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 PDFInfo
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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
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：
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<mn>7</mn>
</msubsup>
<msubsup>
<mi>c</mi>
<mn>2</mn>
<mn>5</mn>
</msubsup>
<mo>-</mo>
<mn>16397095680</mn>
<msup>
<mi>&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>&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>&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>&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>&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>&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>&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>&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>&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>&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>&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>&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>&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>&beta;</mi>
<mn>4</mn>
</msup>
<msubsup>
<mi>c</mi>
<mn>1</mn>
<mn>9</mn>
</msubsup>
<mo>+</mo>
<mn>41460120</mn>
<msup>
<mi>&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>&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>&beta;</mi>
<mn>3</mn>
</msup>
<msubsup>
<mi>c</mi>
<mn>1</mn>
<mn>12</mn>
</msubsup>
<mo>-</mo>
<mn>30678936</mn>
<msup>
<mi>&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>&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>&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>&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>&nu;</mi>
<mo>=</mo>
<mn>2</mn>
<mo>-</mo>
<mfrac>
<mi>b</mi>
<mi>a</mi>
</mfrac>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<munderover>
<mi>&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>&beta;</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mo>/</mo>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<munderover>
<mi>&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>&beta;</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mrow>
With
<mrow>
<mi>&nu;</mi>
<mo>=</mo>
<mn>2</mn>
<mo>-</mo>
<mfrac>
<mi>b</mi>
<mi>a</mi>
</mfrac>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<munderover>
<mi>&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>&beta;</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mo>/</mo>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<munderover>
<mi>&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>&beta;</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>&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>&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>&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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| CN106706169B (en) * | 2017-01-16 | 2019-04-05 | 重庆大学 | The determination method of annular membrane maximum stress under combined load with hard core |
| CN109323923B (en) * | 2018-12-20 | 2021-01-12 | 重庆大学 | Method for determining elastic performance of circular film under limitation of elasticity on maximum deflection |
| CN109323924B (en) * | 2018-12-20 | 2021-01-12 | 重庆大学 | Method for determining maximum stress of circular film under limitation of elasticity on maximum deflection |
| CN111180850B (en) * | 2019-12-31 | 2021-06-11 | 清华大学 | Gradient film |
| CN111442983A (en) * | 2020-03-25 | 2020-07-24 | 重庆大学 | Method for determining elastic strain energy of circular film under transversely uniformly distributed load |
| CN113092040B (en) * | 2021-04-16 | 2022-10-25 | 重庆大学 | Method for determining maximum stress of annular film under transversely uniformly distributed load |
| CN113092039B (en) * | 2021-04-16 | 2022-09-27 | 重庆大学 | Method for determining elastic energy of annular film under transversely uniformly distributed load |
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