Physical and Biological Properties of Agricultural Products Acoustic, Electrical and Optical Properties and Biochemical Property 1. Acoustic and Vibrational Properties 1.1 Acoustics and Vibration Engineering 1.1.1 Vibration Engineering (1) Vibration We can easily find many kinds of phenomena around us that are labeled 'Vibration'. We can literally see vibrations in the swinging head of a bobblehead doll, in the seat of a swing in the park, furniture swaying back and forth during an earthquake, and in the strings of a stringed musical instrument. We experience or feel vibrations with machines that are driven by an engine or electric motor. What is common to these phenomena is an orderly and repetitive movement. For convenience such phenomenon are labeled as 'Vibration'. When vibration is undesirable, it is important to control or isolate it. On the other hand, a stable and accurate vibration, such as that generated by a quartz crystal, is used in electronic watches and personal computers. The discipline of 'vibration engineering', a part of mechanical engineering, has been developed to utilize or control this phenomena of vibration. Vibration engineering is based on the theory of mechanical oscillations (vibrations), and therefore is an important field of study in a broad range of academic disciplines, including not only in mechanical engineering but also in a broad range of other science and engineering disciplines. (2) Period and frequency Vibration, an oscillating phenomenon, is an event that includes both temporal and spatial repetition. When vibrations are periodic and sinusoidal in character, they can Spring Displacement Time Fig.1.1 Vibration of spring 1
be expressed as: where x = displacement A = peak displacement φ = initial phase (constant) f = frequency. x = Asin(2πft+φ) = Asin(ωt+φ) (1.1) The angular speed ω is defined as 'angular frequency'. The angular frequency is presented in radians per second [rad/s]. The horizontal axis in Fig. 1.1 shows time [s] and the vertical axis shows displacement. The units for this differ according to the type of vibration. As in Fig. 1.1, where the vibration of a spring is shown, the vertical axis of displacement is a unit of length [m]. If it is the output of an electronic device, the vertical axis will be a unit of voltage [V]. The time taken for an oscillation to occur is defined as the period. The unit of a period is time [s]. The number of occurrences of a periodic oscillation per unit time (usually per second) is defined as 'frequency'. The unit of frequency is Hertz [Hz], which is equivalent to [1/s]. Such sinusoidal vibration is also referred to as 'simple harmonic motion'. (3) Natural frequency To start the explanation of vibration, we will assume that a mass is suspended from one end of a spring, the other end of the spring is fixed, the spring constant is k [N/m](where k > 0), and the mass is m [kg](where m > 0). If the mass is pulled down x [m] from the initial position, a force -kx [N] acts on the mass. If the upward movement is assumed to be positive, the value of x is negative when the mass is pulled down. Therefore the value of -kx is positive, which means that the force is upward. Conversely, if the mass is pulled up, the value of x becomes positive, and the value of -kx becomes negative. This means that as the spring shortens, the force acting on the mass is downward. When the mass is allowed to move, the acceleration of the mass is expressed by d 2 x/dt 2, and the force applied to the mass is expressed as a product of acceleration and mass. This force equals the pulling force of the spring acting on the mass. Therefore the following equation of motion is generated: m d 2 x dt 2 = -kx (1.2) This equation expresses a simple harmonic motion. Substituting Equation (1.1) into Equation (1.2), we get the following equation: -mω 2 Asin(ωt+φ) = -kasin(ωt+φ) (1.3) 2
As A is not zero, the relation between the angular frequency ω and the spring constant k can be expressed by the equation mω 2 = k. When ω has a positive value, the relationship is expressed by ω 0: ω 0 = m k =2πf 0 (1.4) This angular frequency ω 0 is defined as the 'natural angular frequency', and f 0 is defined as the 'natural frequency'. (4) Free vibration and forced vibration When a theoretical vibration, as described above, continues its motion with a natural angular frequency of ω 0, the sum of the potential energy stored in the spring and the kinetic energy stored in the mass will be conserved (Fig. 1.2a). Displacement (m) Displacement (m) time (s) (a) Vibration without damping (b) Vibration with damping Fig.1.2 Damped vibration A vibration of this type is defined as a 'free vibration'. In reality, the vibration will be subject to friction and pneumatic resistance, suppressing the motion. Therefore, the amplitude of displacement decreases gradually and the motion finally comes to a stop. Such behavior of suppressing the motion is defined as 'attenuation' or 'damping'. The model shown in Fig. 1.3 is equipped with a dashpot to suppress the vibration. It shows a free vibration with viscous damping. Spring k Dashpot c Mass m x Fig.1.3 Mass spring dashpot model 3
The dashpot consists of a piston and a cylinder filled with oil. It produces a viscous force that is proportional to the velocity of the piston relative to the cylinder. The equation of motion of the model is expressed by: where d 2 x dx m +c dt 2 dt +kx =0 c is a coefficient of damping. (1.5) If we add a pulling force on the mass, or a pushing force to the child on the swing, the vibration continues. When a periodic external force (Fcosωt) is applied, we get following equation of motion: d 2 x dx m +c +kx =Fcosωt dt 2 dt (1.6) Such vibration is defined as 'forced vibration'. If the angular frequency ω of the periodic external force equals that of the natural angular frequencyω 0, a large vibration occurs. This phenomenon is called resonance. In the case of the swing, if the pushing frequency equals the natural frequency, a large motion is obtained by a small force because of the resonance. k c Periodic external force F cos ωt m Fig.1.4 Acting periodic external force 1.1.2 Acoustics (1) Sound Sound is a pressure pulsation or a viscosity wave caused by a vibration, and it transmits through the medium, including fluids. The spring constant in the vibration model corresponds to the ratio of the strain change, or the volumetric change, to the pressure change in the sound. Also, the mass in the vibration model corresponds to the density of the medium. Therefore, sound can be understood and dealt with using 4
vibration engineering. However, there are phenomena which are peculiar to sound. A discipline called 'acoustics' has been developed independently to deal with this. Here, some technical terms of acoustics are shown. (2) Technical terms of acoustics 1 Frequency The frequency of a sound has the same definition as that of the vibration. The unit of frequency is [Hz]. 2 Sound pressure An acoustic wave in the air is a vibration of the air, where the pressure pulsation transmits through the air space. The difference between the pressure of silent air, and the air pressure when the sound passes through it is defined as 'sound pressure' [Pa]. 3 Particle velocity Focusing on a small particle in the medium, the vibrating velocity of the particle, in response to the change induced by sound pressure, is defined as particle velocity. Sound pressure is proportional to particle velocity. 4 Sound velocity Sound velocity is the velocity of a sound wave propagating through a medium. It is about 340 m/s in air, and about 1500 m/s in water. (3) Longitudinal wave and transverse wave We will now consider plane wave propagation in a homogeneous, elastic continuum, which is spatially infinite. In general, it is unsuitable to describe wave motion in a continuum using sound pressure, which is a scalar quantity. In such a case, second order tensors of the stress and strain are used. According to the general application of Hooke's law of elasticity, the elastic modulus is defined by a fourth order tensor that shows the linear relationship between the stress tensor and the strain tensor 1),2). We will consider an element of volume as shown in Fig. 1.5. superscript number shows the reference number 5
σ i j is the tensor of stress parallel to the x j axis, acting on the surface perpendicular to x i axis, where i and j are indices from 1 to 3. As the strain is proportional to the stress, if the strain tensor is assumed to be e kl (k and l are indices from 1 to 3) the following formula is obtained 2) : σ ij =c ijkl e kl (1.7) In the above equation, c ijkl is defined as the fourth order elastic modulus tensor, where i, j, k and l are indices from 1 to 3. If no torque acts on the element of volume, σ i j is equal to σ j i. Under this condition, equation (1.7) can be simplified to equation (1.8) using the Voigt notation; i.e. 11 1, 22 2, 33 3, 23 or 32 4, 31 or 13 5, 12 or 21 6. σ m =c mn e n (1.8) where m and n are indices from 1 to 6. Applying this equation to the isotropic (or cubic) medium, c mn can be expressed by equation (1.9). c 11 c 12 c 12 0 0 0 c 12 c 11 c 12 0 0 0 c 12 c 12 c 11 0 0 0 0 0 0 c 44 0 0 0 0 0 0 c 44 0 0 0 0 0 0 c 44 (1.9) If the medium is isotropic, the following equality holds: 2c 44=c 11-c 12 (1.10) Therefore it turns out that the number of the independent moduli of elasticity is two 1). Table 1.1 shows the elastic modulus of equation (1.9) in relation to the moduli, using classical elastic dynamics, i.e., bulk modulus K, Young's modulus E, shear modulus G, Lamé constants λ, μ and Poisson's ratio ν. 6
Table 1.1 Conversion table of elastic modulus for isotropic elastic medium 4) Lamé constants λ Lamé constants μ (=shear modulus G) λ, μ K, G G, ν E, ν E, G λ 2 K - 3 G 2Gν 1-2ν νe (1+ν)(1-2ν) G(E-2G) 3G-E E μ G G 2(1+ν) G bulk modules K 2 λ+ μ K 3 2G(1+ν) 3(1-2ν) E 3(1-2ν) EG 3(3G-E) Young's modulus E (3λ+2μ)μ λ+μ 9KG 3K +G 2(1+ν)G E E Poisson's ratio ν λ 2(λ+μ) 3K-2G 2(3K +G) ν ν E 2G -1 Two types of sound velocity are calculated for this isotropic elastic medium 2) : c 1 = ρ c 11 (1.11) c 2 =c 3 = ρ c 44 where c1 is the sound velocity of the longitudinal wave c2 and c3 are those of the transverse wave. The longitudinal wave is a wave that has the same direction of travel as that of the particle vibration. The transverse wave is a wave that travels perpendicular to the direction of the vibration. Equation (1.12) is a calculus example that gives the sound velocity of the longitudinal wave cl, and that of the transverse wave cs, in the isotropic elastic medium. This equation comes from the relationship shown in Table 1.1. The parameters cl and cs can be calculated using the moduli of classical elastic dynamics. c l = c s = 1 4 ρ (K+ G) 3 G ρ (1.12) Generically, the relationship between sound velocity and elastic moduli can be expressed by the following equation 1) : 7
sound velocity = elastic moduli related to the strain due to the wave density 1.2 Sound Velocity 1.2.1 Definition and characteristics (1) Hammer test The hammer test is a method of the measuring agricultural products. An impulsive mechanical stimulus is applied to an agricultural product, and the sound response is measured. An impulse is theoretically defined as a pulse with infinite height and an infinitesimal time width. As it is impossible to apply such a rigorous impulse stimulus, a short pulse stimulus or the like is substituted for this impulse. The sound response of the hammer test is a kind of impulse response, the dynamic characteristics of which are affected by the object. (2) Measuring method (a) Frequency analysis 5) An evaluation of the quality of a melon is often performed using a hammer test. It is said that if the sound response to such a test is a high tone the melon is immature, and if it is a low tone the melon is mature. Using this characteristic, many so called frequency analysis tests have been established with equipment that measure differences in tone. The principle of frequency analysis is shown in Fig. 1.6. Time- domain Frequency- domain Fig.1.6 Principle of frequency analysis The sound response to a hammer test is captured with a microphone over time, and is observed as a time-domain signal. On the other hand, it is known that an arbitrary waveform (here, a sound elicited by a hammer test) can be duplicated by synthesizing sine waves of different frequencies. In addition, the sound response of the hammer test can be analyzed as sine waves of various frequencies. When the intensity (power spectrum) of each sine wave in the frequency -domain is observed, this is called frequency analysis. The human ear, like a microphone, picks up a time-domain signal, and the signal is automatically changed by the brain into the frequency-domain. This sound is interpreted as a frequency component, with height as tone. 8