CN100580444C - Method for measuring acoustics parameter of viscous-elastic material under medium-high frequency - Google Patents

Abstract

This invention relates to a interim/high frequency viscoelastic material acoustic parameter measurement method, and the major content of the method is using Legendre function to expand the scattering field acoustic dies function of the viscoelastic ball, and compare the Legendre expansion coefficients of the experiment and the Legendre coefficient expression derived from the theoretical formulas, establishing the simultaneous equations, and using optimization algorithms for solving the solution of equations, which is the acoustic parameters of measured material. The method avoids the sound pressure phase measurement, and only need to know the scattering field sound pressure amplitude. The method can simultaneously get material latitudinal/longitudinal wave parameters, no need separately measuring each parameter. It gives more accurately error correction model, to improve the accuracy of parameters measurement, particularly applicable to interim/high frequency viscoelastic material acoustic parameter measurement that difficult to achieve in conventional methods, and with a broad frequencies range.

Description

Viscoelastic material parameters,acoustic measuring method under the medium-high frequency

Technical field

The invention belongs to viscoelastic material parameters,acoustic measuring method under a kind of medium-high frequency in underwater acoustic materials field

Background technology

The method of measuring the viscoelastic material parameters,acoustic at present has multiple, from measuring the theoretical foundation of institute's foundation, can reduce two big classes substantially, one class is to calculate the vibration survey method of its mechanics parameter by the vibration characteristics of measuring material sample, and another kind of is the echo sounding that obtains its mechanics parameter or parameters,acoustic by the measurement to the material acoustics characteristic.

The walking beam method is to measure the classical way of viscoelastic material complex modulus, also is the most general measuring technique, and its principle is by measuring the resonance characteristic at given boundary condition underbeam, deriving the elastic modulus and the dissipation factor of measured material.This method is applicable to based on the flexural vibrations theory measures the low-frequency range (complex modulus in 10～2000Hz) wide temperature ranges.

The sonotubometry method is to utilize sound pipe equipment to carry out the method that small sample is measured, and commonly used have pulse tube method, standing-wave-tube method, two nautical receiving set transfer function method and an aluminum pipe method etc.The sonotubometry method is through people's research of decades, and relevant measurement standard also forms.But this measuring method has its limitation, on the one hand can only mensuration to sound absorption characteristics, be exactly for heterogeneous material on the other hand with certain inner structure, because the sample size of sonotubometry is limited, measurement result can not reflect the whole acoustically effective of sound absorption structure well, simultaneously, the selection of sample backing is also very crucial.For example, the operating frequency range of pulse sound pipe method measurement mechanism is by the vibration mode and the decision of exomonental width of physical dimension, shape and the transducer of sound pipe.For the rigid cylindrical waveguide, produce a radially natural vibration frequency of acoustic pressure node for having the propagation of single plane ripple, the excitation frequency of transducer not to be higher than in the assurance sound pipe.

Free field experiment measuring method also belongs to acoustic method, generally be in big pond, to carry out, such as can the inverting material parameter in big pond by the dull and stereotyped sample oblique incidence sonic reflection coefficient of the material of measuring and transmission coefficient, but during low frequency measurement because the interference of sample edge diffraction can make big must being difficult to of error accept.Jean C.Piquette utilizes sphere material to do the method that sample to be tested is measured complex modulus than proposition early in " the Determination ofthe complex dynamic bulk modulus of elastomers by inverse scattering " that delivered on volume the 5th phase J.Acoust.Soc.Am. in 1985 the 77th.This method has been used for reference Faran elastic ball scattered field computing method, make in theory the scattering coefficient error of the scattering coefficient that calculates and actual measurement reach minimum and calculate the material volume modulus, but the phase information that in derivation, needs scattered field, and the measuring technology of acoustic pressure amplitude is comparatively successful all the time, and the measurement of phase place comparatively speaking, realize that in wider frequency the technical difficulty of measuring is bigger, to in wider frequency range, realize the measurement of material complex modulus, test macro is had relatively high expectations.Subsequently, Jean C.Piquette has designed a kind of test macro of being measured by the insertion phase change that plate produced between transmitter and receiver of sound insulationg board under water, can obtain the vertical characteristics and the shear characteristic of material by the measurement result of oblique incidence.The key of this method is exactly the measurement of the phase change that causes inserting test board between sound source and the receiver, but phase change is very sensitive to the material behavior of plate, and this sensitivity has influenced the accuracy of this measuring technology to a great extent.In the measuring method of existing free field, all need to measure the phase change information of sound field.

Summary of the invention

The objective of the invention is to propose a kind of acoustic measurement method at the last measurement of high frequency (more than the 15kHz) viscoelastic material parameters,acoustic, the method has avoided the measurement of sound pressure phase only need know the sound pressure amplitude of scattered field.This method can provide the horizontal compressional wave parameter of material simultaneously, need not measure respectively.And provide more accurate error correction model, and improve the precision of parameter measurement, be particularly useful for the viscoelastic material parameter measurement on the difficult high frequency of realizing of conventional method.Main thought of the present invention is that the scattered field acoustic pressure modular function of viscoelasticity spheroid is carried out Legendre expansion, the Legendre coefficient expression formula that contain independent variable of the Legendre expansion coefficient that obtains in the experiment with the theoretical formula derivation compared, set up Simultaneous Equations, utilize optimized Algorithm to find the solution being suitable for solution of equations is the material acoustics parameter of being asked.Particular content of the present invention is as follows:

The first step is set up (as shown in Figure 1) viscoelasticity spheroid scattered field model, the computing formula of the scattered field acoustic pressure of derivation viscoelasticity ball target and the Legendre expansion coefficient expression formula of mould thereof.

As shown in Figure 1, point sound source S shines that a radius is ρ for a density in the infinite space in the hydrostatic ₁Spherical viscoelastic body target on, the infinite space Media density is ρ ₃The velocity of sound is c ₃, sound source S is R to centre of sphere O distance ₀, the distance that sound source S arrives acceptance point R is d, the distance that centre of sphere O arrives acceptance point R is r.Sound source S is along z axle emission spherical wave, shines target T and goes up the back and form a series of scattering waves, and the receiving trap at R place is that the center of circle is radius with r with target centre of sphere O, receives everywhere scattering wave and direct wave with the rotation of azimuth angle theta on the xoz plane.p _iRepresent the direct wave acoustic pressure, p _sRepresent the scattering wave acoustic pressure.

Suppose that the incident wave acoustic pressure is

$p_i=\frac{p_0}{d}{e}^{j(\mathrm{\ωt}-\mathrm{kd})}---\left(1\right)$

Wherein: p ₀Be constant, its value is from Sound Source Center S1m place incident acoustic wave sound pressure amplitude.

$d=\sqrt{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C2006101509910011H2.png"\; state="\{\{state\}\}">R</figure-callout>}}_0^2+r^2+2R_{0}r\cos (\mathrm{\&pi;}-\mathrm{\&theta;})}$

As 0＜r＜r ₀The time, incident wave is expressed as following progression form,

$p_i=-j{k}_3{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="C2006101509910011H2.png"\; state="\{\{state\}\}">p</figure-callout>}}_0\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}(2n+1){(-1)}^n{j}_n\left(k_{3}r\right)h_n^{\left(2\right)}\left(k_3{r}_0\right)P_n\left(\cos \mathrm{\&theta;}\right)---\left(2\right)$

Wherein, $k_3=\frac{\mathrm{\&omega;}}{c_3}$ Be the wave number of infinite space medium, ω is an angular velocity.j _n() and h _n ⁽²⁾() is spheric Bessel function, P _n() is Legendre function.

Can know p by symmetric condition _sIrrelevant with the position angle φ under the spherical coordinate system, suppose that scattered field satisfies far field condition, the situation of scattering pressure was separated and can be reduced to this moment

$p_s=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{C}_n{h}_n^{\left(2\right)}\left(k_{3}r\right)P_n\left(\cos \mathrm{\&theta;}\right)---\left(3\right)$

By the boundary condition of liquid and viscoelastic solid, acoustic pressure is continuous on the interface, the normal direction displacement is continuous, tangential stress is zero, can derive scattered field acoustic pressure coefficient C _n, C _nIn contain material parameter information ρ ₁, c ₁, c ₂, α ₁, α ₂, c wherein ₁-longitudinal wave velocity, c ₂-transverse wave velocity, α ₁-compressional wave attenuation coefficient, α ₂-shear wave attenuation coefficient, density of material can directly be measured.Order $\stackrel{\&RightArrow;}{X}=(c_1,c_2,{\mathrm{\&alpha;}}_1,{\mathrm{\&alpha;}}_2),$ Scattered field acoustic pressure modular function | p _s(r, θ) | Legendre coefficient can be written as:

$D_m\left(\stackrel{\&RightArrow;}{X}\right)=\frac{2m+1}{2}\underset{-1}{\overset{1}{\&Integral;}}\left|p_s(r,\mathrm{\&theta;})\right|P_m\left(\cos \mathrm{\&theta;}\right)d\left(\cos \mathrm{\&theta;}\right)$

$=\frac{2m+1}{2}\underset{-1}{\overset{1}{\&Integral;}}\left|\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{C}_n{h}_n^{\left(2\right)}\left(k_{3}r\right)P_n\left(\cos \mathrm{\&theta;}\right)\right|P_m\left(\cos \mathrm{\&theta;}\right)d\left(\cos \mathrm{\&theta;}\right)---\left(4\right)$

Here

Be the result of theoretical derivation, thus D _mIn also contain the material acoustics parameter information.(4) formula has provided the material parameter of target and the relational expression between the scattered field acoustic pressure.

In second step, tank experiments is measured scattered field acoustic pressure p ' _s, obtain the Legendre coefficient of its mould.

Consider that tank experiments measures the signal of gained and comprise many-sided information, can be divided into two parts to it, first is direct wave and sound-source signal reflection wave and reflection wave repeatedly on six walls, is designated as p ₁Second portion is that acoustical signal shines the scattering wave and the reflection of scattering wave on six walls that produce on the ball and involves repeatedly reflection wave, is designated as p ₂The experiment measuring signal is note p _z, can know by the superposition principle of sound wave,

p _z＝p ₁+p ₂

Thereby just have

p ₂＝p _z-p ₁

Given this, experiment is had in mind from the angle of being convenient to signal extraction, is guaranteeing to measure p respectively under the synchronous condition _zAnd p ₁, promptly measure p having under the target conditions _z, do not having to measure p under the target conditions ₁Concrete operations are as follows:

At first, the reference position of determine measuring is carried out mark, and the position during with θ among Fig. 1=0 ° is a reference position, measures the distance of OS, OR;

Then, the continuous signal of several δ pulse signals has been filled in emission, and is synchronous with the synchronizing signal of signal source; Reception has the signal p under the target conditions _z, with target mention treat that the water surface is stable after, receive the signal p under the driftlessness situation ₁With Δ θ=1 ° is that step-length rotation receiving device rests on next measurement point with nautical receiving set, and the measurement above repeating because ball-type target scattering field has symmetry, thereby can arrive the position of θ=180 °, finishes measurement.

At last, two groups of data p of gained will be measured _zAnd p ₁Directly subtract each other according to the method in the measurement scheme, obtain p ₂, again according to the due in of target scattering ripple from p ₂In extract the scattered field acoustic pressure p ' of the target of surveying _s

If the acoustic pressure of the target scattering field that obtains is p ' _s(r, θ), the Legendre expansion formula of its modular function is

$\left|p_s^{\&prime;}(r,\mathrm{\&theta;})\right|=\underset{m=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{D}_m^{\&prime;}{P}_m\left(\cos \mathrm{\&theta;}\right)---\left(5\right)$

Wherein the Legendre coefficient that obtains in the experiment is

$D_m^{\&prime;}=\frac{2m+1}{2}\underset{-1}{\overset{1}{\&Integral;}}\left|p_s^{\&prime;}(r,\mathrm{\&theta;})\right|P_m\left(\cos \mathrm{\&theta;}\right)d\left(\cos \mathrm{\&theta;}\right)---\left(6\right)$

In the 3rd step, set up the mathematical model of measuring method and find the solution the parameters,acoustic that obtains material.

The Legendre expansion coefficient D ' that obtains in the comparative experiments _mThe Legendre coefficient that contains independent variable with the theoretical formula derivation

Expression formula, should have

$|D_m\left(\stackrel{\&RightArrow;}{X}\right)-D_m^{\&prime;}|=0---\left(7\right)$

Like this at m=0,1,2,3,4,5 ... the time can list the system of equations that some equations are formed, the parameters,acoustic that its unknown variable is measured and monitored the growth of standing timber and expected for institute $\stackrel{\&RightArrow;}{X}=(c_1,c_2,{\mathrm{\&alpha;}}_1,{\mathrm{\&alpha;}}_2),$ The mathematical model of material acoustics parameter measurement promptly is the system of equations that is made of (7) formula.Because the system of equations of being separated is the Nonlinear System of Equations of complex forms, this method adopts the genetic algorithm in the optimized Algorithm to separate this system of equations.According to the number of asking unknown number, can determine the value of m.

Here cost function is defined as

$F=\underset{m=0}{\overset{7}{\mathrm{\&Sigma;}}}|D_m\left(\stackrel{\&RightArrow;}{X}\right)-D_m^{\&prime;}|---\left(8\right)$

When what search for Value is during more near the actual parameter value of object ball, D _mJust more near D ' _m, F also just levels off to 0 more.

The 4th step, the contrived experiment error calibrating method.

Range error at bad estimation in the experiment has proposed a kind of range error bearing calibration.Exist as shown in Figure 2 apart from the model after departing from.The vertical depth that hypothetical target ball position O ' departs from former surface level position O is Δ h ₁, the degree of depth that departs from of sound source S ' is Δ h ₂, sound source S ' is Δ R with the horizontal range error of target location O ' ₀, the lateral error of nautical receiving set R and target location O ' is Δ r, | OO ' |=Δ h ₁, | SS ' |=Δ h ₂, | OR|=r+ Δ r, | OS|=R ₀+ Δ R ₀, | RS|=d, | RS ' |=d ₁, | S ' O ' |=r ₀, | O ' R|=r ',

Then have:

$r^{\&prime;}=\sqrt{{\mathrm{\&Delta;h}}_1^2+{(r+\mathrm{\&Delta;r})}^2}---\left(9\right)$

${\mathrm{\&theta;}}^{\&prime;}=\arccos [-\frac{{\mathrm{\&Delta;h}}_1^2-{\mathrm{\&Delta;h}}_1{\mathrm{\&Delta;h}}_2-({\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C2006101509910011H2.png"\; state="\{\{state\}\}">R</figure-callout>}}_0+{\mathrm{\&Delta;R}}_0)(r+\mathrm{\&Delta;r})\cos \mathrm{\&theta;}}{r^{\&prime;}{r}_0}],\mathrm{\&theta;}\&Element;\left[\mathrm{0,180}\right]---\left(10\right)$

Fig. 1 is the ideal model situation, and the measurement situation under the physical presence range error as shown in Figure 2, thereby needs to revise the scattered field acoustic pressure promptly by original p _s(r θ) is modified to p _s(r ', θ '), in (9) formula and (10) formula substitution (3) formula, can get revised scattered field acoustic pressure p _sThe expression formula (11) of (r ', θ ').

$p_s(r^{\&prime;},{\mathrm{\&theta;}}^{\&prime;})=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{C}_n{h}_n^{\left(2\right)}\left(k_3{r}^{\&prime;}\right)P_n\left(\cos {\mathrm{\&theta;}}^{\&prime;}\right)---\left(11\right)$

The variable in (7) formula then

Become ${\stackrel{\&RightArrow;}{X}}^{\&prime;}=(c_1,c_2,{\mathrm{\&alpha;}}_1,{\mathrm{\&alpha;}}_2,{\mathrm{\&Delta;h}}_1,{\mathrm{\&Delta;h}}_2,\mathrm{\&Delta;r},{\mathrm{\&Delta;R}}_0),$ The material parameter that (11) formula of utilization obtains is worked as target exactly and is had Δ h ₁, Δ h ₂, Δ r, Δ R ₀The time measurement result, the result before its numeric ratio is done and revised more presses close to actual value.

The inventive method has overcome the restriction of centering high band in the existing viscoelastic material measurement method of parameters, is particularly useful for broadband measurement.Avoided being difficult to guarantee the sound pressure phase measurement of measuring accuracy.In addition, the measurement of a plurality of parameters be need not to carry out respectively, improved efficiency of measurement.

Description of drawings

The scattered field model that Fig. 1 is desirable

Scattered field model when there is range error in Fig. 2

φ 160mm. aluminium ball scattered field directivity pattern under Fig. 3 frequency f=28kHz

Embodiment

Because also do not have the horizontal compressional wave parameter of viscoelastic material referential data on the medium-high frequency section of rubber-like at present, this method has at first been done method emulation to the resilient material that can consult parameter.

Get the solid aluminum ball of object ball for certain model, radius a=0.080m, the spherical wave of signal source transmission frequency f=28kHz, because the loss of metallic aluminium is extremely little, the dissipation factor of its elastic modulus is 1.0 * 10 ^-3About, it is very little to the influence of scattering pressure, even can ignore, thereby supposes α here ₁=0, α ₂=0.Distance R ₀=1.77m, r=0.95m carries out the simulation calculation of scattered field, and the solid line of Fig. 3 is the simulation result of scattered field directive property for this reason, tries to achieve in Legendre coefficient and substitution (7) formula with this result, utilizes genetic algorithm to carry out parametric inversion.

Under the superincumbent simulated conditions, suppose range error Δ r=0.05m, Δ R ₀=0.1m, Δ h ₁=0.2m, Δ h ₂=0.15m, the dotted line of Fig. 3 be the simulation result of the scattered field directive property under the error for this reason, obtains its Legendre coefficient, and substitution in like manner (7) formula is carried out two kinds of Inversion Calculation before and after the error correction respectively.

Table 1 aluminium ball parameters,acoustic simulation result relatively

That table 1 has provided is error free in hypothesis, have before and after the error correction under the error the material parameter inversion result and with respect to reference value c ₁=6267.9 (m/s), c ₂The inversion accuracy of=3157.2 (m/s).The result is presented under the no range error, the precision of horizontal compressional wave parameter and Young modulus is less than 0.0032%, and when having range error when hypothesis, the precision of material parameter is less than 7.30% before revising, revise the back precision less than 0.64%, more approach reference value through the inversion result after the error correction.

Numerical result shows that the inventive method can realize the measurement of the parameters,acoustic of material on high frequency, in actual measurement, only need measure the acoustic pressure amplitude of scattered field, and the technological means that improves acoustic pressure amplitude measurement precision at present is comparatively ripe, thereby this method can guarantee higher viscoelastic material parameters,acoustic measuring accuracy.

Claims (1)

1. viscoelastic material parameters,acoustic measuring method under the medium-high frequency is characterized in that this method comprises following content:

The first step: set up viscoelasticity ball scattered field model, the computing formula of the scattered field acoustic pressure by derivation viscoelasticity ball obtains the Legendre expansion coefficient expression formula of its mould:

$D_m\left(\stackrel{\&RightArrow;}{X}\right)=\frac{2m+1}{2}\underset{-1}{\overset{1}{\&Integral;}}\left|p_s(r,\mathrm{\&theta;})\right|P_m\left(\cos \mathrm{\&theta;}\right)d\left(\cos \mathrm{\&theta;}\right)$

(4)

$=\frac{2m+1}{2}\underset{-1}{\overset{1}{\&Integral;}}\left|\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{C}_n{h}_n^{\left(2\right)}\left(k_{3}r\right)P_n\left(\cos \mathrm{\&theta;}\right)\right|P_m\left(\cos \mathrm{\&theta;}\right)d\left(\cos \mathrm{\&theta;}\right)$

Wherein, p _s(r θ) is the scattered field acoustic pressure of viscoelasticity ball, C _nBe scattered field acoustic pressure coefficient, C _nIn contain material parameter information ρ ₁, c ₁, c ₂, α ₁, α ₂, ρ ₁Be the density of measured material, c ₁Be longitudinal wave velocity, c ₂Be transverse wave velocity, α ₁Be compressional wave attenuation coefficient, α ₂Be the shear wave attenuation coefficient, $k_3=\frac{\mathrm{\&omega;}}{c_3}$ Be the wave number of infinite space medium, ω is an angular velocity, c ₃Be the velocity of sound of the residing infinite space medium of measured material, r is the distance that the centre of sphere of viscoelasticity ball arrives acceptance point, and θ is the position angle, h _n ⁽²⁾() is spheric Bessel function, P _n() and P _m() is respectively n rank and m rank Legendre function, $\stackrel{\&RightArrow;}{X}(c_1,c_2,{\mathrm{\&alpha;}}_1,{\mathrm{\&alpha;}}_2);$

Second step: measure scattered field acoustic pressure p ' by tank experiments _s(r θ), obtains the Legendre coefficient D ' of its mould _mExpression formula:

$D_m^{\&prime;}=\frac{2m+1}{2}\underset{-1}{\overset{1}{\&Integral;}}\left|p_s^{\&prime;}(r,\mathrm{\&theta;})\right|P_m\left(\cos \mathrm{\&theta;}\right)d\left(\cos \mathrm{\&theta;}\right)---\left(6\right)$

The 3rd step: set up the mathematical model of viscoelastic material parameters,acoustic measuring method under the medium-high frequency and find the solution the parameters,acoustic that obtains material:

The Legendre expansion coefficient D ' that obtains in the comparative experiments _mThe Legendre coefficient that contains independent variable with the theoretical formula derivation Expression formula, should have

$|D_m\left(\stackrel{\&RightArrow;}{X}\right)-D_m^{\&prime;}|=0---\left(7\right)$

Like this at m=0,1,2,3,4,5 ... the time can list the system of equations that some equations are formed, the parameters,acoustic that its unknown variable is measured and monitored the growth of standing timber and expected for institute $\stackrel{\&RightArrow;}{X}=(c_1,c_2,{\mathrm{\&alpha;}}_1,{\mathrm{\&alpha;}}_2),$ The mathematical model of material acoustics parameter measurement promptly is the system of equations that is made of (7) formula;

The 4th step: by setting up the experimental error calibrating patterns, derivation scattered field acoustic pressure correction formula is as follows to improve the parameter measurement precision:

Suppose that the vertical depth that viscoelasticity ball position O ' departs from the former surface level of viscoelasticity ball position O is Δ h ₁, the vertical depth that sound source position S ' departs from the former horizontal level S of sound source is Δ h ₂, sound source position S ' is Δ R with the horizontal range error of viscoelasticity ball position O ' ₀, the horizontal range error of hydrophone position R and viscoelasticity position O ' is Δ r, | OO ' |=Δ h ₁, | SS ' |=Δ h ₂, | OR|=r+ Δ r, | OS|=R ₀+ Δ R ₀, | RS|=d, | RS ' |=d ₁, | S ' O ' |=r ₀, | O ' R|=r ' then has:

$r^{\&prime;}=\sqrt{{\mathrm{\&Delta;h}}_1^2+{(r+\mathrm{\&Delta;r})}^2}---\left(9\right)$

${\mathrm{\&theta;}}^{\&prime;}=\arccos [-\frac{{\mathrm{\&Delta;h}}_1^2-{\mathrm{\&Delta;h}}_1{\mathrm{\&Delta;h}}_2-(R_0+{\mathrm{\&Delta;R}}_0)(r+\mathrm{\&Delta;r})\cos \mathrm{\&theta;}}{r^{\&prime;}{r}_0}],\mathrm{\&theta;}\&Element;\left[\mathrm{0,180}\right]---\left(10\right)$

θ ' is for revising the deflection of back scattered field acoustic pressure, R ₀Be the horizontal range of sound source position S ' with viscoelasticity ball position O ', with the scattered field acoustic pressure by original p _s(r θ) is modified to p _s(r ', θ '), in (9) formula and (10) formula substitution scattered field acoustic pressure formula, can get revised scattered field acoustic pressure p _sThe expression formula of (r ', θ '):

$p_s(r^{\&prime;},{\mathrm{\&theta;}}^{\&prime;})=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{C}_n{h}_n^{\left(2\right)}\left(k_3{r}^{\&prime;}\right)P_n\left(\cos {\mathrm{\&theta;}}^{\&prime;}\right)---\left(11\right)$

The variable in (7) formula then Become the inverting variable

${\stackrel{\&RightArrow;}{X}}^{\&prime;}=\left(c_1{c}_2{\mathrm{\&alpha;}}_1{\mathrm{\&alpha;}}_2{\mathrm{\&Delta;h}}_1{\mathrm{\&Delta;h}}_2\mathrm{\&Delta;r}{\mathrm{\&Delta;R}}_0\right),$ The material parameter that (11) formula of utilization finally obtains is worked as the viscoelasticity ball exactly and is had Δ h ₁, Δ h ₂, Δ r, Δ R ₀The time measurement result.

Images