**2.2.3 Experimental modal analysis of the fruit**

Experimental modal analysis is a technique to obtain natural frequency, mode shapes and damping ratios of an elastic structure by application of a vibrational stimulus to the sample and sensing the resulting vibration at various locations on the sample surface. This section illustrates an experimental modal analysis on a melon fruit (Ehle, 2002). The excitation was a pulse force delivered by an impact hammer. A force transducer at the tip of the hammer measured the force. An accelerometer was used to measure the acceleration response at several points all over the surface of the melon. The measurement points were designed to follow a grid pattern where the distances between adjacent points were almost uniform throughout the surface of the fruit. The melon most closely resembled a sphere. Table 2 and Fig. 18 show the measurement points. A tri-axial accelerometer must be used to sense the important torsional mode (Fig. 8) as well as radial motions.


Table 2. Locations of the points where acceleration was measured on the melon.

Fig. 17. Mode replacing the twisting mode when flesh Young's modulus is 1.5MPa

Experimental modal analysis is a technique to obtain natural frequency, mode shapes and damping ratios of an elastic structure by application of a vibrational stimulus to the sample and sensing the resulting vibration at various locations on the sample surface. This section illustrates an experimental modal analysis on a melon fruit (Ehle, 2002). The excitation was a pulse force delivered by an impact hammer. A force transducer at the tip of the hammer measured the force. An accelerometer was used to measure the acceleration response at several points all over the surface of the melon. The measurement points were designed to follow a grid pattern where the distances between adjacent points were almost uniform throughout the surface of the fruit. The melon most closely resembled a sphere. Table 2 and Fig. 18 show the measurement points. A tri-axial accelerometer must be used to sense the

Layer 1 2 3 4 5 6 7 8 9 Points 1 2-7 8-19 20-37 38-55 56-73 74-85 86-91 92

Angle (deg) 90 67.5 45 22.5 0 -22.5 -45 -67.5 -90

Angle (deg) N/A 60 30 15 15 15 30 60 N/A

Table 2. Locations of the points where acceleration was measured on the melon.

**2.2.3 Experimental modal analysis of the fruit** 

important torsional mode (Fig. 8) as well as radial motions.

Longitudinal

Latitudinal

Fig. 18. Measurement points for the above described testing

Fig. 19. A long elastic cord and rubber bands supporting the melon.

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 323

To reduce random noise, the average of data from ten impacts was used to compute each FRF. The *coherence* is a function in the frequency domain that indicates the quality of the FRF. A perfect coherence is 1.0. A coherence less than that at a given frequency means that the vibration sensed by the accelerometer at that particular frequency is not a linear response to the excitation from the impact hammer. A low coherence in general indicates

excessive noise, nonlinearity, or other causes of bad measurement.

Fig. 21. Frequency response function obtained by hitting the melon rind directly

Any method used to hold the melon is a boundary condition which will alter the vibration modes. However, the free-free boundary condition has been proven to not alter the vibration modes. This boundary condition can be approximated by suspending the melon from a rigid structure with elastic cords. Soft elastic cords are used in the suspension so that the rigid-body vibration of the melon and its suspension has very low natural frequency and does not alter the elastic modes. Figure 19 shows the suspension the setup used for the holding of the sample.

The porous surface of the melon made it difficult to affix the accelerometer to the surface of the melon using conventional methods. To make sure the accelerometer was affixed snugly to the surface of the melon, wax was applied to the bottom of the accelerometer. This allowed for the mating of the accelerometer to the surface of the melon, which would result in better vibration transmission. Also the accelerometer was held in place by a "cradle" made from rubber bands (Figure 20). The width of the rubber bands was approximately 6 mm, just wide enough to cover the top surface of the accelerometer. The rubber band was positioned in such a way as to apply sufficient pressure so that the surface of the accelerometer was always held normal to the surface of the melon. This configuration allowed for an effective and easy-to-move attachment method to install the accelerometer.

Fig. 20. Accelerometer attachment

To excite the fruit a modal hammer with a force transducer was used. A plastic tip was used as opposed to the metal tip. The plastic tip allowed for the force from the hammer to stay within the crucial 20dB range past the cut off frequency of 500 Hz. The accelerometer then measured the resulting accelerations of the surface of the melon. An FFT analyzer transformed the acceleration and force into the frequency domain, and obtained the ratio of acceleration to force, which was the Frequency Response Function (FRF) at each sense point.

Any method used to hold the melon is a boundary condition which will alter the vibration modes. However, the free-free boundary condition has been proven to not alter the vibration modes. This boundary condition can be approximated by suspending the melon from a rigid structure with elastic cords. Soft elastic cords are used in the suspension so that the rigid-body vibration of the melon and its suspension has very low natural frequency and does not alter the elastic modes. Figure 19 shows the suspension the setup used for the

The porous surface of the melon made it difficult to affix the accelerometer to the surface of the melon using conventional methods. To make sure the accelerometer was affixed snugly to the surface of the melon, wax was applied to the bottom of the accelerometer. This allowed for the mating of the accelerometer to the surface of the melon, which would result in better vibration transmission. Also the accelerometer was held in place by a "cradle" made from rubber bands (Figure 20). The width of the rubber bands was approximately 6 mm, just wide enough to cover the top surface of the accelerometer. The rubber band was positioned in such a way as to apply sufficient pressure so that the surface of the accelerometer was always held normal to the surface of the melon. This configuration allowed for an effective and easy-to-move attachment method to install the accelerometer.

To excite the fruit a modal hammer with a force transducer was used. A plastic tip was used as opposed to the metal tip. The plastic tip allowed for the force from the hammer to stay within the crucial 20dB range past the cut off frequency of 500 Hz. The accelerometer then measured the resulting accelerations of the surface of the melon. An FFT analyzer transformed the acceleration and force into the frequency domain, and obtained the ratio of acceleration to force, which was the Frequency Response Function (FRF) at each sense point.

holding of the sample.

Fig. 20. Accelerometer attachment

To reduce random noise, the average of data from ten impacts was used to compute each FRF. The *coherence* is a function in the frequency domain that indicates the quality of the FRF. A perfect coherence is 1.0. A coherence less than that at a given frequency means that the vibration sensed by the accelerometer at that particular frequency is not a linear response to the excitation from the impact hammer. A low coherence in general indicates excessive noise, nonlinearity, or other causes of bad measurement.

Fig. 21. Frequency response function obtained by hitting the melon rind directly

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 325

It is important that the excitation force contain energy at all the frequencies of interest. That means that the force imparted by the hammer on the fruit must be a sharp enough pulse in the time domain. Experimental modal analysis is most commonly done on hard structures such as vehicles, buildings or metal structures. The surface of the fruit is much softer than most other structures for which the hammer tip was designed. Hitting the fruit with a hammer would result in a broad pulse in time domain, which translates into a narrowband excitation in the frequency domain. As a result, the impulse spectrum of the excitation force would not have enough energy to excite vibration modes higher than 200 Hz without its intensity dropping more than 20 dB from its DC magnitude. When that condition is violated, the coherence of the data is very poor as seen in figure 22. To overcome this narrow-band excitation problem, a small metal disk (Figure 23) was attached to the fruit at the point of impact. This resulted in a significantly broader-band

This chapter has presented a few examples of research that has been done to take advantage of the advancement in vibration analysis along with applications to characterize the rheological properties of biomaterials. The literature shows that the rheological properties of biomaterials are associated to quality indicators, specifically for foods to their texture and their sensory evaluation, thus many of the applications described in this chapter deal with food materials. In particular, this chapter has shown examples of application of basic vibration theories to measure the rheology of liquids as well as viscoelastic semi fluids and semi solid materials. The static measurement of modulus, finite element computation of the vibration natural frequencies and mode shapes, and an experimental modal analysis of a

The authors believe that research on testing of biomaterials using vibration methods may

Fig. 23. A stiffening metal disk on the melon rind.

force excitation.

**3. Concluding remarks** 

melon fruit are also described.

help achieve:

Figure 11 shows a typical FRF (top curve) and coherence (bottom curve) from the measurement. Each measurement point in Table 1 resulted in one FRF. A peak at a certain frequency means large vibration at that frequency, which indicates a mode at that frequency. If the FRF at a particular point shows a valley or low response at a modal frequency, it is indicating that the point is a node (point of no motion) of the corresponding mode shape. Using those rules, the mode shapes at any resonant frequency could be visually determined. A modal analysis program uses mathematical algorithms to compute the natural frequency, damping and mode shapes from the FRFs. It was used to analyze data from this test, but may not be necessary if the tester can figure out the modes by careful visual examination of the FRFs.

Fig. 22. An FRF (top) and coherence (bottom) obtained without a local stiffener on the surface of the melon.

Fig. 23. A stiffening metal disk on the melon rind.

It is important that the excitation force contain energy at all the frequencies of interest. That means that the force imparted by the hammer on the fruit must be a sharp enough pulse in the time domain. Experimental modal analysis is most commonly done on hard structures such as vehicles, buildings or metal structures. The surface of the fruit is much softer than most other structures for which the hammer tip was designed. Hitting the fruit with a hammer would result in a broad pulse in time domain, which translates into a narrowband excitation in the frequency domain. As a result, the impulse spectrum of the excitation force would not have enough energy to excite vibration modes higher than 200 Hz without its intensity dropping more than 20 dB from its DC magnitude. When that condition is violated, the coherence of the data is very poor as seen in figure 22. To overcome this narrow-band excitation problem, a small metal disk (Figure 23) was attached to the fruit at the point of impact. This resulted in a significantly broader-band force excitation.
