三家子煤礦主井提升設(shè)備選型設(shè)計(jì)【單繩纏繞式提升機(jī)的設(shè)計(jì)】
三家子煤礦主井提升設(shè)備選型設(shè)計(jì)【單繩纏繞式提升機(jī)的設(shè)計(jì)】,單繩纏繞式提升機(jī)的設(shè)計(jì),三家子,煤礦,提升,晉升,設(shè)備,裝備,選型,設(shè)計(jì),纏繞
對(duì)礦井提升機(jī)鋼絲繩的內(nèi)部阻尼特性進(jìn)行
非平面橫向震動(dòng)分析
概要
本文介紹的工作是為了增加現(xiàn)在礦井提升機(jī)鋼絲繩的疲勞知識(shí)，特別是進(jìn)行周期性非平面橫向震動(dòng)的鋼絲繩線間的國(guó)際電線/鋼絞線在摩擦?xí)r從中損失的內(nèi)部能量。這種摩擦能量損失現(xiàn)在是限制有益的工作生活中使用懸掛鋼絲繩的主要因素之一。
實(shí)驗(yàn)采用的方法指出了鋼絲繩的兩種機(jī)械特性,主要是由于鋼絲繩結(jié)構(gòu)的類型，它們獨(dú)立了振幅和頻率。興趣是集中在曲率的變化率這一主要參數(shù)，它影響了內(nèi)部的阻尼機(jī)理。經(jīng)驗(yàn)結(jié)果顯示,振幅和模態(tài)數(shù)在內(nèi)部的量化損失中起重要的作用,還透露出，由于上升振動(dòng)疲勞指數(shù)潛在的較高水平，一個(gè)關(guān)鍵的曲率半徑在傷害下存在。
介紹
在現(xiàn)代南非黃金深礦井中，傷害的問題由于振動(dòng)疲勞仍在繼續(xù)，它限制了纏繞速度，纏繞深度和有效載荷。直到鋼絲繩的橫向振動(dòng)產(chǎn)生內(nèi)在的內(nèi)能損失這一原理得到充分理解,這樣的傷害將繼續(xù)顯著地影響著南非采礦工作的運(yùn)行成本和效率。
在這一地區(qū)阻礙工程上的突破性進(jìn)展的兩個(gè)主要原因是內(nèi)部阻尼機(jī)制的復(fù)雜性和按時(shí)間邊界情況的鋼絲繩動(dòng)態(tài)響應(yīng)的非線性。到目前為止,這個(gè)問題僅僅的一個(gè)數(shù)學(xué)解決方案顯得很棘手，而且必須要越來越認(rèn)真考慮實(shí)驗(yàn)結(jié)果。因此,這里描述的調(diào)查的主要目標(biāo)是由實(shí)驗(yàn)的方法(由實(shí)驗(yàn)室模擬)確定礦井提升機(jī)鋼絲繩在進(jìn)行大幅度非平面橫向震動(dòng)產(chǎn)生的內(nèi)部損失，這個(gè)振幅是在它的基本的和更高的諧振頻率附近的光譜中。
調(diào)查的范圍被限制在礦井的幾何尺寸中，在南非的深礦采礦工作中這些幾何尺寸很可能在實(shí)踐中遇到,即單繩和布萊爾多繩纏繞系統(tǒng)。鋼絲繩繩從提升滾筒延伸到首輪的這一段長(zhǎng)度,通常被稱為懸鏈線,在實(shí)踐中遭受到最劇烈的橫向振動(dòng),因此這一部分成為這次鋼絲繩調(diào)查的模范。所使用的符號(hào)定義在文章最后。
歷史記錄
在19世紀(jì)50年代初于傳導(dǎo)著結(jié)構(gòu)型鋼絲繩內(nèi)部阻尼特性的基本方面和分析方面的基礎(chǔ)還有這兩個(gè)方面對(duì)橫向震動(dòng)影響的基礎(chǔ)。使用的鋼絲繩是一個(gè)由6根螺旋線圍繞一根單芯線絞成一股而成型的7-電線樣品(0,4 kg.m-I)。所有的組成電線是鍍鋅線或類似化學(xué)成分的電線，公稱直徑大約9.5毫米,總長(zhǎng)度2000毫米,捻?yè)?jù)1270毫米。于的調(diào)查集中在標(biāo)本滯回阻尼特性的測(cè)定，這些標(biāo)本在不受力狀態(tài)下進(jìn)行平面振動(dòng)。
盡管采用的技術(shù)規(guī)范和試驗(yàn)法明顯地遠(yuǎn)離了現(xiàn)代礦井提升鋼絲繩的幾何條件和動(dòng)態(tài)條件,如下從早期的調(diào)查研究得到的觀察值具有重大的作用,并描述在絞線進(jìn)行自由平面振動(dòng)時(shí)內(nèi)部阻尼的基本性質(zhì)。
（1） 金屬絲材料的剛性內(nèi)摩擦很小。
（2） 實(shí)際上,可以假設(shè)只有干摩擦（內(nèi)摩擦)存在。
（3） 與內(nèi)部干摩擦有關(guān)的衰減能量(每個(gè)周期的能量耗散）是一個(gè)振幅線性函數(shù)。
（4） 一個(gè)臨界的振幅似乎存在,它的上面具體的阻尼特性曲線開始極快地上升。
過去的三十年中,看來小獨(dú)立研究更深層次地運(yùn)用了于的首創(chuàng)理論并試圖擴(kuò)大礦井提升鋼絲繩的阻尼特性的現(xiàn)有知識(shí)。然而，許多調(diào)查已經(jīng)處理了大量拉纜的靜態(tài)和動(dòng)態(tài)響應(yīng)。Davenporf 給出了一張這個(gè)領(lǐng)域發(fā)展趨勢(shì)的明細(xì)表。在這張表中他指出于的結(jié)論清楚地確立了等效粘滯阻尼大約是臨界阻尼的20%到70%。雖然對(duì)于簡(jiǎn)單幾何形狀的干性鋼絲繩這一理論可能是正確的,但當(dāng)它應(yīng)用到大量拉纜和礦井提升鋼絲繩中就出現(xiàn)了問題，因?yàn)檫@些鋼絲繩的結(jié)構(gòu)復(fù)雜地多：同軸左旋和右旋螺旋線包含內(nèi)芯線，它會(huì)在塑形區(qū)變形（聚丙烯、劍麻和大麻等含有瀝青基的潤(rùn)滑油)。
利用粘滯阻尼機(jī)制與速率之比的絞纜的簡(jiǎn)化模型明顯在文獻(xiàn)中更加受歡迎主要因?yàn)樗鄬?duì)上解除了構(gòu)想和解答。然而,當(dāng)分析說明了沿著鋼絲繩長(zhǎng)度方向的張力梯度，除了內(nèi)部結(jié)構(gòu)阻尼與振幅和頻率之比,一個(gè)非線性響應(yīng)以拖延和跳躍現(xiàn)象的形式顯現(xiàn)出來。這些現(xiàn)象主要描述了介質(zhì)的響應(yīng)，這一介質(zhì)產(chǎn)生了改變共震頻率的強(qiáng)制震動(dòng)。
Vanderveldr還引用于的文章,并補(bǔ)充說考慮橫向阻尼行為的簡(jiǎn)單的模型不可以被假定。此外,他認(rèn)為至少兩種常見的結(jié)構(gòu)和粘性類型的阻尼必須包含在任何試圖預(yù)測(cè)在絞纜中傳播的橫向波衰減分析中。Vanderveldr通過假設(shè)一個(gè)粘滯阻尼的頻變系數(shù)來克服這一數(shù)學(xué)難題。通過這種方式,并提供激勵(lì)周期,其他類型的內(nèi)部的阻尼機(jī)理現(xiàn)在被假設(shè)包含在阻尼系數(shù)中。他的理論和實(shí)驗(yàn)結(jié)果顯示地特別一致,該處相關(guān)的是被看作補(bǔ)充于的實(shí)驗(yàn)結(jié)果如下。
（a） 對(duì)于一個(gè)金屬芯,內(nèi)部阻尼被拉伸載荷所影響。(徑向力和鏈間力隨著軸向拉力的增高而增強(qiáng)以至于干摩擦阻尼也表現(xiàn)出了增長(zhǎng))。
（b） 對(duì)于非金屬芯,阻尼能力隨著軸向載荷減少而增加。
里值得提到的是,雖然于對(duì)振幅阻尼的依賴性進(jìn)行了評(píng)論,Davenporf和Vanderveldr都沒有明確地認(rèn)為曲率參數(shù)和曲率變化率參數(shù)會(huì)影響能量耗散這一效應(yīng)。Kolsky給出了這個(gè)參數(shù)的數(shù)學(xué)形式:考慮到一水平畸變(大部分)波以x軸方向傳播而y軸方向幾乎不移動(dòng),控制運(yùn)動(dòng)方程可以表現(xiàn)為
一般的解決方案
b和C都是頻變,m是質(zhì)量密度，u是剪切模量，是剪切粘度。注意力集中在末項(xiàng)方程(1),可以清楚地把曲率變化率與剪切粘性聯(lián)系起來。
初步討論
在圖1中，一段鋼絲繩的封皮以基本形式進(jìn)行自由非平面震動(dòng)，它展現(xiàn)了四分之以參數(shù)的一個(gè)完整的循環(huán)?？缍鹊拈L(zhǎng)度是2.L，中跨幅度是S。在一階條件中,根據(jù)任意時(shí)刻震動(dòng)的鋼絲繩描繪出數(shù)學(xué)曲線可以近似為一個(gè)拋物線，這個(gè)拋物線的軸垂直于連接邊界支撐結(jié)構(gòu)的弦。正如迪安所指出的,當(dāng)弦是水平的而且下降距離與跨距之比小于0.02，該曲線的數(shù)學(xué)近似值采用了小的誤差。當(dāng)弦不是水平的時(shí)候，對(duì)稱性會(huì)喪失,而且鋼絲繩在平衡位置會(huì)堅(jiān)持被切去頂端的懸鏈線數(shù)學(xué)微量。然而,對(duì)于相對(duì)較小的下降距離與跨距之比，淺拋物線弧的近似值是足夠地精確和而且在分析中不采用很大的誤差。拋物線和懸鏈線的近似法經(jīng)常出現(xiàn)在文獻(xiàn)中,尤其對(duì)有傾斜跨度的巨大拉索的動(dòng)態(tài)分析。
邊界條件
當(dāng)鋼絲繩直徑相對(duì)于跨度足夠大,而且鋼絲繩振動(dòng)的曲率半徑很小,彎曲應(yīng)力的局部斜度將建立在鋼絲繩中。根據(jù)邊界條件的類型,彎曲應(yīng)力的兩種漸變是可能的。(i)恒定的漸變和(ii)隨不同振動(dòng)模式而變化的時(shí)間相關(guān)的漸變。在接下來的分析中,這兩種類型的漸變都會(huì)被考慮而且是連接震動(dòng)鋼絲繩和支撐物的球鉸式安排的結(jié)果，被僅有控制邊界強(qiáng)加的回轉(zhuǎn)約束條件的類型影響著漸變。
圖1-鋼絲繩封皮進(jìn)行以基本形式進(jìn)行自由非平面橫向震動(dòng)
在這個(gè)例子中滾珠球窩接點(diǎn)以一種方式被約束住，這種方式允許鋼絲繩繞它的幾何中心旋轉(zhuǎn)而且在跨距附近循環(huán)（圖2）。因此,這邊采用的邊界條件允許球形接頭以3個(gè)自由度在套借口內(nèi)自由旋轉(zhuǎn)。這就等于繞跨距旋轉(zhuǎn)的鋼絲繩剛性長(zhǎng)度由支撐結(jié)構(gòu)規(guī)定。圖二顯示了這一鋼絲繩的一個(gè)平面截面的圓軌道發(fā)生在平面y-z的中跨；這邊的跨度采取了正常的頁(yè)面。字母A假定代表鋼絲繩的橫斷面,這兒值得注意的是,字母A繞跨度旋轉(zhuǎn)而且被看作繞相對(duì)于固定在支架上的慣性參考的幾何中心。發(fā)生在字母A頂尖的彎曲應(yīng)力的傾斜度也是同樣顯示在圖二上而且被公認(rèn)為是永遠(yuǎn)恒定不變的t。在圖2中指標(biāo)(c -)和(T +)分別代表相對(duì)壓縮和拉伸的狀態(tài)，這些狀態(tài)發(fā)生在持續(xù)循環(huán)截面的表面上。
從基本的橫梁理論來看這兒的彎曲應(yīng)力是拉伸應(yīng)力,因?yàn)檎駝?dòng)時(shí)頂尖繼續(xù)留在圓截面的最外面的纖維上。約束的性質(zhì)也可以避免中性軸(NA)相對(duì)于固定指標(biāo)A移動(dòng)。在這個(gè)例子中，彎曲應(yīng)力的定值歸因于旋轉(zhuǎn)鋼絲繩的離心效應(yīng)連同彎曲效應(yīng)。
圖2-法生在鋼絲繩不動(dòng)點(diǎn)的彎曲應(yīng)力,旋轉(zhuǎn)運(yùn)動(dòng)
時(shí)變彎曲應(yīng)力
在這個(gè)例子中的邊界條件與上面的那些相似，除了繞橫坐標(biāo)的旋轉(zhuǎn)被限制了。因此,如圖3所示,字母A的引用目前已成為不可以旋轉(zhuǎn)的。這個(gè)事實(shí)通過超過一個(gè)完整周期不改變字母A的垂直方向來被證實(shí)。此外,當(dāng)字母A頂點(diǎn)在跨距周圍完成一個(gè)旋轉(zhuǎn)周期時(shí)，它的引用經(jīng)歷了一次彎曲應(yīng)力的循環(huán)。這里值得注意的是的在被看作相對(duì)于纖維旋轉(zhuǎn)的地方中性軸時(shí)間方向包含了鋼絲繩。在圖3中，發(fā)生在A的頂尖的彎曲應(yīng)力的變化在超過兩個(gè)循環(huán)周期內(nèi)以最快的速度被標(biāo)繪。再次,張應(yīng)力恒定的成分歸因于當(dāng)鋼絲繩氣球到動(dòng)態(tài)的穩(wěn)定的結(jié)構(gòu)時(shí)離心效應(yīng)隨著弧長(zhǎng)的增長(zhǎng)而上升。
圖3-發(fā)生在鋼絲繩不動(dòng)點(diǎn)的彎曲應(yīng)力,旋轉(zhuǎn)運(yùn)動(dòng)
實(shí)驗(yàn)器具
在這次調(diào)查中使用礦井鋼絲繩的規(guī)格是43.5毫米(公稱通徑)與 6 x 32(14/12/6 tri)F和線性質(zhì)量密度800 kg' m-I的結(jié)構(gòu)。圖4中,顯示一段鋼絲繩從不同高度的支撐結(jié)構(gòu)上懸吊下來。邊界條件在支撐結(jié)構(gòu)繞鋼絲繩的中心縱軸純轉(zhuǎn)動(dòng)時(shí)限制鋼絲繩的運(yùn)動(dòng)。最大的止推軸承就是用于這種用途的。
通過一個(gè)放置在下端的液壓千斤頂和固定在電源和上面支撐結(jié)構(gòu)末端的軸承箱體上的鎖緊裝置得到一個(gè)先已決定的張力和鋼絲繩幾何。由懸索規(guī)定的垂直面里的水平轉(zhuǎn)換約束了千斤頂?shù)囊苿?dòng)和較低的支持結(jié)構(gòu)。上面的止推軸承被裝上鉸鏈,以適應(yīng)任何預(yù)期的斜坡,而且一旦上面的軸承的斜坡加以調(diào)整以適應(yīng)鋼絲繩的傾斜，軸承箱體就被鎖定在固定位置上。通過這種方式,兩個(gè)推力軸承受通過他們的軸向中心的純軸向推力(張力)支配。
懸索興奮地旋轉(zhuǎn)下端,通過一臺(tái)電動(dòng)機(jī)，齒輪減速器,一系列的鏈傳動(dòng)裝置,以及一個(gè)雷諾聯(lián)軸器。雷諾聯(lián)軸器位于鋼絲繩的軸伸端和驅(qū)動(dòng)裝置之間,而且有利于隔離激勵(lì)產(chǎn)生的鋼絲繩的動(dòng)態(tài)響應(yīng)。這是令人滿意的因?yàn)橐苑瓷涞目v向和橫向波的形式的機(jī)械成果能夠（給足夠的積累時(shí)間)調(diào)制激勵(lì)頻率和激勵(lì)振幅,特別是在共振條件附近。
電機(jī)的速度由一個(gè)3.7千瓦的三相變頻傳動(dòng)裝置控制,而且由一個(gè)光電的轉(zhuǎn)數(shù)器檢測(cè)。張力的水平分量由一個(gè)內(nèi)聯(lián)的液壓傳感器測(cè)量，它坐落在較低的推力軸承后面而且與鋼絲繩一起旋轉(zhuǎn)。給雷諾聯(lián)軸器應(yīng)用的扭矩由一個(gè)流動(dòng)場(chǎng)測(cè)功機(jī)的功當(dāng)量決定。發(fā)動(dòng)機(jī)、變速箱、和鏈傳動(dòng)裝置被封裝在在一個(gè)單一機(jī)組內(nèi),這一機(jī)組被安裝在耳軸上并且在扭轉(zhuǎn)力矩的下面,這個(gè)機(jī)組可以繞耳軸軸承旋轉(zhuǎn)而且通過移動(dòng)的大量東西達(dá)到平衡。因此,平衡的大量東西的相對(duì)運(yùn)動(dòng)充當(dāng)了應(yīng)用扭矩的一種指示。
為了去除本生鐘擺式擺動(dòng)的主要的測(cè)力計(jì)底座,一個(gè)安裝在底座的伸出臂被浸在車用機(jī)油中，浸沒的部分存在的一個(gè)平漿被放置在正常的震動(dòng)方向。
圖4-測(cè)試裝置布局的等距略圖
結(jié)論
在這次調(diào)查中實(shí)證研究方法的使用引起了為量化復(fù)雜的阻尼機(jī)制而產(chǎn)生的高度全面、有效的技術(shù)，這個(gè)阻尼機(jī)制發(fā)生在所有的諧波模式中進(jìn)行橫向非平面振動(dòng)的礦井吊裝鋼絲繩上。到目前為止按照作者的經(jīng)驗(yàn),沒有以（無(wú)旋的-旋轉(zhuǎn)的)機(jī)械等價(jià)為基礎(chǔ)的理論或?qū)嶒?yàn)證據(jù)已經(jīng)出現(xiàn)在文獻(xiàn)上。因?yàn)榉椒ǖ幕驹碓谟趯?duì)尋找內(nèi)部的摩擦特性的實(shí)際鋼絲繩的測(cè)試,這邊所描述的任何有適合實(shí)驗(yàn)尺寸的鋼絲繩可以受動(dòng)態(tài)測(cè)試。
兩種彎曲型阻尼特性由一種實(shí)驗(yàn)法鑒定。阻尼被口述出來這一精確形式主要以方便計(jì)算。但是,它與在模擬系統(tǒng)中遇到的阻尼相一致,而且定性地符合剪切粘度阻尼這種類型,或者與曲率變化率成正比的這種類型。在這種情況下缺乏大量具體的阻尼的確切性質(zhì)的信息,被鑒定的阻尼特性的類型是合乎情理的:他們有把復(fù)雜簡(jiǎn)單化這個(gè)優(yōu)點(diǎn),并且確保獲得內(nèi)部能量損失的量化地正確的評(píng)價(jià)。
應(yīng)該強(qiáng)調(diào),下面的結(jié)論是基于動(dòng)力學(xué)響應(yīng)的測(cè)試，這個(gè)動(dòng)力學(xué)響應(yīng)是固定建筑的一個(gè)單一礦井提升鋼絲繩上的。結(jié)果,把下面的觀察結(jié)果應(yīng)用于其它具有不同的幾何結(jié)構(gòu)的礦山提升鋼絲繩中可能會(huì)有一些困難。然而,盡管這些潛在的不同,有些定性趨勢(shì)可以概括和總結(jié)如下。
（1） 一個(gè)礦井提升鋼絲繩的內(nèi)能損失可以定性和定量地被描述，通過兩個(gè)實(shí)驗(yàn)確定參數(shù):阻尼性能系數(shù)C1和曲率特性C2。一種發(fā)展的數(shù)學(xué)關(guān)系使它變成可能,這個(gè)數(shù)學(xué)關(guān)系被給予了這兩個(gè)系數(shù)和鋼絲繩的動(dòng)態(tài)環(huán)境(振幅、跨度和頻率),來評(píng)估內(nèi)部能量損失的總數(shù)。
（2） 一個(gè)關(guān)鍵的曲率半徑存在內(nèi)能損失隨著增加的振幅與跨距的比率成直線上升的上方區(qū)域。實(shí)驗(yàn)也有證據(jù)顯示在這一線性地區(qū)的內(nèi)能損失隨著振動(dòng)模態(tài)數(shù)的平方增加。
（3） 對(duì)于曲率半徑小于臨界值的情況,內(nèi)能損失以指數(shù)形式上升,而且不試圖調(diào)查在那個(gè)區(qū)域發(fā)生的內(nèi)能損失。
（4） 對(duì)于典型的礦井裝置,更高的非平面橫向振動(dòng)模式在與不良后果相關(guān)的振動(dòng)疲勞造成的損失上有重大影響。這個(gè)觀察是基于這兒獲得的現(xiàn)有金礦的動(dòng)態(tài)條件的實(shí)驗(yàn)結(jié)果的應(yīng)用。
8
Internal damping characteristics of a mine hoist cable undergoing non-planar transverse vibration
by A.A. MANKOWSKI*
SYNOPSIS
The work described in this paper is an attempt to increase present-day knowledge of fatigue in mine hoisting cables, particularly the internal energy loss arising from interwire/strand friction in a cable undergoing periodic non-planar transverse vibration. Such frictional energy loss is known to be one of the major influences limiting the useful working life of hoisting cables in use today, and is responsible for the large capital outlay required to maintain the high safety factors prescribed by the mining industry.
The experimental method employed identifies two mechanical characteristics of cables that are independent of amplitude and frequency, and are primarily attributed to the type of cable construction. Interest is focused on
the time rate of change of curvature as the major parameter influencing the internal damping mechanism. Empirical results confirm that amplitude and mode number play an important role in quantifying the internal losses, and also reveal that a critical radius of curvature exists below which damage due to vibration fatigue rises exponentially to potentially high levels.
INTRODUCTION
The problem of damage due to vibration fatigue continues to impose limits on winding velocities, depths of wind, and payloads in modern, deep South African gold mines. Until the mechanism of internal energy loss inherent in the transverse vibration of cables is thoroughly understood, such damage will continue to have a marked effect on the running costs and efficiency of South
African mining operations.
Two major reasons impeding engineering breakthroughs in this area are the complex nature of the internal damping mechanism and the nonlinearity of the dynamic response of the cable to time-dependent boundary conditions. To date, a purely mathematical solution to the problem appears intractable, and it has become necessary to give increasingly more serious consideration to experimental results. Accordingly, the primary objective of the investigation described here was to determine experimentally (by laboratory simulation) the internal losses of a mine hoisting cable undergoing non-planar transverse vibration of large amplitude in the spectral neighbourhood of its fundamental and higher harmonic frequencies.
The scope of the investigation was limited to the mine geometries most likely to be encountered in practice in deep South African mining operations, namely the single-drum and the Blair multi-drum winding systems. The length of cable extending from the winding drum to the headsheave, commonly referred to as the catenary, suffers the most violent transverse vibration in practice, and hence served as the section of cable to be modelled in this investigation. The symbols used are defined at the end of the paper.
HISTORICAL NOTE
The groundwork on the fundamental and analytical aspects of the internal damping characteristics of structural cable and their influence on transverse vibration was conducted in the early 1950s by Yul. The cable used was a 7-wire specimen (0,4 kg. m-I) formed by 6 helical wires stranded round a single-core wire. All the constituent wires were zinc-coated and of similar chemical composition, the nominal diameter being approximately 9,5 mm, the overall length 2000 mm, and the lay length 127,0 mm. Yu's investigation concentrated on the determination of hysteretic damping characteristics of a family of these specimens undergoing planar vibration in a state of zero tension.
Although the specifications and experimental method employed were distinctly far removed from the geometry and dynamic conditions of present-day mine hoisting cable, the following observations from that early investigation are relevant and describe the basic nature of the internal damping of stranded cable undergoing free planar vibration.
(1) The solid internal friction of the wire material is small.
(2) For practical purposes, it can be assumed that only dry friction exists (interstrand friction).
(3) The damping capacity (dissipation of energy per cycle) associated with internal dry friction is a linear function of amplitude.
(4) A critical amplitude seems to exist, above which the curve of specific damping capacity begins to rise hyperbolically.
In the past three decades, it appears that little independent research has carried Yu's pioneering efforts further in an attempt to expand present knowledge on the damping characteristics of mine hoisting cable. A number of
investigations, however, have dealt with the static and dynamic response of massive guy cables. A detailed account of developments in this field is given by Davenporf, in which he points out that Yu's conclusions clearly establish an equivalent viscous damping to be of the order of 2 to 7 per cent of critical damping. While this may be true for dry cables of simple geometry, its application to massive guy cables and mine hoisting cables is questionable on the grounds that these cables are much more complex in their construction: concentric left- and right-handed helices containing inner cores that deform
in the plastic regions (polypropylene, sisal, and hemp impregnated with bitumen-based lubrication).
Simplified models of stranded cables employing viscous damping mechanism proportional to velocity are decidedly more popular in the literature mainly because of the relative ease of formulation and solution. However, when the analyses account for tension gradients along the length of a cable in addition to internal structural damping proportional to amplitude and frequency, a nonlinear response manifests itself in the form of drag-out an jump phenomena3. These phenomena primarily describe the response of the medium to forced vibration of varying frequency passing through resonant conditions.
Vanderveldr also cites the work ofYu1, and adds that no simple model taking into account the transverse damping behaviour can be assumed. Furthermore, he contends that at least both the usual structural and viscous types of damping must be included in any analysis attempting to predict the attenuation of transverse waves that are propagated in a stranded cable. Vanderveldr surmounted this mathematical difficulty by assuming a frequency-dependent coefficient of viscous damping. In this way, and providing the excitation is periodic, any other type of internal damping mechanism present is assumed to be contained in the damping coefficient. His
theoretical and experimental results show particularly good agreement and, where relevant, are seen to complement Yu's experimental results as follows.
(a) For a metallic core, the internal damping is affected by the tensile load. (Radial forces and inter-strand stress increase with increasing axial tension so that dry-friction damping also shows an increase.)
(b) For non-metallic cores, the damping capacity increases as the axial loads decrease.
It is worth while mentioning here that, although YUl commented on the dependence of damping on amplitude, neither Davenporf nor Vanderveldr explicitly considered the effect of curvature and its time rate of change as a parameter influencing the dissipation of energy. The mathematical form of this parameter is given by Kolsky: considering a plane distortional (bulk) wave that is propagated in the positive x direction with its particle motion in the y direction, the governing equation of motion can be shown to be
with general solution
where band C are both frequency-dependent, m is the mass density, u the shear modulus, and a the shear viscosity. Attention is drawn to the last term of Equation (1), which clearly associates the time rate of change of curvature with the shear viscosity.
PRELIMINARY DISCUSSION
In Fig. 1 the envelope of a length of cable undergoing free non-planar transverse vibration in the fundamental mode is shown over one complete cycle in increments of one-quarter periods. The length of the span is 2. Land
the amplitude at mid-span is S. To within first-order terms, the mathematical curve traced out by the cable during vibration at anyone instant can be approximated by a parabolic arc having its axis perpendicular to the chord joining the supports at the boundaries. This mathematical approximation of the curve, as pointed out by Dean6, introduces errors that are small when the chord is horizontal and the sag-to-span ratio is less than 0,02. When the chord is not horizontal, symmetry is lost, and the cable will hang in the mathematical trace of a truncated catenary in its equilibrium position. However, for relatively small sag-to-span ratios, the approximation to a shallow parabolic arc is sufficiently accurate and does not introduce significant errors in the analysis. Parabolic-for- catenary approximations are frequent in the literature, particularly for the dynamic analysis of massive guy cables having inclined spans.
Boundary Conditions
When the diameter of the cable is large enough compared with the span, and the radius of curvature of the vibrating cable is sufficiently small, a local gradient in flexure stress will be set up in the cable. Depending on the type of boundary conditions, two gradients in flexure stress are possible: (i) a constant gradient and (ii) a time-dependent gradient varying with the mode of vibration.
In the following analysis, both types of gradients are considered and are the result of ball-and-socket arrangements connecting the vibrating cables to the support, the types of rotational constraints imposed at the boundaries being the sole controlling influence on the gradients.
Fig. 1-Envelope of cable undergoing free non-planar
transverse vibration in the fundamental mode
Constant Gradient in Flexure Stress
In this example the ball-and-socket joints are constrained in a manner that allows the cable to rotate about its geometric centre and revolve round the span (Fig. 2). Thus, the boundary conditions employed here allow the ball joints 3 degrees of rotational freedom within the sockets. This is tantamount to a rigid length of cable whirling round the span defined by the supports. Fig. 2
shows the circular orbit of a plane section of this cable occurring at mid-span in the y-z plane; the span here is taken normal to the page. The letter A is assumed fixed to the transverse section of cable, where, it is noted, the letter A revolves about the span and is seen to rotate about its geometric centre relative to an inertial reference fixed to the supports. The gradient in flexure stress occurring at the apex of the letter A is also shown in Fig. 2 and is seen to be constant for all time t. The indicators (c - ) and (T + ) in Fig. 2 represent the relative compressive and tensile states respectively occurring on the surface of the sections indicated as it continues its cycle.
From basic beam theory the flexure stress here is tensile owing to the fact that the apex remains at the outermost fibres of the circular section during vibration. The nature of the constraints also prevents the neutral axis (NA) from moving relative to the fixed indicator A. The constant value of the flexure stress in this example is attributed to centrifugal effects of the whirling cable combined with the bending effects.
Time-dependent Flexure Stress
The boundary conditions in this example are similar to those above with the exception that rotation about the X axis is constrained. As a consequence, the reference letter A, as shown in Fig. 3, now becomes irrotational. This fact is borne out by the unchanging vertical orientation of the letter A over one complete cycle. Furthermore, the reference of the apex of letter A experiences
a flexure-stress cycle as it completes one revolution round the span. Noteworthy here is the time-dependent orientation of the neutral axis where it is seen to rotate relative to the fibres comprising the cable. The variation in flexure stress occurring at the apex of A is plotted against time over a period of two cycles in Fig. 3. Again, the constant component of tensile stress is attributed to the centrifugal effects arising from the increase in arc length as the cable balloons to a dynamically stable configuration.
Fig. 3-Flexure stress occurring at a fixed point on the cable,
irrotational motion
EXPERIMENTAL APPARATUS
The specifications of the mine cable used in this investigation were 43,5 mm (nominal diameter) with construction 6 x 32(14/12/6 tri)F and linear mass density 8,00 kg' m-I. In Fig. 4, a length of cable is shown suspended from supports of unequal height. The boundary conditions restricted the motion of the cable at the supports to pure rotation about the central longitudinal axis of the cable. Full thrust bearings were used for this purpose.
A predetermined tension and cable geometry were obtained by a hydraulic jack positioned at the lower end and locking devices fixed to the bearing casings at the lower and upper support ends. The movement of the jack and lower support were constrained to horizontal translation in the vertical plane defined by the suspended cable. The upper thrust bearing was hinged to accommodate any desired slope and, once the inclination of the upper bearing
had been adjusted to match the slope of the cable, the bearing casing was locked into position. In this way, both thrust bearings were subject to purely axial thrust (tension) through their axial centres.
The suspended cable was excited by rotating the lower end by an electric motor, gear-reduction transmission, a series of chain drives, and a Reynold coupling. The Reynold coupling was situated between the driven end of the cable and the driving unit, and had the advantage of isolating the dynamic response of the cable from the excitation. This is desirable since mechanical feedback in the form of reflected longitudinal and transverse waves could (given sufficient build-up time) modulate the frequency and amplitude of the excitation, especially in the neighbourhood of resonant conditions.
The speed of the electric motor was controlled by a 3,7 kW three-phase variable-frequency driving unit, and monitored by an electro-optical revolution counter. The horizontal component of tension was measured by an inline hydraulic transducer, which was situated behind the lower thrust bearing and rotated with the cable. The applied torque to the Reynold coupling was determined by a mechanical equivalent of a floating field dynamometer. The motor, transmission, and chain drives were housed in a single unit, which was mounted on trunnions and, under torque reaction, this unit could rotate about the trunnion bearings and be counterbalanced by movable masses. Thus, the relative movement of the balancing masses served as an indication of the applied torque.
To rid the central dynamometer carriage of its natural pendulum-type vibration, an extension arm fixed to the carriage was immersed in motor oil, the immersed section being a flat paddle placed normal to the direction of oscillation.
Fig. 4-lsometric sketch of the layout of the test apparatus
CONCLUSION
The use of an empirical approach in this investigation gave rise to a highly comprehensive and efficient technique for quantifying the complex damping mechanism occurring in mine hoisting cables undergoing non-planar transverse vibration in all the harmonic modes. In the author's experience to date, neither theoretical nor experimental evidence based on the (irrotational-rotational) mechanical equivalence has been encountered in the literature. Because the basis of the method lies in the testing of the actual cable whose internal frictional characteristics are sought, any cable having suitable laboratory dimensions can be subjected to the dynamic test described here.
Two flexural-type damping characteristics were identified by an experimental method. The precise form that the damping took was dictated mainly by convenience in computation. It is, however, consistent with the damping encountered in analogous systems, and conforms qualitively to a type of shear-viscosity damping, or one that is proportional to the time rate of change of curvature. In the absence of a large body of concrete information
on the exact nature of damping in this situation, the type of damping characteristics identified are justified: they have the virtue by being simple and ensure that a quantitatively correct assessment of the internal power loss is achieved.
It should be emphasized that the following conclusions are based on the testing of the dynamic response of a single mine hoisting cable of fixed construction. As a result, there may be some difficulties in the application of the following observations to other mine hoisting cables having different geometric construction. However, in spite of these potential differences, certain qualitative trends can be generalized and summarized as follows.
(1) The internal power loss of a mine hoisting cable can be characterized qualitatively and quantitatively by two experimentally determined parameters: a damping capacity coefficient, Cl, and a curvature characteristic, C2. A mathematical relationship was developed that makes it possible, given these two coefficients and the dynamic environment of the cable (amplitude, span, and frequency), to assess the amount of internal power loss.
(2) A critical radius of curvature exists above which the internal power loss increases linearly with increasing amplitude-to-span ratio. Experimental evidence also shows the losses in this linear region to increase as the square of the mode number of vibration.
(3) For radii of curvature below the critical value, the internal losses rise exponentially, and no attempt was made to investigate the losses occurring in that region.
(4) For typical mine installations, the higher modes of non-planar transverse vibration have a significant influence on the undesirable effects associated with damage from vibration fatigue. This observation is based on the application of the empirical results obtained here to the dynamic conditions of an existing gold mine.
9
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