双功能梯度纳米梁系统振动分析的辛方法

周震寰, 李月杰, 范俊海, 隋国浩, 张俊霖, 徐新生

(大连理工大学 国际计算力学中心 工程力学系;工业装备结构分析国家重点实验室(大连理工大学), 辽宁 大连 116024)

摘要 在辛力学与非局部Timoshenko(铁木辛柯)梁理论的基础上,针对黏弹性介质中的双功能梯度纳米梁系统的自由振动问题,提出了一种全新的解析求解方法 在Hamilton(哈密顿)体系下,位移与广义剪力、转角与广义弯矩互为对偶变量 以对偶变量为基本未知量,Lagrange(拉格朗日)体系下的高阶偏微分控制方程简化为一系列常微分方程 该纳米梁系统的振动问题归结为辛空间下的本征问题,解析频率方程和振动模态可以通过辛本征解和边界条件直接获得 数值结果验证了该方法的正确性与有效性,并针对纳米梁系统的小尺度效应、纳米梁间的相互作用以及黏弹性地基的影响进行了系统的参数分析

Hamilton体系; 辛方法; 双功能梯度纳米梁系统; 自由振动; 解析解

引 言

自从日本学者Lijima发现碳纳米管以来,纳米材料引起了学术界的广泛重视 由双纳米梁构成的复杂纳米梁系统作为一类重要的基础结构具有巨大的发展潜力,并已经成功应用于纳米光机系统(NOMS)中 [1-2] 近年来,为了获得更佳的结构性能,功能梯度型纳米材料被逐渐引入到纳米装置的研发中,如微纳机电系统(MEMS/NAMS) [3] 和原子力显微镜(AFM) [4] 因此,研究双功能梯度纳米梁系统的力学性能具有重要的实际意义

在纳米尺度下,经典连续介质力学理论已经不再适用,分子模拟与实验方法又面临着成本过高及无法处理大尺寸复杂结构的困难 为解决上述问题,Eringen提出了一种可以考虑尺度效应的非局部场理论模型,该模型已经在纳米结构相关研究中获得广泛应用 [5-6] 目前,国内外学者在非局部理论框架下对纳米梁的力学行为进行了系统研究 [7-9] 功能梯度纳米梁自由振动方面也已经积累了大量的研究成果 [10-20] 然而,从现有文献可以发现,解析研究工作还比较少 [11-12, 18] ,解析方法受到求解体系限制均为逆法或半逆法,双功能梯度纳米梁系统的振动问题尚未提及 因此,发展一种能够有效分析双功能梯度纳米梁自由振动问题的解析方法具有重要的理论意义

应用力学中的Hamilton体系理论由钟万勰院士等首次提出 [21] ,并已经成功应用于力学中的各个领域 [22] 本文将Hamilton力学与非局部Timoshenko梁理论相结合,提出一种适用于双功能梯度梁自由振动分析的解析求解方法,求解其自由振动频率及解析振型函数,并研究尺度效应、材料功能梯度分布以及周围黏弹性介质参数对该类纳米梁系统的影响和作用规律

1 非局部场理论

根据Eringen的非局部场理论,应力-应变关系可以表示为

x V

(1)

其中 σ 为应力张量, α 0 ( x - x ′, e 0 a 0 )为非局部函数, e 0 a 0 为表征尺度效应的长度量纲系数,上标“~”表示非局部变量 式(1)中的积分型本构给数学求解带来巨大困难 为解决该问题,Eringen进一步推导出等效的微分型本构,即

(2)

其中 ε 为应变张量, ξ = e 0 a 0 为非局部参数, 2 为Laplace(拉普拉斯)算子

2 功能梯度材料

考虑一个放置于黏弹性介质中的双功能梯度纳米梁系统,其坐标如图1所示 黏弹性介质的刚度和黏弹性系数分别为 K w c 纳米梁之间的Van der Waals(范德华)力、电场力或弹性介质引起的力,均由垂直于纳米梁的一组Winkler弹性介质表示,其刚度系数为 K 两个纳米梁的几何和物理参数完全相同 记梁长为 a ,截面宽为 b ,高度为 h ,截面积为 A ,密度为 ρ ,弹性模量为 E ,剪切模量为 G = E /[2(1+ ν )], ν 为Poisson(泊松)比 对于功能梯度纳米梁,其材料参数沿厚度方向的变化规律为

(3)

其中 P 1 P 2 分别为梁上下表面的材料参数, P 可以表示等效弹性模量 E 、密度 ρ 和剪切模量 G k 为功能梯度指数

图1 放置于黏弹性介质中的双功能梯度纳米梁系统
Fig. 1 A functionally graded double-nanobeam system embedded in viscoelastic medium

3 基 本 方 程

对于非局部Timoshenko梁,其沿 x y z 轴的位移分量分别表示为 u 1 = u 2 =0, u 3 = W ( x , t ), W 为横向位移, θ 为转角 对于自由振动问题,令 W ( x , t )= w ( x )e i ωt ω 为自振频率,则功能梯度纳米梁的自由振动方程可以表示为

(4)

其中,i为虚数单位; 分别为纳米梁的非局部弯矩和剪力; 为质量; p i =- K ( w i - w i )- K w w i - c w i /∂ t i = 1,2代表纳米梁1和2, i ′=2,1 由式(2),功能梯度纳米梁的本构方程可以表示为

(5)

其中,L=1- ξ 2 2 /∂ x 2 为线性算子, G = E /[2(1+ ν )]

非局部内力与经典内力之间的关系为

(6a)

(6b)

其中, J 1 = - E ( z ) z 2 d A J 2 = G ( z )d A K s 为剪切修正系数 将式(6)代入式(4),可得

(7)

根据双纳米梁系统的振动特点,方程(7)可以分为两类子问题,即同向振动和反向振动 w IP = w 1 + w 2 θ IP = θ 1 + θ 2 w OP = w 1 - w 2 θ OP = θ 1 + θ 2 [23] ,则方程(7)可以改写为统一形式:

(8a)

(8b)

其中,当 φ =0时, w = w IP θ = θ IP ,为同向振动问题;当 φ =2时, w = w OP θ = θ OP ,为反向振动问题 方程(8)即为以经典变量表示的双功能梯度纳米梁系统自由振动的控制方程

令S、C和F分别表示简支、固支和自由边界, x =0, a 处的边界条件可以表示为

① S: w =0, M =0, when x =0, a

(9a)

② C: w =0, θ =0, when x =0, a

(9b)

③ F: M =0, Q =0, when x =0, a

(9c)

4 Hamilton体系

为引入双功能梯度纳米梁系统自由振动问题的Hamilton求解体系,定义变量上一点表示对 x 方向的微分,即 并记 q = w , θ T ,则Lagrange密度函数可以表示为 [21]

(10)

其中

根据Legendre(勒让德)变换, q 的对偶变量为

(11)

其中 为广义剪力, 为广义弯矩

由上式可进一步求得

(12)

由式(10)~(12),Hamilton密度函数为

(13)

其中

A 12 =- K s J 2 /[ K s J 2 -( 2 - φK - K w -i ) ξ 2 ], B 11 =- 2 + φK + K w +i

B 22 =- K s J 2 ( 2 - φK - K w -i ) ξ 2 /[ K s J 2 -( 2 - φK - K w -i ) ξ 2 ]- I 2 ω 2 D 11 =1/[ K s J 2 -( 2 - φK - K w -i ) ξ 2 ], D 22 =1/( J 1 - I 2 ω 2 ξ 2 )

由式(13),Hamilton体系下的控制方程可以表示为

(14)

定义全状态向量 Ψ = q T , p T T ,则上式可以写为矩阵形式,即

(15)

式中 为Hamilton矩阵 对应的边界条件可以由式(9)获得

h g Ψ = 0 , when x =0, a ,

(16)

其中 h 为边界指示矩阵,g=S,C和F,

5 辛本征值和本征解

在辛空间下,Hamilton方程(15)可用分离变量方法求解 Ψ ( x )= ψ j e μ j x ,本征方程为

j = μ j ψ j

(17)

其中, μ j ψ j 分别为Hamilton矩阵的本征值和本征向量 由方程(17)可得

μ 4 - γμ 2 + ζ =0,

(18)

其中

方程(18)的4个根为

μ 1,2 =∓

(19)

可以证明,当且仅当 μ 1,2 ≠0, μ 3,4 ≠0, μ 1 μ 3 时,本征值具有物理意义 [23] 此时,辛本征解向量为

(20)

根据式(20),方程(15)的解可以表示为

(21)

其中 d j 为待定系数

6 自 由 振 动

为确定纳米梁的自由振动频率,通解(21)可以进一步表示为等价形式,即

(22)

这里, 为待定系数,它们之间满足

A θj = κ 11 A wn A Mj = κ 12 A wj A Qj = κ 13 A wn ( j =1,2; n =2,1),

(23a)

A θj = κ 21 A wn A Mj = κ 22 A wj A Qj = κ 23 A wn ( j =3,4; n =4,3),

(23b)

其中

下面以两端简支边界条件为例,推导该双纳米梁系统自由振动的频率方程

SS

将通解(22)代入边界条件(16)(g=S),可得

ZA w = 0 ,

(24)

这里, A w = A w 1 , A w 2 , A w 3 , A w 4 T Z 矩阵为自振频率 ω 的函数 根据方程(24)的非零解条件det( Z )=0,可得两端简支非局部双功能梯度纳米梁系统的频率方程:

f 1 ( ω )=0,

(25)

其中

同理,其他边界条件下双功能梯度纳米梁系统的频率方程可推导如下:

CC

(26)

CS

κ 11 f 3 ( ω )- κ 21 f 4 ( ω )=0;

(27)

CF

κ 21 κ 12 κ 14 + κ 22 κ 11 κ 24 +( κ 11 κ 22 κ 14 + κ 21 κ 12 κ 24 ) f 1 ( ω )-

( κ 11 κ 12 κ 24 + κ 21 κ 22 κ 14 ) f 2 ( ω )=0;

(28)

SF

κ 12 κ 24 f 4 ( ω )- κ 22 κ 14 f 3 ( ω )=0;

(29)

FF

(30)

其中

κ 14 = κ 13 + μ ( 2 - φK - K w -i ) ξ 2

7 数 值 算 例

为方便分析和计算,数值算例中均采用无量纲参 为转动惯量

7 . 1 算例1

为验证本文提出方法的正确性和精确性,本小节将计算结果与现有文献数据进行对比 首先,考虑一个两端简支的非局部功能梯度Timoshenko梁,其材料参数为

E 1 =390 GPa, E 2 =210 GPa, ρ 1 =3 960 kg/m 3 ρ 2 =7 800 kg/m 3

ν 1 = 0.24, ν 2 = 0.30, K s =5/6

选用矩形截面,其宽与高均为 b = h =1 000 nm,长为 a =10 000 nm 表1给出了该纳米梁在不同非局部参数和长宽比下对应的无量纲基频 从表中数据可以看出,本文的计算结果与文献[12]的结果吻合得非常好,最大误差仅为1.12% 其次,考虑一个由两端简支的各向同性非局部Timoshenko梁组成的双纳米梁系统,其计算参数为

b =1 nm, h =1 nm, ρ =1 kg/m 3 E =30 MPa, K s =5/6,

表2给出了不同非局部参数下双纳米梁系统的前六阶同向振动频率 从对比结果可以发现,本文的计算结果不仅与现有的非局理论模型结果一致,还与经典连续介质力学的结果(ANSYS, ξ 2 =0)吻合较好 上述表1和表2的数据对比表明,本文提出的辛方法适用于双功能梯度纳米系统的自由振动分析,并可以得到精度较高的结果

表1 两端简支功能梯度纳米梁的基频

Table 1 Fundamental frequencies of an FG nanobeam with 2 ends simply supported

k(ξ2/10-12)/m2a/h20presentref. [12]50presentref. [12]100presentref. [12]009.828 19.829 69.862 99.863 19.867 99.868 019.376 49.377 79.409 59.409 79.414 39.414 328.981 78.982 99.013 49.013 69.018 09.018 038.633 18.634 18.663 48.663 68.667 88.667 80.208.679 18.660 08.708 78.689 58.712 98.693 818.280 18.262 08.308 38.290 18.312 48.294 127.931 67.914 07.958 67.941 17.962 47.944 937.623 87.606 87.649 67.632 77.653 37.636 5107.074 66.967 67.099 06.991 77.102 66.995 216.749 46.647 36.772 76.670 36.776 06.673 626.465 36.367 46.487 66.389 56.490 86.392 736.214 36.120 26.235 76.141 46.238 86.144 4505.983 25.917 26.005 25.938 96.008 45.942 115.708 15.645 25.729 25.665 95.732 25.668 925.467 95.407 55.488 05.427 45.490 95.430 235.255 65.197 55.274 95.216 65.277 75.219 4

表2 两端简支双纳米梁系统的前六阶同向振动频率

Table 2 First 6 in-phase natural frequencies of a double-nanobeam system with 2 ends simply supported

length-width ratioξ2/nm2mode(data in ref. [24] given in parentheses; data in ANSYS given in brackets)123456a/h=1009.707 537.096 278.154 7128.666 0185.318 3245.832 3(9.744 3)(36.840 6)[9.712 2][37.162 5][78.447 5][129.451 7][186.957 0][248.750 2]19.261 231.410 556.875 380.117 4127.792 8169.947 8(9.293 1)(31.236 6)28.871 327.730 346.903 463.096 894.502 0130.078 6(8.899 4)(27.587 0)38.526 925.099 640.825 453.716 383.519 0115.521 8(8.551 7)(24.972 7)a/h=2009.828 138.829 985.661 9148.384 6224.779 4312.618 9(9.838 1)(38.964 5)(85.748 3)[9.832 7][38.900 7][86.006 5][149.416 2][227.144 4][317.191 5]19.709 137.044 877.489 1125.642 2176.775 3227.501 3(9.718 7)(37.161 4)(77.529 1)29.594 235.485 371.282 7110.921 3150.398 7187.613 6(9.603 6)(35.587 5)(71.292 2)39.483 434.107 466.362 8100.398 6133.134 9163.301 6(9.492 4)(34.197 9)(66.351 5)

续表2

length-width ratioξ2/nm2mode(data in ref. [24] given in parentheses; data in ANSYS given in brackets)123456a/h=5009.862 939.371 988.290 7156.235 2242.686 9347.011 1(9.864 5)(39.397 6)(88.414 7)[9.867 5][39.444 7][88.656 3][157.383 7][245.453 9][352.687 8]19.843 539.064 786.762 8151.522 9231.530 2324.703 5(9.845 1)(39.089 7)(86.880 4)29.824 238.764 585.311 6147.212 9221.783 0306.210 1(9.825 8)(38.789 0)(85.423 3)39.805 038.471 283.930 8143.250 9213.171 5290.554 6(9.806 6)(38.495 1)(84.037 2)

7 . 2 算例2

本小节将针对尺度效应、功能梯度材料指数、介质黏弹性参数进行详细讨论 考虑一个放置于黏弹性介质中的双功能梯度纳米梁系统,其计算参数为 E 1 =390 GPa, E 2 =210 GPa, ρ 1 =3 960 kg/m 3 ρ 2 =7 800 kg/m 3 ν 1 = 0.24, ν 2 = 0.30, h = b =1 000 nm, L =10 000 nm, K s =5/6,

表3 不同端部条件下双纳米梁系统的前四阶同向振动频率

Table 3 First 4 in-phase natural frequencies of a functionally graded double-nanobeam

system with various end conditions

end condition(ε2/10-12)/m2mode1234SS06.810 7+3.570 8i26.695 4+3.503 0i56.382 8+3.425 9i92.962 0+3.360 8i16.480 6+3.570 8i22.601 6+3.502 8i41.121 7+3.424 4i58.219 2+3.356 0i26.191 6+3.570 8i19.956 5+3.502 5i33.985 0+3.422 8i46.034 8+3.352 1iCS010.573 4+3.568 3i32.646 4+3.505 0i63.353 6+3.436 7i99.997 4+3.378 7i110.005 2+3.565 8i27.388 0+3.494 9i45.856 4+3.418 1i62.339 6+3.354 1i29.516 4+3.563 7i24.065 5+3.489 1i37.814 8+3.409 6i49.278 3+3.343 5iCC015.039 2+3.569 8i38.782 7+3.509 6i70.233 0+3.449 1i106.790 6+3.396 3i114.196 3+3.562 2i32.247 9+3.486 9i50.483 2+3.413 0i66.263 1+3.353 4i213.476 4+3.556 0i28.176 9+3.474 7i41.542 8+3.397 6i52.336 8+3.336 8iCF01.942 8+3.585 2i14.991 3+3.513 2i39.648 5+3.430 3i72.048 0+3.349 6i11.956 6+3.584 9i14.052 1+3.512 6i32.824 8+3.432 3i51.269 3+3.357 7i21.970 6+3.584 5i13.247 0+3.512 0i28.682 3+3.431 5i42.075 3+3.354 9iFF015.482 8+3.541 9i40.486 2+3.463 5i73.979 0+3.391 2i113.065 5+3.335 3i114.035 2+3.541 6i31.844 9+3.462 8i49.976 5+3.388 4i65.801 4+3.328 3i212.940 6+3.541 3i27.096 3+3.462 1i40.319 0+3.385 7i51.206 1+3.323 0iSF010.718 3+3.522 7i33.319 8+3.428 4i64.989 5+3.337 4i102.877 9 +3.265 8i110.129 1+3.525 1i27.805 2+3.440 5i46.590 2+3.362 3i63.331 9+3.301 8i29.625 3+3.526 9i24.379 2+3.445 6i38.341 3+3.366 9i50.003 6+3.302 9i

首先,考虑尺度效应对该纳米梁系统自由振动的影响 表3和表4分别给出了不同端部条件和非局部参数对应的同向振动和反向振动对应的自由振动频率 计算过程中,黏弹性介质参数为 功能梯度指数为 k =1 从表中数据可以发现,除了CF边界以外,所有边界对应的各阶频率的实部均随着非局部参数的增加而减小,而非局部参数对所有边界对应的频率的虚部影响都很小 该现象与单纳米梁自由振动的特点一致 [25] 此外,在选取相同非局部参数时,反向振动的自振频率明显大于同向振动的结果,该现象应为双纳米梁间相互作用力引起的,相当于改变了纳米梁系统的等效结构刚度 为了直观展示该类双纳米梁系统的自由振动形式,图2和图3分别给出了两端简支功能梯度双纳米梁系统的前四阶同向振动和反向振动的模态

表4 不同端部条件下双纳米梁系统的前四阶反向振动频率

Table 4 First 4 out-of-phase natural frequencies of a functionally graded double-nanobeam system with various end conditions

end condition(ε2/10-12)/m2mode1234SS07.527 7+3.570 8i26.883 7+3.503 0i56.470 2+3.425 9i93.014 0+3.360 7i17.230 4+3.570 8i22.823 6+3.502 8i41.241 4+3.424 4i58.302 1+3.356 0i26.972 5+3.570 8i20.207 5+3.502 5i34.129 6+3.422 8i46.139 5+3.352 0iCS011.048 5+3.568 3i32.800 6+3.505 0i63.431 6+3.436 7i100.046 0+3.378 7i110.505 7+3.565 8i27.571 1+3.494 9i45.963 6+3.418 1i62.417 0+3.354 1i210.041 0+3.563 7i24.273 3+3.489 1i37.944 4+3.409 5i49.375 9+3.343 5iCC015.377 1+3.569 8i38.912 7+3.509 6i70.303 6+3.449 1i106.836 4+3.396 3i114.553 0+3.562 2i32.403 2+3.486 9i50.580 5+3.413 0i66.335 9+3.353 4i213.851 0+3.556 0i28.353 9+3.474 7i41.660 4+3.397 5i52.428 5+3.336 8iCF03.754 5+3.585 2i15.324 9+3.513 2i39.772 9+3.430 3i72.114 9+3.349 6i13.761 5+3.584 8i14.407 5+3.512 5i32.975 0+3.432 3i51.363 5+3.357 6i23.768 7+3.584 4i13.623 3+3.512 0i28.854 1+3.431 5i42.189 9+3.354 9iFF015.808 7+3.541 9i40.609 1+3.463 5i74.045 0+3.391 2i113.108 0+3.335 3i114.393 8+3.541 6i32.001 0+3.462 8i50.074 0+3.388 4i65.874 2+3.328 3i213.328 7+3.541 3i27.279 6+3.462 0i40.439 7+3.385 7i51.299 4+3.322 9iSF011.181 4+3.522 7i33.467 6+3.428 4i65.063 3+3.337 4i102.923 6+3.265 8i110.618 3+3.525 1i27.982 7+3.440 5i46.694 0+3.362 3i63.406 9+3.301 8i210.139 0+3.526 9i24.581 8+3.445 5i38.467 5+3.366 8i50.098 6+3.302 8i

(a) 第一阶(b) 第二阶(c) 第三阶(d) 第四阶
(a) The 1st order(b) The 2nd order(c) The 3rd order(d) The 4th order
图2 前四阶简支同向振动模态
Fig. 2 First 4 in-phase vibration mode shapes

(a) 第一阶(b) 第二阶(c) 第三阶(d) 第四阶
(a) The 1st order(b) The 2nd order(c) The 3rd order(d) The 4th order
图3 前四阶简支反向振动模态
Fig. 3 First 4 out-of-phase vibration mode shapes

其次,表5分析了功能梯度指数对双功能梯度纳米梁系统自由振动的影响 这里,计算参数为 端部条件选取为两端简支 表5中数据变化趋势与表3和表4类似,各阶振动频率的实部和虚部均随着 k 的增加而减小 该现象说明,功能梯度指数对该类功能梯度双纳米梁系统具有显著影响,是该类纳米结构设计中不可忽略的关键因素

最后,表6分析了地基黏弹性参数对双功能梯度纳米梁系统的作用规律 计算参数为 ξ 2 =10 -12 m 2 k =1 从计算结果可以看出,地基黏弹性参数对各阶频率的虚部影响较大,虚部数值均随着 的增大表现出明显的上升趋势,这与其表征阻尼的物理意义相符 地基黏弹性参数对基频的实部影响较大,对高阶频率的实部影响较小 这说明黏弹性地基系数仅对基频造成影响

表5 不同功能梯度指数对应两端简支功能梯度双纳米梁系统的前四阶振动频率

Table 5 First 4 natural frequencies of an SS functionally graded double-nanobeam system with various power-law indexes

kmode1234in-phase vibration09.005 7+4.961 1i31.423 3+4.866 3i57.208 0+4.756 4i81.046 1+4.659 9i0.57.126 7+3.924 9i24.868 1+3.847 6i45.279 0+3.757 9i64.157 0+3.678 6i16.480 6+3.570 8i22.601 6+3.502 8i41.121 7+3.424 4i58.219 2+3.356 0i55.475 7+3.022 4i19.047 0+2.969 7i34.532 8+2.911 0i48.708 4+2.862 2iout-of-phase vibration010.047 2+4.961 1i31.731 5+4.866 2i57.374 0+4.756 4i81.161 0+4.659 9i0.57.950 7+3.924 9i25.111 8+3.847 6i45.410 2+3.757 8i64.247 7+3.678 6i17.230 4+3.570 8i22.823 6+3.502 8i41.241 4+3.424 4i58.302 1+3.356 0i56.111 0+3.022 4i19.235 9+2.969 7i34.635 3+2.910 9i48.779 9+2.862 1i

表6 不同黏性弹性系数对应两端简支功能梯度双纳米梁系统的前四阶振动频率

Table 6 First 4 natural frequencies of an SS functionally graded double-nanobeam system with various viscoelastic coefficients

c-mode1234in-phase vibration07.399 222.871 341.263 858.315 117.390 6+0.357 1i22.868 7+0.350 3i41.262 4+0.342 4i58.314 2+0.335 6i57.180 6+1.785 4i22.804 2+1.751 4i41.228 3+1.712 2i58.291 2+1.678 0i106.480 6+3.570 8i22.601 6+3.502 8i41.121 7+3.424 4i58.219 2+3.356 0iout-of-phase vibration08.064 123.090 841.383 158.397 918.056 2+0.357 1i23.088 1+0.350 3i41.381 6+0.342 4i58.397 0+0.335 6i57.863 9+1.785 4i23.024 3+1.751 4i41.347 7+1.712 2i58.374 0+1.678 0i107.230 4+3.570 8i22.823 6+3.502 8i41.241 4+3.424 4i58.302 1+3.356 0i

8 结 论

本文针对一类双功能梯度纳米梁系统的自由振动问题提出一种全新的解析求解方法 利用Eringen提出的非局部Timoshenko梁理论,以经典变量表示该类双纳米梁系统振动控制微分方程 通过变量代换,将原问题分解为同向振动和反向振动,从而进一步获得统一形式的控制方程 在Hamilton体系下,基本未知量为原变量及其对偶变量组成的全状态向量 传统Lagrange体系下的高阶偏微分振动控制方程转化为一组常微分方程组 因此,分离变量法得以应用,原自由振动问题归结为Hamilton矩阵的本征值和本征解求解问题,并可以直接获得6种端部条件下的解析频率方程和振动模态 对比算例验证了本文提出方法的正确性与精确性,并针对关键影响参数进行了系统分析 研究结果表明,功能梯度指数对双功能梯度纳米梁系统自振频率的实部和虚部影响效果显著,而尺度效应仅对自振频率实部影响较大,介质黏弹性参数仅对自振频率虚部作用效果明显 上述结论可以为该类纳米梁系统的设计提供科学指导,并为其安全评估提供可靠依据 此外,本文提出的辛方法可以推广至其他纳米结构的动力分析中,为相关领域提供解析求解方法

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A Symplectic Approach for Free Vibration of Functionally Graded Double-Nanobeam Systems Embedded in Viscoelastic Medium

ZHOU Zhenhuan, LI Yuejie, FAN Junhai, SUI Guohao,ZHANG Junlin, XU Xinsheng

( State Key Laboratory of Structural Analysis for Industrial Equipment ( Dalian University of Technology ); Department of Engineering Mechanics , International Research Center for Computational Mechanics , Dalian University of Technology , Dalian , Liaoning 116024, P . R . China )

Abstract: A new analytical approach was proposed for free vibration of functionally graded (FG) double-nanobeam systems (DNBSs) embedded in viscoelastic medium under the framework of symplectic mechanics and the nonlocal Timoshenko beam theory. In the Hamiltonian system, the dual variables of the displacement and the rotation angle are the generalized shear force and bending moment, respectively. The high-order governing partial differential equations in the classical Lagrangian system were simplified into a set of ordinary differential equations through introduction of an unknown vector composed of the fundamental variables and their dual variables. The free vibration of DNBSs was finally reduced to an eigenproblem in the symplectic space. Analytical frequency equations and vibration mode functions were directly obtained with the symplectic eigensolutions and boundary conditions. Numerical results verify the accuracy and efficiency of the presented method. A systematic parametric study on the small size effect, the interaction between the double nanobeams and the viscoelastic foundation influence, was also provided.

Key words: Hamiltonian system; symplectic method; functionally graded double-nanobeam system; free vibration; analytical solution

Foundation item: The National Natural Science Foundation of China(11672054); The National Basic Research Program of China(973 Program)(2014CB046803); The National Key R&D Program of China(2016YFB0201600)

ⓒ 应用数学和力学编委会,ISSN 1000-0887

文章编号 1000-0887(2018)10-1159-13

作者简介: 周震寰(1983—),男,副教授,博士(通讯作者. E-mail: zhouzh@dlut.edu.cn ).

基金项目: 国家自然科学基金(11672054);国家重点基础研究发展计划(973计划)(2014CB046803);国家重点研发计划(2016YFB0201600);辽宁省自然科学基金(20470540186);中央高校基本科研业务费(DUT17LK57)

修订日期: 2018-05-18

收稿日期: 2018-04-23;

DOI: 10.21656/1000-0887.390130

文献标志码: A

中图分类号 O326

引用本文 / Cite this paper: 周震寰, 李月杰, 范俊海, 隋国浩, 张俊霖, 徐新生. 双功能梯度纳米梁系统振动分析的辛方法[J]. 应用数学和力学, 2018, 39 (10): 1159-1171.ZHOU Zhenhuan, LI Yuejie, FAN Junhai, SUI Guohao, ZHANG Junlin, XU Xinsheng. A symplectic approach for free vibration of functionally graded double-nanobeam systems embedded in viscoelastic medium[J]. Applied Mathematics and Mechanics , 2018, 39 (10): 1159-1171.