude that in similar cases when peaks are narrow and with the top part of the parabolic shape, the position of the corrected peak can be well estimated by the approximation method.

4.

Conclusions

Corrections of AFP should be based on the true peak position after corrections. This position can be calculated using Eqs. (6a) or (6b). The solutions of the equations in special conditions are given by a numerical solution based on the data. The peak shift after corrections strongly depends on the peak width and shape. For narrow sharp peaks, the shift is low and the corrected peak positions can be determined using parabolic approximation of the top part of the obtained peak. The analytical solution of Eqs. (6a) and (6b) is given in these cases. R E F E R E N C E S

Abbott J A; Bachmann G S; Childers N F; Fitzgerald J V; Matusik J F (1968). Sonic techniques for measuring texture of fruits and vegetables. Food Technology, 22, 635

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Blahovec J; Akimoto H; Sakurai N (2007). Laser Doppler forced vibrology of soft agricultural products. Applied Rheology, 17(2), 25111 (1–7) Cooke J R (1970). A theoretical analysis of the resonance of intact apples. ASAE Paper No. 70-345, St Joseph, Michigan De Baerdemaeker J; Jancsok P T; Verlinden B E (2002). Firmness and softenning of fruits and vegetables. In: Physical Methods in Agriculture: Approach to Precision and Quality (Blahovec J; Kutilek M, eds), pp 343–357. Kluwer Academic/Plenum Publishers, New York Muramatsu N; Sakurai N; Wada N; Yamamoto R; Tanaka K; Asakura T; Ishikawa-Takano Y; Nevins D J (1997). Critical comparison of an accelerometer and Laser Doppler vibrometer for measuring fruit firmness. HortTechnology, 7, 434–438 ORIGIN Getting Started Manual (2002) Version 7. OriginLab Corporation. Northampton, USA.

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