基于虛擬材料復模量非均勻分布的螺栓連接薄板結構半解析建模

劉曉峰; 孫偉; 孫悅

doi:10.13374/j.issn2095-9389.2020.04.20.005

基于虛擬材料復模量非均勻分布的螺栓連接薄板結構半解析建模

doi: 10.13374/j.issn2095-9389.2020.04.20.005

劉曉峰^{1, 2},
孫偉^{1, 2, ,},
孫悅^{1, 2}

1.
東北大學機械工程與自動化學院，沈陽 110819
2.
東北大學航空動力裝備振動及控制教育部重點實驗室，沈陽 110819

基金項目: 中央高校基本科研業務費專項資金資助項目（N180312012）

詳細信息

通訊作者:
E-mail：weisun@mail.neu.edu.cn

中圖分類號: TH131.3；O327
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出版歷程
- 收稿日期: 2020-04-20
- 刊出日期: 2021-06-25

Semianalytical modeling of a bolted thin plate structure based on nonuniform distributions of the complex modulus of a virtual material

LIU Xiao-feng^{1, 2},
SUN Wei^{1, 2
, ,},
SUN Yue^{1, 2}

1.
School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819, China
2.
Key Laboratory of Vibration and Control of Aero-Propulsion Systems, Ministry of Education of China, Northeastern University, Shenyang 110819, China

More Information

Corresponding author: E-mail: weisun@mail.neu.edu.cn

摘要

摘要: 基于非均勻分布的虛擬材料模擬螺栓連接薄板搭接部分的力學特性，其中虛擬材料的材料參數用復模量表示，可直接生成復剛度矩陣以表示搭接部分的剛度及阻尼特性，省卻了常規建模中生成結合部阻尼矩陣的步驟，在保證模型精確性的基礎上簡化了建模流程，以此建立了螺栓連接薄板結構的半解析模型并對其進行了動力學分析。首先描述了建模理念，將虛擬材料分別假定了三種復模量非均勻分布形式模擬螺栓搭接部分的力學特性，提出用反推辨識技術確定虛擬材料儲能模量與耗能模量的方法。接著，基于能量法并用正交多項式假定模態，推導了螺栓連接薄板的半解析分析模型，并創新性地給出了求解半解析模型任意錘擊點與拾振點處頻響函數的公式。最后，以一個具體的螺栓連接薄板結構為對象進行了實例研究，結果表明：用所創建的半解析模型計算出的各階仿真固有頻率與實驗測得的各階固有頻率的誤差均在5%以內，計算得到的各階仿真模態振型以及頻響函數曲線與實測值均較為接近，從而證明了利用復模量非均勻分布的虛擬材料模擬螺栓搭接部分可有效簡化螺栓結合部建模，亦可達到較高的仿真計算精度。
- 虛擬材料 /
- 復模量 /
- 非均勻分布 /
- 螺栓連接薄板 /
- 半解析建模
Abstract: The simulation of bolt joints affects the analysis accuracy of the dynamic characteristics of the whole structure in the dynamic modeling of bolted connection structures. In this study, the mechanical properties of the bolted thin-plate lap joint were simulated based on a nonuniformly distributed virtual material. The parameters of the virtual material were expressed based on a complex modulus, and the complex stiffness matrix can be directly generated to express the stiffness and damping characteristics of the lap joint. The steps used to generate a joint damping matrix in conventional modeling were omitted, and the modeling process was simplified to ensure model accuracy. We established a semianalytical model of a bolted thin plate structure to enable its dynamic analysis. In this study, we first described the modeling concept. The virtual material was assumed to have three types of nonuniform complex modulus distributions to simulate the mechanical properties of the bolted lap joint. We proposed a method for determining the storage modulus and energy dissipation modulus of the virtual material using a reverse identification technique. Based on the energy method and the assumed modes of orthogonal polynomials, we derived a semianalytical model of bolted thin plates and develop an innovative formula for solving the frequency response function at any hammering point and the vibration point of the semianalytical model. Finally, we conducted a case study on a bolted thin plate structure. Results show that the deviation between the simulated natural frequencies calculated using the semianalytical model and the experimental natural frequencies are less than 5%. Further, the calculated model shapes and frequency-response-function curves are close to those obtained based on the measured values. These results prove that a virtual material with a nonuniform complex modulus distribution can effectively simplify the modeling of a bolted joint and achieve high simulation accuracy.
- virtual material /
- complex modulus /
- nonuniform distribution /
- bolted thin plate /
- semianalytical modeling

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圖 1 虛擬材料儲能模量非均勻分布。（a）三維結構示意；（b）二維結構示意

Figure 1. Nonuniform distribution of the storage modulus of a virtual material: (a) 3D structure diagram; (b) 2D structure diagram

下載: 全尺寸圖片幻燈片

圖 2 虛擬材料儲能模量（耗能模量）辨識流程

Figure 2. Identification process of the storage modulus (loss modulus) of the virtual material

下載: 全尺寸圖片幻燈片

圖 3 螺栓連接薄板結構實物圖

Figure 3. Real structure of the bolted thin-plate structure

下載: 全尺寸圖片幻燈片

圖 4 實測與仿真頻響函數對比

Figure 4. Comparison of the frequency response functions obtained based on the measured and simulated data

下載: 全尺寸圖片幻燈片

表 1 螺栓連接薄板結構中板的相關材料及幾何參數

Table 1. Material and geometric parameters of the plate in a bolted thin plate structure

Length/ mm	Width/ mm	Thickness/ mm	Elastic modulus/ GPa	Density/ （kg·m^?3）	Poisson’s ratio
120	120	2.4	200	7930	0.3

下載: 導出CSV

表 2 反推法辨識獲得的各分布狀態下的虛擬材料儲能模量

Table 2. Storage modulus of the virtual material for each distribution obtained using the inverse identification technique

Complex modulus distribution of virtual materials	Maximum storage modulus, E_D/Pa
Linear distribution	7.6 × 10¹⁰
Parabola distribution	1.0835 × 10¹¹
Sinusoidal distribution	1 × 10¹¹
Uniform distribution	2.195 × 10¹⁰

下載: 導出CSV

表 3 各虛擬材料儲能模量分布模型固有頻率與實驗固有頻率對比

Table 3. Comparison of the natural frequencies obtained using the virtual-material storage modulus distribution model and the experiment

Order	Natural frequencies of text/Hz	Natural frequencies of parabola distribution/Hz	Error/%	Natural frequencies of linear distribution/Hz	Error/%	Natural frequencies of sinusoidal distribution/Hz	Error/%	Natural frequencies of uniform distribution/Hz	Error/%
1	55.537	56.933	2.51364	56.931	2.510038	56.9321	2.512019	56.9319	2.511659
2	197.793	189.6264	?4.12866	192.3208	?2.76663	189.9875	?3.9463	198.1551	0.18307
3	311.441	318.5184	2.272469	332.0858	6.6288	320.419	2.882729	357.1359	14.67209
4	619.969	616.2225	?0.6043	621.7815	0.292353	616.9306	?0.49009	635.7655	2.54795
5	881.62	865.2161	?1.86065	877.6824	?0.44663	866.8424	?1.67619	905.9436	2.758966

下載: 導出CSV

表 4 各非均勻分布形式中仿真與實驗前5階固有頻率的均方根誤差

Table 4. Root mean square error (RMSE) of the first five natural experimental and simulated frequencies in various nonuniform distributions

Nonuniform distribution	RMSE value
Linear distribution	3.411172
Parabola distribution	2.543903
Sinusoidal distribution	2.578504

下載: 導出CSV

表 5 實驗與仿真前5階振型對照

Table 5. Comparison of the first five experimental and simulated vibration modes

Order	Vibration modes by testing	Vibration modes by simulation
1
2
3
4
5

下載: 導出CSV

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