CN105675484B - The determination method of the annular membrane elasticity energy of center band rigid plate under uniform load - Google Patents

The determination method of the annular membrane elasticity energy of center band rigid plate under uniform load Download PDF

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CN105675484B
CN105675484B CN201610266626.6A CN201610266626A CN105675484B CN 105675484 B CN105675484 B CN 105675484B CN 201610266626 A CN201610266626 A CN 201610266626A CN 105675484 B CN105675484 B CN 105675484B
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msup
mrow
beta
msub
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CN105675484A (en
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何晓婷
练永盛
杨志欣
郭莹
孙俊贻
郑周练
蔡珍红
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Jiashan Guohong Environmental Protection Technology Co Ltd
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Chongqing University
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01NINVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N19/00Investigating materials by mechanical methods

Abstract

The invention discloses the determination method of the annular membrane elasticity energy of center band rigid plate under uniform load:Use clamping device of the inside radius for a, it is h by thickness, the annular membrane for the rigid plate that Young's modulus of elasticity E, Poisson's ratio ν, center band radius are b fixes to clamp, one outer radius of formation is a, the axial symmetry annular membrane for the center band rigid plate that the periphery that inside radius is b fixes to clamp, a uniform load q is laterally applied to it, standing balance analysis and Energy Balance Analysis based on this On Axisymmetric Deformation of A, and using uniform load q measured value, then it can determine that the elasticity of the annular membrane can U.

Description

The determination method of the annular membrane elasticity energy of center band rigid plate under uniform load
Technical field
The present invention relates to the determination method of the annular membrane elasticity energy of center band rigid plate under uniform load.
Background technology
The analytic solutions of the annular membrane On Axisymmetric Deformation of A of center band rigid plate under Uniform Loads, to sensor with And the development of instrument, instrument is significant.Because film is flexible material, typically exhibited out under Uniform Loads compared with Big amount of deflection, thus its problem on deformation has stronger non-linear, these nonlinear problems are generally difficult to Analytical Solution.The present invention It is directed to the analysis research of the annular membrane On Axisymmetric Deformation of A of center band rigid plate under uniform load, obtains the problem Analytic solutions, and the determination method of the annular membrane elasticity energy of center band rigid plate under uniform load is given on this basis.
The content of the invention
The determination method of the annular membrane elasticity energy of center band rigid plate under uniform load:Use clamping of the inside radius for a Thickness is h by device, the annular membrane for the rigid plate that Young's modulus of elasticity E, Poisson's ratio ν, center band radius are b is fixed Clamping, one outer radius of formation is a, the axial symmetry annular membrane for the center band rigid plate that the periphery that inside radius is b fixes to clamp, A uniform load q is laterally applied to it, the standing balance analysis based on this On Axisymmetric Deformation of A, it is possible to obtain this and ask The analytic solutions of topic, because caused heat consumption is very small during deformation of thin membrane, therefore uniform load can be approximately considered The elasticity that lotus q work done is completely converted into film can, then according to principle of energy balance, is obtained using standing balance analysis The analytic solutions obtained, then the parsing relation that can obtain the elasticity energy U and uniform load q of the annular membrane are
Wherein,
And β=(a+b)/2a, c1And c2Value by equation
With
It is determined that c0Value by equation
It is determined that.
So, as long as accurately measuring applied uniform load q value, it is possible to after determining annular membrane deformation Elasticity can U.Wherein, all parameters all use the International System of Units.
Brief description of the drawings
Fig. 1 is the loading organigram of the annular membrane for the center band rigid plate that uniform load following peripheral fixes to clamp, Wherein, 1- annular membranes, 2- rigid plates, 3- clamping devices, and a represent clamping device inside radius and annular membrane it is outer Radius, b represent the radius of rigid plate and the inside radius of annular membrane, and r represents radial coordinate, and w (r) represents the horizontal seat at point r Mark, q represent horizontal uniform load, wmRepresent the maximum defluxion of film.
Embodiment
Technical scheme is described in further detail below in conjunction with the accompanying drawings:
As shown in figure 1, with inside radius a=20mm clamping device, by thickness h=0.06mm, Young's modulus of lasticity E =7.84MPa, Poisson's ratio ν=0.47, the center band radius b=5mm rubber film of rigid plate fix to clamp, formed one it is outer The axial symmetry annular membrane for the center band rigid plate that radius a=20mm, inside radius b=5mm periphery fix to clamp, to its transverse direction Apply uniform load q, and measure q=0.01MPa.Using the method given by the present invention, pass through equation
With
C can then be obtained1=-0.7026607165, c2=-0.6927450924, wherein,
β=(a+b)/2a=0.625,
Then, then by equationC can be obtained0=0.3710002899.Finally, by equation
The elasticity that the annular membrane can then be obtained can U=50.2088 × 10-3J。

Claims (1)

1. the determination method of the annular membrane elasticity energy of center band rigid plate under uniform load, it is characterised in that:Using inside radius By thickness it is h, the annular for the rigid plate that Young's modulus of elasticity E, Poisson's ratio ν, center band radius are b for a clamping device Film fixes to clamp, and one outer radius of formation is a, the axial symmetry for the center band rigid plate that the periphery that inside radius is b fixes to clamp Annular membrane, a uniform load q is laterally applied to it, and measures applied uniform load q value, determined by below equation The elasticity of the annular membrane can U:
<mrow> <mi>U</mi> <mo>=</mo> <mi>&amp;pi;</mi> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mi>a</mi> <mn>4</mn> </msup> <msup> <mi>q</mi> <mn>4</mn> </msup> </mrow> <mrow> <mi>h</mi> <mi>E</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mn>10</mn> </munderover> <mrow> <mo>&amp;lsqb;</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> <msub> <mi>c</mi> <mi>n</mi> </msub> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>a</mi> <mo>-</mo> <mi>b</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>c</mi> <mi>n</mi> </msub> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>a</mi> <mo>-</mo> <mi>b</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <msub> <mi>c</mi> <mi>n</mi> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>b</mi> <mo>-</mo> <mi>a</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>n</mi> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mo>,</mo> </mrow>
Wherein,
<mrow> <msub> <mi>c</mi> <mn>3</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> <mfrac> <mn>1</mn> <mrow> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mrow> <mo>(</mo> <mn>8</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>10</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>3</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msub> <mi>c</mi> <mn>4</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> <mfrac> <mn>1</mn> <mrow> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mrow> <mo>(</mo> <mn>48</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>-</mo> <mn>120</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>90</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>16</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>21</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>8</mn> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c</mi> <mn>5</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>120</mn> </mfrac> <mfrac> <mn>1</mn> <mrow> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>3</mn> </msubsup> </mrow> </mfrac> <mo>(</mo> <mn>384</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>-</mo> <mn>1440</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>1896</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>232</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>1032</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>3</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>312</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>198</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>4</mn> </msubsup> <mo>-</mo> <mn>101</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <mo>+</mo> <mn>8</mn> <msubsup> <mi>c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c</mi> <mn>6</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>720</mn> </mfrac> <mfrac> <mn>1</mn> <mrow> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>4</mn> </msubsup> </mrow> </mfrac> <mo>(</mo> <mn>3840</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> <mo>-</mo> <mn>19200</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>36720</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>-</mo> <mn>3552</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>33360</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>3</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>8296</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>14340</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>4</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>6208</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>2340</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <mo>+</mo> <mn>344</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>1489</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>8</mn> </msubsup> <mo>-</mo> <mn>204</mn> <msubsup> <mi>c</mi> <mn>1</mn> <mn>11</mn> </msubsup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c</mi> <mn>7</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2520</mn> </mfrac> <mfrac> <mn>1</mn> <mrow> <msup> <mi>&amp;beta;</mi> <mn>9</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> </mrow> </mfrac> <mo>(</mo> <mn>23040</mn> <msup> <mi>&amp;beta;</mi> <mn>9</mn> </msup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>6</mn> </msubsup> <mo>-</mo> <mn>144000</mn> <msup> <mi>&amp;beta;</mi> <mn>8</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> <mo>+</mo> <mn>362880</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>-</mo> <mn>39664</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>469800</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>3</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>100944</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>328140</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>4</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>124644</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>116910</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>5416</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>66120</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>8</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>16605</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>7468</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>11</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>12708</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>9</mn> </msubsup> <mo>+</mo> <mn>2524</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>12</mn> </msubsup> <mo>-</mo> <mn>86</mn> <msubsup> <mi>c</mi> <mn>1</mn> <mn>15</mn> </msubsup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c</mi> <mn>8</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>20160</mn> </mfrac> <mfrac> <mn>1</mn> <mrow> <msup> <mi>&amp;beta;</mi> <mn>10</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> </mrow> </mfrac> <mo>(</mo> <mn>322560</mn> <msup> <mi>&amp;beta;</mi> <mn>10</mn> </msup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>7</mn> </msubsup> <mo>-</mo> <mn>2419200</mn> <msup> <mi>&amp;beta;</mi> <mn>9</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> <msubsup> <mi>c</mi> <mn>2</mn> <mn>6</mn> </msubsup> <mo>+</mo> <mn>7580160</mn> <msup> <mi>&amp;beta;</mi> <mn>8</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> <mo>-</mo> <mn>543744</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>12821760</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>3</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>2455488</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>12602520</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>4</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>-</mo> <mn>4317408</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>7175700</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>156816</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>3689808</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>8</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>2186730</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>357560</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>11</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>1531056</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>9</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>274995</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <mo>+</mo> <mn>265844</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>12</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>246591</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <mo>-</mo> <mn>7136</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>15</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>64454</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>13</mn> </msubsup> <mo>+</mo> <mn>4508</mn> <msubsup> <mi>c</mi> <mn>1</mn> <mn>16</mn> </msubsup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c</mi> <mn>9</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>181440</mn> </mfrac> <mfrac> <mn>1</mn> <mrow> <msup> <mi>&amp;beta;</mi> <mn>12</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> </mrow> </mfrac> <mo>(</mo> <mn>5160960</mn> <msup> <mi>&amp;beta;</mi> <mn>12</mn> </msup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>8</mn> </msubsup> <mo>-</mo> <mn>45158400</mn> <msup> <mi>&amp;beta;</mi> <mn>11</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> <msubsup> <mi>c</mi> <mn>2</mn> <mn>7</mn> </msubsup> <mo>+</mo> <mn>169344000</mn> <msup> <mi>&amp;beta;</mi> <mn>10</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>6</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>10916352</mn> <msup> <mi>&amp;beta;</mi> <mn>9</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>6</mn> </msubsup> <mo>-</mo> <mn>354654720</mn> <msup> <mi>&amp;beta;</mi> <mn>9</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>3</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> <mo>+</mo> <mn>61770240</mn> <msup> <mi>&amp;beta;</mi> <mn>8</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> <mo>+</mo> <mn>452571840</mn> <msup> <mi>&amp;beta;</mi> <mn>8</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>4</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>142403904</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>-</mo> <mn>359432640</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>4495680</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>171002304</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>8</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>173093760</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>14640192</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>11</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>-</mo> <mn>112679604</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>9</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>46131120</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>17513808</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>12</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>38590524</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>5205060</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>8</mn> </msubsup> <mo>-</mo> <mn>392216</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>15</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>9120816</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>13</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>5362677</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>11</mn> </msubsup> <mo>+</mo> <mn>547624</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>16</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>1744380</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>14</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>188706</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>17</mn> </msubsup> <mo>+</mo> <mn>3568</mn> <msubsup> <mi>c</mi> <mn>1</mn> <mn>20</mn> </msubsup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c</mi> <mn>10</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>181440</mn> </mfrac> <mfrac> <mn>1</mn> <mrow> <msup> <mi>&amp;beta;</mi> <mn>13</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>8</mn> </msubsup> </mrow> </mfrac> <mo>(</mo> <mn>92897280</mn> <msup> <mi>&amp;beta;</mi> <mn>13</mn> </msup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>9</mn> </msubsup> <mo>-</mo> <mn>92897280</mn> <msup> <mi>&amp;beta;</mi> <mn>12</mn> </msup> <msub> <mi>c</mi> <mn>1</mn> </msub> <msubsup> <mi>c</mi> <mn>2</mn> <mn>8</mn> </msubsup> <mo>+</mo> <mn>4058449920</mn> <msup> <mi>&amp;beta;</mi> <mn>11</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>7</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>238970880</mn> <msup> <mi>&amp;beta;</mi> <mn>10</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>7</mn> </msubsup> <mo>-</mo> <mn>10149027840</mn> <msup> <mi>&amp;beta;</mi> <mn>10</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>3</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>6</mn> </msubsup> <mo>+</mo> <mn>1630158336</mn> <msup> <mi>&amp;beta;</mi> <mn>9</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>6</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>15980146560</mn> <msup> <mi>&amp;beta;</mi> <mn>9</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>4</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> <mo>-</mo> <mn>4676078592</mn> <msup> <mi>&amp;beta;</mi> <mn>8</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> <mo>-</mo> <mn>16397095680</mn> <msup> <mi>&amp;beta;</mi> <mn>8</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>5</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>131654016</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>5</mn> </msubsup> <mo>+</mo> <mn>7304043456</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>8</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>10942646400</mn> <msup> <mi>&amp;beta;</mi> <mn>7</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>6</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>564117120</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>11</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>4</mn> </msubsup> <mo>-</mo> <mn>6702770736</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>9</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>-</mo> <mn>4571471520</mn> <msup> <mi>&amp;beta;</mi> <mn>6</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>7</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>949114656</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>12</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>3610117836</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>10</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>1083264840</mn> <msup> <mi>&amp;beta;</mi> <mn>5</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>8</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>18404544</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>15</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>3</mn> </msubsup> <mo>-</mo> <mn>783565920</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>13</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>1055710800</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>11</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>110837160</mn> <msup> <mi>&amp;beta;</mi> <mn>4</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>9</mn> </msubsup> <mo>+</mo> <mn>41460120</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>16</mn> </msubsup> <msubsup> <mi>c</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>317333772</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>14</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>129215439</mn> <msup> <mi>&amp;beta;</mi> <mn>3</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>12</mn> </msubsup> <mo>-</mo> <mn>30678936</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>17</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>50420286</mn> <msup> <mi>&amp;beta;</mi> <mn>2</mn> </msup> <msubsup> <mi>c</mi> <mn>1</mn> <mn>15</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mn>484984</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>20</mn> </msubsup> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>7457610</mn> <msubsup> <mi>&amp;beta;c</mi> <mn>1</mn> <mn>18</mn> </msubsup> <mo>-</mo> <mn>316628</mn> <msubsup> <mi>c</mi> <mn>1</mn> <mn>21</mn> </msubsup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
And β=(a+b)/2a, c1And c2Value by equation
<mrow> <mi>&amp;nu;</mi> <mo>=</mo> <mn>2</mn> <mo>-</mo> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> <mrow> <mo>&amp;lsqb;</mo> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mn>10</mn> </munderover> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>c</mi> <mi>i</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> <mo>-</mo> <mi>&amp;beta;</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mo>/</mo> <mrow> <mo>&amp;lsqb;</mo> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>10</mn> </munderover> <msub> <mi>jc</mi> <mi>j</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> <mo>-</mo> <mi>&amp;beta;</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mrow>
With
<mrow> <mi>&amp;nu;</mi> <mo>=</mo> <mn>2</mn> <mo>-</mo> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> <mrow> <mo>&amp;lsqb;</mo> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mn>10</mn> </munderover> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>c</mi> <mi>i</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mi>&amp;beta;</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mo>/</mo> <mrow> <mo>&amp;lsqb;</mo> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>10</mn> </munderover> <msub> <mi>jc</mi> <mi>j</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mi>&amp;beta;</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mrow>
It is determined that c0Value by equation
<mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>10</mn> </munderover> <msub> <mi>c</mi> <mi>j</mi> </msub> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> <mi>j</mi> </msup> </mrow>
It is determined that all parameters all use the International System of Units.
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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101923043A (en) * 2010-08-04 2010-12-22 重庆大学 Accurate measurement method for interface energy release rate of coating film-substrate structure
CN102365315A (en) * 2009-03-27 2012-02-29 旭硝子株式会社 Hard coating composite, and resin substrate having a hard coat layer
CN104833576A (en) * 2015-05-22 2015-08-12 哈尔滨工业大学 Testing device and method for determining bending breaking strength of aggregate-asphalt mortar interface under pulling-shearing mixed modal

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR101158107B1 (en) * 2010-02-25 2012-06-22 한국기계연구원 Method for measuring elastic modulus of thin film sample with dual section, method measuring for thermal expansion modulus of thin film sample with dual section, thin film sample for measuring elastic modulus and hermal expansion modulus and apparatus for measuring elastic modulus and thermal expansion modulus

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102365315A (en) * 2009-03-27 2012-02-29 旭硝子株式会社 Hard coating composite, and resin substrate having a hard coat layer
CN101923043A (en) * 2010-08-04 2010-12-22 重庆大学 Accurate measurement method for interface energy release rate of coating film-substrate structure
CN104833576A (en) * 2015-05-22 2015-08-12 哈尔滨工业大学 Testing device and method for determining bending breaking strength of aggregate-asphalt mortar interface under pulling-shearing mixed modal

Non-Patent Citations (3)

* Cited by examiner, † Cited by third party
Title
不同模量弹性力学问题研究进展;何晓婷;《重庆建筑大学学报》;20051231;第136-141页 *
均布荷载下受有预加张力圆薄膜的轴对称变形;何晓婷;《重庆大学学报》;20100131;第109-112页 *
拉压不同模量弹性结构分析中的对称性问题;何晓婷;《重庆大学学报》;20080731;第735-739页 *

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