二维功能梯度壁板热颤振本征问题的精确解

代林桐; 邢誉峰

doi:10.13700/j.bh.1001-5965.2020.0351

二维功能梯度壁板热颤振本征问题的精确解

doi: 10.13700/j.bh.1001-5965.2020.0351

代林桐,
邢誉峰^,

北京航空航天大学固体力学所, 北京 100083

基金项目:

国家自然科学基金 11672019

详细信息

通讯作者:
邢誉峰, E-mail: xingyf@buaa.edu.cn

中图分类号: V221⁺.3;TB553
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- 被引次数: 0
出版历程
- 收稿日期: 2020-07-21
- 录用日期: 2020-08-21
- 网络出版日期: 2021-10-20

Exact solutions of thermal flutter of two-dimensional functionally graded panel

DAI Lintong,
XING Yufeng^,

Institute of Solid Mechanics, Beihang University, Beijing 100083, China

Funds:

National Natural Science Foundation of China 11672019

More Information

Corresponding author: XING Yufeng, E-mail: xingyf@buaa.edu.cn

摘要

摘要:
为了得到二维功能梯度壁板热颤振的精确解并揭示颤振机理，根据经典薄板理论及一阶活塞理论，建立了超声速气流下二维功能梯度壁板的本征控制微分方程并求得了精确解，根据得到的本征根对颤振机理进行了分析。针对功能梯度材料（FGM）的不同体积分数，分别研究了壁板在恒温场及非线性温度场下的颤振边界随马赫数的变化规律，并比较了2种温度场下的结果。通过分析简支、固支及其组合边界情况下的壁板颤振特性，从数学角度发现颤振现象的发生是由于挠度的一阶导数导致刚度非对称，且功能梯度材料能够有效提高热环境下壁板的颤振边界，同时利用ABAQUS软件对功能梯度壁板的振动特性进行了模拟，进一步验证了所提方法的有效性。
- 超声速 /
- 功能梯度材料(FGM) /
- 热颤振 /
- 本征根 /
- 精确解
Abstract:
For achieving the thermal flutter exact solutions of two-dimensional functionally graded panel and revealing the mechanism of the thermal flutter, based on the classical thin plate theory and the first-order piston theory, the characteristic governing differential equation of two-dimensional functionally graded panel in supersonic flow is established and exact solutions are obtained. Through the analysis of the eigenvalues, the mechanism of the panel flutter is investigated. According to different volume fraction of Functionally Graded Materials (FGM), the flutter boundary changes with Mach number in constant temperature field and nonlinear temperature field are studied respectively, and the results in two temperature fields are compared. By analyzing the flutter characteristics of panels with simply supported, fixed and integrated edges, it can be concluded that the flutter phenomenon is caused by the first-order derivative of deflection which leads to the asymmetry of system stiffness, and FGM can effectively improve the flutter boundary of the panel in the thermal environment. Meanwhile, the vibration properties of the FGM panel is simulated with ABAQUS, further validating the effectiveness of the present method.
- supersonic /
- Functionally Graded Materials (FGM) /
- thermal flutter /
- eigenvalue /
- exact solution

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图 1 两端简支壁板几何模型

Figure 1. Geometric model of simply supported panel at both ends

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图 2 临界动压值随梯度指数变化曲线

Figure 2. Critical dynamic pressure versus gradient index

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图 3 临界颤振频率随温度的变化曲线

Figure 3. Critical flutter frequency versus temperature

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图 4 两种热环境下频率随马赫数的变化曲线

Figure 4. Frequency versus Mach number in two thermal environments

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表 1 简支和固支边界条件

Table 1. Boundary conditions for simple support and clamp

BCs(边界条件)	x=0或x=a
简支(S)	w=0，
固支(C)	w=0,

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表 2 二维壁板颤振频率方程和本征函数

Table 2. Eigensolutions of two-dimensional panel flutters

BCs	频率方程	本征函数的系数
简支-简支(SS) ϕ(0)=0 ϕ(1)=0 ϕ″(0)=0 ϕ″(1)=0
固支-固支(CC) ϕ(0)=0 ϕ(1)=0 ϕ′(0)=0 ϕ′(1)=0
简支-固支(SC) ϕ(0)=0 ϕ″(0)=0 ϕ(1)=0 ϕ′(1)=0	4ϑα₁β₁sinh 2ϑ-β₁(4ϑ²-α₁²-β₁²)sin α₁cosh β₁-α₂(4ϑ²+α₁²+β₁²)cos α₁sinh β₁=0

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表 3 功能梯度材料弹性常数

Table 3. Elastic constants of FGM

材料名称	组成成分	E/GPa	υ	ρ/(kg·m^-3)
陶瓷金属	Si₃N₄(氮化硅陶瓷)	322	0.24	2 370
陶瓷金属	SUS304(不锈钢)	207	0.32	8 166

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表 4 分层为5层时密度和弹性模量

Table 4. Density and modulus of elasticity for 5 layers

坐标方向	厚度方向坐标/(10^-4m)	密度/(kg·m^-3)	弹性模量/(10¹¹Pa)
z	-8.000	8 165.942	2.070
z	-4.000	8 151.916	2.073
z	0	7 984.875	2.106
z	4.000	7 191.866	2.263
z	8.000	4 743.520	2.749

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表 5 分层为10层时密度和弹性模量

Table 5. Density and modulus of elasticity for 10 layers

坐标方向	厚度方向坐标/(10^-4m)	密度/(kg·m^-3)	弹性模量/(10¹¹Pa)
z	-9.000	8 165.998	2.070
z	-7.000	8 165.560	2.070
z	-5.000	8 160.340	2.071
z	-3.000	8 135.558	2.076
z	-1.000	8 059.047	2.091
z	1.000	7 874.296	2.127
z	3.000	7 493.496	2.203
z	5.000	6 790.582	2.343
z	7.000	5 594.284	2.580
z	9.000	3 681.166	2.959

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表 6 不同边界条件下FGM板频率

Table 6. Frequency of FGM plate under different boundary conditions

边界条件	模态阶数	频率/Hz
边界条件	模态阶数	精确解	ABAQUS(5层)	ABAQUS(10层)
SS	1	131.679 3	129.345 7	131.010 7
SS	2	523.997 7	517.596 2	521.925 4
CC	1	296.864 6	293.355 6	295.806 1
CC	2	818.792 2	808.520 3	815.306 1
SC	1	204.444 7	202.048 4	203.851 7
SC	2	663.293 8	654.707 9	660.551 3

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表 7 本文解与Galerkin方法结果的对比

Table 7. Comparison of result between present method and Galerkin method

方法	固有频率(Ma=2)		颤振参数
方法	ω₁/Hz	ω₂ /Hz	ω_f /Hz	Ma_f
Galerkin	650.725 8	2 130.318 9	1 773.443 7	7.177 4
本文	651.975 4	2 130.663 1	1 759.928 3	6.989 6

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表 8 不同边界下FGM板的临界颤振频率和临界动压的比较

Table 8. Comparison of flutter frequency and critical dynamic pressure of FGM plate under different boundary conditions

T/K	n	SS		CC		SC
T/K	n	λ^*	ω^*	λ^*	ω^*	λ^*	ω^*
T_t=300 T_b=300	1	434.22	26.0	788.45	41.7	594.19	33.3
	5	388.51	21.1	719.89	34.1	548.49	27.3
	50	354.23	19.0	651.33	30.6	491.35	24.5
T_t=500 T_b=300	1	399.94	24.8	754.17	40.7	571.34	32.4
	5	365.66	20.2	685.61	33.2	514.21	26.3
	50	319.95	17.9	617.05	29.7	457.07	23.4
T_t=500 T_b=500	1	365.66	23.4	708.46	39.3	525.63	30.9
	5	331.38	18.9	639.90	31.9	479.93	25.1
	50	285.67	16.6	571.34	28.5	422.79	22.3

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