Article Request Page ASABE Journal Article ## Analysis of Pear Dynamic Characteristics Based on Harmonic Response

W. Zhang, D. Cui, Z. Liu, Y. Ying## Published in

Transactions of the ASABE59(6): 1905-1913 (doi: 10.13031/trans.59.11845). Copyright 2016 American Society of Agricultural and Biological Engineers.Submitted for review in March 2016 as manuscript number PRS 11845; approved for publication by the Processing Systems Community of ASABE in September 2016.

The authors are

Wen Zhang,Doctoral Candidate,Di Cui,Associate Professor,Zihao Liu,Doctoral Candidate, andYibin Ying, ASABE Member,Professor, College of Biosystems Engineering and Food Science, Zhejiang University, Hangzhou, China.Corresponding author:Yibin Ying, 866 Yuhangtang Road, Hangzhou 310058, China; phone: +86-571-88982885; e-mail: ybying@zju.edu.cn or yingyb@zju.edu.cn.

The dynamic characteristics of fruits are correlated with their mechanical and structural properties. A finite element (FE) harmonic response analysis was used to investigate the dynamic characteristics of pears. First, the pear dynamic characteristics were measured by the acoustic vibration method using a laser Doppler vibrometer (LDV). A sample pear was placed vertically on the vibration stage and excited with a sweeping sine wave signal. The LDV was used to measure the response signal from the top of the pear. A three-dimensional model of the pear was then established by a machine vision-based modeling system. In the FE model, a force excitation was applied to the pear model from the bottom of the pear, and the vertical displacement at the top of the pear model was calculated. The resonance frequencies were extracted from the frequency response curve, and then the corresponding mode shapes were obtained. The results showed that the resonance frequencies of the FE models agreed well with the measured frequencies, indicating that harmonic response analysis is feasible for investigating the pear dynamic characteristics. All of the mode shapes showed a large deformation in the longitudinal direction. Therefore, it would be better to measure these modes from the top of the pear by the LDV method when the pear is placed vertically on the vibration stage. In addition, the effects of material properties and fruit shape on pear dynamic characteristics were investigated. The resonance frequencies increased with increasing Young’s modulus and decreased with increasing Poisson’s ratio, density, and mass. It was also found that there were strong linear relationships between the resonance frequencies and height/diameter (h/d) ratio. The obtained results could provide guidance for pear texture evaluation by the LDV method.Abstract.

Dynamic characteristics, Fruit shape, Harmonic response analysis, Laser Doppler vibrometer, Modes. Keywords.Nondestructive detection of pear texture is important because texture represents the freshness or ripeness of pears. Various techniques have been adopted to nondestructively detect pear texture, including the visible/near-infrared spectroscopy (Vis/NIRS) technique (Liu et al., 2008; Nicolaï et al., 2008), hyperspectral imaging technique (Qin and Lu, 2008), electronic nose technique (Zhang et al., 2008), and acoustic technique (De Belie et al., 2000; Gomez et al., 2005; Murayama et al., 2006; Wang et al., 2004). The acoustic vibration method is one of most effective and accurate methods for pear texture evaluation. A laser Doppler vibrometer (LDV) has been used to measure vibration characteristics of fruit because of its advantages of noncontact measurement and insensitivity to ambient noise (Muramatsu et al., 1997; Terasaki et al., 2006; Zhang et al., 2014a). Proper application of the LDV method for pear texture evaluation requires better understanding of the dynamic characteristics of pears on the vibration stage. The dynamic characteristics of fruits mainly include the resonance frequency, damping ratio, and mode shape.

In most existing studies, finite element (FE) modal analysis is used to analyze the dynamic characteristics of fruit. The resonant frequencies and corresponding mode shapes can be obtained by modal analysis. Modal analysis has been used for various fruit, including spherical fruit (Huarng et al., 1993; Kimmel et al., 1992) and non-spherical fruit (Abbaszadeh et al., 2014; Chen and De Baerdemaeker, 1993a; Cherng, 2000). The vibration of a homogeneous elastic sphere can be classified into two modes (Cooke and Rand, 1973; Lamb, 1881): the torsional mode, in which the radial displacement component vanishes everywhere (no volume change), and the spheroidal mode, in which the volume is not constant and the displacement vector requires three independent components. However, not every mode obtained by modal analysis can be detected in experimental measurement. This is because a singe-axial accelerometer and LDV can measure the vibration displacement in only one direction.

In the LDV method for fruit texture detection, the fruit is usually placed on the vibration stage and excited with a sweeping sine wave signal. The resulting vibration is a forced vibration. However, modal analysis is generally used for analysis of free vibration. FE harmonic response analysis is used to predict the steady-state dynamic response of a structure subjected to sinusoidally varying loads. Therefore, FE harmonic response analysis can be used to simulate the fruit vibration on the vibration stage. The theoretical resonance frequencies can be obtained from the frequency response curve, and their corresponding vibration shapes can be calculated. Furthermore, the mode shapes help determine the sense direction of the LDV.

The shape of the fruit also has an effect on its dynamic characteristics. However, the effect of the fruit shape is ignored in most fruit texture measurements with the LDV method. Modal analysis has been applied to evaluate the effect of shape on the free vibration of fruits (Chen and De Baerdemaeker, 1993b; Jancsok et al., 2001; Lu and Abbott, 1996). However, few researchers have investigated the effect of shape on the resonant frequencies of fruit excited by forced vibration.

Therefore, the objectives of this investigation were: (1) to simulate pear vibration on the vibration stage by FE harmonic response analysis, (2) to investigate pear deformation at each resonance frequency, and (3) to explore the effect of material properties and fruit shape on pear dynamic characteristics, especially for resonance frequencies.

Materials and MethodsPears (

Pyrus pyrifoliacv. ‘Hosui’) were hand harvested at the orchard of Sanshui Fruit Co., Ltd., in Hangzhou, China. The samples were stored at 4°C and 90% relative humidity. The pears were placed in the experimental environment for 24 h before the test. First, ten samples were used to measure the dynamic characteristics with the LDV method, and then their three-dimensional geometrical models were established for harmonic response analysis. A total of 102 samples with different shapes were then selected and modeled to investigate the effect of fruit shape on pear dynamic characteristics.Finally, 60 additional samples (table 1) were used for fruit texture evaluation with the FE model. Young’s modulus was first estimated with the FE model, and then the fruit texture was measured with the puncture test.

Table 1. Morphological properties of the tested pear samples ( n= 60).Mass

(m, g)Height ^{[a]}

(h, mm)Diameter ^{[a]}

(d, mm)Shape

Index^{[b]}

(SI)Stiffness

(N mm^{-1})Mean 299.57 73.84 91.49 0.81 11.27 Maximum 387.29 80.26 102.88 0.89 13.87 Minimum 230.61 68.75 84.22 0.74 8.95 SD 39.94 3.20 2.48 0.03 1.33

^{[a]}Height and diameter were obtained from three-dimensional models.

^{[b]}Shape index is the ratio of height to diameter (h/d).

Experimental Determination of Resonance FrequencyThe experimental setup to measure the vibration spectra of the pears is shown in figure 1 and was described in detail in our previous studies (Zhang et al., 2014a, 2014b). The sample pear was placed on the middle of the vibration stage and mechanically excited with a sweeping sine wave signal (frequency ranging from 200 to 2000 Hz) with a constant acceleration amplitude of 1

g. The signal started at 200 Hz to prevent pear movement on the vibration stage. The response signal at the top of the pear was optically sensed with the LDV (LV-S01, Sunny Instruments Singapore Pte., Ltd., Singapore). A reflective film was attached to the top of the pear to enhance the reflected signal strength, thereby acquiring more accurate results. At the same time, an accelerometer (752A12, Meggitt’s Endevco Corp., San Juan Capistrano, Cal.) attached to the vibration stage was used to measure the excitation signal of the pear. Frequency response curves were determined after fast Fourier transform (FFT) was applied to the excitation and response signals. The second to fifth resonance frequencies (f_{2}tof_{5}) were extracted for further analysis.

[Please see the PDF for this figure.] Figure 1. Schematic of the experimental setup for measuring the vibration spectra of pears. The setup includes control elements (black

arrows) and signal acquisition elements (blue and red arrows).

Texture MeasurementThe puncture test was performed with a texture analyzer (TA-XT2i, Stable Micro Systems, Ltd., Godalming, UK) at four peeled sites at evenly spaced intervals along the equatorial plane. A cylindrical probe with a diameter of 5 mm was inserted into the sample to a penetration depth of 10 mm at a loading speed of 1 mm s

^{-1}. Stiffness (Stif), calculated using equation 1, was determined from the force-deformation curve (Zhang et al., 2015a, 2015b):(1)

where

F= force at the rupture point (N)_{rup}

d_{1}= deformation at 60%Fbefore the rupture point (mm)_{rup}

d_{2}= deformation atF(mm)._{rup}

Geometric Examination and Model GenerationA machine vision-based modeling system was used to establish the three-dimensional models of the pears. Each pear was cut into halves, and one half was placed in a sealed box with its stem-calyx horizontal. Images were taken from the top view with an industrial camera (DVP-30GC03E, Imaging Source Europe, Bremen, Germany). The contour of the pear was obtained from the image by a series of algorithms, including gray-scale transformation, threshold segmentation, and contour extraction (fig. 2). A total of 2500 to 3000 points on the pear contour were obtained, so only some key points on the contour were used. One point of every 40 points was chosen, and about 60 to 80 total points were obtained for every pear contour. The coordinate data of the obtained key points of the contour were then imported into ANSYS (ver. 12.0, ANSYS Inc., Canonsburg, Pa.). The contour of the pear was reconstructed using the Spline option in the software. The pear was assumed to be axisymmetric, and half of the contour was used to establish the pear model. The three-dimensional model of the pear was obtained by rotating half the contour along the stem-calyx axis.

[Please see the PDF for this figure.] Figure 2. Pear geometry modeling procedure.

Harmonic Response AnalysisANSYS Workbench can perform harmonic response analysis of a structure, determining the steady-state sinusoidal response to sinusoidal varying loads acting at a specified frequency. The equation of motion for harmonic response analysis is shown in equation 2:

(-?

^{2}[M] +i?[C] + [K])({u_{1}} +i{u_{2}}) = ({F_{1}} +i{F_{2}}) (2)where

[

M] = mass matrix[

C] = damping matrix[

K] = stiffness matrix[

u] = displacement vector[

F] = external load vector.As shown in figure 3, the FE model was created based on the geometrical description of the pear model to be analyzed. First, some assumptions were made to simplify the FE model. The pear model was assumed to be linear elastic, isotropic, and homogeneous (Jancsok et al., 2001; Song et al., 2006). Although multilayer FE models have been used for some fruits, it was found that the variations of Young’s modulus for the pericarp and core did not change much of the natural frequencies (Chen and De Baerdemaeker, 1993a; Lu and Abbott, 1996). Therefore, the effects of the pericarp and core on the dynamic characteristics were neglected, and only a single material was contained in the FE model. Next, a free meshing strategy was adopted for mesh generation. The structural element SOLID 187 was used for the pear model. The element is defined by ten nodes with three degrees of freedom at each node. SOLID 187 is well suited to model irregular meshes. Third, sinusoidal excitation was applied to the pear model. In the experimental test, the pear was placed with its stem upward on the vibration stage, and a sine wave signal with constant acceleration amplitude of 1

g(g= 9.8 m s^{-2}) was applied across a range of frequencies (200 to 2000 Hz). To simulate the pear vibration in the FE model, the force excitation with the same frequency range (200 to 2000 Hz) was applied to the pear model from the bottom of the pear. The acceleration input is not supported in harmonic response analysis in ANSYS, so the acceleration load was converted into a force load by equation 3:

F=ma(3)where

F= excitation force (N)

m= mass of the pear sample (kg)

a= acceleration load (m s^{-2}).

[Please see the PDF for this figure.] Figure 3. Finite element (FE) model of a pear. The pear model was assumed to be linear elastic, isotropic, and homogeneous. Sinusoidal excitation was applied to the pear model from the bottom, and the vertical displacement ( y-axis) at the top of the model was calculated.Because of its detection principle, the LDV can measure only the velocity component parallel to the laser beam, so the vibration in the vertical direction was measured with the LDV method in this study. Therefore, the vertical displacement (

y-axis) at the top of the pear model was calculated in the FE model. The resonance frequencies were extracted from the frequency response curve, and then the corresponding mode shapes were obtained. The modes corresponding tof_{2}tof_{5}are called modes 2 to 5 in this study. In addition, the model updating technique was adopted to tune the sec-ond resonance frequency (f_{2}) of the FE model to match that of the experimental results (Cherng, 2000; Jancsok et al., 2001; Nourain et al., 2005).

Dynamic Characteristic Analysis of FE ModelTen samples were used to compare the experimental and FE results. The density was assumed to be 975 kg m

^{-3}based on a preliminary experiment in which the mass and volume of ten samples were measured, and Poisson’s ratio was assumed to be 0.3. Young’s modulus was estimated by the model updating technique.To investigate the pear dynamic characteristics and their correlation with material properties, different values of Young’s modulus (2 to 12 MPa), Poisson’s ration (0.2 to 0.4), and density (850 to 1100 kg m

^{-3}) were taken while keeping the other conditions constant (table 2). To explore the effect of pear mass on dynamic characteristics, several pear models with the same geometric shape were constructed by proportionally changing the contour using the Scale option in the software. The height/diameter (h/d) ratio (Chen and De Baerdemaeker, 1993b; Jancsok et al., 2001) was used as the shape index, and 102 pear models with different shapes were established to investigate the relationships between theh/dratio and pear dynamic characteristics. When investigating the effects of mass and fruit shape, the values of the material parameters remained constant (table 2).

Table 2. Material parameter settings when investigating the factors affecting pear dynamic characteristics. The other conditions were kept constant when investigating the effect of one parameter. Affecting Factors Young’s Modulus

(MPa)Poisson’s

RatioDensity

(kg m^{-3})Young’s modulus - 0.3 975 Poisson’s ratio 10 - 975 Density 10 0.3 - Mass 10 0.3 975 Fruit shape 10 0.3 975

Results

Table 3. Comparisons of experimental and modeled resonance frequencies ( f_{2}tof_{5}, Hz) of ten tested pears. Young’s modulus was estimated by the model updating technique.Pear Laser Doppler Vibrometer (LDV) Method Finite Element (FE) Model E_{est}^{[a]}

(MPa)f_{2}f_{3}f_{4}f_{5}f_{2}f_{3}f_{4}f_{5}1 680 1020 1300 1580 680 1025 1345 1642 11.4 2 700 1080 1340 1640 700 1057 1404 1693 11.6 3 680 1030 1290 1590 680 1024 1349 1637 11.3 4 710 1090 1330 1655 710 1073 1399 1696 10.1 5 740 1125 1410 1710 740 1117 1474 1775 15.4 6 660 1015 1295 1575 660 995 1310 1592 9.9 7 735 1110 1390 1685 735 1106 1472 1756 14.1 8 680 1030 1310 1590 680 1026 1346 1633 11.9 9 645 990 1260 1545 645 970 1284 1559 12.4 10 735 1115 1395 1690 735 1108 1452 1753 12.7

^{[a]}Eis the estimated Young’s modulus._{est}Figures 4a and 4b show typical frequency response curves obtained by the LDV method and the FE model, respectively, for the same sample. The waveforms obtained by the two methods are basically the same; however, the energy is obviously different. This was because both the excitation and response signals were used to obtain the frequency response curve in the LDV method, but only the response at the top of the pear was analyzed in the FE model.

[Please see the PDF for this figure.]

Firm pears usually exhibit four or five resonance frequencies in the frequency range of 200 to 2000 Hz. However, the higher resonance frequencies may be missing in relatively soft pears, which is consistent with the finding of Kimmel et al. (1992). Table 3 summarizes the second to fifth resonance frequencies (

f_{2}tof_{5}) measured by the LDV method and those obtained by the FE models. Young’s modulus was estimated so that thef_{2}of the FE model matched that measured by the LDV method. The remaining resonance frequencies of the FE models were then evaluated according to the estimated Young’s modulus (E). The results showed that the resonance frequencies of the FE models agreed well with the measured values. Although there were some differences in the resonance frequencies, all the differences were less than 6%._{est}

Mode Shapes

Table 4. Mode shapes corresponding to the second to fifth resonance frequencies ( f_{2}tof_{5}) of the pear. For every mode, five shape images are used to show the deformation process.[Please see the PDF for this figure.] The mode shapes corresponding to

f_{2}tof_{5}are shown in table 4. According to the definition of the two vibration modes, it was concluded that all four mode shapes belong to the spheroidal mode. The mode shape corresponding tof_{2}is the so-called oblate-prolate mode. In this mode, the pear model extends and contracts in two mutually perpendicular directions simultaneously. During extension or contraction in the transverse direction, the pear contracts or extends in the longitudinal direction. Two nodal lines exist in this mode. The mode shapes corresponding tof_{3}tof_{5}are analogous to that off_{2}. These modes show simultaneous extension and contraction in two mutually perpendicular directions, but the mode shape becomes more complex with increasing mode number. There are three and four nodal lines in modes 3 and 4, respectively.

Effects of Material Properties and Fruit Shape on Pear Dynamic CharacteristicsFigure 5 shows the effect of Young’s modulus on

f_{2}tof_{5}. The resonance frequencies increased with increasing Young’s modulus. It was also found that the difference in these resonance frequencies diminished as the value of Young’s modulus decreased, indicating that the locations of the resonance peaks were getting closer as the pear became riper and softer. When the square of the resonance frequency (f^{2}) was used as they-coordinate (fig. b), there was a strong linear relationship betweenf^{2}and Young’s modulus. The result confirms the stiffness factor (f^{2}m^{2/3}) proposed by Cooke (1972).Figure 6 shows the effect of Poisson’s ratio on

f_{2}tof_{5}. All four resonance frequencies decreased with an increase in Poisson’s ratio. However, the effect of Poisson’s ratio on the resonance frequencies was obviously less than that of Young’s modulus. The changes in the resonance frequencies were less than 7% when Poisson’s ratio ranged from 0.2 to 0.4.Figure 7 shows the effect of density on

f_{2}tof_{5}. The different densities resulted in different masses. To avoid the effect of mass on the resonance frequency, a normalization process for the resonance frequency was applied (Jancsok et al., 2001). The eigenfrequencies of a theoretical object with different size and mass but the same material properties and shape were calculated. The mass of the theoretical object (m_{0}) was 100 g. The normalized frequency (f_{n}) (Jancsok et al., 2001) was calculated using equation 4:

f_{n}= (m/m_{0})^{1/3}f(4)where

f_{n}= normalized frequency (Hz)

f= frequency to be normalized (Hz)

m= mass of the pear (g).

[Please see the PDF for this figure.] Figure 5. Effect of Young’s modulus (2 to 12 MPa) on the second to fifth resonance frequencies ( f_{2}tof_{5}) of pear. Poisson’s ratio was 0.3, and density was 975 kg m^{-3}.The resonance frequencies decreased with increasing pear density. The changes in normalized frequencies were less than 5% when the density ranged from 850 to 1100 kg m

^{-3}. In addition, the range of pear density is much smaller in practice. The density of the ‘Hosui’ pears used in this study was usually between 950 and 1000 kg m^{-3}. Therefore, the effect of density on the resonance frequency was also obviously less than that of Young’s modulus.Figure 8 shows the effect of mass on

f_{2}tof_{5}. Five pear models with the same geometric shape but different sizes were established. The resonance frequencies decreased with increasing mass, which is consistent with previous studies (Chen and De Baerdemaeker, 1993a, 1993c).Figure 9 shows the relationship between the

h/dratio and normalized frequencies (f_{2n}tof_{5n}) of the 102 modeled pear samples with different shapes. Because these samples had different masses, the resonance frequencies were also normalized to avoid the effect of mass. It is clearly shown that there was a linear relationship between theh/dratio and the normalized frequencies: the higher theh/dratio, the lower the resonance frequencies for all four modes. In addition,f_{2n}had the strongest linear relationship with theh/dratio. The coefficient of determination (R^{2}) decreased with increasing mode number.

[Please see the PDF for this figure.] Figure 9. Effect of height/diameter ( h/d) ratio on the second to fifth resonance frequencies (f_{2}tof_{5}) of pear. Young’s modulus was 10 MPa, Poisson’s ratio was 0.3, and density was 975 kg m^{-3}. The normalized frequency (f_{n}) was calculated to avoid the effect of mass on the resonance frequency.

[Please see the PDF for this figure.]

Figure 10. Correlations between (a) elasticity index (EI) and stiffness (Stif) and (b) estimated Young’s modulus (E) and_{est}Stif).

Fruit Texture EvaluationFigure 10 shows the correlations between elasticity index (

EI) andStif, and betweenEand_{est}Stif. TheEIvalue was calculated usingf_{2}^{2}m^{2/3}(Zhang et al., 2014a). A good linear correlation was found betweenEIandStif, which is consistent with our previous findings (Zhang et al., 2014a, 2015b).Ealso showed a good linear correlation with_{est}Stifand had a higher correlation coefficient withStifthanEI. This is because the effect of fruit shape was considered in the FE model. This result verified that the fruit shape can help improve the performance of the prediction model for fruit texture evaluation.

DiscussionIn most existing studies on the analysis of fruit dynamic characteristics, modal analysis was used to investigate the free vibration of the fruit. In these studies, the fruit was struck by a stick or pendulum and then vibrated freely (Chen and De Baerdemaeker, 1993a; Jancsok et al., 2001). However, in the LDV method, the fruit is placed on the vibration stage and excited with a sweeping sine wave signal. It is difficult to find mode shapes corresponding to the resonance frequencies measured by the LDV method because too many modes are obtained by the modal analysis. In addition, forced vibration is adopted in the LDV method. Therefore, harmonic response analysis was used to simulate pear vibration on the vibration stage. The primary advantages of this approach are that: (1) the mode shapes (pear deformation) help determine the sense direction of the LDV, (2) the importance of each factor affecting pear vibration characteristics can be obtained, and (3) the effect of some factors (such as fruit shape) that are difficult to obtain in practice can be simulated in the FE model.

To investigate the feasibility of FE harmonic response analysis, the resonance frequencies

f_{2}tof_{5}were mainly studied. The first resonance peak was found to be approximately 100 Hz in our previous studies. However, the first resonance frequency (f_{1}) is hard to measure, especially for firm pears, because the high gain in the low frequency region causes bouncing. The vibration displacement of the vibration stage is large in the low-frequency region, even at a small excitation level; thus, there is a loss of contact between the pear and the vibration stage. In addition, the mode corresponding tof_{1}was found to be a rigid body (Chen and De Baerdemaeker, 1993b; Kimmel et al., 1992). Therefore,f_{1}was not investigated in this study, and the frequency ranged from 200 Hz to 2000 Hz. The good agreement forf_{2}tof_{5}obtained with the two methods (table 3) indicated that harmonic response analysis is feasible for investigating the dynamic characteristics of pears on the vibration stage. The difference in the resonance frequencies between the two methods mainly resides in several aspects: (1) the geometric configurations of the real pear and the pear model were different, (2) the effects of the pericarp and core were neglected and only a single material was considered, and (3) the pear model was assumed to be linear elastic, isotropic, and homogeneous.The mode shape corresponding to

f_{2}was found to be the oblate-prolate mode. The oblate-prolate mode was also found in pineapples (Chen and De Baerdemaeker, 1993a, 1993c), apples (Chen and De Baerdemaeker, 1993c), cantaloupes, watermelons (Cherng, 2000), and pears with non-spherical shapes (Jancsok et al., 2001) by model analysis. As shown in table 4, all four mode shapes showed large deformations in the longitudinal direction. Therefore, it would be easier to measure these modes when the laser beam is paral-lel to the longitudinal direction. This also explains why the four modes can be easily detected from the top of the pear by the LDV method.The effects of material properties and fruit shape on the resonance frequencies of pears were explored in this study. To avoid the effect of mass on the resonance frequency, the normalized frequency was calculated when investigating the effect of density and fruit shape. The effect of Young’s modulus is consistent with the results obtained by modal analysis (Chen and De Baerdemaeker, 1993a, 1993c; Dewulf et al., 1999; Lu and Abbott, 1996; Song et al., 2006). The results also showed that the effects of Young’s modulus and mass were much greater than that of Poisson’s ratio and density. In addition, Poisson’s ratio and density are difficult to measure in practice. Therefore, Poisson’s ratio and density are generally not considered in nondestructive evaluation of fruit texture.

The shape of the fruit has an effect on its dynamic characteristics. However, how the fruit shape affects the dynamic characteristics of fruit on the vibration stage measured by the LDV method has not been investigated yet. Strong linear relationships were found between the resonance frequencies and

h/dratio in this study. In addition, the improved results of the prediction model (fig. 10) indicated that a combination of the LDV method and a machine vision system may give a more accurate result for pear texture evaluation in practical applications.

ConclusionThree-dimensional models of pears were established by a machine vision-based modeling system, and pear dynamic characteristics were investigated. FE harmonic response analysis was adopted instead of the modal analysis used in most existing studies because forced vibration was applied to the pear on the vibration stage. The resonance frequencies of the FE models agreed well with the measured values, indicating that harmonic response analysis is feasible for investigating pear dynamic characteristics. The approach provides a new way to simulate fruit vibration on the vibration stage.

The mode shapes corresponding to

f_{2}tof_{5}showed large deformations in the longitudinal direction. Therefore, it would be better to measure these modes from the top of the pear with the LDV method. The effect of fruit shape on pear dynamic characteristics is often ignored in the LDV method. Therefore, the strong linear relationships between theh/dratio and resonance frequencies obtained in this study would help reduce the effect of fruit shape on pear texture evaluation with the LDV method.

AcknowledgementsThe authors gratefully acknowledge the support of the program by the National Natural Science Foundation of China (Grant No. 31571764) and by the Jiangsu Key Laboratory of Physical Processing of Agricultural Products (Grant No. JAPP2014-1). Any opinions, findings, and conclusions expressed in this publication are those of the authors and do not necessarily reflect the views of Zhejiang University. The trade and manufacturer names are necessary to factually report on the available data.

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