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使用聲發(fā)射傳感器的低速軸承故障診斷
摘 要
本文提出了一種新的低速軸承故障診斷方法。這種基于聲學(xué)發(fā)射(AE)的技術(shù)從外差頻率降低方法開始,其以與振動(dòng)為中心的方法相當(dāng)?shù)乃俾蕦?duì)AE信號(hào)進(jìn)行采樣。然后,對(duì)采樣的AE信號(hào)進(jìn)行時(shí)間同步重采樣,以解釋可能的軸速波動(dòng)和軸承滑移。重新取樣方法夠根據(jù)軸穿越時(shí)間來分割A(yù)E信號(hào),使得均勻數(shù)據(jù)點(diǎn)的數(shù)量可用于計(jì)算單個(gè)頻譜平均值,用于提取特征并評(píng)估軸承故障診斷的眾多條件指標(biāo)(CI)。與現(xiàn)有的基于平均值的降噪方法不同,該方法需要計(jì)算每個(gè)軸承故障類型的多個(gè)平均值,所提出的方法僅計(jì)算所有軸承故障類型的一個(gè)平均值。所提出的技術(shù)通過在軸承試驗(yàn)臺(tái)上使用種子故障鋼軸承的AE信號(hào)進(jìn)行驗(yàn)證。本文的結(jié)果表明,采用低采樣AE信號(hào)并結(jié)合所提出的方法可以有效地提取條件指標(biāo),以診斷在10 Hz以下的多個(gè)低速軸上所有四種軸承故障類型。
關(guān)鍵詞:軸承故障,診斷聲,發(fā)射傳感器,低速
1. 緒論
大多數(shù)重型設(shè)備和機(jī)械設(shè)備至少包含某種類型的軸承。另外,因?yàn)檩S承在非理想的工作條件下使用通常是很常見的,所以可能會(huì)出現(xiàn)問題并導(dǎo)致過早失效。軸承失效時(shí),故障可能導(dǎo)致嚴(yán)重的停機(jī)時(shí)間,成本,以及生產(chǎn)力下降的可能性。最近的一項(xiàng)研究提出了一種使用神經(jīng)網(wǎng)絡(luò)和時(shí)域/頻域振動(dòng)方法的滾動(dòng)軸承故障診斷方法[1]。這項(xiàng)研究表明神經(jīng)網(wǎng)絡(luò)可以幫助使用振動(dòng)數(shù)據(jù)診斷各種電機(jī)軸承故障?由軸承試驗(yàn)臺(tái)產(chǎn)生。當(dāng)使用統(tǒng)計(jì)時(shí)間特征時(shí),神經(jīng)網(wǎng)絡(luò)也被證明對(duì)于狀態(tài)監(jiān)測(cè)是有效的[2]。另一項(xiàng)研究主要集中在旋轉(zhuǎn)機(jī)械上,并使用基于模型的方法來檢測(cè)和診斷機(jī)械故障[3]。該研究開發(fā)了一種非線性濾波技術(shù),以解決由于諸如不平衡,剛度變化以及轉(zhuǎn)子軸承系統(tǒng)的阻尼等因素導(dǎo)致的復(fù)雜的非線性振動(dòng)響應(yīng)。上述技術(shù)的驗(yàn)證通過使用低信號(hào)噪聲環(huán)境模擬。參考[4]提出了基于定子電流監(jiān)測(cè)的原位軸承故障的故障檢測(cè)方法。 在本文中,采用噪聲消除和統(tǒng)計(jì)過程控制技術(shù)相結(jié)合的方法提取軸承故障特征,以檢測(cè)由于缺乏潤滑的軸承的退化而失控的樣本。一研究已經(jīng)提出了兩步數(shù)據(jù)挖掘方法來分類塑料軸承中的缺陷[5]。 這項(xiàng)研究有效地使用了經(jīng)驗(yàn)?zāi)J椒纸猓‥MD)來提取時(shí)域條件指標(biāo)(CI),這些指標(biāo)被用作監(jiān)督學(xué)習(xí)算法的輸入來分類軸承缺陷。 其他研究已經(jīng)通過使用局部和非局部保持投影[6],跟蹤比線性判別分析[7]或功率譜密度(PSD)分析了放大器和頻率電流解調(diào)軸承信號(hào),提出了有效的軸承缺陷技術(shù)對(duì)于直驅(qū)風(fēng)力渦輪機(jī)[8]。
在許多行業(yè)中,使用低速旋轉(zhuǎn)機(jī)器是成功操作的主要原因。這些機(jī)器可以在鋼鐵和造紙廠,生物應(yīng)用和風(fēng)力渦輪機(jī)中找到。因此,在這些應(yīng)用中監(jiān)測(cè)軸承,軸和齒輪對(duì)于低速設(shè)備的正確維護(hù)至關(guān)重要。在低速軸承故障診斷文獻(xiàn)中,低速被認(rèn)為在0.33 Hz到10 Hz的范圍內(nèi)[9],而顯著較低的速度閾值被認(rèn)為是一個(gè)單獨(dú)的分類范圍。例如,速度低于0.5赫茲被認(rèn)為是“超低”[10],如果低于0.83赫茲被認(rèn)為是“極低”[11]。 在本文中,將討論軸速度落在[9]中描述的低速范圍內(nèi)的軸承故障診斷問題。
迄今為止,使用振動(dòng)分析對(duì)滾動(dòng)軸承和其他旋轉(zhuǎn)設(shè)備進(jìn)行狀態(tài)監(jiān)測(cè)是一種已建立的技術(shù)和行業(yè)標(biāo)準(zhǔn)。一項(xiàng)研究調(diào)查了振幅解調(diào)振動(dòng)信號(hào)的參數(shù)模型的使用以及由此產(chǎn)生的頻譜,用于軸承故障檢測(cè)和診斷1 Hz軸速缺陷的滾子軸承[12]。在該論文中,信號(hào)處理技術(shù)首先用于檢測(cè)軸承缺陷,然后用頻譜對(duì)缺陷進(jìn)行分類。該方法雖然有效,但需要對(duì)頻譜進(jìn)行視覺檢查以實(shí)現(xiàn)故障診斷。后來,另一項(xiàng)研究開發(fā)了低速技術(shù)系統(tǒng)來測(cè)量由低速旋轉(zhuǎn)機(jī)械引起的振動(dòng)[9]。該系統(tǒng)的核心是將機(jī)器的高頻噪聲與感興趣的低頻特征分開,并使用低速轉(zhuǎn)子和變速箱的結(jié)果來驗(yàn)證。最近,對(duì)于低速應(yīng)用,已經(jīng)建立了一個(gè)有缺陷的軸承振動(dòng)信號(hào)的一般模型,并且已經(jīng)表明包絡(luò)自相關(guān)可以在有故障的軸承中觀察到,但是在健康的軸承情況下可以觀察到[13]。其他人則試圖開發(fā)新的加速度計(jì),并表明借助共振解調(diào)技術(shù)可以檢測(cè)到低速滾動(dòng)軸承故障[14]。然而,在低速應(yīng)用中使用振動(dòng)是有限的,因?yàn)槭褂脗鹘y(tǒng)的基于加速度計(jì)的監(jiān)測(cè)系統(tǒng)可能無法檢測(cè)到以這種速度發(fā)生的故障所產(chǎn)生的能量變化。因此,研究人員已經(jīng)研究了在低速條件下使用聲發(fā)射(AE)傳感器和應(yīng)變儀進(jìn)行組分監(jiān)測(cè)。
基于AE的研究已經(jīng)顯示了對(duì)早期故障檢測(cè)的有希望的結(jié)果,最近的研究探索了它們?cè)诘退佥S承監(jiān)測(cè)中的應(yīng)用。早期的一項(xiàng)調(diào)查研究了低速監(jiān)測(cè)軸承[15]。在這項(xiàng)研究中,提出了比較加速度,沖擊脈沖換能器,聲發(fā)射和加速度測(cè)量的結(jié)果。在得出的其他結(jié)論中,有人提到AE能夠在低至0.17 Hz的速度下清楚地檢測(cè)到外部軌道軸承缺陷。但是,也有人提到,觀測(cè)到的AE響應(yīng)無法解釋,而且由于信號(hào)沒有在每轉(zhuǎn)一次的基礎(chǔ)上嚴(yán)格重復(fù),因此無法使用平均方法。另一項(xiàng)研究報(bào)道了在0.0083 Hz至0.083 Hz的極低軸速下使用AE監(jiān)測(cè)滾動(dòng)軸承的情況[11]。在這項(xiàng)研究中發(fā)現(xiàn),當(dāng)軸承以非常低的速度旋轉(zhuǎn)時(shí),AE測(cè)量對(duì)于檢測(cè)軸承故障非常敏感,而加速度包絡(luò)僅限于在10Hz的最低軸速度下檢測(cè)故障。這項(xiàng)研究利用了AE脈沖計(jì)數(shù),并指出使用這種方法可以使數(shù)據(jù)易于管理。參考[16]探索了AE在軸承低速監(jiān)測(cè)中的應(yīng)用。本研究采用K均值聚類方法對(duì)球面滾子軸承上的線缺陷進(jìn)行分類。雖然取得了有希望的結(jié)果,但該研究集中于100 kHz至1 MHz的頻率范圍,并且還依賴于數(shù)據(jù)挖掘技術(shù),由于模型的訓(xùn)練和測(cè)試需要大量數(shù)據(jù),因此這種技術(shù)可能非常耗時(shí)。參考 [17]也依賴于數(shù)據(jù)挖掘方法,提出了基于相關(guān)向量機(jī)和支持向量機(jī)的不同分類技術(shù)以0.33 Hz至1.33 Hz的軸速診斷軸承故障。在另一項(xiàng)研究中,使用AE傳感器研究低頻滾動(dòng)軸承在100 kHz頻率范圍內(nèi)的早期故障檢測(cè)[18]。本研究使用不同的濾波器評(píng)估了多個(gè)時(shí)域條件指標(biāo)頻帶來確定可以區(qū)分健康和有故障的內(nèi)部軸承信號(hào)的最佳參數(shù)。然而,本研究的目標(biāo)是確定可用于評(píng)估獲取的AE信號(hào)的有效濾波器頻帶范圍和時(shí)域條件指標(biāo)。此外,僅觀察到內(nèi)圈故障,并且不能使用本文中介紹的相應(yīng)濾波器帶和CI來確定所有四種軸承故障類型的診斷。后來,參考文獻(xiàn)[10]提出了一種使用AE包絡(luò)波形的診斷方法,并以小于1.67 Hz的速度獲得結(jié)果。這項(xiàng)研究證實(shí),外圈爆震的周期性可以在低至0.17 Hz的速度下捕獲。最近,其他研究調(diào)查了AE及其用于低速軸和推力球軸承的狀態(tài)監(jiān)測(cè)[19,20]。這些研究證明了使用AE來檢測(cè)裂紋萌生的能力并且還主要關(guān)注AE源位置的調(diào)查。此外,上述實(shí)驗(yàn)著重于1.2 Hz的單軸轉(zhuǎn)速以及在負(fù)載和缺乏潤滑條件下測(cè)試的軸承狀況。
盡管如此,上述高頻AE信號(hào)伴隨著高采樣率。此外,對(duì)于低速軸承故障診斷應(yīng)用,數(shù)據(jù)采集需要相對(duì)較長的時(shí)間來捕獲機(jī)械故障頻率。高采樣率與長數(shù)據(jù)采樣限制的結(jié)合實(shí)際應(yīng)用基于AE的方法的可行性。最近,已經(jīng)表明,通過使用外差電路來降低AE傳感器的頻率范圍,可以使用低成本數(shù)據(jù)采集(DAQ)系統(tǒng)來在速率與基于振動(dòng)的技術(shù)相當(dāng)。這種方法已成功應(yīng)用于齒輪分析[21-23]以及增材制造應(yīng)用[24]。此外,基于時(shí)間同步重采樣的頻譜平均方法相結(jié)合,它已被證明對(duì)軸向速度為30-60 Hz的所有四種軸承故障類型進(jìn)行診斷是有效的[25,26]。然而,被發(fā)現(xiàn)對(duì)上述高速應(yīng)用有效的時(shí)域信號(hào)和統(tǒng)計(jì)特征組合對(duì)于本研究中使用的低速數(shù)據(jù)無效。在本文中,提出了一種有助于使用低采樣AE信號(hào)的方法,其使用連續(xù)的AE時(shí)間信號(hào)可管理。結(jié)果發(fā)現(xiàn),新的分析信號(hào)和不同的狀態(tài)指標(biāo)組合對(duì)評(píng)估的低轉(zhuǎn)速有效。因此,上述外差式模擬電路與這種新的信號(hào)處理方法結(jié)合使用,以處理在低采樣率下采集的AE數(shù)據(jù),滑行速率,并在2-10 Hz的軸速下診斷所有四種軸承故障類型。文獻(xiàn)中未提供所有四種軸承故障類型的診斷結(jié)果。這與低采樣率結(jié)合提供了在工業(yè)中實(shí)際實(shí)施基于AE的軸承監(jiān)測(cè)方法的可能性。
在本文的其余結(jié)構(gòu)如下。 第2部分提供了方法的詳細(xì)解釋。 在第3節(jié)中,討論了用于驗(yàn)證方法的種子故障測(cè)試和實(shí)驗(yàn)裝置的細(xì)節(jié)。 第4節(jié)介紹了種子故障測(cè)試的軸承故障診斷結(jié)果,第5節(jié)總結(jié)了本文。
2. 方法
圖1描繪了所提出的方法的概述。 首先,基于外差的頻率降低技術(shù)用于以與基于振動(dòng)的方法相當(dāng)?shù)乃俾什蓸覣E信號(hào),同時(shí)獲取轉(zhuǎn)速計(jì)信號(hào)。 其次,采樣的AE信號(hào)是用時(shí)間同步重新采樣的轉(zhuǎn)速計(jì)信號(hào)過零時(shí)間戳。 接下來,重采樣信號(hào)被頻譜平均并用于計(jì)算軸承故障診斷的CI。
圖1.方法學(xué)概述
該方法將分為4個(gè)部分。 第2.1節(jié)討論基于外差的AE信號(hào)采樣技術(shù)。然后,在第2.2節(jié)中討論軸承基本缺陷頻率。 接下來是對(duì)2.3節(jié)中的時(shí)間同步平均(TSA),時(shí)間同步重新采樣(TSR)和頻譜平均方法的回顧。 最后,2.4節(jié)解釋了軸承故障診斷的條件指標(biāo)的計(jì)算。
2.1 使用外差的采樣頻率降低技術(shù)
以AE為中心的技術(shù)的一個(gè)缺點(diǎn)是實(shí)際的計(jì)算負(fù)擔(dān)。 由于AE傳感器輸出信號(hào)的頻率通常高達(dá)幾MHz,所以基于AE的方法通常伴隨著高達(dá)幾個(gè)至10 MHz的采樣率。 本文采用一種采樣頻率降低技術(shù)用于降低與信號(hào)相關(guān)的能量,從而可以利用與振動(dòng)方法相當(dāng)?shù)牟蓸勇省?這種方法已被有效地用于齒輪分析[21-23],增材制造監(jiān)測(cè)[24]以及軸速30 Hz及以上軸承故障的診斷[25-26]。這種方法是計(jì)算量很大,因?yàn)檩^少的數(shù)據(jù)需要收集并存儲(chǔ)在計(jì)算機(jī)上,并最終降低伴隨數(shù)據(jù)采集的成本。
外差的概念早已在通信領(lǐng)域得到應(yīng)用。在無線電中,典型調(diào)幅信號(hào)的載波信號(hào)的頻率通常高達(dá)幾MHz,而調(diào)制到該載波信號(hào)的音頻信號(hào)通常具有低至幾kHz的頻率。通過解調(diào),放大調(diào)制信號(hào)頻率被降低,這使得音頻可以以低得多的速率被采集。 其結(jié)果不僅是采樣速率的降低,而且還是要處理數(shù)據(jù)所需的計(jì)算能力。
本文中使用的AE信號(hào)解調(diào)器與射頻正交解調(diào)器類似:將載波頻率切換到基帶,然后進(jìn)行低通濾波。這里應(yīng)用的方法稱為外差法。 在數(shù)學(xué)上,外差基于三角函數(shù)。 對(duì)于頻率為f 1的兩個(gè)信號(hào)和f 2,它可以寫成如下
(1)
其中f1是AE載波頻率,f2是解調(diào)器的參考信號(hào)頻率。
然后,如圖3所示,調(diào)制信號(hào)被低通濾波以抑制頻率(f 1 + f 2)處的高頻圖像。
接下來提供應(yīng)用于原始AE信號(hào)的外差法的詳細(xì)對(duì)話。 普遍認(rèn)為,放大調(diào)制是AE信號(hào)調(diào)制的主要形式。 盡管頻率和相位調(diào)制可能存在在AE信號(hào)中,它們被認(rèn)為是微不足道的,在此不再贅述。 幅度調(diào)制功能由提供
(2)
其中Ua是調(diào)制信號(hào),Um是載波信號(hào)幅度,ωc是載波信號(hào)頻率,m是調(diào)制系數(shù),x是感興趣的信號(hào)。 對(duì)于振幅Xm和頻率X,假設(shè)x可以表示為:
(3)
注意假設(shè)信號(hào)x的頻率x通常遠(yuǎn)小于載波信號(hào)的頻率ωc。然后,利用外差技術(shù),調(diào)制信號(hào)將乘以單位幅度參考信號(hào)cos(ωct)。 接下來提供結(jié)果Uo:
(4)
(5)
由于Um沒有包含與調(diào)制信號(hào)相關(guān)的任何有用信息,所以將其設(shè)置為0或通過去趨勢(shì)刪除。 從(5)可以看出,只有與感興趣信號(hào)有關(guān)的部分12 mXm cos Xt將在低通濾波后保留,而頻率為2ωc的高頻關(guān)聯(lián)分量將被去除。
解調(diào)步驟的添加實(shí)現(xiàn)了將信號(hào)頻率向下移位到接近振動(dòng)信號(hào)的頻率范圍的10秒kHz的目的。
2.2 軸承的基本缺陷頻率
當(dāng)軸承以恒定的速度旋轉(zhuǎn)時(shí),它的AE信號(hào)可以通過周期性特征來理論上表征。 一般來說,有4個(gè)基本的缺陷頻率來描述這個(gè)運(yùn)動(dòng)。 4個(gè)缺陷頻率為:基本列車頻率(FTF),球旋轉(zhuǎn)頻率(BSF),球通頻率外(BPFO)和球通頻率內(nèi)部(BPFI)。 這些頻率分別代表籠,球,外圈和內(nèi)圈的缺陷頻率[27]。 缺陷頻率定義如下:
其中De是滾動(dòng)元件直徑,Dp是節(jié)圓直徑,Z是滾動(dòng)元件的數(shù)量,?是以度數(shù)表示的接觸角,x是以Hz為單位的軸的旋轉(zhuǎn)速度。 圖4顯示了6205-2RS鋼球軸承的繪制。此外,參數(shù)和計(jì)算的缺陷倍頻器 6205-2RS分別在表1和2中提供。
在給定的軸速下,受監(jiān)測(cè)軸承的頻譜理論上應(yīng)該包含與存在或不存在軸承故障頻率有關(guān)的峰值。由于信號(hào)中存在機(jī)械噪聲,這些峰值往往難以觀察到。因此,已經(jīng)開發(fā)了信號(hào)處理技術(shù),例如平均方法,以幫助減少這種噪聲并提高信噪比。測(cè)量方程式中顯示的軸承缺陷頻率。 (6)-(9)通常用于基于窄帶和邊帶的分析方法。由于這些軸承故障頻率的振幅與軸的順序和齒輪嚙合頻率相比非常小,因此使用傅里葉分析直接從窄帶讀取軸承故障是很困難的。為了克服常規(guī)窄帶分析的先前缺點(diǎn),已經(jīng)開發(fā)了軸承包絡(luò)分析(BEA)。盡管BEA方法已經(jīng)建立,但是選擇合適的包絡(luò)解調(diào)頻帶(例如系統(tǒng)諧振頻率)是隱藏的或非常復(fù)雜。根據(jù)最近的BEA論文[28,29],不恰當(dāng)?shù)拇翱谶x擇可能會(huì)影響診斷性能。本文提出的方法不同于那些基于直接軸承頻率讀取的方法。通過應(yīng)用韋爾奇的頻譜平均方法并提取故障特征作為條件指標(biāo),故障軸承信號(hào)在統(tǒng)計(jì)上是可分的,并且盡管操作參數(shù)發(fā)生了變化(例如軸速度),但這些趨勢(shì)仍保持不變。在下面的章節(jié)中,討論了本文采用的平均方法。
圖2 兩個(gè)正弦信號(hào)的相乘
圖3 通過頻域?yàn)V波來提取外差信號(hào)
圖4 6205-2RS球軸承的圖紙
表1 6205-2RS鋼球軸承的參數(shù)s
2.3 AE信號(hào)的頻譜平均
時(shí)間同步平均(TSA)是用于提取周期波形的有效方法,并且在齒輪故障診斷中對(duì)齒嚙合振動(dòng)有多種應(yīng)用[30,31]。此外,TSA已經(jīng)被有效地用于處理軸承的振動(dòng)信號(hào)包絡(luò)故障診斷[32,33]。該概念是計(jì)算任何感興趣波形的連續(xù)周期的總體平均值。 這樣可以顯著降低噪聲,并增強(qiáng)表示平均波形周期的信號(hào)。 Braun [34]正式表示了在間隔nT采樣的信號(hào)x(t)的TSA y(nT)如下:
(10)
平均期間由mT表示。 關(guān)于TSA的更多細(xì)節(jié)可以在[30]中找到。
雖然TSA已被廣泛應(yīng)用于齒輪故障分析[35-38],迄今為止的文獻(xiàn)僅包含有限的軸承故障診斷應(yīng)用[32,33,39]。要成功應(yīng)用TSA,需要了解感興趣的重復(fù)頻率或沒有噪聲的同步信號(hào)。因此,應(yīng)用TSA進(jìn)行軸承分析的一個(gè)缺點(diǎn)是,這需要計(jì)算每個(gè)軸承故障類型的TSA。而且,所有應(yīng)用程序TSA的軸承分析基于振動(dòng)信號(hào)的量化,并且未能報(bào)告診斷籠狀缺陷的能力。目前的實(shí)施方案僅出現(xiàn)內(nèi)圈,外圈或球故障診斷的結(jié)果。此外,這些研究集中在相同故障類型的多個(gè)碎片或損壞的嚴(yán)重程度,沒有顯示出診斷所有四種軸承故障類型的能力。此外,盡管TSA可以改善包絡(luò)分析技術(shù)[33],但仍然需要對(duì)每個(gè)被調(diào)查的軸承故障類型進(jìn)行多次平均。此外,要成功將TSA應(yīng)用于AE信號(hào)進(jìn)行軸承分析,需要觸發(fā)信號(hào)必須與所有軸承故障類型同步。而且,AE信號(hào)的大數(shù)據(jù)量和非平穩(wěn)行為使TSA的直接計(jì)算和實(shí)時(shí)狀態(tài)監(jiān)測(cè)不切實(shí)際。在這篇論文中,基于外差的AE DAQ系統(tǒng),TSR和頻譜平均被用來克服這個(gè)問題上述挑戰(zhàn)并成功應(yīng)用平均方法這導(dǎo)致提取用于軸承故障診斷的特征。
在工業(yè)運(yùn)行中,軸承可能會(huì)出現(xiàn)速度和潛在滑動(dòng)的波動(dòng)。 因此,基于軸承幾何形狀的故障率可能不準(zhǔn)確。 在參考文獻(xiàn) [40],已經(jīng)表明,通過使用基于軸旋轉(zhuǎn)的同步重采樣技術(shù),可以考慮軸速的潛在波動(dòng),并且減少了光譜拖尾的影響。 通過在觸發(fā)信號(hào)旋轉(zhuǎn)之間重新采樣偶數(shù)個(gè)點(diǎn),獲得更好的快速傅立葉變換(FFT)結(jié)果。 在本文中,類似的時(shí)間同步重采樣方法是使用軸零交叉時(shí)間(ZCT)完成的,并且在下文中正式提出。
在形式上,重采樣過程是通過將一個(gè)軸旋轉(zhuǎn)中的多個(gè)數(shù)據(jù)點(diǎn)插入L個(gè)數(shù)據(jù)點(diǎn)中來實(shí)現(xiàn)的,從而:其中L是ZCT之間的插值點(diǎn)的數(shù)量,r是重新采樣時(shí)軸間交叉點(diǎn)之間的平均點(diǎn)數(shù)。 一旦L被確定,每個(gè)段包含用于FFT計(jì)算的相等數(shù)量的數(shù)據(jù)點(diǎn),并且可以執(zhí)行頻譜平均。
本文提出的方法將TSR方法與頻譜平均相結(jié)合,以計(jì)算單個(gè)平均值,從而可以提取軸承故障診斷的有效條件指標(biāo)。 這有效地消除了需要計(jì)算多個(gè)TSA。 如圖5所示,是對(duì)頻譜平均方法的綜述。 頻譜平均值的計(jì)算需要對(duì)數(shù)據(jù)進(jìn)行分段,以便可以計(jì)算每個(gè)分段的傅立葉變換。 然后,實(shí)施平方數(shù)量級(jí)的平均集合平均值。
圖5 頻譜平均方法
為了實(shí)現(xiàn)頻譜平均,首先必須切斷信號(hào)。因此,軸ZCT用于對(duì)AE信號(hào)進(jìn)行分段和重采樣。然后,獲得各部分的平方幅度譜的平均值,其導(dǎo)致用于評(píng)估統(tǒng)計(jì)特征的譜平均值。換句話說,軸旋轉(zhuǎn)之間的持續(xù)時(shí)間被用作平均AE信號(hào)的截面函數(shù)。此外,數(shù)據(jù)段長度L由重采樣之后軸的ZCT之間的數(shù)據(jù)點(diǎn)的數(shù)量確定。通過將分段重新采樣到L個(gè)數(shù)據(jù)點(diǎn),相同數(shù)量的數(shù)據(jù)點(diǎn)用于FFT計(jì)算。因此,本文提出的方法利用軸旋轉(zhuǎn)的ZCT對(duì)AE信號(hào)進(jìn)行分段和重新采樣,并計(jì)算各部分傅立葉變換平方幅度的總體平均值。這種方法提供了計(jì)算一個(gè)頻譜平均值的能力,并有效地診斷可能發(fā)生的任何軸承故障,同時(shí)消除了多次TSA計(jì)算的必要性。此外,計(jì)算單個(gè)平均值的需求也減少了硬件上的計(jì)算負(fù)擔(dān),使在線分析和維護(hù)決策成為可能。在計(jì)算AE信號(hào)的頻譜平均值之后,計(jì)算和評(píng)估各種軸承故障特征。
2.4 軸承故障診斷的條件指標(biāo)
文獻(xiàn)中有許多軸承故障狀態(tài)指示器,用于量化加速度計(jì)信號(hào)以幫助承載故障診斷。最近的一些研究已經(jīng)開發(fā)出有效的CIs,通過量化AE信號(hào)來完成軸承故障診斷[25,26,42,43]??捎玫腃I之間的主要區(qū)別在于計(jì)算方法。例如,CI可以從信號(hào)的時(shí)域和頻域中提取。此外,CI可以從原始信號(hào)或由信號(hào)處理技術(shù)(如TSA或光譜平均)處理的信號(hào)中提取。以前,頻譜平均值的平方幅度的反傅里葉變換與RMS和峰值CI相結(jié)合已被用于AE和振動(dòng)信號(hào)在30 Hz和更高的軸速下成功診斷軸承故障[25,26,44]。但是,由于上述分析信號(hào)和CIs在這些研究中測(cè)試的結(jié)果對(duì)低速分析無效,因此這些工作留待將來調(diào)查。本文使用平方光柵的對(duì)數(shù)的傅立葉逆變換,tude譜平均結(jié)果來研究新CIs的潛力,該新CI可以明確診斷低速應(yīng)用中的所有四種軸承故障類型。通過在傅立葉逆變換中引入一個(gè)對(duì)數(shù),被分析的信號(hào)類似于參考文獻(xiàn)中所使用的功率倒譜。 [45]進(jìn)行齒輪分析。還應(yīng)該提到的是,盡管發(fā)現(xiàn)日志的引入對(duì)于低速應(yīng)用是有效的,但它對(duì)[25,26]中使用的高速數(shù)據(jù)不是有效的。此外,用于高速調(diào)查的配置項(xiàng)對(duì)本文提供的低速數(shù)據(jù)無效。參考文獻(xiàn)中使用的一些低速軸承配合比。 [17]對(duì)高速應(yīng)用沒有效果,但是發(fā)現(xiàn)在本文中測(cè)試的軸速下工作結(jié)合新的分析信號(hào)。形式上,用于CI計(jì)算的輸入信號(hào)給出如下:
(11)
輸入信號(hào)x是用于CI計(jì)算的時(shí)域信號(hào),是由(19)得到的頻譜平均結(jié)果的對(duì)數(shù)的傅立葉逆變換。
可以使用能源操作員(EO)來計(jì)算CI。 EO是一個(gè)
自相關(guān)函數(shù)的殘差類型[46]。 在離散化的形式下,數(shù)學(xué)公式如下:
(12)
還使用(20)中獲得的信號(hào)的幅度調(diào)制(AM)信號(hào)或希爾伯特包絡(luò)來計(jì)算CI。 AM通過以下方式正式獲得:
(13)
其中是數(shù)據(jù)集x的Hilbert包絡(luò)(20)獲得。
在本文中,為軸承故障診斷探索了許多可行的CI。 表3中提供了被調(diào)查的CIs的定義。 (RMS),峰值,峰值因子(CF),峰度(Kurt),偏度(Skew),峰峰值(p2p),香農(nóng)熵,Log熵,直方圖上限(UB), 和直方圖下限(LB)。 使用((20),(21)和(22))計(jì)算出的信號(hào)對(duì)每個(gè)CI進(jìn)行評(píng)估。
表3 CI的定義
3.實(shí)驗(yàn)設(shè)置
本節(jié)介紹用于評(píng)估所提出的基于AE傳感器的低速軸承故障診斷方法的實(shí)驗(yàn)裝置。圖6和圖7描述了用于進(jìn)行含鋼種子故障測(cè)試的軸承試驗(yàn)臺(tái)的兩個(gè)觀點(diǎn)。 測(cè)試裝置的機(jī)制以及AE傳感器和轉(zhuǎn)速計(jì)的位置。物理聲學(xué)公司(PAC)寬頻帶(WD)型AE傳感器的工作范圍為125 kHz至1 MHz,使用瞬間膠軸向安裝在軸承箱的表面。
在實(shí)驗(yàn)期間使用6205-2RS型鋼球軸承。 在上述鋼軸承上播種了四種故障類型:內(nèi)圈和外圈故障,滾動(dòng)體故障和籠式故障(見圖8)。 兩個(gè)軸承座圈故障是通過用金剛石尖端砂輪鉆頭刮擦內(nèi)外鋼制滾道表面以覆蓋滾珠接觸表面而產(chǎn)生的。 兩場(chǎng)比賽的種子斷層大約1/16英寸寬,1/250英寸深。 通過在其中一個(gè)鋼球位置切割鋼保持架,然后使用金剛石尖端砂輪鉆頭產(chǎn)生一個(gè)約為鋼球體積的20%的小凹陷,從而產(chǎn)生滾動(dòng)元件故障損壞。 對(duì)于籠狀缺陷,鋼籠被切割在兩個(gè)球位置之間。 切口的大小約為球直徑的50%。 對(duì)于所有種子故障測(cè)試,軸承密封和潤滑脂均已拆除,并在執(zhí)行故障后進(jìn)行更換。
此外,在進(jìn)行種子故障測(cè)試時(shí),故意將種子的尺寸盡可能小以緊密模擬傳播階段的故障。 當(dāng)故障傳播時(shí),缺陷尺寸可能會(huì)很小,隨著時(shí)間的推移會(huì)變得更加重要。 盡管本文中使用的種子斷裂的大小并不像初始斷層那樣小,但文獻(xiàn)中廣泛報(bào)道了使用AE傳感器的一個(gè)優(yōu)點(diǎn)是能夠檢測(cè)早期損傷進(jìn)展中的初期斷層 階段。
圖9給出了解調(diào)板,電源,功能發(fā)生器和取樣裝置。 解調(diào)板執(zhí)行AE傳感器信號(hào)與從函數(shù)發(fā)生器輸出的參考信號(hào)的相乘。 這允許實(shí)施基于外差的采樣頻率降低方法。 兩個(gè)信號(hào)都作為輸入,輸出是兩個(gè)信號(hào)的乘積。 在將信號(hào)信息轉(zhuǎn)移到較低的頻率范圍之后,包絡(luò)的低頻輸出將被轉(zhuǎn)換到采樣板,同時(shí)濾除高頻成分。 為了降低AE傳感器信號(hào)頻率,確定其中心載波頻率并將其設(shè)置為用于解調(diào)的參考信號(hào)頻率。
圖6 軸承試驗(yàn)臺(tái)
使用函數(shù)發(fā)生器實(shí)現(xiàn)的增加的啁啾函數(shù)用于記錄系統(tǒng)的輸出。 發(fā)現(xiàn)中心AE信號(hào)載波頻率為400kHz,因此用作解調(diào)參考頻率。 在研究期間使用的采樣設(shè)備是一個(gè)低頻數(shù)據(jù)采集板,能夠處理高達(dá)250 kS / s的采樣頻率。
圖7 軸承試驗(yàn)臺(tái)的前視圖
圖8 含鋼種子的故障測(cè)試
NI Labview signal express用于所有信號(hào)采集,以100 kHz采樣率采集連續(xù)AE信號(hào)。 健康和種子故障軸承在5軸速度下測(cè)試:2 Hz,4 Hz,6 Hz,8 Hz和10 Hz。 在每個(gè)調(diào)查的軸速下,收集5個(gè)樣本,每個(gè)軸承類型總共25個(gè)樣本。 為了確保樣本包含至少200軸平均轉(zhuǎn)速的計(jì)算,對(duì)于上述軸速度,分別記錄信號(hào)100,50,33,25和20秒。 同樣重要的是要注意,由于電機(jī)控制的限制,最低的測(cè)試軸轉(zhuǎn)速為2 Hz,并且在實(shí)驗(yàn)期間沒有施加負(fù)載。 為了一致性,AE傳感器被放置在相同的軸向位置以進(jìn)行所有數(shù)據(jù)采集。
圖9 解調(diào)和采樣設(shè)備
4. 結(jié)果
本節(jié)介紹種子故障測(cè)試的驗(yàn)證結(jié)果。 外差后,AE采集信號(hào)以100 kHz的采樣率采集。 然后,收集到的信號(hào)被時(shí)間同步重新采樣并進(jìn)行頻譜平均。 結(jié)果用于CI計(jì)算。 盡管共調(diào)查了30個(gè)CI,但有4個(gè)顯示可以清楚診斷所有四種軸承故障類型。 如圖10所示的是軸速度下AM Shannon熵熵值的平均值。 每個(gè)點(diǎn)表示每個(gè)測(cè)試軸速度下每個(gè)軸承5個(gè)樣本的平均值。 隨著每個(gè)平均CI值是95%的置信度誤差欄。
如圖10所示,AM香農(nóng)的熵CI清楚地區(qū)分了所有四種軸承故障類型。 由于誤差線沒有重疊,故障模式相互分離以及健康的軸承類型在統(tǒng)計(jì)上顯著。 另外,隨著軸轉(zhuǎn)速的增加,所有軸承信號(hào)似乎都包含相似的CI值上升趨勢(shì)。 另一個(gè)有趣的觀察結(jié)果是故障模式的順序與參考文獻(xiàn)中的高速診斷結(jié)果一致。[25,26]。 接下來,圖11顯示了軸轉(zhuǎn)速的平均EO香農(nóng)熵CI值。
圖10 平均AM Shannon熵的軸速(Hz)
圖11中的結(jié)果與圖10中的結(jié)果相同。每個(gè)點(diǎn)對(duì)應(yīng)于每個(gè)軸速度下5個(gè)采集樣本的平均CI值,以及95%的誤差線。 該結(jié)果驗(yàn)證了EO香農(nóng)熵CI的軸承故障診斷能力。 在確認(rèn)軸承故障類型分離的統(tǒng)計(jì)顯著性的任何誤差線之間沒有重疊。 雖然診斷已經(jīng)完成,但分離似乎沒有使用AM香農(nóng)熵CI所觀察到的顯著。 另外,軸承類型順序與圖10一樣保持不變,以及上述參考文獻(xiàn)中的高速軸承結(jié)果[25,26]。 下面的圖12顯示了軸轉(zhuǎn)速的平均香農(nóng)熵結(jié)果。
如圖12所示,香農(nóng)熵CI也明確地分離了所有軸承故障類型。 趨勢(shì)與圖1和圖2中觀察到的相似。 10和11,并且分離在統(tǒng)計(jì)上顯著,沒有95%誤差條的重疊。 此外,分離的出現(xiàn)似乎大于圖11所示的結(jié)果,盡管不如圖10所示的結(jié)果顯著??傊腥齻€(gè)結(jié)果證實(shí)了不同形式的 香農(nóng)的熵CI。 接下來,圖13顯示了軸速度的平均直方圖下限結(jié)果。
如圖13所示,直方圖下界CI也明確區(qū)分了所有四種軸承故障類型和健康的軸承箱。 這種分離在統(tǒng)計(jì)上也是顯著的,其結(jié)果包含誤差棒的零重疊。 另外,與前面的圖一樣,觀察到相同的軸承類型的順序。 需要注意的一點(diǎn)是,在2Hz的軸速下觀察到最高的CI值,并且隨著軸速度增加到10Hz,趨勢(shì)向下。
總之,圖10-13展示了AM香農(nóng)熵,EO香農(nóng)熵,香農(nóng)熵和直方圖下限用于低速軸承診斷時(shí)與基于外差的AE DAQ結(jié)合的能力以及所提出的信號(hào)處理方法。 這些結(jié)果證實(shí),使用低采樣AE數(shù)據(jù)可以在低于10 Hz的速度下實(shí)現(xiàn)故障診斷。 還應(yīng)該注意的是,對(duì)于圖1和圖2中呈現(xiàn)的香農(nóng)熵CI結(jié)果。 10-12,故障模式的趨勢(shì)和分離隨軸轉(zhuǎn)速的降低而增加。 因此,該方法有可能在軸速低于此處所述的情況下診斷軸承故障模式。 結(jié)果提供在圖。 10-13證實(shí)本文提出的基于AE的新型診斷方法已經(jīng)得到驗(yàn)證。
圖11 軸轉(zhuǎn)速的平均EO香農(nóng)熵(Hz)
圖12 軸轉(zhuǎn)速的平均香農(nóng)熵(Hz)
圖13 軸轉(zhuǎn)速的平均直方圖下限(Hz)
5. 結(jié)論
在本文中,介紹了一種新的低速軸承故障診斷方法。這種基于AE的方法從外差技術(shù)開始,允許AE信號(hào)以與基于振動(dòng)的方法相當(dāng)?shù)乃俾蔬M(jìn)行采樣。然后,使用輸入觸發(fā)信號(hào),低采樣AE信號(hào)被時(shí)間同步重采樣,以考慮軸速和軸承滑動(dòng)的潛在波動(dòng)。重采樣方法還提供根據(jù)軸ZCT分割A(yù)E信號(hào)的能力,從而可以使用偶數(shù)個(gè)數(shù)據(jù)點(diǎn)來計(jì)算單個(gè)頻譜平均值,該均值用于提取特征并評(píng)估大量的軸承故障診斷的CI。所提出的方法使用從軸承試驗(yàn)臺(tái)上的種子故障鋼軸承收集的AE信號(hào)進(jìn)行驗(yàn)證。在所測(cè)試的CI中,AM香農(nóng)熵,EO香農(nóng)熵,香農(nóng)熵和直方圖下界對(duì)于低速軸承故障診斷是有效的。特別是,對(duì)于香農(nóng)熵的CIs,軸承斷層的分離隨軸轉(zhuǎn)速的降低而增加。這表明該方法診斷軸承故障的可能性低于本文所介紹的軸速?;诋?dāng)前時(shí)間同步平均的軸承分析方法需要計(jì)算每個(gè)軸承故障類型的多個(gè)平均值。另外,所有用于低速軸承分析的基于AE的方法都需要高采樣率,并且最重視故障檢測(cè)或缺乏診斷所有軸承故障類型的能力。本文的結(jié)果表明,低采樣AE信號(hào)與本文提出的方法相結(jié)合可以用于在低于10 Hz的低速下診斷所有四種軸承故障類型。
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Low speed bearing fault diagnosis using acoustic emission sensorsBrandon Van Hecke,Jae Yoon,David HeDepartment of Mechanical and Industrial Engineering,The University of Illinois at Chicago,Chicago,IL 60607,United Statesa r t i c l ei n f oArticle history:Received 3 December 2014Received in revised form 28 October 2015Accepted 29 October 2015Keywords:Bearing faultDiagnosisAcoustic emission sensorLow speeda b s t r a c tIn this paper,a new methodology for low speed bearing fault diagnosis is presented.This acoustic emis-sion(AE)based technique starts with a heterodyne frequency reduction approach that samples AE signalsat a rate comparable to vibration centered methodologies.Then,the sampled AE signal is time syn-chronously resampled to account for possible fluctuations in shaft speed and bearing slippage.Theresampling approach is able to segment the AE signal according to shaft crossing times such that an evennumber of data points are available to compute a single spectral average which is used to extract featuresand evaluate numerous condition indicators(CIs)for bearing fault diagnosis.Unlike existing averagingbased noise reduction approaches that require the computation of multiple averages for each bearingfault type,the presented approach computes only one average for all bearing fault types.The presentedtechnique is validated using the AE signals of seeded fault steel bearings on a bearing test rig.The resultsin this paper have shown that the low sampled AE signals in combination with the presented approachcan be utilized to effectively extract condition indicators to diagnose all four bearing fault types at mul-tiple low shaft speeds below 10 Hz.?2015 Published by Elsevier Ltd.1.IntroductionMost heavy equipment and mechanical devices contain at leastsome type of bearing.Additionally,because it is typical for bear-ings to be utilized in less than ideal working conditions,problemsmay arise and result in premature failures.When a bearing fails,the malfunction can lead to significant downtime,elevated mainte-nance costs,and the potential for a decrease in productivity.Onerecent study presented a rolling bearing fault diagnosis approachusing neural networks and a time/frequency-domain vibrationapproach 1.This study showed that neural networks can aid inthe diagnosis of various motor bearing faults using vibration datagenerated from a bearing test rig.Neural networks have also beenshown to be effective for condition monitoring when usingstatistical-time features 2.Another study focused on rotatingmachinery in general and used a model-based approach for thedetection and diagnosis of mechanical faults 3.This researchdeveloped a nonlinear filtering technique to address complex non-linear vibration responses due to factors such as unbalance,changes in stiffness,and damping of the rotor bearing system.Validation of the aforementioned technique was accomplishedvia the use low signal-to-noise environment simulations.Ref.4presented a fault detection approach based on stator currentmonitoring for in situ bearing faults.In this paper,bearing fault fea-tures were extracted using a combination of noise cancellation andstatistical process control techniques to detect out of control sam-ples due to the degradation of lubrication starved bearings.Oneresearch has presented a two-step data mining approach to classifydefects in plastic bearings 5.This study effectively used empiricalmode decomposition(EMD)to extract time domain condition indi-cators(CIs)which were used as inputs to a supervised learningalgorithm to classify bearing defects.Other studies have presentedeffective bearing defect techniques through the use of local andnonlocal preserving projection 6,trace ratio linear discriminantanalysis 7,or the power spectral density(PSD)analysis of ampli-tude and frequency current-demodulated bearing signals fordirect-drive wind turbines 8.In many industries,the use of low speed rotating machines is astaple for successful operation.Such machines can be found insteel and paper mills,biological applications,and wind turbines.Thus,the monitoring of bearings,shafts and gears in such applica-tions is critical for the proper maintenance of low speed equip-ment.In the low speed bearing fault diagnosis literature,lowspeed has been referred to as in the range from 0.33 Hz to 10 Hz9,whereas significantly lower speed thresholds have been con-sidered as a separate classification range.For example,speedsbelow 0.5 Hz have been viewed as ultra low”10 and if below0.83 Hz considered extremely low”11.In this paper,the bearinghttp:/dx.doi.org/10.1016/j.apacoust.2015.10.0280003-682X/?2015 Published by Elsevier Ltd.Corresponding author.E-mail address:davidheuic.edu(D.He).Applied Acoustics 105(2016)3544Contents lists available at ScienceDirectApplied Acousticsjournal homepage: diagnostic problem with a shaft speed falling within the lowspeed range described in 9 will be addressed.To date,the condition monitoring of rolling element bearingsand other rotating equipment using vibratory analysis is an estab-lished technique and the industry standard.One study investigatedthe use of parametric models of amplitude demodulated vibrationsignals,and the resulting frequency spectra,for bearing faultdetection and diagnosis of roller bearings with defects at a shaftspeed of 1 Hz 12.In that paper,a signal processing techniquewas first used to detect bearing defects and the frequency spectrawere then used to classify the defects.Although effective,thismethodology required visual inspection of the frequency spectrato achieve fault diagnosis.Later,another study developed a lowspeed technology system to measure vibrations resulting fromlow speed rotating machinery 9.This system was centered onseparating the high frequency noise of the machine from the lowfrequency signatures of interest,and validation was presentedusing results from a low speed rotor and gearbox.More recently,for low speed applications,a general model of faulty bearing vibra-tion signals has been established and it was shown that envelope-autocorrelation can be observed in a faulty bearing,but not in thehealthy bearing case 13.Others have sought to develop newaccelerometers and have shown that with the aid of the resonancedemodulation technique,low speed rolling bearing faults can bedetected 14.However,the use of vibration is limited at low speedapplications because the change in energy generated from faults atsuch speeds may not be detectable using traditional accelerometerbased monitoring systems.Thus,researchers have investigated theuse of acoustic emission(AE)sensors and strain gauges for compo-nent monitoring at low speed conditions.AE based studies have shown promising results for incipientfault detection,and recent studies have explored their use forlow speed bearing monitoring.One early investigation looked atthe monitoring of bearings at low speeds 15.In this study,resultscomparing acceleration,shock pulse transducer,acoustic emission,and jerk measurements were presented.Among other conclusionsdrawn,it was mentioned that AE resulted in clear detection of anouter track bearing defect at speeds as low as 0.17 Hz.However,it was also mentioned that the observed AE response could notexplained and that averaging methodologies could not be utilizedbecause the signals did not repeat exactly on a once per revolutionbasis.Another study reported the use of AE for monitoring rollingelement bearings at extremely low shaft speeds from 0.0083 Hz to0.083 Hz 11.In this study it was found that AE measurement isquite sensitive for detecting bearing faults when the bearing isrotating at an extremely low speed,whereas the accelerationenvelope was limited to detecting faults at the lowest shaft speedof 10 Hz.This study utilized an AE pulse count and noted that usingsuch an approach allowed the data to be manageable.Ref.16explored the application of AE for the low speed monitoring ofbearings.This study used the K-means clustering approach to clas-sify line defects on a spherical roller bearing.Although promisingresults were achieved,the study focused on the frequency rangeof 100 kHz to 1 MHz and also relied on a data mining techniquewhich can be time consuming due to the large data requirementfor training and testing of the models.Ref.17 also relied on a datamining approach,presenting a different classification techniquebased on relevance vector machines and support vector machinesto diagnose bearing faults at shaft speeds ranging from 0.33 Hzto 1.33 Hz.In another study,an AE sensor was used to investigatethe incipient fault detection of low speed rolling element bearingsin the frequency range up to 100 kHz 18.This study evaluated anumber of time domain condition indicators using different filterbands to determine the best parameters that can distinguishbetween healthy and faulty inner race bearing signals.However,the goal of this study was to determine effective filter band rangesand time domain condition indicators that can be used to evaluatethe acquired AE signals.Moreover,only an inner race fault wasobserved and the diagnosis of all four bearing fault types couldnot be confirmed using the respective filter band and CIs presentedin the paper.Later,Ref.10 proposed a diagnostic method usingthe AE envelope waveform and obtained results at speeds less than1.67 Hz.This study confirmed that the periodicity of outer ringflaking could be captured at speeds as low as 0.17 Hz.Recently,other studies investigated AE and its use for the condition monitor-ing of low speed shafts and thrust ball bearings 19,20.Thesestudies demonstrated the ability to use AE to detect crack initiationand growth and also mainly focused on the investigation of AEsourcelocation.Moreover,theaforementionedexperimentfocused on a single shaft speed of 1.2 Hz and tested bearing condi-tion under loading and starved lubricating conditions.Nonetheless,the aforementioned high frequency AE signals areaccompanied by high sampling rates.Moreover,for low speedbearing fault diagnostic applications,data acquisitions need to berelatively long to capture the mechanical defect frequencies.Thecombination of high sampling rates with long data samples limitsthe feasibility of practical application of AE based approaches.Recently,it has been shown that by using a heterodyne circuit todownshift the frequency range of an AE sensor,a low cost dataacquisition(DAQ)system can be utilized to sample AE sensors ata rate comparable to vibration based techniques.This approachhas been successfully applied for gear analysis 2123 as well asfor an additive manufacturing application 24.Additionally,withthe combination of a time synchronous resampling based spectralaverage approach,it has been shown to be effective to diagnose allfour bearing fault types for shaft speeds of 3060 Hz 25,26.How-ever,the time domain signal and statistical feature combinationthat was found to be effective for the aforementioned high speedapplications were not effective for the low speed data used in thisstudy.In this paper,an approach that facilitates the use of lowsampled AE signals is presented which make the use of continuousAE time signals manageable.It was found that the combination of anew analysis signal and different condition indicators were effec-tive for the evaluated low shaft speeds.Thus,the aforementionedheterodyne analog circuit is used in combination with this new sig-nal processing approach to process AE data acquired at a low sam-pling rate,and diagnose all four bearing fault types at shaft speedsfrom 210 Hz.The diagnosis of all four bearing fault types at thepresented shaft speeds has not been presented in literature.That,in combination with the low sampling rate provides merit for thepossibility of a practical implementation of an AE based bearingmonitoring approach in industry.The remainder of the paper is structured as follows.Section 2provides a detailed explanation of the methodology.In Section 3,the details of the seeded fault tests and experimental setup usedto validate the methodology are discussed.Section 4 presents thebearing fault diagnosis results from the seeded fault tests and Sec-tion 5 concludes the paper.2.The methodologyFig.1 depicts an overview of the presented methodology.First,aheterodyne based frequency reduction technique is used to samplean AE signal at a rate comparable to vibration based approacheswith the simultaneous acquisition of a tachometer signal.Second,the sampled AE signal is time synchronously resampled using thetachometer signals zero crossing time stamp.Next,the resampledsignal is spectrally averaged and used to compute CIs for bearingfault diagnosis.The methodology will be presented in 4 sections.Section 2.1discusses the heterodyne based AE signal sampling technique.36B.Van Hecke et al./Applied Acoustics 105(2016)3544Then,in Section 2.2 a discussion on the bearing fundamental defectfrequencies is covered.It is followed by a review of time syn-chronous average(TSA),time synchronous resampling(TSR),andthe spectral averaging approach in Section 2.3.Finally,the calcula-tion of condition indicators for bearing fault diagnosis is explainedin Section 2.4.2.1.The sampling frequency reduction technique using heterodyneOne shortcoming of AE centered techniques is the substantialcomputational burden.Because the frequency of the AE sensor out-put signal is typically as high as several MHz,AE based approachesare ordinarily accompanied with sampling rates as high as severalto 10 MHz.In this paper,a sampling frequency reduction techniqueis utilized to down shift the energy related to the signal so that asampling rate comparable to vibration methods can be utilized.This type of approach has effectively been utilized for gear analysis2123,additive manufacturing monitoring 24,and for the diag-nosis of bearing faults at shaft speeds of 30 Hz and above 25,26.This approach is computationally substantial because less dataneeds to be collected and stored on the computer and ultimatelyreduces the cost accompanied with data acquisition.The concept of heterodyne has long been employed in the com-munications field.In radio,the frequency of the carrier signal of atypical amplitude modulated signal is often as high as several MHzwhereas the audio signal modulated to that carrier signal often hasa frequency as low as a few kHz.With demodulation,the ampli-tude modulated signal frequency is reduced which allows theaudio to be acquired at a much lower rate.The result is not onlya reduction in sampling rate,but also the required computationalpower for data to be processed.The AE signal demodulator used in this paper works similarly toa radio quadrature demodulator:shifting the carrier frequency tobaseband,followed by low pass filtering.The approach appliedhere is called heterodyne.Mathematically,heterodyning is basedon a trigonometric identity.For two signals with frequency f1and f2,respectively,it could be written as the following:sin 2pf1tsin 2pf2t 12cos 2pf1?f2t?12cos2pf1 f2t?1where f1is the AE carrier frequency and f2is the demodulators ref-erence signal frequency.Forinstance,letf1 3 Hzandf2 4 Hz,andnotey1 sin2p3tandy2 sin2p4t:Then,theirmultiplicationY y1y2,is presented in Fig.2.Then,as expressed in Fig.3,the modulated signal is low pass fil-tered to reject the high frequency image at frequency(f1 f2).A detailed dialogue of the heterodyne approach applied on theraw AE signal is provided next.It is generally accepted that ampli-tude modulation is the major form of modulation for AE signals.Although,frequency and phase modulation are potentially existentin the AE signal,they are considered trivial and will not be dis-cussed herein.The amplitude modulation function is provided by(2).Ua Um mxcosxct2where Uais the modulated signal,Umis the carrier signal amplitude,xcis the carrier signal frequency,m is the modulation coefficient,and x is the signal of interest.With an amplitude Xmand frequencyX,assume that x can be expressed as:x XmcosXt3Note that it is presumed that the frequencyxof the signal x isnormally much smaller than the frequencyxcof the carrier signal.Then,with the heterodyne technique,the modulated signal will bemultiplied by a unit amplitude reference signal cosxct.The resultUois provided next:Uo Um mxcosxctcosxct Um mx1212cos2xct?4Next,after substituting Eq.(3)into Eq.(4):Uo12Um12mXmcosXt 12Umcos2xct14mXmcos2xcXt cos2xc?Xt?5Because Umdoes not enclose any useful information associatedwith the modulated signal,it is set as 0,or removed via de-trending.From(5),it can be understood that only the part relatedto the signal of interest,12mXmcosXt,will remain after low pass fil-tering,while the high frequency associated components aroundfrequency 2xcwill be removed.The addition of the demodulation step achieves the purpose ofdown shifting the signal frequency to 10 s of kHz,which is close tothe frequency range of vibration signals.Hence,any data acquisi-tion board with a low sampling rate should be able to samplepre-processed AE data.2.2.Discussion on the bearing fundamental defect frequenciesAs a bearing rotates at a constant speed,its AE signal can be the-oretically characterized by a periodical property.Generally,thereare 4 fundamental defect frequencies to describe this motion.The 4 defect frequencies are:fundamental train frequency(FTF),ball spin frequency(BSF),ball pass frequency outer(BPFO),and ballpass frequency inner(BPFI).These frequencies respectively repre-sent the defect frequencies of the cage,ball,outer race,and innerrace 27.The defect frequencies are defined as:FTF x=2 1?De=Dpcos?6BSF xDp=2Def1?De=Dpcos?2g7BPFO xZ=2 1?De=Dpcos?8BPFI xZ=2 1 De=Dpcos?9where Deis the rolling elements diameter,Dpis the pitch diameter,Z is the number of rolling elements,is the contact angle indegrees,andxis the rotational speed of the shaft in Hz.The draw-ing of a 6205-2RS steel ball bearing is shown in Fig.4.Moreover,theparameters and the calculated defect frequency multipliers of the6205-2RS are provided in Tables 1 and 2 respectively.At a given shaft speed,the frequency spectra of the monitoredbearing should theoretically contain peaks that are related to thepresence or absence of a bearing defect frequency.These peaksare often difficult to observe due to mechanical noise present inthe signal.Thus,signal processing techniques,such as averagingFig.1.Overview of the methodology.B.Van Hecke et al./Applied Acoustics 105(2016)354437approaches,have been developed to help reduce such noise andincrease signal to noise ratio.Measuring the bearing defect fre-quencies shown in Eqs.(6)(9)are usually utilized by narrowbandand sideband based analysis methods.Because the amplitude ofthose bearing defect frequencies are very small compared to shaftorder and gear mesh frequency,detecting bearing faults by directreading from narrowband using Fourier analysis is difficult.Toovercome the preceding drawback of the conventional narrowbandanalysis,bearing envelope analysis(BEA)has been developed.Although the BEA method is well established,the selection of aFig.2.The multiplication of two sinusoid signals.Fig.3.Extracting the heterodyned signal by frequency domain filtering.38B.Van Hecke et al./Applied Acoustics 105(2016)3544proper envelope demodulation band(e.g.system resonance fre-quency)is either hidden or very complicated.An improper windowselection could compromise the diagnostic performance accordingto the recent BEA papers 28,29.The method presented in thispaper is different from those direct bearing frequency readingbased method.By applying Welchs spectral averaging methodand extracting fault features as condition indicators,faulty bearingsignals become statistically separable and those trends are main-tained although the operating parameter is altered(e.g.shaftspeed).In the following section,the averaging approach employedin this paper is discussed.2.3.Spectral averaging of AE signalTime synchronous averaging(TSA)is a validated approach forthe extraction of periodic waveforms and has multiple appli
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