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基于Green函数法的Timoshenko曲梁强迫振动分析

赵翔 周扬 邵永波 刘波 周仁

赵翔, 周扬, 邵永波, 刘波, 周仁. 基于Green函数法的Timoshenko曲梁强迫振动分析[J]. 工程力学, 2020, 37(11): 12-27. doi: 10.6052/j.issn.1000-4750.2019.11.0708
引用本文: 赵翔, 周扬, 邵永波, 刘波, 周仁. 基于Green函数法的Timoshenko曲梁强迫振动分析[J]. 工程力学, 2020, 37(11): 12-27. doi: 10.6052/j.issn.1000-4750.2019.11.0708
Xiang ZHAO, Yang ZHOU, Yong-bo SHAO, Bo LIU, Ren ZHOU. ANALYTICAL SOLUTIONS FOR FORCED VIBRATIONS OF TIMOSHENKO CURVED BEAM BY MEANS OF GREEN’S FUNCTIONS[J]. Engineering Mechanics, 2020, 37(11): 12-27. doi: 10.6052/j.issn.1000-4750.2019.11.0708
Citation: Xiang ZHAO, Yang ZHOU, Yong-bo SHAO, Bo LIU, Ren ZHOU. ANALYTICAL SOLUTIONS FOR FORCED VIBRATIONS OF TIMOSHENKO CURVED BEAM BY MEANS OF GREEN’S FUNCTIONS[J]. Engineering Mechanics, 2020, 37(11): 12-27. doi: 10.6052/j.issn.1000-4750.2019.11.0708

基于Green函数法的Timoshenko曲梁强迫振动分析

doi: 10.6052/j.issn.1000-4750.2019.11.0708
基金项目: 国家自然科学基金项目(11702230);工程结构安全评估与防灾技术四川省青年科技创新团队项目(2019JDTD0017)
详细信息
    作者简介:

    周 扬(1991−),男,成都人,硕士生,从事土木工程结构抗震研究(E-mail: 263729126@qq.com)

    邵永波(1973−),男,成都人,教授,博士,博导,从事钢结构稳定性与疲劳研究(E-mail: ybshao@swpu.edu.cn)

    刘 波(1981−),男,重庆人,工程师,硕士,从事油气田开发研究(E-mail: liu_bo2011@petrochina.com.cn)

    周 仁(1986−),男,重庆人,工程师,硕士,从事油气田开发研究(E-mail: zhouren@petrochina.com.cn)

    通讯作者:

    赵 翔(1982−),男,成都人,副教授,博士,硕导,从事土木工程结构抗震研究(E-mail: zhaoxiang_swpu@126.com)

  • 中图分类号: O326

ANALYTICAL SOLUTIONS FOR FORCED VIBRATIONS OF TIMOSHENKO CURVED BEAM BY MEANS OF GREEN’S FUNCTIONS

  • 摘要: 该文运用Green函数法求解了Timoshenko曲梁在强迫振动下的解析解,通过分析曲梁截面的力学平衡,建立了Timoshenko曲梁的振动方程。依次采用分离变量法和Laplace变换法,对不同边界的Timoshenko曲梁求解出了相应的Green函数。并且通过引入两个特征参数来考虑阻尼对强迫振动的影响。数值计算中,验证了该解析解的有效性,并对其中涉及的各种重要物理参数的影响进行了研究。研究结果表明:通过将半径R设置为无穷大,可以简化为Timoshenko直梁振动模型,在此基础上,将剪切修正因子κ设置为无穷大,可以退化为Prescott直梁振动模型,最后再把转动惯量γ设置为0,可退化为Euler-Bernoulli直梁振动模型。该文给出的数值结果验证了所得解的有效性。
  • 图  1  两端简支的曲梁在x=L/2处受到简谐力作用

    Figure  1.  A simply-supported curved beam subject to harmonic force at x=L/2

    图  2  简支TB在中点处作用单位简谐集中力的无量纲化的挠度

    Figure  2.  The dimensionless deflection subject to the external unit simple harmonic concentrated force at the middle section of simply supported TB

    图  3  固支TCB在中点处作用单位集中力的无量纲化的挠度

    Figure  3.  The dimensionless deflection subject to the external unit concentrated force at the middle section of fixed-fixed TCB

    图  4  以外激励频率Ω1为自变量的无量纲化挠度g(1/2,1/2)

    Figure  4.  The dimensionless deflection g(1/2, 1/2) as a function of the dimensionless frequency Ω1 of external dynamic force as independent variable

    图  5  不同半径的TCB以外激励频率Ω1为自变量的无量纲化挠度g(1/2,1/2)

    Figure  5.  The dimensionless deflection g(1/2, 1/2) as a function of the dimensionless frequency Ω1 of external dynamic force as independent variable for TCB with different radiuses

    图  6  在外激频Ω1=0.5作用下,EB、PB、TB和TCB(R=5)的无量纲化挠度g(ξ,1/2)

    Figure  6.  The dimensionless deflection g(ξ,1/2) of EB, PB, TB, and TCB (R=5) corresponding to Ω1=0.5 of external dynamic force.

    图  7  不同半径下,TCB的无量纲化挠度g(ξ,1/2)

    Figure  7.  The dimensionless deflection g(ξ,1/2) of TCB with different radiuses

    图  8  以半径R为自变量的TCB的无量纲化挠度g(1/2,1/2)

    Figure  8.  The dimensionless deflection g(1/2,1/2) of TCB with radius R as independent variable

    图  9  以阻尼比ζ1为自变量的TCB的无量纲化位移g(1/2, 1/2)

    Figure  9.  The dimensionless deflection g(1/2,1/2) of TCB with damping ratio ζ1 as independent variable

    图  10  以阻尼比ζ2为自变量的无量纲化挠度g(1/2, 1/2)

    Figure  10.  The dimensionless deflection g(1/2,1/2) of TCB with damping ratio ζ2 as independent variable

    表  1  式(11)中不同曲梁模型的系数

    Table  1.   Coefficients of different curved beam models from equation (11)

    ECB PCB TCB(ND) TCB(D)
    ${a_1}$ ${\varOmega ^2}\dfrac{\mu }{{EA}}{\rm{+}}\dfrac{{\rm{2}}}{{{R^2}}}$ ${\varOmega ^2}\left( {\dfrac{\gamma }{{EI}}+\dfrac{\mu }{{EA}}} \right){\rm{+}}\dfrac{{\rm{2}}}{{{R^2}}}$ ${\varOmega ^2}\left( {\dfrac{\gamma }{{EI}}+\dfrac{\mu }{{\kappa AG}}+\dfrac{\mu }{{EA}}} \right){\rm{+}}\dfrac{{\rm{2}}}{{{R^2}}}$ $ - {\rm{i}}\varOmega \left( {\dfrac{{{c_1}}}{{\kappa AG}}{\rm{+}}\dfrac{{{c_2}}}{{EI}}} \right){\rm{+}}{\varOmega ^2}\left( {\dfrac{\gamma }{{EI}}+\dfrac{\mu }{{\kappa AG}}+\dfrac{\mu }{{EA}}} \right){\rm{+}}\dfrac{{\rm{2}}}{{{R^2}}}$
    ${a_2}$ ${\varOmega ^2}\left( { - \dfrac{\mu }{{{R^2}EA}} - \dfrac{\mu }{{EI}}} \right){\rm{+}}\dfrac{1}{{{R^4}}}$ $\begin{array}{l} {\varOmega ^2}\left( {\dfrac{ {2\gamma } }{ { {R^2}EI} } - \dfrac{\mu }{ { {R^2}EA} } - \dfrac{\mu }{ {EI} } } \right) + \\ {\varOmega ^4}\dfrac{ {\mu \gamma } }{ { {E^2}AI} }{\rm{+} }\dfrac{1}{ { {R^4} } } \end{array}$ $\begin{array}{l} {\varOmega ^2}\left( {\dfrac{ {2\gamma } }{ { {R^2}EI} } - \dfrac{\mu }{ { {R^2}EA} } - \dfrac{\mu }{ {EI} } - \dfrac{\mu }{ { {R^2}\kappa AG} } } \right) +\\{\varOmega ^4}\left( {\dfrac{ {\mu \gamma } }{ {\kappa AGEI} }{\rm{+} }\dfrac{ {\mu \gamma } }{ { {E^2}AI} }+\dfrac{ { {\mu ^2} } }{ {\kappa {A^2}GE} } } \right){\rm{+} }\dfrac{1}{ { {R^4} } } \end{array}$ $\begin{array}{l} {\rm{i} }\varOmega \left( {\dfrac{ { {c_1} } }{ { {R^2}EA} }+\dfrac{ { {c_1} } }{ {EI} } - \dfrac{ {2{c_2} } }{ { {R^2}EI} } } \right){\rm{+} } \\{\varOmega ^2}\left( {\dfrac{ {2\gamma } }{ { {R^2}EI} } - \dfrac{\mu }{ { {R^2}EA} } - \dfrac{\mu }{ {EI} } - \dfrac{\mu }{ { {R^2}\kappa AG} } - \dfrac{ { {c_1}{c_2} } }{ {\kappa AGEI} } } \right) - \\ {\rm{i} }{\varOmega ^3}\left( {\dfrac{ {\gamma {c_1} } }{ {\kappa AGEI} }{\rm{+} }\dfrac{ {\mu {c_2} } }{ {\kappa AGEI} }{\rm{+} }\dfrac{ {\mu {c_2} } }{ { {E^2}AI} }{\rm{+} }\dfrac{ {\mu {c_1} } }{ {\kappa {A^2}GE} } } \right)+ \\{\varOmega ^4}\left( {\dfrac{ {\mu \gamma } }{ {\kappa AGEI} }{\rm{+} }\dfrac{ {\mu \gamma } }{ { {E^2}AI} }+\dfrac{ { {\mu ^2} } }{ {\kappa {A^2}GE} } } \right){\rm{+} }\dfrac{1}{ { {R^4} } } \end{array}$
    ${a_3}$ ${\varOmega ^2}\dfrac{\mu }{{{R^2}EI}} - {\varOmega ^4}\dfrac{{{\mu ^2}}}{{{E^2}AI}}$ $\begin{array}{l} {\varOmega ^2}\left( {\dfrac{\gamma }{ { {R^4}EI} }+\dfrac{\mu }{ { {R^2}EI} } } \right) - \\ {\varOmega ^4}\left( {\dfrac{ { {\mu ^2} } }{ { {E^2}AI} }{\rm{+} }\dfrac{ {\gamma \mu } }{ { {R^2}{E^2}AI} } } \right) \\ \end{array}$ $\begin{array}{l} {\varOmega ^2}\left( {\dfrac{\gamma }{ { {R^4}EI} }+\dfrac{\mu }{ { {R^2}EI} } } \right)+{\varOmega ^6}\dfrac{ { {\mu ^2}\gamma } }{ {\kappa {E^2}{A^2}GI} } \\ - {\varOmega ^4}\left( {\dfrac{ { {\mu ^2} } }{ { {E^2}AI} }{\rm{+} }\dfrac{ {\gamma \mu } }{ { {R^2}{E^2}AI} }{\rm{+} }\dfrac{ {\mu \gamma } }{ { {R^2}\kappa AGEI} } } \right) \end{array}$ $\begin{array}{l} - \dfrac{ { {\rm{i} }\varOmega {c_2} } }{ { {R^4}EI} }{\rm{+} }{\varOmega ^2}\left( {\dfrac{\gamma }{ { {R^4}EI} }+\dfrac{\mu }{ { {R^2}EI} }+\dfrac{ { {c_1}{c_2} } }{ { {R^2}{E^2}AI} } } \right)+ \\ {\rm{i} }{\varOmega ^3}\left( {\dfrac{ {\mu {c_2} } }{ { {R^2}\kappa AGEI} }+\dfrac{ {\mu {c_2} } }{ { {R^2}{E^2}AI} }+\dfrac{ {\gamma {c_1} } }{ { {R^2}{E^2}AI} }+\dfrac{ {\mu {c_1} } }{ { {E^2}AI} } } \right) - \\ {\varOmega ^4}\left( {\dfrac{ { {\mu ^2} } }{ { {E^2}AI} }{\rm{+} }\dfrac{ {\gamma \mu } }{ { {R^2}{E^2}AI} }{\rm{+} }\dfrac{ {\mu \gamma } }{ { {R^2}\kappa AGEI} }{\rm{+} }\dfrac{ {\mu {c_1}{c_2} } }{ {\kappa {E^2}{A^2}GI} } } \right) - \\ {\rm{i} }{\varOmega ^5}\left( {\dfrac{ {\mu \gamma {c_1} } }{ {\kappa {E^2}{A^2}GI} }{\rm{+} }\dfrac{ { {\mu ^2}{c_2} } }{ {\kappa {E^2}{A^2}GI} } } \right)+{\varOmega ^6}\dfrac{ { {\mu ^2}\gamma } }{ {\kappa {E^2}{A^2}GI} } \end{array}$
    ${b_1}$ 0 0 $\dfrac{1}{{\kappa AG}}$ $\dfrac{1}{{\kappa AG}}$
    ${b_2}$ $\dfrac{1}{{EI}}+\dfrac{1}{{{R^2}EA}}$ $\dfrac{1}{{EI}}+\dfrac{1}{{{R^2}EA}}$ $\dfrac{1}{{EI}} - {\varOmega ^2}\left( {\dfrac{\gamma }{{\kappa AGEI}}{\rm{+}}\dfrac{\mu }{{\kappa {A^2}GE}}} \right)+\dfrac{1}{{{R^2}EA}}$ $\dfrac{1}{{EI}}+\dfrac{{{\rm{i}}\varOmega {c_2}}}{{\kappa AGEI}} - {\varOmega ^2}\left( {\dfrac{\gamma }{{\kappa AGEI}}{\rm{+}}\dfrac{\mu }{{\kappa {A^2}GE}}} \right)+\dfrac{1}{{{R^2}EA}}$
    ${b_3}$ ${\varOmega ^2}\dfrac{\mu }{{{E^2}AI}}$ ${\varOmega ^2}\left( {\dfrac{\mu }{{{E^2}AI}}+\dfrac{\gamma }{{{R^2}{E^2}AI}}} \right)$ ${\varOmega ^2}\left( {\dfrac{\mu }{{{E^2}AI}}+\dfrac{\gamma }{{{R^2}{E^2}AI}}} \right) - \dfrac{{\gamma \mu {\varOmega ^4}}}{{\kappa {E^2}{A^2}GI}}$ $ - \dfrac{{{\rm{i}}\varOmega {c_2}}}{{{R^2}{E^2}AI}}{\rm{+}}{\varOmega ^2}\left( {\dfrac{\mu }{{{E^2}AI}}+\dfrac{\gamma }{{{R^2}{E^2}AI}}} \right)+ \dfrac{{{\rm{i}}\mu {\varOmega ^3}{c_2}}}{{\kappa {E^2}{A^2}GI}} - \dfrac{{\gamma \mu {\varOmega ^4}}}{{\kappa {E^2}{A^2}GI}}$
    注:D和ND分别是阻尼和无阻尼的缩写。
    下载: 导出CSV

    表  2  EB、PB和TCB的边界条件

    Table  2.   Boundary conditions of EB、PB and TCB

    BC Pinned Fixed Free
    EB $\begin{array}{l} W( {0/L} ) = 0;\\ W''( {0/L} ) = 0 \end{array}$ $\begin{array}{l} W( {0/L} ) = 0;\\ W'( {0/L} ) = 0 \end{array}$ $\begin{array}{l} W''( {0/L} ) = 0;\\ W'''( {0/L} ) = 0 \end{array}$
    PB $\begin{array}{l} W( {0/L} ) = 0;\\ W''( {0/L} ) = 0 \end{array}$ $\begin{array}{l} W( {0/L} ) = 0;\\ W'( {0/L} ) = 0 \end{array}$ $\begin{array}{l} W''( {0/L} ) = 0;\\ {\left. {( {W'''{\rm{ + }}{\lambda _0}W'} )} \right|_{x = 0/L}} = 0 \end{array}$
    TCB(ND) $\begin{array}{l} {\left. W \right|_{x = 0/L;{c_1}{c_2} = 0}} = 0;\\ {\left. {( {{\lambda _1}{W^{( 4 )}} + {\lambda _2}W''} )} \right|_{x = 0/L;{c_1}{c_2} = 0}} = 0;\\ {\left. {( {{\lambda _3}{W^{( 5 )}} + {\lambda _4}W''' + {\lambda _5}W'} )} \right|_{x = 0/L;{c_1}{c_2} = 0}} = 0 \end{array}$ $\begin{array}{l} {\left. W \right|_{x = 0/L;{c_1}{c_2} = 0}} = 0;\\ {\left. {( {{\lambda _6}W'''{\rm{ + }}{\lambda _7}W'} )} \right|_{x = 0/L;{c_1}{c_2} = 0}} = 0;\\ {\left. {( {{\lambda _8}{W^{( 5 )}} + {\lambda _9}W''' + {\lambda _{10}}W'} )} \right|_{x = 0/L;{c_1}{c_2} = 0}} = 0 \end{array}$ $\begin{array}{l}{\left. {\left( {RW''{\rm{ + } }\dfrac{1}{R}W} \right)} \right|_{x = 0/L;{c_1} = 0;{c_2} = 0} } = 0;\\ {\left. {( { {\lambda _1}{W^{( 4 )} } + {\lambda _{11} }W'' + {\lambda _{12} }W} )} \right|_{x = 0/L;{c_1}、{c_2} = 0} } = 0;\\ {\left. {( { {\lambda _{13} }{W^{( 5 )} } + {\lambda _{14} }W''' + {\lambda _{15} }W'} )} \right|_{x = 0/L;{c_1}、{c_2} = 0} } = 0; \end{array}$
    TCB(D) $\begin{array}{l} {\left. W \right|_{x = 0/L}} = 0;\\ {\left. {( {{\lambda _1}{W^{( 4 )}} + {\lambda _2}W''} )} \right|_{x = 0/L}} = 0;\\ {\left. {( {{\lambda _3}{W^{( 5 )}} + {\lambda _4}W''' + {\lambda _5}W'} )} \right|_{x = 0/L}} = 0 \end{array}$ $\begin{array}{l} {\left. W \right|_{x = 0/L}} = 0;\\ {\left. {( {{\lambda _6}W'''{\rm{ + }}{\lambda _7}W'} )} \right|_{x = 0/L}} = 0;\\ {\left. {( {{\lambda _8}{W^{( 5 )}} + {\lambda _9}W''' + {\lambda _{10}}W'} )} \right|_{x = 0/L}} = 0 \end{array}$ $\begin{array}{l} {\left. {\left( {RW''{\rm{ + } }\dfrac{1}{R}W} \right)} \right|_{x = 0/L} } = 0;\\ {\left. {( { {\lambda _1}{W^{( 4 )} } + {\lambda _{11} }W'' + {\lambda _{12} }W} )} \right|_{x = 0/L} } = 0;\\ {\left. {( { {\lambda _{13} }{W^{( 5 )} } + {\lambda _{14} }W''' + {\lambda _{15} }W'} )} \right|_{x = 0/L} } = 0; \end{array}$
    注:λi (i=0, 1, 2, ···, 15)的表达式参见附录4中表4
    下载: 导出CSV

    表  3  文中所涉及的物理量符号说明表

    Table  3.   Related physical properties

    符号 含义说明
    N(s, t) 曲梁截面轴向力
    Q(s, t) 曲梁截面剪力
    M(s, t) 曲梁截面弯矩
    p(s, t) 外部荷载
    v(s, t) 曲梁轴向位移
    w(s, t) 曲梁径向位移
    ψ(s, t) 曲梁转角
    t 时间
    R 曲梁半径
    A 曲梁横截面面积
    I 曲梁横截面静距
    E 弹性模量
    G 剪切模量
    μ 单位长度曲梁质量
    γ 转动惯量
    κ 剪切修正因子
    c1 平动阻尼
    c2 转动阻尼
    P(s) 外部荷载分布
    W(s) 稳态径向位移
    V(s) 稳态轴向位移
    Ψ(s) 稳态转角
    Ω 外激力频率
    x 曲梁的任意截面位置
    x0 外部荷载的作用位置
    L 曲梁长度
    G(x, x0) Green函数
    δ(·) 狄拉克函数
    $ \hat W$(s, x0) Laplace变换后的稳态位移
    si (i=1, 2, ···, 6) 特征方程的根
    H(·) 单位阶跃函数
    Ω0 Euler-Bernoulli直梁的一阶固有频率 ${\varOmega _0} = {\pi ^2}\sqrt {EI/\rho A} /{L^2}$
    $w_{\rm{max}}^s $ 简支梁的中截面x0=L/2处受到单位力作用产生的最大静挠度 $w_{\rm{max}}^s $=L3/(48EI)
    下载: 导出CSV
  • [1] 孙皆宜. 曲梁的应用及研究[J]. 科技风, 2018, (16): 100-102.

    Sun Jieyi. Application and research of curved beams [J]. Technology Wind, 2018, (16): 100−102. (in Chinese)
    [2] 周诗俊, 王金安. 曲线隧道盾构引起地表沉降分析[J]. 地下空间与工程学报, 2007, 3(5): 909 − 913.

    Zhou Shijun, Wang Jinan. The surface settlement analysis of curved tunnel shield [J]. Chinese Journal of Underground Space and Engineering, 2007, 3(5): 909 − 913. (in Chinese)
    [3] 张泽文, 罗章华, 黄叹生. 移动荷载作用下曲线桥梁的动态响应实桥分析[J]. 福建交通科技, 2019(1): 49 − 53.

    Zhang Zhewen, Luo Zhanghua, Huang Tansheng. Dynamic response analysis of curved bridges under moving load [J]. Fujian Jiaotong Keji, 2019(1): 49 − 53. (in Chinese)
    [4] 孙广俊, 李鸿晶, 王通等. 基于dqm的曲梁平面外固有振动特性及参数分析[J]. 工程力学, 2013, 30(12): 220 − 227.

    Sun Guangjun, Li Hongjing, Wang Tong, et al. Out-of-plane natural vibration characteristic and parameter analysis of curved girders based on DQM [J]. Engineering Mechanics, 2013, 30(12): 220 − 227. (in Chinese)
    [5] 魏双科. 曲线梁桥的固有振动特性及地震反应分析[D]. 南京: 南京工业大学, 2006.

    Wei Shuangke. Natural vibration characteristics and seismic response analysis of curved girder bridges [D]. Nanjing: Nanjing Tech University, 2006. (in Chinese)
    [6] 闫磊, 李青宁, 赵花静, 等. 非规则曲线桥梁漂浮抗震体系理论及试验研究[J]. 土木建筑与环境工程, 2018, 40(4): 103 − 110.

    Yan Lei, Li Qingning, Zhao Huajing, et al. Theoretical and experimental investigation on irregular curved bridge with floating system [J]. Journal of Civil, Architectural and Environmental Engineering, 2018, 40(4): 103 − 110. (in Chinese)
    [7] 周彦良, 戴俊, 张晓君. 地震波输入方向对曲线隧道地震响应的影响分析[J]. 铁道建筑, 2013(8): 65 − 67. doi: 10.3969/j.issn.1003-1995.2013.08.21

    Zhou Yanliang, Dai Jun, Zhang Xiaojun. Influence of seismic wave input direction on seismic response of curved tunnel [J]. Railway Engineering, 2013(8): 65 − 67. (in Chinese) doi: 10.3969/j.issn.1003-1995.2013.08.21
    [8] 周彦良, 戴俊, 曹东. 曲线隧道的震害机理及抗震分析方法探讨[J]. 公路交通技术, 2013(2): 107 − 110. doi: 10.3969/j.issn.1009-6477.2013.02.026

    Zhou Yanliang, Dai Jun, Cao Dong. Probe into mechanisms of seismic disasters of curved tunnels and seismic analysis methods [J]. Technology of Highway and Transport, 2013(2): 107 − 110. (in Chinese) doi: 10.3969/j.issn.1009-6477.2013.02.026
    [9] 何燕丽, 赵翔. 曲梁压电俘能器强迫振动的格林函数解[J]. 力学学报, 2019, 51(4): 1 − 12.

    He Yanli, Zhao Xiang. Closed-form solutions for force vibration of curved piezoelectric energy harvesters by means of Green's function [J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(4): 1 − 12. (in Chinese)
    [10] Radenkovic G, Borkovic A. Linear static isogeometric analysis of an arbitrarily curved spatial bernoulli-euler beam [J]. Computer Methods in Applied Mechanics and Engineering, 2018, 341: 360 − 396. doi: 10.1016/j.cma.2018.07.010
    [11] Fallah N, Ghanbari A. A displacement finite volume formulation for the static and dynamic analyses of shear deformable circular curved beams [J]. Scientia Iranica, 2018, 25(3): 999 − 1014.
    [12] Tufekci E, Aya S A, Oldac O. In-plane static analysis of nonlocal curved beams with varying curvature and cross-section [J]. International Journal of Applied Mechanics, 2016, 8(1): 1650010-1 − 1650010-28.
    [13] 李卓庭, 宋郁民. 曲梁几何方程推导[J]. 工程力学, 2019, 36(增刊 1): 12 − 16.

    Li Zhuoting, Song Yumin. Geometric equation derivation of curved beam [J]. Engineering Mechanics, 2019, 36(Suppl 1): 12 − 16. (in Chinese)
    [14] 叶康生, 姚葛亮. 平面曲梁有限元静力分析的p型超收敛算法[J]. 工程力学, 2017, 34(11): 31 − 38, 63.

    Ye Kangsheng, Yao Geliang. A p-type superconvergent recovery method for fe static analysis of planar curved beams [J]. Engineering Mechanics, 2017, 34(11): 31 − 38, 63. (in Chinese)
    [15] Lee S Y, Yan Q Z. Exact static analysis of in-plane curved timoshenko beams with strong nonlinear boundary conditions [J]. Mathematical Problems in Engineering, 2015, 2015: 1 − 13.
    [16] Lee S Y, Yan Q Z. An analytical solution for out-of-plane deflection of a curved timoshenko beam with strong nonlinear boundary conditions [J]. Acta Mechanica, 2015, 226(11): 3679 − 3694. doi: 10.1007/s00707-015-1410-7
    [17] 李晓伟, 何光辉. Timoshenko组合梁动力特性与瞬态响应的求积元分析[J]. 振动与冲击, 2018, 37(18): 257 − 265.

    Li Xiaowei, He Guanghui. Quadrature element analysis on dynamic characteristics and transient responses of Timoshenko composite beams [J]. Journal of Vibration and Shock, 2018, 37(18): 257 − 265. (in Chinese)
    [18] 叶康生, 殷振炜. 平面曲梁面内自由振动有限元分析的p型超收敛算法[J]. 工程力学, 2019, 36(5): 31 − 39, 55.

    Ye Kangsheng, Yin Zhenwei. A p-type superconvergent recovery method for fe analysis of in-plane free vibration of planar curved beams [J]. Engineering Mechanics, 2019, 36(5): 31 − 39, 55. (in Chinese)
    [19] Lee J. In-plane free vibration analysis of curved timoshenko beams by the pseudospectral method [J]. Journal of Mechanical Science and Technology, 2003, 17(8): 1156 − 1163.
    [20] Lee J. Out-of-plane free vibration analysis of curved timoshenko beams by the pseudospectral method [J]. International Journal of Precision Engineering and Manufacturing, 2004, 5(2): 53 − 59.
    [21] Liu H L, Zhu X F, Yang D X. Isogeometric method based in-plane and out-of-plane free vibration analysis for timoshenko curved beams [J]. Structural Engineering and Mechanics, 2016, 59(3): 503 − 526. doi: 10.12989/sem.2016.59.3.503
    [22] Lv X H, Shi D Y, Wang Q S, Liang Q. A unified solution for the in-plane vibration analysis of multi-span curved timoshenko beams with general elastic boundary and coupling conditions [J]. Journal of Vibroengineering, 2016, 18(2): 1071 − 1087.
    [23] Calim F F. Dynamic response of curved timoshenko beams resting on viscoelastic foundation [J]. Structural Engineering and Mechanics, 2016, 59(4): 761 − 774. doi: 10.12989/sem.2016.59.4.761
    [24] Li X Y, Zhao X, Li Y H. Green's functions of the forced vibration of timoshenko beams with damping effect [J]. Journal of Sound and Vibration, 2014, 333(6): 1781 − 1795. doi: 10.1016/j.jsv.2013.11.007
    [25] 陈宵寒, 吴太红, 李翔宇. 铁木辛柯纳米梁简谐强迫振动的格林函数解[J]. 四川理工学院学报(自然科学版), 2018, 31(4): 20 − 28.

    Chen Xiaohan, Wu Taihong, Li Xiangyu. Green's function for the harmonic forced vibration of Timoshenko nanobeam [J]. Journal of Sichuan University of Science & Engineering (Natural Science Edition), 2018, 31(4): 20 − 28. (in Chinese)
    [26] Caddemi S, Calio I, Cannizzaro F. Closed-form solutions for stepped timoshenko beams with internal singularities and along-axis external supports [J]. Archive of Applied Mechanics, 2013, 83(4): 559 − 577.
    [27] Failla G, Santini A. On euler–bernoulli discontinuous beam solutions via uniform-beam green’s functions [J]. International Journal of Solids and Structures, 2007, 44(22): 7666 − 7687.
    [28] Tseng Y P, Huang C S, Lin C J. Dynamic stiffness analysis for in-plane vibrations of arches with variable curvature [J]. Journal of Sound & Vibration, 1997, 207(1): 15 − 31.
    [29] Erturk A, Inman D J. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters [J]. Journal of Vibration & Acoustics, 2008, 130(4): 1257 − 1261.
    [30] Manoach E, Ribeiro P. Coupled, thermoelastic, large amplitude vibrations of timoshenko beams [J]. International Journal of Mechanical Sciences, 2004, 46(11): 1589 − 1606. doi: 10.1016/j.ijmecsci.2004.10.006
    [31] Ditkin V A, Prudinkov A P. Formulaire pour le calcul opérationnel [M]. Paris: Masson Cie, 1967.
    [32] Timoshenko S. Strength of materials [M]. New York: D. Van Nostrand Company, 1930.
    [33] Abu-Hilal M. Forced vibration of euler–bernoulli beams by means of dynamic green functions [J]. Journal of Sound and Vibration, 2003, 267(2): 191 − 207. doi: 10.1016/S0022-460X(03)00178-0
    [34] Bishop R E D, Johnson D C. The mechanics of vibration [M]. Cambridge: Cambridge University Press, 2011.
    [35] Majkut L. Free and forced vibrations of timoshenko beams described by single difference equation [J]. Journal of Teoretical & Applied Mechanics, 2009, 47(1): 193 − 210.
    [36] Huang T C. The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions [J]. Journal of Applied Mechanics, 1961, 28(4): 579 − 584. doi: 10.1115/1.3641787
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出版历程
  • 收稿日期:  2019-11-29
  • 修回日期:  2020-04-08
  • 刊出日期:  2020-11-25

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