Wu et al 2019a - Revision history

Rimni at 14:58, 15 December 2020

2020-12-15T14:58:41Z

Rimni at 12:26, 11 December 2020

2020-12-11T12:26:58Z

Rimni at 12:26, 11 December 2020

2020-12-11T12:26:08Z

Rimni: /* 4. Numerical analysis */

2020-12-04T11:44:38Z

‎4. Numerical analysis

Rimni: /* 4.1 Effect of water head */

2020-12-04T11:30:51Z

‎4.1 Effect of water head

Rimni: /* 3. Stability analysis */

2020-12-03T14:26:52Z

‎3. Stability analysis

Rimni: /* 2. Derivation of the nonlinear differential equation of motion */

2020-12-03T13:58:52Z

‎2. Derivation of the nonlinear differential equation of motion

Rimni at 13:43, 3 December 2020

2020-12-03T13:43:43Z

Rimni: /* 2. Derivation of the nonlinear differential equation of motion */

2020-12-03T13:24:28Z

‎2. Derivation of the nonlinear differential equation of motion

Show changes

Rimni at 10:04, 8 January 2020

2020-01-08T10:04:48Z

@@ Line 671: / Line 671: @@
 ====4.2 Effects of excitation frequency ratio and initial end bending moment ====
-Figure 6 depicts the relationship curves plotted between the vibration amplitude and the excitation frequency ratio when subjected to the initial end bending moment of different water heads. As shown in the figure, at <math>H_0</math>, the peak value of parametric resonance only appears at <math>\lambda_d=1</math>  and <math>\lambda_d=2</math>. This phenomenon conforms to the current design code “Design Code for Steel Gate in Water Resources and Hydropower Projects (SL74-2013)” (2013). Figure 7 shows that when the initial end bending moment is not considered, the beat vibration phenomenon can be observed, and the beat amplitude value increases over time at the subharmonic parametric resonance. However, considering the initial end bending moment (Figure 6), the amplitude peak shifts and the arm response amplitudes increase with the initial end bending moment level and the resonance region become larger, which facilitates parametric resonance excitation. According to the design retaining water height of the gate, 80 m is the critical value between high head gates and super high head gates, and maximum existing design head is 163 m (Table 1). The maximum displacements of parametric resonance are 1.44 m and 2.20 m at water heads of 80 m and 163 m, respectively. Through the above analysis, it is further elucidated that the parametric resonance excitation condition given in Chinese code is a special case in the parametric resonance region, which only considers the parametric resonance occurring at nearly twice the natural frequency. In order to obtain a higher accuracy response of the arm, the proposed calculation method of the instability regions in this study should be adopted.
+[[#img-6|Figure 6]] depicts the relationship curves plotted between the vibration amplitude and the excitation frequency ratio when subjected to the initial end bending moment of different water heads. As shown in the figure, at <math>H_0</math>, the peak value of parametric resonance only appears at <math>\lambda_d=1</math>  and <math>\lambda_d=2</math>. This phenomenon conforms to the current design code “Design Code for Steel Gate in Water Resources and Hydropower Projects (SL74-2013)” (2013). [[#img-7|Figure 7]] shows that when the initial end bending moment is not considered, the beat vibration phenomenon can be observed, and the beat amplitude value increases over time at the subharmonic parametric resonance. However, considering the initial end bending moment ([[#img-6|Figure 6]]), the amplitude peak shifts and the arm response amplitudes increase with the initial end bending moment level and the resonance region become larger, which facilitates parametric resonance excitation. According to the design retaining water height of the gate, 80 m is the critical value between high head gates and super high head gates, and maximum existing design head is 163 m ([[#tab-1|Table 1]]). The maximum displacements of parametric resonance are 1.44 m and 2.20 m at water heads of 80 m and 163 m, respectively. Through the above analysis, it is further elucidated that the parametric resonance excitation condition given in Chinese code is a special case in the parametric resonance region, which only considers the parametric resonance occurring at nearly twice the natural frequency. In order to obtain a higher accuracy response of the arm, the proposed calculation method of the instability regions in this study should be adopted.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 678: / Line 679: @@
 |  style="text-align: center;vertical-align: top;padding:10px;"| [[Image:Draft_Wu_507251901-image59.png|300px]]
 |-
-|  style="text-align: center;font-size: 75%;"|(a) ''H ''< 100m
+|  style="text-align: center;font-size: 75%;"|(a) <math>H < 100</math> m
-|  style="text-align: center;font-size: 75%;"|(b) ''H ''≥ 100 m
+|  style="text-align: center;font-size: 75%;"|(b) <math>H \ge 100</math> m
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" style="padding:10px;"| '''Figure 6.''' Vibration amplitude–excitation frequency ratio curves
 |}
+<div id='img-7'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 690: / Line 691: @@
 |  style="text-align: center;vertical-align: top;padding:10px;"| [[File:Review_342535525234_2662_7-2.png]]
 |-
-|  style="text-align: center;font-size: 75%;"|(a) ''λ'' = 1
+|  style="text-align: center;font-size: 75%;"|(a) <math>\lambda  = 1</math>
-|  style="text-align: center;font-size: 75%;"|(b) ''λ'' = 2
+|  style="text-align: center;font-size: 75%;"|(b) <math>\lambda = 2</math>
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" style="padding:10px;"| '''Figure 7.''' Displacement time–history curves
 |}
 ====4.3 Effect of excitation amplitude====

← Older revision		Revision as of 12:26, 11 December 2020
Line 557:		Line 557:
	====4.1 Effect of water head====		====4.1 Effect of water head====

−	According to the Chinese and American codes, the layout of the main beam adopts equivalent load method, and the hydrostatic force <math>H_0</math> is a function of water head. [[#tab-1\|Table 1]] summarizes the Chinese super high head radial gates (<math>> 80</math> m). We first chose the initial disturbance <math>V(0)=10^{−4}</~~math~~> m, excitation amplitude <math>A_d = 0.01</math> m, time <math>t = 500</math> s, damping coefficient <math>\mu = 0</math>, and water head <math>H = 0</math> as a basic case. [[#img-5\|Figure 5]] shows the influence of water head on the vibration amplitudes, <math>\lambda_d</math> denotes the ratio of the external excitation frequency to the natural frequency.	+	According to the Chinese and American codes, the layout of the main beam adopts equivalent load method, and the hydrostatic force <math>H_0</math> is a function of water head. [[#tab-1\|Table 1]] summarizes the Chinese super high head radial gates (<math>> 80</math> m). We first chose the initial disturbance <math>V(0)=10</math><sup>{−4}</sup> m, excitation amplitude <math>A_d = 0.01</math> m, time <math>t = 500</math> s, damping coefficient <math>\mu = 0</math>, and water head <math>H = 0</math> as a basic case. [[#img-5\|Figure 5]] shows the influence of water head on the vibration amplitudes, <math>\lambda_d</math> denotes the ratio of the external excitation frequency to the natural frequency.

← Older revision		Revision as of 12:26, 11 December 2020
Line 557:		Line 557:
	====4.1 Effect of water head====		====4.1 Effect of water head====

−	According to the Chinese and American codes, the layout of the main beam adopts equivalent load method, and the hydrostatic force <math>H_0</math> is a function of water head. [[#tab-1\|Table 1]] summarizes the Chinese super high head radial gates (<math>> 80</math> m). We first chose the initial disturbance <math>V(0) = 10^{−4}</math>m, excitation amplitude <math>A_d = 0.01</math> m, time <math>t = 500</math> s, damping coefficient <math>\mu = 0</math>, and water head <math>H = 0</math> as a basic case. [[#img-5\|Figure 5]] shows the influence of water head on the vibration amplitudes, <math>\lambda_d</math> denotes the ratio of the external excitation frequency to the natural frequency.	+	According to the Chinese and American codes, the layout of the main beam adopts equivalent load method, and the hydrostatic force <math>H_0</math> is a function of water head. [[#tab-1\|Table 1]] summarizes the Chinese super high head radial gates (<math>> 80</math> m). We first chose the initial disturbance <math>V(0)=10^{−4}</math> m, excitation amplitude <math>A_d = 0.01</math> m, time <math>t = 500</math> s, damping coefficient <math>\mu = 0</math>, and water head <math>H = 0</math> as a basic case. [[#img-5\|Figure 5]] shows the influence of water head on the vibration amplitudes, <math>\lambda_d</math> denotes the ratio of the external excitation frequency to the natural frequency.

@@ Line 542: / Line 542: @@
 ==4. Numerical analysis==
-In this section, the nonlinear responses of the radial gate arm are investigated. As a example, considering a radial gate arm with box-section, the cross-sectional dimensions are shown in Figure 4. The arm length ''L ''= 16 m, mass per unit length ''m ''= 386 kg/m, and modulus of elasticity ''E ''= 210 GPa.
+In this section, the nonlinear responses of the radial gate arm are investigated. As a example, considering a radial gate arm with box-section, the cross-sectional dimensions are shown in [[#img-4|Figure 4]]. The arm length <math>L = 16</math> m, mass per unit length <math>m = 386</math> kg/m, and modulus of elasticity <math>E = 210</math> GPa.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 556: / Line 557: @@
 ====4.1 Effect of water head====
-According to the Chinese and American codes, the layout of the main beam adopts equivalent load method, and the hydrostatic force <math>H_0</math> is a function of water head. [[#tab-1|Table 1]] summarizes the Chinese super high head radial gates (<math>> 80</math> m). We first chose the initial disturbance <math>V(0) = 10^{−4}</math>m, excitation amplitude <math>A_d = 0.01</math> m, time <math>t = 500</math> s, damping coefficient <math>\mu = 0</math>, and water head <math>H = 0</math> as a basic case. The Figure 5 shows the influence of water head on the vibration amplitudes, <math>\lambda_d</math> denotes the ratio of the external excitation frequency to the natural frequency.
+According to the Chinese and American codes, the layout of the main beam adopts equivalent load method, and the hydrostatic force <math>H_0</math> is a function of water head. [[#tab-1|Table 1]] summarizes the Chinese super high head radial gates (<math>> 80</math> m). We first chose the initial disturbance <math>V(0) = 10^{−4}</math>m, excitation amplitude <math>A_d = 0.01</math> m, time <math>t = 500</math> s, damping coefficient <math>\mu = 0</math>, and water head <math>H = 0</math> as a basic case. [[#img-5|Figure 5]] shows the influence of water head on the vibration amplitudes, <math>\lambda_d</math> denotes the ratio of the external excitation frequency to the natural frequency.
@@ Line 569: / Line 570: @@
 |  style="text-align: center;"|Xiaowan
 |  style="text-align: center;"|Lantsang
-|  style="text-align: center;"|5.0×7.0
+|  style="text-align: center;"|5.0<math>\times</math>7.0
 |  style="text-align: center;"|163.00
 |-
@@ Line 575: / Line 576: @@
 |  style="text-align: center;"|Shuibuya
 |  style="text-align: center;"|Qing River
-|  style="text-align: center;"|6.0×7.0
+|  style="text-align: center;"|6.0<math>\times</math>7.0
 |  style="text-align: center;"|154.00
 |-
@@ Line 581: / Line 582: @@
 |  style="text-align: center;"|Xiaolangdi
 |  style="text-align: center;"|Yalong River
-|  style="text-align: center;"|4.8×4.8
+|  style="text-align: center;"|4.8<math>\times</math>4.8
 |  style="text-align: center;"|140.00
 |-
@@ Line 587: / Line 588: @@
 |  style="text-align: center;"|Jinping-I
 |  style="text-align: center;"|Yalong River
-|  style="text-align: center;"|5.0×6.0
+|  style="text-align: center;"|5.0<math>\times</math>6.0
 |  style="text-align: center;"|133.00
 |-
@@ Line 593: / Line 594: @@
 |  style="text-align: center;"|Laxiwa
 |  style="text-align: center;"|Yellow River
-|  style="text-align: center;"|4.0×6.0
+|  style="text-align: center;"|4.0<math>\times</math>6.0
 |  style="text-align: center;"|132.00
 |-
@@ Line 599: / Line 600: @@
 |  style="text-align: center;"|Pubugou
 |  style="text-align: center;"|Dadu River
-|  style="text-align: center;"|6.5×8.0
+|  style="text-align: center;"|6.5<math>\times</math>8.0
 |  style="text-align: center;"|126.28
 |-
@@ Line 660: / Line 661: @@
 When the principal parametric resonance occurs, the amplitude of the arm increases with the water head, and the maximum displacement reaches 0.57 m. In the case of subharmonic parametric resonance, the change is contrasting, and the maximum amplitude is 0.18 m. Overall, both the types of parametric resonance responses change slowly, so supposes the axial hydrostatic force generated by 80 m water head in further analysis.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
 |style="padding:10px;"| [[Image:Draft_Wu_507251901-picture- 1.svg|600px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding:10px;"| '''Figure 5.''' Vibration amplitude–water head curves
+| colspan="1" style="padding:10px;"| '''Figure 5'''. Vibration amplitude–water head curves
 |}
 ====4.2 Effects of excitation frequency ratio and initial end bending moment ====
-Figure 6 depicts the relationship curves plotted between the vibration amplitude and the excitation frequency ratio when subjected to the initial end bending moment of different water heads. As shown in the figure, at ''H ''= 0, the peak value of parametric resonance only appears at ''λ''<sub>d</sub> = 1 and ''λ''<sub>d</sub> = 2. This phenomenon conforms to the current design code “Design Code for Steel Gate in Water Resources and Hydropower Projects (SL74-2013)” (2013). Figure 7 shows that when the initial end bending moment is not considered, the beat vibration phenomenon can be observed, and the beat amplitude value increases over time at the subharmonic parametric resonance. However, considering the initial end bending moment (Figure 6), the amplitude peak shifts and the arm response amplitudes increase with the initial end bending moment level and the resonance region become larger, which facilitates parametric resonance excitation. According to the design retaining water height of the gate, 80 m is the critical value between high head gates and super high head gates, and maximum existing design head is 163 m (Table 1). The maximum displacements of parametric resonance are 1.44 m and 2.20 m at water heads of 80 m and 163 m, respectively. Through the above analysis, it is further elucidated that the parametric resonance excitation condition given in Chinese code is a special case in the parametric resonance region, which only considers the parametric resonance occurring at nearly twice the natural frequency. In order to obtain a higher accuracy response of the arm, the proposed calculation method of the instability regions in this study should be adopted.
+Figure 6 depicts the relationship curves plotted between the vibration amplitude and the excitation frequency ratio when subjected to the initial end bending moment of different water heads. As shown in the figure, at <math>H_0</math>, the peak value of parametric resonance only appears at <math>\lambda_d=1</math>  and <math>\lambda_d=2</math>. This phenomenon conforms to the current design code “Design Code for Steel Gate in Water Resources and Hydropower Projects (SL74-2013)” (2013). Figure 7 shows that when the initial end bending moment is not considered, the beat vibration phenomenon can be observed, and the beat amplitude value increases over time at the subharmonic parametric resonance. However, considering the initial end bending moment (Figure 6), the amplitude peak shifts and the arm response amplitudes increase with the initial end bending moment level and the resonance region become larger, which facilitates parametric resonance excitation. According to the design retaining water height of the gate, 80 m is the critical value between high head gates and super high head gates, and maximum existing design head is 163 m (Table 1). The maximum displacements of parametric resonance are 1.44 m and 2.20 m at water heads of 80 m and 163 m, respectively. Through the above analysis, it is further elucidated that the parametric resonance excitation condition given in Chinese code is a special case in the parametric resonance region, which only considers the parametric resonance occurring at nearly twice the natural frequency. In order to obtain a higher accuracy response of the arm, the proposed calculation method of the instability regions in this study should be adopted.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"

@@ Line 556: / Line 556: @@
 ====4.1 Effect of water head====
-According to the Chinese and American codes, the layout of the main beam adopts equivalent load method, and the hydrostatic force ''H''<sub>0</sub> is a function of water head. Table 1 summarizes the Chinese super high head radial gates (> 80 m). We first chose the initial disturbance ''V''(0) = 10<sup>−4</sup> m, excitation amplitude ''A''<sub>d</sub> = 0.01 m, time ''t ''= 500 s, damping coefficient ''μ = ''0, and water head ''H ''= 0 as a basic case. The Figure 5 shows the influence of water head on the vibration amplitudes, ''λ''<sub>d</sub> denotes the ratio of the external excitation frequency to the natural frequency.
+According to the Chinese and American codes, the layout of the main beam adopts equivalent load method, and the hydrostatic force <math>H_0</math> is a function of water head. [[#tab-1|Table 1]] summarizes the Chinese super high head radial gates (<math>> 80</math> m). We first chose the initial disturbance <math>V(0) = 10^{−4}</math>m, excitation amplitude <math>A_d = 0.01</math> m, time <math>t = 500</math> s, damping coefficient <math>\mu = 0</math>, and water head <math>H = 0</math> as a basic case. The Figure 5 shows the influence of water head on the vibration amplitudes, <math>\lambda_d</math> denotes the ratio of the external excitation frequency to the natural frequency.
-<div class="center" style=" font-size: 75%;">'''Table 1.''' Super high head radial gates in China</div>
+<div class="center" style=" font-size: 75%;">'''Table 1'''. Super high head radial gates in China</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:80%;"
+<div id='tab-1'></div>
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
+|  style="text-align: center;"|1
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-aling:top;"|'''Engineering'''
+|  style="text-align: center;"|Xiaowan
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-aling:top;"|'''Location'''
+|  style="text-align: center;"|Lantsang
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-aling:top;"|'''Size of face plane'''
+|  style="text-align: center;"|5.0×7.0
 |  style="text-align: center;"|163.00
-'''width-height (m)'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-aling:top;"|'''Water head (m)'''
-|-
-|  style="border-top: 1pt solid black;text-align: center;"|1
-|  style="border-top: 1pt solid black;text-align: center;"|Xiaowan
-|  style="border-top: 1pt solid black;text-align: center;"|Lantsang
-|  style="border-top: 1pt solid black;text-align: center;"|5.0×7.0
-|  style="border-top: 1pt solid black;text-align: center;"|163.00
 |-
 |  style="text-align: center;"|2
@@ Line 655: / Line 650: @@
 |  style="text-align: center;"|83.00
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;"|15
+|  style="text-align: center;"|15
-|  style="border-bottom: 1pt solid black;text-align: center;"|Goupitan
+|  style="text-align: center;"|Goupitan
-|  style="border-bottom: 1pt solid black;text-align: center;"|Wujiang River
+|  style="text-align: center;"|Wujiang River
-|  style="border-bottom: 1pt solid black;text-align: center;"|10.0×9.0
+|  style="text-align: center;"|10.0×9.0
-|  style="border-bottom: 1pt solid black;text-align: center;"|81.00
+|  style="text-align: center;"|81.00
 |}

@@ Line 348: / Line 348: @@
 |}
+where <math>T_0 = t</math>, <math>T_1 = t\varepsilon</math>, <math>T_2 = t\varepsilon^2</math>. Substituting Equation (18) into Equation (17) and letting the coefficients <math>\varepsilon</math>, <math>\varepsilon^2</math> and <math>\varepsilon^3</math> equal zero, the following equations are obtained:
-where ''T''<sub>0</sub> = ''t'', ''T''<sub>1</sub> = ''tε'', ''T''<sub>2</sub> = ''tε''<sup>2</sup>. Substituting Equation (18) into Equation (17) and letting the coefficients ''ε'', ''ε''<sup>2</sup> and ''ε''<sup>3</sup> equal zero, the following equations are obtained:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 394: / Line 393: @@
 |}
+where  <math>\overline{A}</math> is the complex conjugate of <math>A</math>.
-where  <math>\overline{A}</math> is the complex conjugate of ''A''.
 Substituting Equation (22) into Equation (20), one gets:
@@ Line 410: / Line 408: @@
-where ''cc'' indicates the complex conjugate part. Eliminating the secular terms yields:
+where <math>cc</math> indicates the complex conjugate part. Eliminating the secular terms yields:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 441: / Line 439: @@
-Through practical observation [10], when excitation frequency ''ω''<sub>d</sub> approaches 2 times of  the structure frequency ''ω''<sub>a</sub>, the parametric resonance may occur. A detuning parameter ''σ ''is introduced to quantify the deviation of ''ω''<sub>d </sub>from 2''ω''<sub>a</sub>, and ''ω''<sub>d</sub> is described by:
+Through practical observation [10], when excitation frequency <math>\omega_d</math> approaches 2 times of  the structure frequency <math>\omega_a</math>, the parametric resonance may occur. A detuning parameter ''σ ''is introduced to quantify the deviation of <math>\omega_d</math> from 2<math>\omega_a</math> and <math>\omega_d</math> is described by:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 467: / Line 465: @@
-Then, by expressing the function ''A'' in the polar form as:
+Then, by expressing the function <math>A</math> in the polar form as:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 480: / Line 478: @@
-where ''α''(''t'') and ''θ''(''t'') are real functions. By substituting this transformation into Equation (27), and separating the resulting real and imaginary parts yield:
+where <math>\alpha (t)</math> and <math>\theta (t)</math> are real functions. By substituting this transformation into Equation (27), and separating the resulting real and imaginary parts yield:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 497: / Line 495: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\alpha \dot{\gamma }=\sigma \alpha -\displaystyle\frac{7b^2{\alpha }^3}{12{\omega }_\mbox{a}^3}-</math><math>\displaystyle\frac{3c{\alpha }^3}{8{\omega }_\mbox{a}}+\left[\displaystyle\frac{bf_2}{2\left({\omega }_\mbox{d}^2-{\omega }_\mbox{a}^2\right)}+\right. </math><math>\left. \displaystyle\frac{f_1}{4}\right]U_\mbox{d}\alpha \hbox{ sin }\gamma </math>
+| <math>\alpha \dot{\gamma }=\sigma \alpha -\displaystyle\frac{7b^2{\alpha }^3}{12{\omega }_\mbox{a}^3}-</math><math>\displaystyle\frac{3c{\alpha }^3}{8{\omega }_\mbox{a}}+\left[\displaystyle\frac{bf_2}{2\left({\omega }_\mbox{d}^2-{\omega }_\mbox{a}^2\right)}+ \displaystyle\frac{f_1}{4}\right]U_\mbox{d}\alpha \hbox{ sin }\gamma </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (30)
@@ Line 503: / Line 501: @@
-where ''γ'' = ''ε''<sup>2</sup>''σT''<sub>0</sub> - ''θ''. The stationary solution equation ''α''<sub>s</sub> and ''γ''<sub>s</sub> can be obtained from Equations (29) and (30) by letting  <math>\dot{\alpha }=\dot{\gamma }=0</math> . The vibration amplitude can be expressed as:
+where <math>\gamma = \varepsilon^2 \sigma T_0- \theta</math>. The stationary solution equation ''α''<sub>s</sub> and ''γ''<sub>s</sub> can be obtained from Equations (29) and (30) by letting  <math>\dot{\alpha }=\dot{\gamma }=0</math>. The vibration amplitude can be expressed as:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 514: / Line 512: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (31)
 |}
 For ''α''<sub>s</sub> is a real number, from Equation (31), the stability boundary is given by:

@@ Line 123: / Line 123: @@
 |}
-where <math>s</math> represents the lagrange coordinate by the arc length, <math>m</math> is the mass per unit length of the arm and the damping coefficient <math>\mu</math>. When <math>\alpha = 90°</math>, the arm is upright, same as layout of [[#img-2|Figure 2]].
+where <math>s</math> represents the lagrange coordinate by the arc length, <math>m</math> is the mass per unit length of the arm and the damping coefficient <math>\mu</math>. When <math>\alpha = 90^\circ</math>, the arm is upright, same as layout of [[#img-2|Figure 2]].
 According to the geometric deformation and stress state of the arm model, the following results are obtained:
@@ Line 137: / Line 137: @@
 |}
-where ''H''<sub>0</sub> and ''h'' represent the axial force and additional axial hydrodynamic force, respectively.
+where <math>H_0</math> and <math>h</math> represent the axial force and additional axial hydrodynamic force, respectively.
-By Taylor formula, Equation (5) is expanded at d''η''/d''x ''= 0:
+By Taylor formula, Equation (5) is expanded at <math>d\eta /dx= 0</math>:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 225: / Line 225: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>v\left(0,t\right)=0</math> ,  <math>v\left(L,t\right)=0</math> ,  <math>u\left(0,t\right)=D_\mbox{d}cos\alpha </math> ,  <math>u\left(L,t\right)=0</math> .
+| <math>v\left(0,t\right)=0</math> ,  <math>v\left(L,t\right)=0</math> ,  <math>u\left(0,t\right)=D_\mbox{d}\cos\alpha </math> ,  <math>u\left(L,t\right)=0</math> .
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (11)
@@ Line 242: / Line 242: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)
 |}
 where <math>V\left(t\right)</math> is the displacement amplitude. According to early analysis, Liu et al.  pointed out that, for radial gate arm vibration induced by flow excitation, the first mode is dominated [11]. Hence, We can use the single mode approximation with neglecting modal interaction.
@@ Line 293: / Line 292: @@
 <math>2\xi {\omega }_\mbox{a}=\frac{\mu }{m}</math>,  <math>{\omega }_\mbox{a}^2={\omega }_0^2\left(1+\lambda \right)</math>,  <math>{\omega }_0^2=\frac{{\pi }^2H_0}{mL_{}^2}+\frac{EI{\pi }^4}{mL_{}^4}</math>,
-<math>\lambda =\left[-\frac{{\pi }^2H_0}{2L}\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)-\right. </math><math>\left. \frac{{\pi }^2H_0M_0^2}{2E^2I^2L^2}\left(\frac{L^3}{6}-\right. \right. </math><math>\left. \left. \frac{L^3}{4{\pi }^2}\right)-\frac{M_0^2H_0{\pi }^2}{4REI}-\right. </math><math>\left. \frac{2L^3M_0^2A}{{\pi }^4EI^2}\right]/\left[\left(\frac{{\pi }^2H_0}{mL_{}^2}+\right. \right. </math><math>\left. \left. \frac{EI{\pi }^4}{mL_{}^4}\right)\left(-\right. \right. </math><math>\left. \left. \frac{mL}{2}\right)\right]</math>,
+<math>\lambda =\left[-\frac{{\pi }^2H_0}{2L}\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)-\right. </math><math>\left. \frac{{\pi }^2H_0M_0^2}{2E^2I^2L^2}\left(\frac{L^3}{6}-\right. \right. </math><math>\left. \left. \frac{L^3}{4{\pi }^2}\right)-\frac{M_0^2H_0{\pi }^2}{4REI}-\right. </math><math>\left. \frac{2L^3M_0^2A}{{\pi }^4EI^2}\right]/\left[\left(\frac{{\pi }^2H_0}{mL_{}^2}+\right. \right. </math><math>\left. \left. \frac{EI{\pi }^4}{mL_{}^4}\right)\left(- \frac{mL}{2}\right)\right]</math>,
 <math>a=-\frac{{\pi }^2EA}{L^3}\left[\frac{L}{2}\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)+\right. </math><math>\left. \frac{M_0^2}{2E^2I^2}\left(\frac{L^3}{6}-\frac{L^3}{4{\pi }^4}\right)+\right. </math><math>\left. \frac{M_0^2L^2}{4REI}\right]/\left(-\frac{mL}{2}\right)</math>,
@@ Line 299: / Line 298: @@
 <math>b=\left\{\begin{array}{c}
 \\
-\end{array}\left(\frac{-2{\pi }^2M_0A}{{\pi }^3IL}\right)\left[\frac{L}{2}\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)+\right. \right. </math><math>\left. \left. \frac{M_0^2}{2E^2I^2}\left(\frac{L^3}{6}-\right. \right. \right. </math><math>\left. \left. \left. \frac{L^3}{4{\pi }^2}\right)+\right. \right. </math><math>\left. \left. \frac{M_0^2L^2}{4REI}\right]-\frac{M_0{\pi }^2A}{4IL^2}\left[\frac{L^2}{\pi }\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)+\right. \right. </math><math>\left. \left. \frac{M_0^2}{2E^2I^2}\left(\frac{L^4}{\pi }-\right. \right. \right. </math><math>\left. \left. \left. \frac{6L^4}{{\pi }^3}\right)+\right. \right. </math><math>\left. \left. \frac{M_0^2L^2}{REI\pi }\right]\right\}/\left(-\right. </math><math>\left. \frac{mL}{2}\right)</math>,
+\end{array}\left(\frac{-2{\pi }^2M_0A}{{\pi }^3IL}\right)\left[\frac{L}{2}\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)+\right. \right. </math><math>\left. \left. \frac{M_0^2}{2E^2I^2}\left(\frac{L^3}{6}-\right. \right. \right. </math><math>\left. \left. \left. \frac{L^3}{4{\pi }^2}\right)+\right. \right. </math><math>\left. \left. \frac{M_0^2L^2}{4REI}\right]-\frac{M_0{\pi }^2A}{4IL^2}\left[\frac{L^2}{\pi }\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)+\right. \right. </math><math>\left. \left. \frac{M_0^2}{2E^2I^2}\left(\frac{L^4}{\pi }-\right. \right. \right. </math><math>\left. \left. \left. \frac{6L^4}{{\pi }^3}\right)+ \frac{M_0^2L^2}{REI\pi }\right]\right\}/\left(- \frac{mL}{2}\right)</math>,
 <math>c=\left(\frac{-EA{\pi }^4}{4L^4}\right)\left[\frac{L}{2}\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)+\right. </math><math>\left. \frac{M_0^2}{2E^2I^2}\left(\frac{L^3}{6}-\frac{L^3}{4{\pi }^2}\right)+\right. </math><math>\left. \frac{M_0^2L^2}{4REI}\right]/\left(-\frac{mL}{2}\right)</math>,
-<math>d=\left(-\frac{AM_0}{LI}\right)\left[\frac{L^2}{\pi }\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)+\right. </math><math>\left. \frac{M_0^2}{2E^2I^2}\left(\frac{L^4}{\pi }-\right. \right. </math><math>\left. \left. \frac{6L^4}{{\pi }^3}\right)+\frac{M_0^2L^2}{REI\pi }\right]/\left(-\right. </math><math>\left. \frac{mL}{2}\right)</math>.
+<math>d=\left(-\frac{AM_0}{LI}\right)\left[\frac{L^2}{\pi }\left(1\mbox{+}\frac{M_0^2}{2R^2}\right)+\right. </math><math>\left. \frac{M_0^2}{2E^2I^2}\left(\frac{L^4}{\pi }-\right. \right. </math><math>\left. \left. \frac{6L^4}{{\pi }^3}\right)+\frac{M_0^2L^2}{REI\pi }\right]/\left(- \frac{mL}{2}\right)</math>.
 where <math>\xi</math> is the damping ratio, <math>{\omega }_0</math> corresponds to the first-order natural frequency and <math>\lambda</math> is the arm parameter of the end moment.

@@ Line 48: / Line 48: @@
 ==2. Derivation of the nonlinear differential equation of motion==
-Three dimensional overview of the radial gate is shown in [[#img-1|Figure 1]], the gate contains a face plate to retain water, two major crossbeams, four inclined arms with box-sectional area, major vertical beams and minor beams [13]. The anchor of radial gate arm is attached to piers, which can be regard as a semi-rigid connection. In accordance with Chinese design code SL74-2013 [1], the structure can be simplified as an extended beam, both ends constrained by water seal in <math>y</math> direction, and assembled with arms by welding. The water pressure can pass to piers through face plane, main beams and arms, while, the load acting on main beam can be considered as uniform hydrostatic pressure <math>q</math> by equivalent calculation ([[#img-2|Figure 2]]). Then on the base of these, further analyze the arm vibration under local coordinate, the model as shown in [[#img-3|Figure 3]], with the length of the arm <math>L</math>, the transverse displacement <math>v(x,t)</math>, flow-induced excitation frequency <math>\omega_d</math>, the tangential hydrostatic force <math>T_0= q (L_b + 2c)/2</math>, and the additional tangential hydrodynamic force <math>\tau </math>. Due to parametric vibration is induced by harmonic load, so <math>\tau </math> can be assumed as the function of harmonic displacement of flow excitation <math>D''<sub>d</sub> = A_d \cos\omega_d t</math>.
+Three dimensional overview of the radial gate is shown in [[#img-1|Figure 1]], the gate contains a face plate to retain water, two major crossbeams, four inclined arms with box-sectional area, major vertical beams and minor beams [13]. The anchor of radial gate arm is attached to piers, which can be regard as a semi-rigid connection. In accordance with Chinese design code SL74-2013 [1], the structure can be simplified as an extended beam, both ends constrained by water seal in <math>y</math> direction, and assembled with arms by welding. The water pressure can pass to piers through face plane, main beams and arms, while, the load acting on main beam can be considered as uniform hydrostatic pressure <math>q</math> by equivalent calculation ([[#img-2|Figure 2]]). Then on the base of these, further analyze the arm vibration under local coordinate, the model as shown in [[#img-3|Figure 3]], with the length of the arm <math>L</math>, the transverse displacement <math>v(x,t)</math>, flow-induced excitation frequency <math>\omega_d</math>, the tangential hydrostatic force <math>T_0= q (L_b + 2c)/2</math>, and the additional tangential hydrodynamic force <math>\tau </math>. Due to parametric vibration is induced by harmonic load, so <math>\tau </math> can be assumed as the function of harmonic displacement of flow excitation <math>D_d= A_d \cos\omega_d t</math>.
 <div id='img-1'></div>

@@ Line 786: / Line 786: @@
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