{"defaultlang":"zh","titlegroup":{"articletitle":[{"lang":"zh","data":[{"name":"text","data":"连续转速状态下卫星反作用飞轮微振动参数识别"}]},{"lang":"en","data":[{"name":"text","data":"Identification method of micro-vibration parameters of satellite reaction wheelsunder continuous rotating speed"}]}]},"contribgroup":{"author":[{"name":[{"lang":"zh","surname":"杨","givenname":"林","namestyle":"eastern","prefix":""},{"lang":"en","surname":"YANG","givenname":"Lin","namestyle":"eastern","prefix":""}],"stringName":[],"aff":[{"rid":"aff1","text":"1"},{"rid":"aff2","text":"2"}],"role":["first-author"],"bio":[{"lang":"zh","text":["杨　林（1989-），男，山东汶上人，副研究员，博士，主要从事卫星结构和机构设计，卫星动力学优化方面的研究。E-mail： yanglincas@sdu.edu.cn"],"graphic":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225304&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225322&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225317&type=","width":"22.01333618","height":"32.00402832","fontsize":""}],"data":[[{"name":"text","data":"杨　林"},{"name":"text","data":"（1989-），男，山东汶上人，副研究员，博士，主要从事卫星结构和机构设计，卫星动力学优化方面的研究。E-mail： "},{"name":"text","data":"yanglincas@sdu.edu.cn"}]]}],"email":"yanglincas@sdu.edu.cn","deceased":false},{"name":[{"lang":"zh","surname":"王","givenname":"岩松","namestyle":"eastern","prefix":""},{"lang":"en","surname":"WANG","givenname":"Yan-song","namestyle":"eastern","prefix":""}],"stringName":[],"aff":[{"rid":"aff1","text":"1"},{"rid":"aff2","text":"2"}],"role":["corresp"],"corresp":[{"rid":"cor1","lang":"en","text":"E-mail： wangyansong@sdu.edu.cn","data":[{"name":"text","data":"E-mail： wangyansong@sdu.edu.cn"}]}],"bio":[{"lang":"zh","text":["王岩松（1992-），男，辽宁朝阳人，工程师，2017年于北京理工大学获得硕士学位，主要从事飞行器结构设计与分析、结构动力学等方面的研究。E-mail： wangyansong@sdu.edu.cn"],"graphic":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225327&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225337&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225332&type=","width":"22.01333618","height":"32.00405884","fontsize":""}],"data":[[{"name":"text","data":"王岩松"},{"name":"text","data":"（1992-），男，辽宁朝阳人，工程师，2017年于北京理工大学获得硕士学位，主要从事飞行器结构设计与分析、结构动力学等方面的研究。E-mail： "},{"name":"text","data":"wangyansong@sdu.edu.cn"}]]}],"email":"wangyansong@sdu.edu.cn","deceased":false},{"name":[{"lang":"zh","surname":"魏","givenname":"磊","namestyle":"eastern","prefix":""},{"lang":"en","surname":"WEI","givenname":"Lei","namestyle":"eastern","prefix":""}],"stringName":[],"aff":[{"rid":"aff1","text":"1"},{"rid":"aff2","text":"2"}],"role":[],"deceased":false},{"name":[{"lang":"zh","surname":"胡","givenname":"自强","namestyle":"eastern","prefix":""},{"lang":"en","surname":"HU","givenname":"Zi-qiang","namestyle":"eastern","prefix":""}],"stringName":[],"aff":[{"rid":"aff1","text":"1"},{"rid":"aff2","text":"2"}],"role":[],"deceased":false}],"aff":[{"id":"aff1","intro":[{"lang":"zh","label":"1","text":"山东大学 前沿交叉科学青岛研究院，山东 青岛 266200","data":[{"name":"text","data":"山东大学 前沿交叉科学青岛研究院，山东 青岛 266200"}]},{"lang":"en","label":"1","text":"Institute of Frontier and Interdisciplinary， Shandong University， Qingdao 266200， China","data":[{"name":"text","data":"Institute of Frontier and Interdisciplinary， Shandong University， Qingdao 266200， China"}]}]},{"id":"aff2","intro":[{"lang":"zh","label":"2","text":"山东大学 空间科学研究院，山东 威海 264209","data":[{"name":"text","data":"山东大学 空间科学研究院，山东 威海 264209"}]},{"lang":"en","label":"2","text":"Institute of Space Sciences， Shandong University， Weihai 264209， China","data":[{"name":"text","data":"Institute of Space Sciences， Shandong University， Weihai 264209， China"}]}]}]},"abstracts":[{"lang":"zh","data":[{"name":"p","data":[{"name":"text","data":"由于制造和装配误差，反作用飞轮产生的谐波干扰是导致光学遥感卫星微振动现象出现的主要因素，微振动现象严重阻碍了光学遥感卫星向更高分辨率方向发展。为了在设计时更有效地抑制光学遥感卫星微振动现象，需要较好地把握反作用飞轮微振动相关参数。本文提出了一种反作用飞轮连续转速状态下的参数辨识方法，即在时频域内获取飞轮扰振变化特性。首先利用时频分析获取非稳定干扰力信号的时频分布；然后利用加扇形窗的线性最小二乘估计方法辨识反作用飞轮的倍频特征；其后建立与飞轮转速相关的参数化模型，并利用加权非线性最小二乘估计方法辨识出参数化模型的待估参数，其后将模型参数转化为模态参数；再结合模态参数和倍频参数，获取飞轮干扰力或力矩的转速-频率峰值点。最后通过一组反作用飞轮测试试验验证本文所提出的方法，试验结果显示，连续转速状态下的辨识结果更接近反作用飞轮真实工作状态，该方法准确掌握了飞轮干扰力和力矩变化趋势，相对于传统固定转速状态下的测试结果平均误差小于1%，结果具有良好的精度。"}]}]},{"lang":"en","data":[{"name":"p","data":[{"name":"text","data":"Owing to manufacturing and assembling errors， the harmonic interference generated by a reaction wheel assembly （RWA） is the primary factor leading to the micro-vibration of optical remote sensing satellites， which seriously hinders the development of high-resolution optical remote sensing satellites. To achieve more effective micro-vibration suppression in the design of optical remote sensing satellites， it is necessary to accurately grasp the relevant parameters of RWA micro-vibrations. This paper proposes a parametric identification method for RWA at continuous rotating speeds. The micro-vibration characteristics of the RWA are obtained in the time-frequency domain. First， a time-frequency analysis is used to obtain a time-frequency distribution of non-stationary disturbing force/moment signals. Then， a linear least squares （LLS） estimator with a sector window is established to identify the harmonic characteristics. Subsequently， a speed-dependent parameterized model is built to describe the RWA model； further， using a weighted nonlinear least-square （WNLS） estimator， the parameters of the parameterized model are estimated and transformed into modal parameters. By combining modal parameters and harmonic characteristics， the speed-frequency peaks of the disturbing forces/moments can be estimated. Experiments are performed to verify the proposed method. The results demonstrate that the proposed method presents practical trends of disturbing forces/moments. Compared with the results obtained under traditional fixed-speed states， the errors are less than 1%， confirming the accuracy of the proposed method."}]}]}],"keyword":[{"lang":"zh","data":[[{"name":"text","data":"反作用飞轮"}],[{"name":"text","data":"参数辨识"}],[{"name":"text","data":"微振动"}],[{"name":"text","data":"时频分析"}]]},{"lang":"en","data":[[{"name":"text","data":"reaction wheel assembly"}],[{"name":"text","data":"parameter identification"}],[{"name":"text","data":"micro-vibration"}],[{"name":"text","data":"time-frequency analysis"}]]}],"highlights":[],"body":[{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"1 引　言"}],"level":"1","id":"s1"}},{"name":"p","data":[{"name":"text","data":"随着航天技术的快速发展，光学遥感卫星正呈现出更高指向精度和更高分辨率的特点。近些年国际上的商业遥感卫星已经达到优于0.3 m的分辨率"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"1","type":"bibr","rid":"R1","data":[{"name":"text","data":"1"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"。对于高分辨率遥感卫星来说，光学载荷对于卫星上的微小扰动十分敏感，微振动对成像质量的影响已经变得不可忽略。在反作用飞轮微振动作用下，光学遥感卫星在成像时会出现图像扭曲和模糊等问题"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"2","type":"bibr","rid":"R2","data":[{"name":"text","data":"2"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"。"}]},{"name":"p","data":[{"name":"text","data":"为降低卫星微振动的影响，主要采取的手段包括：降低干扰源的影响、优化星体传力路径以及隔振等方法"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"blockXref","data":{"data":[{"name":"xref","data":{"text":"3","type":"bibr","rid":"R3","data":[{"name":"text","data":"3"}]}},{"name":"text","data":"-"},{"name":"xref","data":{"text":"5","type":"bibr","rid":"R5","data":[{"name":"text","data":"5"}]}}],"rid":["R3","R4","R5"],"text":"3-5","type":"bibr"}},{"name":"text","data":"］"}]},{"name":"text","data":"。反作用飞轮是航天器的姿态控制系统的执行机构之一，其性能影响着航天器姿态控制的精度。由于反作用飞轮在工作时受到转子动静不平衡、轴承缺陷、飞轮结构模态放大等因素的影响"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"6","type":"bibr","rid":"R6","data":[{"name":"text","data":"6"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"，会激发出一系列复杂的谐波扰动，所以反作用飞轮是微振动的主要干扰源之一。对反作用飞轮力学机理以及其运转过程中的干扰特性的研究，对航天器微振动技术以及其抑制方法的研究具有重要的意义。对反作用飞轮的研究方法主要包括飞轮理论模型的建模"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"7","type":"bibr","rid":"R7","data":[{"name":"text","data":"7"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"，飞轮干扰特性的经验模型的建立以及飞轮干扰特性及结构模态参数识别等"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"blockXref","data":{"data":[{"name":"xref","data":{"text":"8","type":"bibr","rid":"R8","data":[{"name":"text","data":"8"}]}},{"name":"text","data":"-"},{"name":"xref","data":{"text":"11","type":"bibr","rid":"R11","data":[{"name":"text","data":"11"}]}}],"rid":["R8","R9","R10","R11"],"text":"8-11","type":"bibr"}},{"name":"text","data":"］"}]},{"name":"text","data":"。通过飞轮参数识别方法可以直观地反应出飞轮干扰特性和结构模态的分布特点，并且能够反应出理论建模不能反应的真实特性，为后续飞轮的隔振方法研究、整星的微振动分析和试验提供基础。"}]},{"name":"p","data":[{"name":"text","data":"当前应用范围较广的是恒定转速的飞轮参数识别方法，通常是利用一系列固定转速的干扰力/力矩数据，分析飞轮的干扰特性及模态参数"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"12","type":"bibr","rid":"R12","data":[{"name":"text","data":"12"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"。此类方法仅能分析恒定转速下飞轮的模态参数和干扰力特性，为了得到飞轮在一个转速区间的干扰特性，需要通过大量的试验来获得不同转速情况下试验数据"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"13","type":"bibr","rid":"R13","data":[{"name":"text","data":"13"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"，大大增加了工作量；并且，反作用飞轮在实际工作过程中，尤其在卫星姿态运动过程中，其转速并不是完全恒定的"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"14","type":"bibr","rid":"R14","data":[{"name":"text","data":"14"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"，飞轮的干扰力为一段非平稳信号，此时利用传统的傅里叶变换方法得到的模态特性和干扰力特性将因转速的变化而变得不稳定。所以常规的恒转速下飞轮微振动参数识别方法是不能准确反应在轨工作状态的。"}]},{"name":"p","data":[{"name":"text","data":"本文将建立一种非恒定转速运动状态下反作用飞轮的参数识别方法，可以利用非平稳信号识别出飞轮非恒定转速下的干扰力倍频特性以及模态参数。首先利用时频分析，将试验得到的飞轮非恒定转速下的干扰力非平稳信号，转化为时频分布函数，并通过转速与时间的对应关系，将时频分布函数转化为转频分布函数；之后建立飞轮的倍频特性识别方法，识别出飞轮的干扰力分布；最后建立非恒定转速飞轮的参数化的结构模态参数识别方法，识别出飞轮的结构模态。本文将通过干扰力测试验证所提出的方法，首先进行一次非恒定转速的干扰力测试得到飞轮干扰力试验数据，利用以上方法识别出该飞轮的干扰力分布特性和模态参数。通过结合倍频特性和模态参数，可以得到飞轮的转速-频率峰值点，预测飞轮干扰力最大值的分布情况。该方法将在保证一定精度的前提下，大大提升飞轮参数识别的效率。"}]}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"2 反作用飞轮倍频特性识别"}],"level":"1","id":"s2"}},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"2.1　动力学模型"}],"level":"2","id":"s2a"}},{"name":"p","data":[{"name":"text","data":"反作用飞轮的动力学模型可以表示为"}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（1）"}],"data":[{"name":"math","data":{"math":"Mq¨2+C+Gq˙+Kq=Q","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225350&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225346&type=","width":"44.45000076","height":"4.06400013","fontsize":""}}},{"name":"text","data":"."}],"id":"DF1"}},{"name":"p","data":[{"name":"text","data":"通过求解特征方程，可以得到反作用飞轮的无阻尼固有频率，可以表示为"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"13","type":"bibr","rid":"R13","data":[{"name":"text","data":"13"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（2）"}],"data":[{"name":"math","data":{"math":"ωr=kr/mf,ωa=ka/mfωψ,ϑ=±Ωξ2+Ωξ22+ωθ2","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225359&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225354&type=","width":"46.99000168","height":"17.94933319","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF2"}},{"name":"p","data":[{"name":"text","data":"其中："},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ωr","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225369&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225364&type=","width":"3.47133350","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"和"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ωa","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225383&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225374&type=","width":"3.64066648","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"分别对应径向平动和轴向平动，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ωψ,ϑ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225393&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225388&type=","width":"5.84200001","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"对应径向摇摆，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Ω","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225403&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225398&type=","width":"2.45533323","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"指飞轮转速，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ξ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225419&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225407&type=","width":"1.60866666","height":"3.64066648","fontsize":""}}}]},{"name":"text","data":"指极惯性矩与径向惯性矩之比。从"},{"name":"xref","data":{"text":"式（2）","type":"disp-formula","rid":"DF2","data":[{"name":"text","data":"式（2）"}]}},{"name":"text","data":"中可以发现，飞轮的摇摆模态是与飞轮转子速度相关的。"}]}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"2.2　时频分析原理"}],"level":"2","id":"s2b"}},{"name":"p","data":[{"name":"text","data":"对于固定转速的飞轮，其干扰力/力矩时间响应信号是一组平稳信号，通过Fourier变换即可得到当前转速下飞轮各个方向干扰力的频谱。但当飞轮的转速不断变化时，飞轮干扰力信号的频谱特性是随时间变化的。因此传统的Fourier变换不再适应变转速飞轮干扰力/力矩频域特性分析。"}]},{"name":"p","data":[{"name":"text","data":"时频分析方法是一种时变非平稳信号处理方法，能够将非平稳信号分解为由时间和频率二元变量决定的时频分布函数，以描述信号在不同时间和频率下的幅值分布密度"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"15","type":"bibr","rid":"R15","data":[{"name":"text","data":"15"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"。时频分析方法可以直观的体现信号的频率变化特性，广泛应用于故障诊断和模态参数识别等领域。目前应用最为广泛的时频分析方法主要包括短时Fourier变换（STFT）、小波变换（WT）、Wigner-Ville分布（WVD）等。以下将采用短时Fourier变换作为飞轮响应信号的时频分析方法。"}]},{"name":"p","data":[{"name":"text","data":"短时Fourier变换的频谱可以表示为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（3）"}],"data":[{"name":"math","data":{"math":"STFTt,f=∫-∞+∞xτhτ-te-j2πfτdτ=∫-∞+∞xτht,ωτdτ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225430&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225424&type=","width":"66.80200195","height":"15.66333389","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF3"}},{"name":"p","data":[{"name":"text","data":"其中："},{"name":"inlineformula","data":[{"name":"math","data":{"math":"xτ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225438&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225434&type=","width":"6.85800028","height":"3.97933340","fontsize":""}}}]},{"name":"text","data":"为被分析的时间信号，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"h(t)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225447&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225442&type=","width":"6.77333355","height":"3.55599999","fontsize":""}}}]},{"name":"text","data":"为窗函数，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ht,ωτ=hτ-te-j2πfτ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225455&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225451&type=","width":"36.23733139","height":"4.06400013","fontsize":""}}}]},{"name":"text","data":"为窗函数对应的基函数。"}]},{"name":"p","data":[{"name":"text","data":"之后利用转速随着时间变化的函数，可以将时频分布谱的时间轴转化为转速轴，得到短时Fourier变换在转速-频率平面上的分布"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"STFTr,f","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225465&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225460&type=","width":"17.69533348","height":"4.82599974","fontsize":""}}}]},{"name":"text","data":"，功率谱密度函数可以表示为"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Gr,f=STFTr,f2","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225474&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225469&type=","width":"38.01533508","height":"6.26533318","fontsize":""}}}]},{"name":"text","data":"。"}]}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"2.3　倍频特性识别"}],"level":"2","id":"s2c"}},{"name":"p","data":[{"name":"text","data":"飞轮的干扰力由一系列复杂的谐波和噪声构成。这些谐波包括基础谐波、超谐波和次谐波，其中基础谐波即为一阶谐波，而超谐波和次谐波对应的频率均可以表示为一阶谐波频率的倍数，即倍频。本节将通过建立线性最小二乘方法，识别出飞轮各个方向干扰力或力矩的主要倍频。"}]},{"name":"p","data":[{"name":"text","data":"定义飞轮倍频特性的线性最小二乘估计的费用函数为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（4）"}],"data":[{"name":"math","data":{"math":"𝓁LSθ=∑i=1NrNfεLSri,fi,p2","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225484&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225479&type=","width":"40.97866821","height":"9.31333351","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF4"}},{"name":"p","data":[{"name":"text","data":"其中：系数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"p","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225493&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225488&type=","width":"1.86266661","height":"3.64066648","fontsize":""}}}]},{"name":"text","data":"为待估计的倍频系数，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ri","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225502&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225498&type=","width":"2.28600001","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"为第"},{"name":"italic","data":[{"name":"text","data":"i"}]},{"name":"text","data":"个转频采样点的转速，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"fi","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225512&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225507&type=","width":"1.86266661","height":"4.57200003","fontsize":""}}}]},{"name":"text","data":"为第"},{"name":"italic","data":[{"name":"text","data":"i"}]},{"name":"text","data":"个转频采样点的频率值，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Nr","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225520&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225515&type=","width":"3.81000018","height":"3.81000018","fontsize":""}}}]},{"name":"text","data":"和"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Nf","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225529&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225524&type=","width":"3.64066648","height":"3.81000018","fontsize":""}}}]},{"name":"text","data":"分别为转速点数和频率点数，误差函数可以表示为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（5）"}],"data":[{"name":"math","data":{"math":"εLSri,fi,p=wifri,p-fi=wirip60-fi","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225537&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225533&type=","width":"52.49333191","height":"16.34066582","fontsi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window"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225581&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225590&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225584&type=","width":"75.01467133","height":"52.23933029","fontsize":""}]}},{"name":"p","data":[{"name":"text","data":"将第2.2节得到的时频分布函数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Gr,f","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225599&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225595&type=","width":"10.58333302","height":"4.82599974","fontsize":""}}}]},{"name":"text","data":"写为向量形式："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（7）"}],"data":[{"name":"math","data":{"math":"G=Gr1,f1⋯Gr1,fNf⋯GrNr,fNfT","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225609&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225604&type=","width":"77.80866241","height":"7.11199999","fontsize":""}}},{"name":"text","data":"."}],"id":"DF7"}},{"name":"p","data":[{"name":"text","data":"并令权函数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"wi=hsiGi","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225618&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225613&type=","width":"15.91733360","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"，则误差函数可以表示为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（8）"}],"data":[{"name":"math","data":{"math":"εLS=w11⋮w1Nfw21⋮⋮wNrNf.*r1⋮r1r2⋮⋮rNrp60-f1⋮fNff1⋮⋮fNf=U-V","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225629&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225624&type=","width":"66.80200195","height":"40.04733276","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF8"}},{"name":"p","data":[{"name":"text","data":"其中：符号"},{"name":"inlineformula","data":[{"name":"math","data":{"math":".*","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225637&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225634&type=","width":"2.87866688","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"表示两向量元素相乘，待估参数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"p","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225645&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225641&type=","width":"1.86266661","height":"3.64066648","fontsize":""}}}]},{"name":"text","data":"的最小二乘解可以表示为"}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（9）"}],"data":[{"name":"math","data":{"math":"p=UTU-1UTV","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225653&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225649&type=","width":"29.04066658","height":"5.16466665","fontsize":""}}},{"name":"text","data":"."}],"id":"DF9"}},{"name":"p","data":[{"name":"text","data":"通过改变扇形窗函数的分布，即可识别出不同扇形窗内的倍频特性。"}]}]}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"3 反作用飞轮模态参数识别"}],"level":"1","id":"s3"}},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"3.1　参数化的矩阵分式模型建立"}],"level":"2","id":"s3a"}},{"name":"p","data":[{"name":"text","data":"本节将建立一种与转速相关的参数化的飞轮结构模态参数识别方法，首先建立飞轮的参数化矩阵分式模型"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"16","type":"bibr","rid":"R16","data":[{"name":"text","data":"16"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"，利用加权非线性最小二乘方法估计出模型的待估参数，最后将矩阵分式模型参数转化为模态参数。"}]},{"name":"p","data":[{"name":"text","data":"飞轮的动力学模型可以写为如下形式的参数化的右矩阵分式模型："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（10）"}],"data":[{"name":"math","data":{"math":"Ĝkr,f=Bkr,fA-1r,f","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225661&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225657&type=","width":"42.41799927","height":"4.91066647","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF10"}},{"name":"p","data":[{"name":"text","data":"其中："},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Bk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225669&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225665&type=","width":"3.64066648","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"为分子系数矩阵，"},{"name":"italic","data":[{"name":"text","data":"k"}]},{"name":"text","data":"表示第"},{"name":"italic","data":[{"name":"text","data":"k"}]},{"name":"text","data":"个输出信号，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"A","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225675&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225672&type=","width":"2.79399991","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"为分母系数矩阵，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"r","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225680&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225678&type=","width":"1.52399993","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"表示转速变量，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"f","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225686&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225683&type=","width":"1.10066664","height":"3.64066648","fontsize":""}}}]},{"name":"text","data":"表示频率变量。对于单参考情况，可以简化为公分母模型："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（11）"}],"data":[{"name":"math","data":{"math":"Ĝkr,f=Bkr,fAr,f","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225690&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225689&type=","width":"29.46399879","height":"10.66799927","fontsize":""}}},{"name":"text","data":"."}],"id":"DF11"}},{"name":"p","data":[{"name":"text","data":"将分母系数和分子系数定义为如"},{"name":"xref","data":{"text":"式（12）","type":"disp-formula","rid":"DF12","data":[{"name":"text","data":"式（12）"}]}},{"name":"text","data":"所示的多项式形式："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（12）"}],"data":[{"name":"math","data":{"math":"Bk=∑i=0nr∑j=0nfbk,i,jφi,j(r,f)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225696&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225694&type=","width":"38.35400009","height":"9.73666668","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF12"}},{"name":"dispformula","data":{"label":[{"name":"text","data":"（13）"}],"data":[{"name":"math","data":{"math":"A=∑i=0nr∑j=0nfai,jφi,j(r,f)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225703&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225699&type=","width":"35.47533417","height":"9.73666668","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF13"}},{"name":"p","data":[{"name":"text","data":"其中："},{"name":"inlineformula","data":[{"name":"math","data":{"math":"φi,j(r,f)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225709&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225706&type=","width":"13.54666710","height":"4.65666676","fontsize":""}}}]},{"name":"text","data":"表示与频率和转速相关的基函数，采用正交多项式与"},{"name":"italic","data":[{"name":"text","data":"z"}]},{"name":"text","data":"域多项式混合形式"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"φi,jr,ω=pirzj","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225715&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225712&type=","width":"29.63333321","height":"4.06400013","fontsize":""}}}]},{"name":"text","data":"，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"bk,i,j","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225723&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225720&type=","width":"6.01133347","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"和"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ai,j","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225728&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225726&type=","width":"4.14866638","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"为多项式系数，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"nr","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225738&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225733&type=","width":"2.87866688","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"和"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"nf","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225746&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225742&type=","width":"2.62466669","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"分别为转速多项式和频率多项式的阶数"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"17","type":"bibr","rid":"R17","data":[{"name":"text","data":"17"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"。"}]},{"name":"p","data":[{"name":"text","data":"将多项式系数写成向量形式如下："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（14）"}],"data":[{"name":"math","data":{"math":"Bk,j=bk,0,jbk,1,j⋯bk,nr,jT∈C(nr+1)×1 Aj=a0,ja1,j⋯anr,jT∈C(nr+1)×1","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225752&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225748&type=","width":"66.71733856","height":"14.90133286","fontsize":""}}},{"name":"text","data":"."}],"id":"DF14"}},{"name":"p","data":[{"name":"text","data":"进一步可写为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（15）"}],"data":[{"name":"math","data":{"math":"βk=Bk,0T,Bk,1T,...,Bk,nfTT∈C(nr+1)(nf+1)×1α=A0T,A1T,...,AnfTT∈C(nr+1)(nf+1)×1","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225758&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225756&type=","width":"62.82266617","height":"14.56266594","fontsize":""}}},{"name":"text","data":"."}],"id":"DF15"}},{"name":"p","data":[{"name":"text","data":"则右矩阵分式模型的所有待估参数写成以下形式："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（16）"}],"data":[{"name":"math","data":{"math":"θ=β1T,β2T,...,βNoT,αTT∈C(nr+1)(nf+1)(No+1)×1","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225765&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225763&type=","width":"70.10399628","height":"6.34999990","fontsize":""}}},{"name":"text","data":"."}],"id":"DF16"}}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"3.2　加权非线性最小二乘法处理"}],"level":"2","id":"s3b"}},{"name":"p","data":[{"name":"text","data":"本小节利用加权非线性最小二乘估计方法辨识出参数化模型的待估参数，将模型参数转化为模态参数。加权非线性最小二乘估计的费用函数可以表示为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（17）"}],"data":[{"name":"math","data":{"math":"𝓁WNLS=∑k=1No∑l=1Nr∑m=1Nfεk(rl,fm,θ)Hεk(rl,fm,θ)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225773&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225769&type=","width":"64.60066223","height":"9.31333351","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF17"}},{"name":"p","data":[{"name":"text","data":"式中，符号H表示矩阵的共轭转置，第"},{"name":"italic","data":[{"name":"text","data":"k"}]},{"name":"text","data":"个输出信号的误差函数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"εk(rl,fm,θ)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225779&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225776&type=","width":"17.27199936","height":"4.65666676","fontsize":""}}}]},{"name":"text","data":"可以表示为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（18）"}],"data":[{"name":"math","data":{"math":"εk(rl,fm,θ)=wkrl,fmBk(rl,fm,βk)A(rl,fm,α)-Gk(rl,fm)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225784&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225782&type=","width":"73.57533264","height":"9.73666668","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF18"}},{"name":"p","data":[{"name":"text","data":"式中令权函数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"wkrl,fm=1/Gkrl,fm","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225788&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225786&type=","width":"41.31733322","height":"6.18066692","fontsize":""}}}]},{"name":"text","data":"，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Gk(rl,fm)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225791&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225789&type=","width":"14.30866623","height":"4.65666676","fontsize":""}}}]},{"name":"text","data":"为测量信号的转-频谱，则各转速-频率点的方程误差函数，可以写成向量形式："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（19）"}],"data":[{"name":"math","data":{"math":"εWNLSθ=ε1(θ)ε2(θ)⋮εNo(θ)","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225795&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225793&type=","width":"30.64933395","height":"21.92866707","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF19"}},{"name":"dispformula","data":{"label":[{"name":"text","data":"（20）"}],"data":[{"name":"math","data":{"math":"εk(θ)=εk(t1,ω1,θ)⋮εk(tNτ,ω1,θ)εk(t1,ω2,θ)⋮εk(tNτ,ωNf,θ)∈CNτNf×1","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225798&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225797&type=","width":"51.39266586","height":"33.78200150","fontsize":""}}},{"name":"text","data":"."}],"id":"DF20"}},{"name":"p","data":[{"name":"text","data":"加权非线性最小二乘方法的待估参数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"θ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225801&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225800&type=","width":"1.94733334","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"需要通过迭代的方式计算，采用Gauss-Newton算法，待估参数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"θ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225806&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225803&type=","width":"1.94733334","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"的增量"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Δθ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225809&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225807&type=","width":"4.40266657","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"可由如"},{"name":"xref","data":{"text":"式（21）","type":"disp-formula","rid":"DF21","data":[{"name":"text","data":"式（21）"}]}},{"name":"text","data":"的线性方程进行求解："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（21）"}],"data":[{"name":"math","data":{"math":"ReJHθnJθnΔθn=-ReJHθnεθn","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225813&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225811&type=","width":"67.90267181","height":"5.67266655","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF21"}},{"name":"p","data":[{"name":"text","data":"其中："},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Jθn","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225818&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225816&type=","width":"8.46666718","height":"4.91066647","fontsize":""}}}]},{"name":"text","data":"表示第"},{"name":"italic","data":[{"name":"text","data":"n"}]},{"name":"text","data":"个迭代步的Jacobi矩阵，Jacobi矩阵"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"J","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225822&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225820&type=","width":"1.77800000","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"的含义是误差函数对待估参数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"θ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225827&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225825&type=","width":"1.94733334","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"的偏导："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（22）"}],"data":[{"name":"math","data":{"math":"J=X1Y1X2Y2⋱⋮XNoYNo","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225831&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225829&type=","width":"46.48200226","height":"20.31999969","fontsize":""}}},{"name":"text","data":"."}],"id":"DF22"}},{"name":"p","data":[{"name":"text","data":"Jacobi矩阵中对应的分块矩阵如"},{"name":"xref","data":{"text":"式（23）","type":"disp-formula","rid":"DF23","data":[{"name":"text","data":"式（23）"}]}},{"name":"text","data":"所示："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（23）"}],"data":[{"name":"math","data":{"math":"Xk=φr1,f1⊗wkr1,f1Ar1,f1⋮φrNr,f1⊗wkrNr,f1ArNr,f1⋮φrNr,fNf⊗wkrNr,fNfArNr,fNf","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225835&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225833&type=","width":"52.32399750","height":"54.86400223","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF23"}},{"name":"dispformula","data":{"label":[{"name":"text","data":"（24）"}],"data":[{"name":"math","data":{"math":"Yk=φr1,f1⊗wkr1,f1Bkr1,f1Ar1,f12⋮φrNr,f1⊗wkrNr,f1BkrNr,f1ArNr,f12⋮φrNr,fNf⊗wkrNr,fNfBkrNr,fNfArNr,fNf2","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225840&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225838&type=","width":"66.63265991","height":"59.01266479","fontsize":""}}},{"name":"text","data":" ，"}],"id":"DF24"}},{"name":"p","data":[{"name":"text","data":"其中"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"⊗","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225842&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225841&type=","width":"3.72533321","height":"3.55599999","fontsize":""}}}]},{"name":"text","data":"表示kronecker积，"}]},{"name":"dispformula","data":{"label":[],"data":[{"name":"math","data":{"math":"φrl,fm=φ0,0rl,fm⋯φnr,0rl,fm⋯φnr,nfrl,fm","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225844&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225843&type=","width":"74.25267029","height":"12.36133385","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF25"}},{"name":"p","data":[{"name":"text","data":"（25）"}]},{"name":"p","data":[{"name":"text","data":"则"},{"name":"xref","data":{"text":"式（21）","type":"disp-formula","rid":"DF21","data":[{"name":"text","data":"式（21）"}]}},{"name":"text","data":"可以表示为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（26）"}],"data":[{"name":"math","data":{"math":"P1Q1P2Q2⋱⋮PNoQNoQ1HQ2H⋯QNoH∑k=1NoRkΔβ1Δβ2⋮ΔβNoΔα=-ReX1Hε1ReX2Hε2⋮ReXNoHεNo∑k=1NoReYkHεk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225847&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225846&type=","width":"86.95266724","height":"37.42266846","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF26"}},{"name":"p","data":[{"name":"text","data":"其中："},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Pk=ReXkHXk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225850&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225849&type=","width":"24.97666740","height":"4.91066647","fontsize":""}}}]},{"name":"text","data":"，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Qk=ReXkHYk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225852&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225851&type=","width":"25.14599991","height":"4.91066647","fontsize":""}}}]},{"name":"text","data":"，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Rk=ReYkHYk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225854&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225853&type=","width":"25.14599991","height":"4.91066647","fontsize":""}}}]},{"name":"text","data":"。跟据"},{"name":"xref","data":{"text":"式（26）","type":"disp-formula","rid":"DF26","data":[{"name":"text","data":"式（26）"}]}},{"name":"text","data":"的前"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"No","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225856&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225855&type=","width":"4.06400013","height":"3.81000018","fontsize":""}}}]},{"name":"text","data":"行可得："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（27）"}],"data":[{"name":"math","data":{"math":"PkΔβk+QkΔα=ReXkHεk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225859&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225857&type=","width":"42.84133148","height":"4.91066647","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF27"}},{"name":"p","data":[{"name":"text","data":"根据"},{"name":"xref","data":{"text":"式（26）","type":"disp-formula","rid":"DF26","data":[{"name":"text","data":"式（26）"}]}},{"name":"text","data":"的第"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"No+1","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225862&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225860&type=","width":"10.75266743","height":"3.81000018","fontsize":""}}}]},{"name":"text","data":"行，可得："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（28）"}],"data":[{"name":"math","data":{"math":"∑k=1NoQkHΔβk+RkΔα=-∑k=1NoReYkHεk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225865&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225864&type=","width":"60.79066849","height":"9.31333351","fontsize":""}}},{"name":"text","data":"."}],"id":"DF28"}},{"name":"p","data":[{"name":"text","data":"结合"},{"name":"xref","data":{"text":"式（27）","type":"disp-formula","rid":"DF27","data":[{"name":"text","data":"式（27）"}]}},{"name":"text","data":"和"},{"name":"xref","data":{"text":"式（28）","type":"disp-formula","rid":"DF28","data":[{"name":"text","data":"式（28）"}]}},{"name":"text","data":"消去"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Δβk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225869&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225866&type=","width":"5.58799982","height":"4.57200003","fontsize":""}}}]},{"name":"text","data":"，可得到缩减正则方程如"},{"name":"xref","data":{"text":"式（29）","type":"disp-formula","rid":"DF29","data":[{"name":"text","data":"式（29）"}]}},{"name":"text","data":"："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（29）"}],"data":[{"name":"math","data":{"math":"∑k=1NoRkQkHPk-1QkΔα=-∑k=1NoReYkHεk-QkHPk-1ReXkHεk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225871&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225870&type=","width":"59.35132980","height":"20.31999969","fontsize":""}}},{"name":"text","data":"."}],"id":"DF29"}},{"name":"p","data":[{"name":"text","data":"定义矩阵："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（30）"}],"data":[{"name":"math","data":{"math":"D=∑k=1NoRkQkHPk-1Qkb=-∑k=1NoReYkHεk-QkHPk-1ReXkHεk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225873&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225872&type=","width":"65.87066650","height":"20.31999969","fontsize":""}}},{"name":"text","data":"."}],"id":"DF30"}},{"name":"p","data":[{"name":"text","data":"则通过求解"},{"name":"xref","data":{"text":"式（31）","type":"disp-formula","rid":"DF31","data":[{"name":"text","data":"式（31）"}]}},{"name":"text","data":"的方程即可得到每一次迭代的"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Δα","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225875&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225874&type=","width":"4.57200003","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"和"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Δβk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225877&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225876&type=","width":"5.58799982","height":"4.57200003","fontsize":""}}}]},{"name":"text","data":"："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（31）"}],"data":[{"name":"math","data":{"math":"DΔα=bΔβk=Pk-1ReXkHεk-QkΔα","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225879&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225878&type=","width":"48.76799774","height":"10.32933331","fontsize":""}}},{"name":"text","data":"."}],"id":"DF31"}},{"name":"p","data":[{"name":"text","data":"将每步得到的"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Δα","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225881&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225880&type=","width":"4.57200003","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"和"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Δβk","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225883&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225882&type=","width":"5.58799982","height":"4.57200003","fontsize":""}}}]},{"name":"text","data":"合写成"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"Δθ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225885&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225884&type=","width":"4.40266657","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"，并对待估参数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"θ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225888&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225887&type=","width":"1.94733334","height":"2.79399991","fontsize":""}}}]},{"name":"text","data":"进行更新，即："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（32）"}],"data":[{"name":"math","data":{"math":"θn+1=θn+Δθn","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225890&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225889&type=","width":"25.06133270","height":"3.81000018","fontsize":""}}},{"name":"text","data":"."}],"id":"DF32"}},{"name":"p","data":[{"name":"text","data":"迭代终止条件为："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（33）"}],"data":[{"name":"math","data":{"math":"θm+1-θmθm≤ζ or kiter≥kmax","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225892&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225891&type=","width":"45.21199799","height":"9.56733322","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF33"}},{"name":"p","data":[{"name":"text","data":"其中："},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ζ","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225894&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225893&type=","width":"1.52399993","height":"3.64066648","fontsize":""}}}]},{"name":"text","data":"为相对误差的阈值，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"kiter","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225896&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225895&type=","width":"4.48733330","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"为迭代次数，"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"kmax","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225898&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225897&type=","width":"5.24933338","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"为给定的最大迭代次数。"}]},{"name":"p","data":[{"name":"text","data":"待估参数的初始值"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"θ0","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225900&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225899&type=","width":"3.04799986","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"可由线性最小二乘估计方法所估计出的参数给定"},{"name":"sup","data":[{"name":"text","data":"［"},{"name":"xref","data":{"text":"18","type":"bibr","rid":"R18","data":[{"name":"text","data":"18"}]}},{"name":"text","data":"］"}]},{"name":"text","data":"。"}]}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"3.3　模态参数计算"}],"level":"2","id":"s3c"}},{"name":"p","data":[{"name":"text","data":"模态参数可以通过求解如"},{"name":"xref","data":{"text":"式（34）","type":"disp-formula","rid":"DF34","data":[{"name":"text","data":"式（34）"}]}},{"name":"text","data":"广义伴随矩阵的特征值获得："}]},{"name":"dispformula","data":{"label":[{"name":"text","data":"（34）"}],"data":[{"name":"math","data":{"math":"Ac(rl)=-Anf+1,rl-1Anf,rl-Anf+1,rl-1Anf-1,rl⋯-Anf+1,rl-1A2,rl-Anf+1,rl-1A1,rl10⋯0001⋯00⋮⋮⋱0000010","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225902&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225901&type=","width":"138.76866150","height":"25.31533432","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF34"}},{"name":"dispformula","data":{"label":[{"name":"text","data":"（35）"}],"data":[{"name":"math","data":{"math":"Ai,rj=Airj,:","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225904&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225903&type=","width":"23.70666695","height":"4.91066647","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF35"}},{"name":"dispformula","data":{"label":[{"name":"text","data":"（36）"}],"data":[{"name":"math","data":{"math":"Ai=Prαi-1(nr+1)+1:i(nr+1),:i=1,2,...,nω","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225906&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225905&type=","width":"67.90267181","height":"10.83733368","fontsize":""}}},{"name":"text","data":"，"}],"id":"DF36"}},{"name":"dispformula","data":{"label":[{"name":"text","data":"（37）"}],"data":[{"name":"math","data":{"math":"Pr=p1r1p2r1⋯pnrr1p1r2p2r2⋯pnrr2⋮⋮⋱⋮p1rNrp2rNr⋯pnrrNr","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225908&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225907&type=","width":"57.48866653","height":"26.07733345","fontsize":""}}},{"name":"text","data":"."}],"id":"DF37"}},{"name":"p","data":[{"name":"text","data":"则模态频率"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"fr","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225910&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225909&type=","width":"2.11666679","height":"4.57200003","fontsize":""}}}]},{"name":"text","data":"和阻尼比"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"ξr","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225912&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225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试验与分析"}],"level":"1","id":"s4"}},{"name":"p","data":[{"name":"text","data":"本节将通过连续转速状态下的反作用飞轮干扰力/力矩测试，来验证本文提出的方法。同时增加一组反作用飞轮固定转速下的干扰力/力矩测试，通过传统的傅里叶变换方法进行分析，并与本文所提出方法的结果进行对比。"}]},{"name":"p","data":[{"name":"text","data":"根据第2节和第3节内容，编写分析程序，程序伪代码如"},{"name":"xref","data":{"text":"图2","type":"fig","rid":"F2","data":[{"name":"text","data":"图2"}]}},{"name":"text","data":"所示。"}]},{"name":"fig","data":{"id":"F2","caption":[{"lang":"zh","label":[{"name":"text","data":"图 2"}],"title":[{"name":"text","data":"分析程序伪代码"}]},{"lang":"en","label":[{"name":"text","data":"Fig.2"}],"title":[{"name":"text","data":"Pseudo-code of analysis program"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225921&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225923&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225922&type=","width":"160.02000427","height":"143.21366882","fontsize":""}]}},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"4.1　试验工况"}],"level":"2","id":"s4a"}},{"name":"p","data":[{"name":"text","data":"如"},{"name":"xref","data":{"text":"图3","type":"fig","rid":"F3","data":[{"name":"text","data":"图3"}]}},{"name":"text","data":"所示，试验系统由飞轮、kistler测力台、试验工装、信号采集系统、控制器、电源和计算机组成。将飞轮通过试验工装连接在测力台上，飞轮方向如"},{"name":"xref","data":{"text":"图4","type":"fig","rid":"F4","data":[{"name":"text","data":"图4"}]}},{"name":"text","data":"所示定义。设定飞轮力矩为0.5 N·m，测试飞轮转速从0转匀加速转到2 000转的过程，在测得干扰力信号的同时，记录飞轮转速随时间变化的曲线。测试的现场图如"},{"name":"xref","data":{"text":"图5","type":"fig","rid":"F5","data":[{"name":"text","data":"图5"}]}},{"name":"text","data":"所示。"}]},{"name":"fig","data":{"id":"F3","caption":[{"lang":"zh","label":[{"name":"text","data":"图3"}],"title":[{"name":"text","data":"试验系统示意图"}]},{"lang":"en","label":[{"name":"text","data":"Fig.3"}],"title":[{"name":"text","data":"Schematic diagram of experimental system"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225924&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225926&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225925&type=","width":"75.01467133","height":"47.24399948","fontsize":""}]}},{"name":"fig","data":{"id":"F4","caption":[{"lang":"zh","label":[{"name":"text","data":"图4"}],"title":[{"name":"text","data":"飞轮安装方向坐标定义"}]},{"lang":"en","label":[{"name":"text","data":"Fig.4"}],"title":[{"name":"text","data":"Flywheel installation direction definition"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225927&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225929&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225928&type=","width":"75.01467133","height":"47.49800110","fontsize":""}]}},{"name":"fig","data":{"id":"F5","caption":[{"lang":"zh","label":[{"name":"text","data":"图5"}],"title":[{"name":"text","data":"测试现场图"}]},{"lang":"en","label":[{"name":"text","data":"Fig.5"}],"title":[{"name":"text","data":"Photo of experimental site"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225930&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225932&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225931&type=","width":"75.01467133","height":"55.66833115","fontsize":""}]}},{"name":"p","data":[{"name":"text","data":"试验得到的飞轮的转速时间历程曲线如"},{"name":"xref","data":{"text":"图6","type":"fig","rid":"F6","data":[{"name":"text","data":"图6"}]}},{"name":"text","data":"所示，飞轮各个方向干扰力/力矩的时间历程曲线如"},{"name":"xref","data":{"text":"图7","type":"fig","rid":"F7","data":[{"name":"text","data":"图7"}]}},{"name":"text","data":"所示。可以发现，连续转速状态下飞轮的干扰力信号是一条幅值呈上升趋势的非稳态信号。此时利用传统的傅里叶变换方法进行分析时，干扰力在频域上的频率和幅值都将是不稳定的，因此需要将连续转速的干扰力信号转化的时频域上进行分析。"}]},{"name":"fig","data":{"id":"F6","caption":[{"lang":"zh","label":[{"name":"text","data":"图6"}],"title":[{"name":"text","data":"转速时间历程曲线"}]},{"lang":"en","label":[{"name":"text","data":"Fig.6"}],"title":[{"name":"text","data":"Time 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signals"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225936&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225938&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225937&type=","width":"160.02000427","height":"106.17199707","fontsize":""}]}}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"4.2　干扰力/力矩时频分析"}],"level":"2","id":"s4b"}},{"name":"p","data":[{"name":"text","data":"利用第2.2节的方法将干扰力信号转化到时频域，得到干扰力和力矩的转频谱如"},{"name":"xref","data":{"text":"图8","type":"fig","rid":"F8","data":[{"name":"text","data":"图8"}]}},{"name":"text","data":"所示。图中可以发现明显的飞轮结构模态和倍频特性，可以通过本文建立的方法进行识别。"}]},{"name":"fig","data":{"id":"F8","caption":[{"lang":"zh","label":[{"name":"text","data":"图8"}],"title":[{"name":"text","data":"干扰力和力矩的转频谱"}]},{"lang":"en","label":[{"name":"text","data":"Fig.8"}],"title":[{"name":"text","data":"Speed-frequency 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moments"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225939&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225941&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225940&type=","width":"160.02000427","height":"92.24433136","fontsize":""}]}}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"4.3　倍频特性识别"}],"level":"2","id":"s4c"}},{"name":"p","data":[{"name":"text","data":"利用第2.3节建立的方法对飞轮倍频特性进行识别，识别出的各方向干扰力或力矩的倍频特性进行统计如"},{"name":"xref","data":{"text":"表1","type":"table","rid":"T1","data":[{"name":"text","data":"表1"}]}},{"name":"text","data":"和"},{"name":"xref","data":{"text":"图9","type":"fig","rid":"F9","data":[{"name":"text","data":"图9"}]}},{"name":"text","data":"所示。"}]},{"name":"table","data":{"id":"T1","caption":[{"lang":"zh","label":[{"name":"text","data":"表1"}],"title":[{"name":"text","data":"飞轮各方向干扰力/力矩倍频统计"}]},{"lang":"en","label":[{"name":"text","data":"Tab.1"}],"title":[{"name":"text","data":"Summary of main harmonic factors"}]}],"note":[],"table":[{"head":[[{"align":"center","style":"border-top:solid;border-bottom:solid;","data":[{"name":"p","data":[{"name":"text","data":"干扰力"}]},{"name":"p","data":[{"name":"text","data":"/力矩"}]}]},{"align":"center","style":"border-top:solid;border-bottom:solid;","data":[{"name":"text","data":"倍频数"}]}]],"body":[[{"align":"center","data":[{"name":"italic","data":[{"name":"text","data":"F"},{"name":"sub","data":[{"name":"text","data":"x"}]}]}]},{"align":"center","data":[{"name":"text","data":"1.0; 10.7; 12.1; 12.8; 16.2; 18.2; 21.1; 24.3"}]}],[{"align":"center","data":[{"name":"italic","data":[{"name":"text","data":"F"},{"name":"sub","data":[{"name":"text","data":"y"}]}]}]},{"align":"center","data":[{"name":"text","data":"1.0; 10.7; 12.1; 12.8; 16.2; 18.4; 21.0; 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24.3;"}]}]],"foot":[]}],"graphics":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225943&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225942&type=","width":"76.89949799","height":"36.09209061","fontsize":""}}},{"name":"fig","data":{"id":"F9","caption":[{"lang":"zh","label":[{"name":"text","data":"图9"}],"title":[{"name":"text","data":"倍频辨识结果（各直线表示各倍频辨识结果）"}]},{"lang":"en","label":[{"name":"text","data":"Fig.9"}],"title":[{"name":"text","data":"Identification results of harmonic characteristics (Lines denote the identification results)"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225944&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225946&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225945&type=","width":"160.02000427","height":"103.63199615","fontsize":""}]}}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"4.4　飞轮模态参数识别"}],"level":"2","id":"s4d"}},{"name":"p","data":[{"name":"text","data":"利用第3节建立的方法辨识飞轮的模态参数，设置转速多项式参数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"nr=1","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225948&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225947&type=","width":"10.32933331","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"，频率多项式参数，最大迭代次数"},{"name":"inlineformula","data":[{"name":"math","data":{"math":"kmax=10","graphicsData":{"small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225950&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225949&type=","width":"14.56266594","height":"3.72533321","fontsize":""}}}]},{"name":"text","data":"，辨识出的转频谱如"},{"name":"xref","data":{"text":"图10","type":"fig","rid":"FG1","data":[{"name":"text","data":"图10"}]}},{"name":"text","data":"所示。辨识的随转速变化的模态频率和阻尼比如"},{"name":"xref","data":{"text":"图11","type":"fig","rid":"FG2","data":[{"name":"text","data":"图11"}]}},{"name":"text","data":"所示，图中，红色矩形符号表示飞轮的轴向平动模态，绿色圆形符号表示飞轮的摇摆模态，蓝色菱形符号表示飞轮的径向平动模态（彩图见期刊电子版）。所辨识出的飞轮模态可以为飞轮的仿真分析、模型校核以及飞轮系统隔振设计等提供参考。"}]},{"name":"figgroup","data":{"id":"FG1","caption":[{"lang":"zh","label":[{"name":"text","data":"图10"}],"title":[{"name":"text","data":"参数化估计的转频谱"}]},{"lang":"en","label":[{"name":"text","data":"Fig.10"}],"title":[{"name":"text","data":"Parametric 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spectra"}]}],"note":[],"layout":"1;2;3;","grid":[[{"name":"fig","data":{"id":"FG1-1","caption":[{"lang":"zh","title":[]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225951&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225953&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225952&type=","width":"75.01467133","height":"54.52533722","fontsize":""}]}}],[{"name":"fig","data":{"id":"FG1-2","caption":[{"lang":"zh","title":[]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225954&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225956&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225955&type=","width":"75.01467133","height":"57.86967087","fontsize":""}]}}],[{"name":"fig","data":{"id":"FG1-3","caption":[{"lang":"zh","title":[]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225958&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225960&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225959&type=","width":"75.01467133","height":"52.36633301","fontsize":""}]}}]]}},{"name":"figgroup","data":{"id":"FG2","caption":[{"lang":"zh","label":[{"name":"text","data":"图11"}],"title":[{"name":"text","data":"模态频率和阻尼比"}]},{"lang":"en","label":[{"name":"text","data":"Fig. 11"}],"title":[{"name":"text","data":"Modal frequency and damping ratio"}]}],"note":[],"layout":"1;2;","grid":[[{"name":"fig","data":{"id":"FG2-1","caption":[{"lang":"zh","title":[]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225961&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225963&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225962&type=","width":"75.01464844","height":"65.78601074","fontsize":""}]}},{"name":"fig","data":{"id":"FG2-2","caption":[{"lang":"zh","title":[]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225964&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225966&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225965&type=","width":"75.01464844","height":"69.00335693","fontsize":""}]}}]]}}]},{"name":"sec","data":[{"name":"sectitle","data":{"title":[{"name":"text","data":"4.5　固定转速试验结果"}],"level":"2","id":"s4e"}},{"name":"p","data":[{"name":"text","data":"在进行变转速飞轮干扰力测试的同时，进行固定转速的干扰力测试作为对比试验。从0转到2 000转每隔100转进行一组测试，共进行21组试验。通过傅里叶变换，得到飞轮的干扰力/力矩随频率和转速变化的瀑布图如"},{"name":"xref","data":{"text":"图12","type":"fig","rid":"F10","data":[{"name":"text","data":"图12"}]}},{"name":"text","data":"所示。"}]},{"name":"fig","data":{"id":"F10","caption":[{"lang":"zh","label":[{"name":"text","data":"图12"}],"title":[{"name":"text","data":"固定转速试验瀑布图"}]},{"lang":"en","label":[{"name":"text","data":"Fig.12"}],"title":[{"name":"text","data":"Waterfall plots of the fixed-speed tests"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225967&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225972&type=","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10225969&type=","width":"160.02000427","height":"89.91600037","fontsize":""}]}},{"name":"p","data":[{"name":"text","data":"通过结合飞轮模态频率和倍频特性，可以预测干扰力或力矩的峰值点位置。如"},{"name":"xref","data":{"text":"图13","type":"fig","rid":"F11","data":[{"name":"text","data":"图13"}]}},{"name":"text","data":"所示（彩图见期刊电子版），将飞轮轴向模态频率和倍频曲线显示在同一张图上，两曲线交点即为干扰力转频峰值点。通过"},{"name":"xref","data":{"text":"表2","type":"table","rid":"T2","data":[{"name":"text","data":"表2"}]}},{"name":"text","data":"给出了连续转速试验和固定转速试验的轴向干扰力的转频峰值点及其相对误差，可以发现两种方式得到的峰值点误差最大为4.3%，平均误差经计算小于1%。并且固定转速试验得到的峰值点只能出现在试验测试设定的转速上，容易忽略最大峰值点所出现的转速。而连续转速方法可以得到干扰力（或力矩）的平滑变化趋势。此外，连续转速试验的试验量仅为固定转速试验的4.8%。"}]},{"name":"fig","data":{"id":"F11","caption":[{"lang":"zh","label":[{"name":"text","data":"图13"}],"title":[{"name":"text","data":"轴向干扰力峰值分布(红色实线表示模态频率，黑色虚线表示倍频曲线，蓝色点表示参考试验的峰值点)"}]},{"lang":"en","label":[{"name":"text","data":"Fig.13"}],"title":[{"name":"text","data":"Distribution 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