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Design of missile vibration spectrum based on inverse pseudo-excitation method
LYU Weimin, XIAO Yang, FANG Dengjian
The 7 th Department, Naval Aeronautical Engineering Institute, Yantai 264001, China
Received: 2016-04-26; Accepted: 2016-07-16; Published online: 2016-08-26 15:21
Corresponding author. LYU Weimin, E-mail:lwm_yt@sina.com
Abstract: Aimed at the low design precision and larger gap with actual load profile in missile life extension test, a missile vibration spectrum design based on inverse pseudo-excitation method was put forward. First, four-degree-of-freedom vibration model of missile-suspension system was established, and the system frequency response was inferred. Then, the time-domain response data was converted into the response matrix of power spectral density by means of Fourier transform, and harmonic response was constructed based on the decomposition of response matrix of power spectral density. Finally, the system excitation power spectrum was gained based on pseudo-excitation method. At the same time, the effect of response noise and system damping on identification precision was studied. The experimental results show that the inverse pseudo-excitation method has high identification precision and good robustness. When there is no noise, the identification error is 2.39%, and it reaches 3.21% when there is 30% noise. The design means of vibration spectrum is also applicable to other equipment.
Key words: missile vibration spectrum     inverse pseudo-excitation method     power spectral density     damping     identification precision

1 逆虚拟激励法基本原理

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)
2 导弹-悬架系统四自由度振动模型

 图 1 导弹-悬架系统四自由度振动模型 Fig. 1 Four-degree-of-freedom vibration model of missile-suspension system

 (8)

3 基于逆虚拟激励法的振动谱识别算法 3.1 数据预处理

 (9)

Z1(t)的自相关函数为

 (10)

 (11)

 (12)

3.2 振动谱识别模型

 (13)

 (14)

 (15)

 (16)

 (17)

 (18)

 (19)

 (20)

3.3 振动谱设计流程

1) 根据式(15) 建立导弹-悬架系统动力学方程，计算频率响应矩阵H

2) 对实测得到的导弹时域振动响应数据进行预处理，按式(10)~式(12) 计算得到系统位移响应功率谱矩阵Syy

3) 将响应功率谱矩阵Syy按式(2) 进行分解，构造如式(16) 的虚拟简谐响应

4) 根据式(17) 计算各虚拟激励，并按式(18) 第2个等式计算激励谱矩阵

5) 采用一定的功率谱处理方法，求得激励谱矩阵上限曲线，设计出导弹运输振动试验谱。

4 实验仿真 4.1 仿真流程

 变量 m1/kg m2/kg m3/kg k1/(kN·m-1) k2/(kN·m-1) k3/(kN·m-1) k4/(kN·m-1) c1/(Ns·m-1) c2/(Ns·m-1) l1/m l2/m A/m wo/(rad·s-1) v/(m·s-1) 数值 1 000 40 45 18 24 190 220 350 600 3 2 20 30 50

4.2 仿真结果分析

 图 2 运输车前后轮及导弹质心垂直位移PSD Fig. 2 Vertical displacement PSD of front-rear wheels of vehicle and missile centroid

1) 零噪声识别误差分析。零噪声条件下，采用逆虚拟激励法，由上述响应功率谱反求得到运输车前后轮所受激励谱，并与给定激励谱进行对比，如图 3图 4所示。可知，前轮识别得到的功率谱在低频段误差较小，中高频段( > 100Hz)由于系统共振误差出现一定波动，Error=2.39%；后轮识别得到的功率谱与实际功率谱基本吻合，Error=1.96%。由此可见，根据逆虚拟激励法，由系统响应反求系统激励的方法是可行的，具有较高精度。

 图 3 前轮识别功率谱密度及误差曲线 Fig. 3 Identified PSD and identification error curves of front wheels
 图 4 后轮识别功率谱密度及误差曲线 Fig. 4 Identified PSD and identification error curves of rear wheels

2) 加入30%响应噪声识别误差分析。以前轮为例，利用MATLAB中“randn”命令，在系统响应数据中加入30%的高斯白噪声，得到前轮识别功率谱及其误差曲线，如图 5所示。可知，在低频段识别精度较高，中高频段( > 80Hz)由于高频噪声影响误差较大，Error=3.21%，证明该方法具有一定抗噪能力。

 图 5 30%噪声条件下前轮识别功率谱密度及误差曲线 Fig. 5 Identified PSD and identification error curves of front wheels under condition of 30% noise

3) 小阻尼条件下识别误差分析。以前轮为例，分别取原系统阻尼c1c2的60%及30%，利用逆虚拟激励法求得前轮激励功率谱及误差曲线，如图 6图 7所示。可知，系统阻尼降为原来的60%时，功率谱识别误差Error=2.48%，30%时Error=4.49%。由此可见，降低系统阻尼后，功率谱识别精度有所降低，尤其在系统固有频率附近，误差较大。这是由于较小的阻尼导致系统刚度矩阵在固有频率附近出现病态，可通过增加测点数量和提高响应数据精度的方法克服。

 图 6 60%系统阻尼前轮识别功率谱密度及误差曲线 Fig. 6 Identified PSD and identification error curves of front wheels under condition of 60% system damping
 图 7 30%系统阻尼前轮识别功率谱密度及误差曲线 Fig. 7 Identified PSD and identification error curves of front wheels under condition of 30% system damping
5 结论

1) 根据导弹公路运输特点，建立了导弹-悬架四自由度振动模型。

2) 基于逆虚拟激励法求得导弹运输过程路面激励谱，给出了导弹运输振动谱设计流程。

3) 讨论了噪声及阻尼对振动谱精度的影响，通过仿真验证了本文算法具有一定抗噪能力和较好的鲁棒性。

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#### 文章信息

LYU Weimin, XIAO Yang, FANG Dengjian

Design of missile vibration spectrum based on inverse pseudo-excitation method

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(5): 872-879
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0335