JPH0254491B2 - - Google Patents

Description

[Detailed Description of the Invention] The present invention is a viscoelasticity measurement method suitable for measuring the complex modulus of elasticity in an audio frequency range using a forced vibration method, in particular a method in which vibration displacement is applied simultaneously from both ends of a sample. Related to viscoelasticity measurement method.

In conventional viscoelasticity measurements, measurements are mainly performed using the forced vibration method in the low frequency band of several hundred Hz or less, and by the ultrasonic absorption method in the high frequency band of several KHz or more. At present, no highly accurate measurement method has been developed for the intermediate frequency band, that is, the audible frequency band from several hundred Hz to several KHz, which is considered to be most useful in practice.

Therefore, to describe the conventional viscoelasticity measurement method (for low frequency bands) that uses the same forced vibration method as the present invention, as shown in Fig. Holding the chucks 4 and 5, a vibrator 6 is connected to one end of one of the connecting rods 2 to forcibly apply vibration strain, and the displacement detector 7 detects the elongation Δl of the sample 1 due to this strain. At the same time, the force F is detected by a force detector 8 connected to one end of the other connecting rod 3, and the complex modulus of elasticity E ^* is calculated. Here, 9 represents the equivalent mass of the force detector 8, and 10 (shaded) represents fixed.

Regarding the calculation method at this time, sample 1
If the length of is l and its cross-sectional area is A, then the strain γ is γ
=Δl/l, and the stress σ is σ=F/A, so the complex modulus of elasticity E ^* calculated from the following equation (1) (including phase) is used.

E ^* = σ / γ = Fl / AΔl ...(1) As you can see, the strain γ in this equation (1)
is a value using elongation Δl at the left end of sample 1, and stress σ is a value using detection force F at the right end, so strain γ is uniform everywhere in the sample, and everything is static. Equation (1) only holds true when sample 1 is in a balanced state, in other words when it is pulled (or compressed) with a constant force, but it also holds true in viscoelastic measurements that apply vibrational strain. Since the measurement error is relatively small in a band where the excitation frequency is low, formula (1) has conventionally been used as an approximation formula.

Examining this in a little more detail, we find that, as is natural in viscoelasticity measurements, since vibrational strain is applied by the vibrator 6 as shown in Figure 1, the force propagates within the sample 1 as a wave. Become. As is well known, this wave propagation is an elastic wave that propagates through a solid.
It follows the wave equation:

ρ∂ ² ξ／∂t ² =E∂ ² ξ／∂x ² +η∂ ³ ξ／∂t∂x ² ...
...(2) Here, ξ is the displacement, x is the distance from the left end measured in the length direction of the sample 1, t is the time, ρ is the density, E is the elastic modulus, and η is the viscosity coefficient.

Therefore, stress σ and strain γ are functions of position x and time t, that is, σ(x, t) and γ(x, t), and are no longer constant, and the complex modulus of elasticity E ^* is the same. Since it must be calculated using the values of stress σ and strain γ at a location, for example, at a distance x, it must be calculated using the following equation (3).

E ^* = σ (x, t) / γ (x, t) ...(3) However, if we consider point x inside sample 1, we can measure the stress σ (x, t) at this point. This is practically impossible, and the complex modulus of elasticity E ^* cannot be calculated.

Furthermore, as shown in Fig. 1, if a forced vibration, ^e.g. Since it can be regarded as an elastic body with a constant k and a mass m, we can form a coupled vibration system with sample 1 and solve the wave equation (2) under these boundary conditions (details omitted). ),
The displacement function ξ(x, t) becomes very complex, and the stress σ and strain γ (=∂ξ/∂x) are extremely difficult to calculate mathematically, and as a result, the complex modulus of elasticity is
Calculating E ^* is difficult.

For this reason, conventionally, when the excitation angular frequency ω is small and the frequency is less than several hundred Hz, equation (1) is inevitably used as an approximation equation, and at best the force detector 8
The resonance angular frequency ω _r of the force detector 8, i.e., ω _r =√
Measurements were carried out by designing k/m so that ω _r ≫ω with respect to the excitation angular frequency ω.

However, as the excitation angular frequency ω of the vibrator 6 becomes relatively high and approaches the audible frequency band mentioned above, the strain σ becomes uniform depending on the location, apart from the influence of the coupled vibration caused by the force detector 8. The problem is that using equation (1) increases the measurement error and makes it impossible to approximate the complex viscoelastic modulus E ^* using equation (1).

Therefore, the present invention analyzes the stress and strain occurring in the sample as waves, and as a result, the two ends of the sample are
The mechanical impedance Z (θ ₁ ) seen from one end in the first vibration state is measured when vibrational displacement is applied simultaneously with a phase difference θ ₁ using two vibrators, and this is used to calculate the complex elasticity. The present invention provides _a new measurement method for measuring the ratio E ^* , and more specifically, the at least Either measure the mechanical impedance Z (θ _o ) in one different vibration state and combine this with the mechanical impedance Z (θ ₁ ), or activate the exciter at one end and fix the other end of the sample. Provides a new measurement method for determining the complex modulus of elasticity E ^* by measuring the mechanical impedance Z _fix in yet another vibration state and combining it with the mechanical impedance Z (θ ₁ ). It is something to do.

The present invention will be explained below with reference to the drawings, but first the principle of the present invention will be explained.

Now, as shown in FIG.
(See Fig. 3) are connected and vibrated from both ends at the same time with the same amplitude of phase difference θ. In sample 11, since the above-mentioned wave equation (2) holds in this case as well, the vibrators 16A and 16B If the excitation caused by is given as vibration displacement e ^iwt , e ^i(wt- 〓 ⁾ of unit size, the boundary conditions at this time are x = 0, ξ = e ^iwt (left end) ...(4) x =l, ξ=e ^i(wt- 〓 ⁾ (right end) ...(5) Therefore, solving equation (2) with this, ξ(x, t) = {cosω/C ^* x+N(θ) sinω／C ^* x
} xe ^iwt ...(6) is obtained. However, here, N(θ)=e ^-i 〓−cos(ωl/C ^* )/sin(ωl/C ^* )
...(7) E ^* =E+iωη...(9).

However, from these equations we can directly calculate the strain γ(x,
t) (=∂ξ/∂x) and stress σ(x, t) to calculate the complex modulus of elasticity E ^* is too complicated and practically difficult.

Therefore, when calculating the mechanical impedance Z(θ) seen from the left end of sample 11, this Z(θ) is generally Z(θ)=F/[∂ξ/∂t] _x=0 ... (10), so the power F in this case is
Formula (3) is modified and rearranged, F=-AE ^* [∂ξ／∂x] _x=0 ...(11) to find Z(θ), Z(θ)=-A√ρE ^* /iN(θ) ...(12)

From the above formula, the mechanical impedance Z(θ)
Even if the value of can be measured directly, for example with an impedance head, there is C ^* in N(θ) and C ^*
Since there is E ^* in , the complex modulus of elasticity E* cannot be easily calculated.

Therefore, according to the method discovered by the present invention, the phase difference θ of the displacement applied to the sample 11 by both the vibrators 16A and 16B is 0, π/2, π,
The mechanical impedance Z (θ) seen from the left end of sample 11 when 3π/2, ..., is Z
(0), Z (π/2), Z (π), Z (3π/2), Z
(…
), and the mechanical impedance when the left vibrator 16A is activated and the displacement of the right end of the sample 11 is fixed to zero is Z _FIX ,
To find these values, first enter each value into Z(θ) in equation (12) and find Z(θ). Z(0)=-A√ρE ^* /itan(ωl /2C ^* ) ...(13) Z [π/2] = A√ρE ^* /i×i+cos(ωl/C ^* )/
sin(ωl／C ^* ) ...(14) Z(π)=A√ρE ^* /icot(ωl/2C ^* ) ...(15) Z[3π/2]=−A√ρE ^* /ii−cos (ωl／C ^* )
^{/sin(ωl/C*)...(16} ) is obtained. Next, regarding Z _FIX , I will only introduce the results as it will be complicated, but if the wave equation (2) is set to excite the left end of sample 11 (e ^iwt ) and fix the right end, then
The boundary conditions at this time are x = 0, ξ = e ^iwt (left end) ... (17) x = l, ξ = 0 (right end) ... (18), so solve equation (2), Mechanical impedance seen from the left end of sample 11 in the same way as above
When we find Z _FIX , we get Z _FIX = A√ρE ^* / icot (ωl / C ^* ) ... (19), and we discovered that the following algebraic relationship holds between them. That is, E ^* = Z (0) Z (π) / ρA ² ... (20) E ^* = {2Z _FIX - Z (π)} Z (π) / ρA ² ... (21) E ^* = {2Z _FIX −Z(0))}Z(0)/ρA ² ...(22) E ^* = {Z(π/2)+Z(3π/2)−Z(π)}Z(π)/ρA ² ... …(23), and the formula is simplified.

As a result, if the mechanical impedance Z (θ) or Z _FIX seen from the left end of sample 11 is measured, formula (20)
By using any of ~(23), the interior point x
The complex modulus of elasticity E ^* can be easily calculated without measuring the stress σ(x, t) and strain γ(x, t) at .

Next, an example using this measurement method will be described with reference to FIG. Completed vibrator 16A,
16B are connected.

Here, 12 and 13 are connecting rods similar to those in FIG. 1, 14 and 15 are chucks, 20 (hatched) is fixed, 21 is a differential circuit, and 22 is an arithmetic circuit. Further, although the displacement detectors 17A and 17B are illustrated in a directly connected state for convenience, they may be constructed in a non-contact manner using optical ones.

In such a configuration, to start a measurement,
The vibration displacement e ^iwt of the vibrators 16A and 16B is
e ^i(wt- 〓 ⁾ , but when calculating the complex modulus of elasticity E ^* using equation (20), Z
Since it is sufficient to find (0) and Z(π), first, the phase difference θ is set to θ=0, and the mechanical impedance Z(0) as seen from the left end of the sample 11 is determined.

To find this, the force F at the left end at this time is
is measured by the force detector 18, and at the same time, the displacement ξ at the left end is also measured by the displacement detector 17A. Therefore, this displacement ξ is partially differentiated with respect to time t by the differentiating circuit 21 to obtain (∂ξ/∂ t) By determining _x=0 , inputting this output together with the output of the force detector 18 into the arithmetic circuit 22, and calculating according to equation (10), the mechanical impedance Z(0) can be determined.

Similarly, the mechanical impedance Z(π) when the phase difference θ is θ=π and vibration displacement is applied is determined, and this is input into the arithmetic circuit 22 to calculate the previously determined Z(0).
By calculating with Equation (20) together with, the complex modulus of elasticity E ^*
can be calculated.

In addition, as can be seen from equations (20) to (23), the complex modulus of elasticity can be determined only by the mechanical impedance seen from the left end.
Since E ^* can be calculated, there is no need for a displacement detector or force detector at the right end of sample 11, but
7B is arranged for use as a monitor for controlling the phase of the vibrator 16B. In addition, in the above example, the displacement detector 17A and the force detector 18
Of course, it is also possible to directly measure the mechanical impedance by connecting the left end of the sample 11 directly to the vibrator 16A via an impedance head.

In addition, when calculating the complex modulus of elasticity E ^* using equation (21), it is sufficient to calculate Z (π) and Z _FIX , so as above, this time the vibration is applied with the phase difference θ ₁ = π. vessel 16
A and 16B are simultaneously excited to find the mechanical impedance Z (π), and then the right end of the sample 11 is fixed, or the displacement of the right end of the sample 11 is zero [ξ (l, t) by the vibrator 16B. )=0] and calculate the mechanical impedance Z _FIX seen from the left end when the vibrator 16A is activated, it is possible to calculate the complex modulus of elasticity E ^* using equation (21).

In this way, in the present invention, since the complex modulus of elasticity is determined using the results of analyzing the vibrational displacement of the sample as a wave, it is possible to perform more accurate viscoelasticity measurements even in the above-mentioned audio frequency band. It has the advantage of

[Brief explanation of drawings]

Fig. 1 is a schematic explanatory diagram showing a conventional example, Fig. 2 is an explanatory diagram for explaining the principle of the present invention, and Fig. 3 is a schematic explanatory diagram showing an embodiment of the present invention using the same principle. . 1, 11... Sample, 2, 3, 12, 13... Connecting rod, 4, 5, 14, 15... Chack, 6, 1
6A, 16B... vibrator, 7, 17A, 17B...
...Displacement detector, 8, 18A, 18B... Force detector, 9... Equivalent mass, 21... Differential circuit, 22...
...Arithmetic circuit, ξ...Displacement, ω...Excitation angular frequency,
θ...Phase angle, x...Distance from the left edge of the sample, l...
...sample length, t...time.

Claims (1)

[Claims] 1 Both ends of the sample 11 are connected to a pair of vibrators 16A, 1
6B is used to simultaneously excite the sample 11 with a phase difference θ ₁ of the displacement applied to the sample 11 and apply vibrational displacement.
The first mechanical impedance Z(θ ₁ ) and _at least one other mechanical impedance measured separately are used to determine the complex modulus of elasticity. Viscoelasticity measurement method. 2. One side of the sample 11 when the other mechanical impedance is simultaneously excited using the vibrators 16A and 16B with a phase difference θ _o different from the phase difference θ ₁ , and a vibrational displacement caused by this is applied. Claim 1 which is the mechanical impedance Z (θ _o ) seen from the end
Viscoelasticity measurement method described in section. 3 Mechanical impedance Z FIX as seen from the one end in another vibration state when the other mechanical impedance operates the vibrator 16A at one end of the sample 11 and fixes the other end of the sample ₁₁ .
A method for measuring viscoelasticity according to claim 1.