裝配圖車(chē)梁加工用翻轉(zhuǎn)臺(tái)的設(shè)計(jì)
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對(duì)振動(dòng)偵查和測(cè)量的一種實(shí)用方法物理原則和偵查技術(shù) 作者:John Wilson, 動(dòng)態(tài)顧問(wèn), LLC這篇論文論述振動(dòng)物理、彈簧質(zhì)量系統(tǒng)的動(dòng)力學(xué),阻止、位移、速度和加速度,并且查出和測(cè)量這些物產(chǎn)傳感器的操作原理。振動(dòng)擺動(dòng)由振動(dòng)或作用在機(jī)構(gòu)的力的變化引起振動(dòng)的擺動(dòng)。振動(dòng)行動(dòng)反向。由于我們將看到,這振蕩可能是在經(jīng)過(guò)若干時(shí)間有價(jià)值的周期連續(xù)不斷的或者可能間斷的。它可能是周期性或非周期性, 那就是說(shuō),它可能或者可能不呈現(xiàn)一規(guī)則的周期的重復(fù)。動(dòng)擺的本質(zhì)取決于力量的本質(zhì)駕駛它和結(jié)構(gòu)被駕駛。運(yùn)動(dòng)是一個(gè)矢量,呈現(xiàn)一個(gè)方向和一個(gè)量。振動(dòng)的方向通常被描述依據(jù)一些獨(dú)立的坐標(biāo)系(典型地笛卡爾的或者直角的)其運(yùn)動(dòng)的方向被稱(chēng)作坐標(biāo)軸。這些坐標(biāo)軸的正交座標(biāo)系的原點(diǎn)是被任意地被定義在一些適當(dāng)?shù)牡奈恢?。機(jī)構(gòu)的多數(shù)振動(dòng)的響應(yīng)可以用當(dāng)做單自由度彈簧質(zhì)量系統(tǒng)模型,并且許多振動(dòng)傳感器使用他們的一個(gè)彈簧質(zhì)量系統(tǒng)當(dāng)做轉(zhuǎn)導(dǎo)機(jī)構(gòu)的機(jī)械部分。除外形尺寸之外,一個(gè)彈簧質(zhì)量系統(tǒng)可以用彈簧的剛度K,和質(zhì)量M,或者質(zhì)量的重量W等性能參數(shù)來(lái)阿描述。這些特征不僅決定來(lái)這機(jī)構(gòu)的靜態(tài)特性(靜變位d),而且決定來(lái)它的動(dòng)態(tài)特性。如果g 是重力的加速度:F = MAW = MgK = F/d = W/dd = F/K = W/K = Mg/K一個(gè)彈簧質(zhì)量系統(tǒng)的動(dòng)力學(xué)一個(gè)彈簧質(zhì)量系統(tǒng)的動(dòng)力學(xué)的可以被體系的特性在自由振動(dòng)及有效的振動(dòng)表示。自由振動(dòng) 自由振動(dòng)被那情況情形哪里那彈簧是偏斜于是釋放以及允許到自由地?fù)u擺。例子包括一個(gè)跳板、一個(gè)跳簧跨接管,以及一個(gè)擺或搖擺偏斜以及留某事給自由地振動(dòng)處理。兩個(gè)特征特性應(yīng)該注意。 第一、阻尼在那體系表示原因的那振幅的那振蕩到減少將來(lái)。 那包括市區(qū)及郊區(qū)的那阻尼、那更快的那振幅隨時(shí)間減小。(只要彈性極限不是超過(guò)),那頻率或時(shí)期的那振蕩無(wú)關(guān)原始的大小原始的偏轉(zhuǎn)的的。 那自然地發(fā)生頻率的那自由振動(dòng)被呼叫那自然頻率fn:受迫振動(dòng) 受迫振動(dòng)當(dāng)能量是連續(xù)地被加到那彈簧質(zhì)量系統(tǒng)由申請(qǐng)振動(dòng)的力在一些受迫振動(dòng)頻率時(shí)的情形ff. 兩個(gè)二例子連續(xù)地推一個(gè)孩子上去一個(gè)搖擺和一失衡旋轉(zhuǎn)電機(jī)元件。如果提供充足的能量到克服那阻尼是,那動(dòng)作就會(huì)延續(xù)長(zhǎng)達(dá)那激勵(lì)延續(xù)之久。受迫振動(dòng)可以取自勵(lì)的或外部地激發(fā)振動(dòng)的形式。自激振動(dòng)發(fā)生在激發(fā)力是產(chǎn)生在或上去那懸掛質(zhì)量的時(shí)候;外部地激發(fā)振動(dòng)發(fā)生在激發(fā)力作用于彈簧的時(shí)候。這是那情形、例如:、當(dāng)那基礎(chǔ)對(duì)此那彈簧附屬于是移動(dòng)時(shí)。傳導(dǎo)能力 當(dāng)基礎(chǔ)正在振動(dòng),而且力整個(gè)彈簧被傳輸?shù)街兄沟馁|(zhì)量時(shí)候,質(zhì)量的動(dòng)作將會(huì)是來(lái)自基礎(chǔ)的動(dòng)作差積。 我們將會(huì)認(rèn)為基礎(chǔ)的動(dòng)作是輸入,I, 和質(zhì)量的動(dòng)作響應(yīng), R. 比率半徑/我被定義為傳輸度,Tr:Tr = R/I共振 在力頻率好低于體系的固有頻率,RI, 和 Tr1。由于作用力的頻率接近那固有頻率,由于共振,所以傳遞率增加。共振是在機(jī)械系統(tǒng)中的量的存儲(chǔ)。在力頻率接近那固有頻率、能量是存儲(chǔ)和積聚、導(dǎo)致增加響應(yīng)振幅。阻尼也增加由于增加響應(yīng)振幅、然而,并且最后那能量為阻尼所吸收、每一周期、等于能量增加由激振力,并且平衡狀態(tài)到達(dá)。我們發(fā)現(xiàn)當(dāng)fffn.時(shí)最大傳遞率發(fā)生,這個(gè)情況被稱(chēng)作共振。隔振 如果激振力頻率超過(guò)fn,R降低。當(dāng)ff = 1.414 fn, R = I 或Tr = 1時(shí),在比較高的頻率R I 或Tr 1。在頻率當(dāng)0.1英寸,到使他們成為現(xiàn)實(shí)的。一束對(duì)準(zhǔn)在一個(gè)反射面上光束在強(qiáng)度或者角度的的變化能被使用當(dāng)做一距離指示從震源的角度之上方面。如果該探測(cè)儀器是足夠快的,變化的距離也可以被測(cè)定。最靈敏的、準(zhǔn)確的和精密的測(cè)定距離或位移的光學(xué)裝置是激光干擾儀。利用這個(gè)儀器,一束反射激光束間雜有原來(lái)的入射光束。這由相位差形成的干涉圖樣可以測(cè)量位移下至1 MHz 震動(dòng)加速度儀。最現(xiàn)代的PR傳感器是用單個(gè)碎片硅制造的。一般說(shuō)來(lái),造型整體傳感器的優(yōu)點(diǎn)從一個(gè)單一的材料塊是更好的穩(wěn)定性,較少熱量的失配在部分之間,并且較高的可靠性。欠阻尼的 PR 加速度儀容易不比 PE 裝置高低不平。 單一晶體矽能有特別的降伏強(qiáng)度,特別地以高的應(yīng)變率,但是它是然而一個(gè)脆的事物。 矽的內(nèi)磨擦非常低,因此,諧振擴(kuò)大可能是比較高的超過(guò)對(duì)于 PE 傳動(dòng)器。 兩者的這些功能成為它的比較易脆性的因素, 雖然如果適當(dāng)?shù)卦O(shè)計(jì)而且安裝他們被規(guī)律性用測(cè)量震動(dòng)很好上述的 100,000 g 。他們通常有較寬的頻帶寬度勝于 PE 傳動(dòng)器 (比較相似實(shí)物大小范圍的模型), 連同較小的非線性,零的移位和磁滯特性。 因?yàn)樗麄冇兄绷麟姺磻?yīng),他們?cè)趯⒁a(chǎn)生長(zhǎng)期計(jì)量時(shí)才使用。在 PR 加速度儀的一個(gè)典型獨(dú)石矽可察元件中,1 毫米角尺矽芯片合并整個(gè)的彈簧,質(zhì)量和四個(gè)臂的 PR 應(yīng)變計(jì)橋總成。 感知器經(jīng)由各向異性的浸蝕和顯微機(jī)械加工技術(shù)是利用一個(gè)單一晶體矽做成的。 應(yīng)變計(jì)被本來(lái)平的矽一個(gè)雜物的圖案造形。 溝流的后來(lái)浸蝕釋放規(guī)并且同時(shí)地定義如只是最初厚度的矽區(qū)域的質(zhì)量。橋路可以由放置并聯(lián)補(bǔ)償電阻或者級(jí)數(shù)用任何這木頭支架平衡了,做相配的或者這阻抗值及價(jià)值的變化用溫度的修正。補(bǔ)償是一種藝術(shù); 因?yàn)?PR 傳動(dòng)器能有非線性特性, 用激發(fā)來(lái)自它被制作或校正的條件差積操作它是不受勸告的。 舉例來(lái)說(shuō), PR 靈敏度只有大約成比例激發(fā), 通常是一個(gè)固定的電壓或, 在一些外殼, 定流中有一些性能利益。因?yàn)闊岬男阅軐?huì)大體上和激發(fā)電壓的變化,在靈敏度和激發(fā)之間沒(méi)有一個(gè)精密的比例。 另外的預(yù)防在處理電壓驅(qū)動(dòng)的橋方面, 特別地有低的電阻那些, 是確認(rèn)橋拿適當(dāng)?shù)募ぐl(fā)。 輸入熔斷絲的級(jí)數(shù)電阻擔(dān)任一個(gè)分壓器。注意這輸入導(dǎo)線有低電阻,或者那一六線的大小是制成的(用讀出線在這橋梁趨于允許這激勵(lì)被校準(zhǔn))所以這橋梁獲得這特有的激勵(lì)。恒定電流激勵(lì)工作沒(méi)有這些用串聯(lián)電阻的問(wèn)題。然而, PR 傳動(dòng)器通常被補(bǔ)整傲慢的固定電壓激發(fā)并且不可能用定流給被需要的性能。 PR 橋的平衡是它的健康最敏感衡量, 而且通常是傳動(dòng)器的總不確定度的占優(yōu)勢(shì)的功能。 平衡,有時(shí)叫做了偏向, 零偏位 , 或 ZMO( 零可測(cè)量產(chǎn)量,和 0 g 的產(chǎn)量),能被通常是熱的特性或在內(nèi)部或外面地誘導(dǎo)了感知器的應(yīng)變變化的一些效應(yīng)改變。傳動(dòng)器外殼設(shè)計(jì)嘗試隔離來(lái)自外面的應(yīng)變 , 像是熱的暫態(tài),基本的應(yīng)變或固定轉(zhuǎn)矩的感知器。 內(nèi)部的應(yīng)變變化,舉例來(lái)說(shuō),環(huán)氧基樹(shù)脂蠕升,容易成為長(zhǎng)期的不穩(wěn)定的因素。所有的這些比較對(duì)于錒加倍的裝置因?yàn)樗麄冊(cè)谥绷髡呒颖秱鲃?dòng)器的較寬頻帶中更時(shí)常發(fā)生,通常低周波效應(yīng)對(duì)直流傳動(dòng)器是更重要的。一些PR設(shè)計(jì),尤其是高靈敏度傳感器,是設(shè)計(jì)有阻尼延長(zhǎng)頻帶和過(guò)量程的能力。 阻尼系數(shù)0.7 是考慮過(guò)的理想。 如此的設(shè)計(jì)時(shí)常使用油或一些其他的粘滯液體。 二個(gè)特性聽(tīng)寫(xiě)技術(shù)是有用的只有在相對(duì)地低周波: 阻尼軍隊(duì)成比例流過(guò)速度,而且適當(dāng)?shù)牧髁克俣缺唤逵捎么蟮奈灰票昧黧w達(dá)到。 這是在那敏感的傳動(dòng)器的一個(gè)快樂(lè)的巧合他們?cè)诘偷募铀俣阮l率操作位移足夠大哪里。粘滯阻尼可以有效地除去共振放大率,延長(zhǎng)過(guò)量程的能力,并且比加倍有效帶寬。然而,因?yàn)榫彌_液的粘性是一溫度的強(qiáng)函數(shù),傳感器的有用的溫度范圍實(shí)質(zhì)上是受限制的。可變電容 VC傳感器是通常平行板空隙電容器其中的設(shè)計(jì)運(yùn)動(dòng)垂直于電鍍層。在一些設(shè)計(jì)中屏從一個(gè)邊緣被把建成懸臂式,因此,動(dòng)作實(shí)際上是轉(zhuǎn)動(dòng); 其他的屏在圓周的周?chē)恢г? 當(dāng)做在一個(gè)彈網(wǎng)中。 由于加速度的在 VC 元件的電容方面的改變被一對(duì)目前檢波器感覺(jué)皈依者進(jìn)入電壓產(chǎn)量之內(nèi)的變化。許多VC傳感器是微電機(jī)一致地在一間隔一點(diǎn)點(diǎn)微米厚的趨于允許空氣減震中間插進(jìn)的腐蝕劑硅片。事實(shí)是空氣粘度變化由只有一點(diǎn)百分比在一寬的工作溫度范圍提供一頻率響應(yīng)比是可完成的用油阻尼PR設(shè)計(jì)更堅(jiān)固的上方。在一VC加速度記錄器中,一個(gè)高頻振蕩器給VC元件提供必要的激勵(lì)。電容變化被這檢流器檢測(cè)。輸出電壓與電容變化成正比因此,趨于加速度。這結(jié)合的超程停留在這間隔可以提高高低不平的在這靈敏的方向,雖然阻力趨于過(guò)量程的在橫向必須信任單獨(dú)地靠這懸浮的力量,按現(xiàn)狀對(duì)全部的其他的傳感器設(shè)計(jì)沒(méi)有超程停止來(lái)說(shuō)是正確的。一些設(shè)計(jì)可以繼續(xù)存在極其大加速度過(guò)量程的工況是 1000倍的測(cè)量范圍。一臺(tái)典型微電機(jī)VC加速度記錄器的傳感器是由三硅元件膠合到一起形成的密封的裝配。元件中的二個(gè)是空氣介質(zhì),平行板積蓄器的電極。 中央的元件用化學(xué)被蝕刻造形被薄又易曲手指中止的一個(gè)硬的中央質(zhì)量。 阻尼特性被位于質(zhì)量之上的孔氣體流量控制。VC傳感器可以提供好傳感器的特色測(cè)定類(lèi)型論述初期的中許多:大的過(guò)量程的,直流電響應(yīng),低阻抗的輸出端,和單純的外部信號(hào)工況。缺點(diǎn)是成本并且以那在板子上調(diào)節(jié)的增加錯(cuò)綜度按規(guī)定尺寸制作關(guān)聯(lián)。 同時(shí), 高頻電容檢波電路被用,而且一些高頻載波通常在產(chǎn)量信號(hào)上出現(xiàn)。它是通常連達(dá)到(即,1000倍)比輸出信號(hào)的頻率高三數(shù)量級(jí)也不被注意到。伺服系統(tǒng)(力平衡) 雖然伺服加速度計(jì)是主要地使用在慣性制導(dǎo)系統(tǒng),但是一些他們的工作特性必然使他們?cè)谝欢ǖ恼駝?dòng)應(yīng)用中是合乎需要的。所有的在先前被描述的加速度儀類(lèi)型是開(kāi)環(huán)裝置在哪一產(chǎn)量由于可察元件的撓曲被直接地讀。在倍力器中-控制, 或閉合回路,加速度儀, 撓曲信號(hào)被用當(dāng)一個(gè)身體上地驅(qū)動(dòng)或再平衡返回平衡位的質(zhì)量電路的反饋。 倍力器加速度儀制造業(yè)者建議仰賴位移 (也就是,晶體和 piezoresistive 元件的繃皮操作) 時(shí)常生產(chǎn)一個(gè)產(chǎn)量信號(hào)的開(kāi)環(huán)儀器引起非線性錯(cuò)誤。在閉合回路中設(shè)計(jì),內(nèi)部的位移被試驗(yàn)過(guò)的質(zhì)量電再平衡保持極端小,將非線性減到最少。 除此之外,閉合回路設(shè)計(jì)被說(shuō)有較高的精確度勝于開(kāi)環(huán)打字。 然而,期間精確度的定義改變。以傳感器制造商校核。伺服加速度計(jì)可以使兩個(gè)基本幾何結(jié)構(gòu)的其中任何一個(gè):線的(例如,擴(kuò)音器)和擺動(dòng)的(儀表的測(cè)量機(jī)構(gòu))。振動(dòng)的幾何結(jié)構(gòu)是商業(yè)的設(shè)計(jì)中應(yīng)用最廣泛的。直到最近,伺服機(jī)構(gòu)是主要地以電磁原則為基礎(chǔ)。力通常被藉由在一個(gè)磁場(chǎng)之前經(jīng)過(guò)在質(zhì)量上的線圈駕駛電流提供。 在和一個(gè)電磁的再平衡機(jī)構(gòu)的下垂倍力器加速度儀中,下垂的質(zhì)量發(fā)展對(duì)試驗(yàn)過(guò)的質(zhì)量和那應(yīng)用的加速度的產(chǎn)品轉(zhuǎn)矩比例項(xiàng)。 質(zhì)量的動(dòng)作被位感知器 ( 典型地電容的感知器) 發(fā)現(xiàn), 送一個(gè)誤差訊號(hào)給伺服系統(tǒng)。 誤差訊號(hào)引起對(duì)產(chǎn)量的倍力器放大器對(duì)轉(zhuǎn)矩電動(dòng)機(jī)的一個(gè)反饋電流,發(fā)展相等在量中到來(lái)自下垂的質(zhì)量加速度產(chǎn)生的轉(zhuǎn)矩一個(gè)反對(duì)轉(zhuǎn)矩。輸出端是激勵(lì)電流它本身(或者交叉一輸出端電阻器)作用的,與偏轉(zhuǎn)環(huán)傳感器相似,跟外加力成比例因此趨于加速度。和開(kāi)環(huán)傳感器的高低不平的彈簧元件相反,再平衡壓入回路加速度記錄器的箱體中主要地有關(guān)電的并且只有當(dāng)有動(dòng)力提供時(shí)存在。當(dāng)能實(shí)行的和大多數(shù)的阻尼被提供透過(guò)電子學(xué)的時(shí)候,彈簧在敏感的方向中是如易壞的。不像獨(dú)自地仰賴可察元件 (s) 的特性其他的直流- 響應(yīng)加速度儀,它是閉合回路設(shè)計(jì)的反饋電子學(xué)控制使存偏見(jiàn)穩(wěn)定性。因此伺服加速度計(jì)傾向于提供較少零點(diǎn)飄移,是我們?cè)谡駝?dòng)測(cè)量中使用他們的主要的理由。一般說(shuō)來(lái),他們有一個(gè)1000赫茲的有效帶寬的并且被設(shè)計(jì)成以比較地低加速度級(jí)并且極低頻元件方式使用在應(yīng)用。 A Practical Approach to Vibration Detection and MeasurementPhysical Principles and Detection TechniquesBy: John Wilson, the Dynamic Consultant, LLCThis tutorial addresses the physics of vibration; dynamics of a spring mass system; damping; displacement, velocity, and acceleration; and the operating principles of the sensors that detect and measure these properties.Vibration is oscillatory motion resulting from the application of oscillatory or varying forces to a structure. Oscillatory motion reverses direction. As we shall see, the oscillation may be continuous during some time period of interest or it may be intermittent. It may be periodic or nonperiodic, i.e., it may or may not exhibit a regular period of repetition. The nature of the oscillation depends on the nature of the force driving it and on the structure being driven. Motion is a vector quantity, exhibiting a direction as well as a magnitude. The direction of vibration is usually described in terms of some arbitrary coordinate system (typically Cartesian or orthogonal) whose directions are called axes. The origin for the orthogonal coordinate system of axes is arbitrarily defined at some convenient location.Most vibratory responses of structures can be modeled as single-degree-of-freedom spring mass systems, and many vibration sensors use a spring mass system as the mechanical part of their transduction mechanism. In addition to physical dimensions, a spring mass system can be characterized by the stiffness of the spring, K, and the mass, M, or weight, W, of the mass. These characteristics determine not only the static behavior (static deflection, d) of the structure, but also its dynamic characteristics. If g is the acceleration of gravity:F = MAW = MgK = F/d = W/dd = F/K = W/K = Mg/KDynamics of a Spring Mass SystemThe dynamics of a spring mass system can be expressed by the systems behavior in free vibration and/or in forced vibration. Free Vibration. Free vibration is the case where the spring is deflected and then released and allowed to vibrate freely. Examples include a diving board, a bungee jumper, and a pendulum or swing deflected and left to freely oscillate.Two characteristic behaviors should be noted. First, damping in the system causes the amplitude of the oscillations to decrease over time. The greater the damping, the faster the amplitude decreases. Second, the frequency or period of the oscillation is independent of the magnitude of the original deflection (as long as elastic limits are not exceeded). The naturally occurring frequency of the free oscillations is called the natural frequency, fn: (1) Forced Vibration. Forced vibration is the case when energy is continuously added to the spring mass system by applying oscillatory force at some forcing frequency, ff. Two examples are continuously pushing a child on a swing and an unbalanced rotating machine element. If enough energy to overcome the damping is applid, the motion will continue as long as the excitation continues. Forced vibration may take the form of self-excited or externally excited vibration. Self-excited vibration occurs when the excitation force is generated in or on the suspended mass; externally excited vibration occurs when the excitation force is applied to the spring. This is the case, for example, when the foundation to which the spring is attached is moving. Transmissibility. When the foundation is oscillating, and force is transmitted through the spring to the suspended mass, the motion of the mass will be different from the motion of the foundation. We will call the motion of the foundation the input, I, and the motion of the mass the response, R. The ratio R/I is defined as the transmissibility, Tr:Tr = R/I Resonance. At forcing frequencies well below the systems natural frequency, RI, and Tr1. As the forcing frequency approaches the natural frequency, transmissibility increases due to resonance. Resonance is the storage of energy in the mechanical system. At forcing frequencies near the natural frequency, energy is stored and builds up, resulting in increasing response amplitude. Damping also increases with increasing response amplitude, however, and eventually the energy absorbed by damping, per cycle, equals the energy added by the exciting force, and equilibrium is reached. We find the peak transmissibility occurring when fffn. This condition is called resonance.Isolation. If the forcing frequency is increased above fn, R decreases. When ff = 1.414 fn, R = I and Tr = 1; at higher frequencies R I and Tr 1. At frequencies when R I, the system is said to be in isolation. That is, some of the vibratory motion input is isolated from the suspended mass. Effects of Mass and Stiffness Variations. From Equation (1) it can be seen that natural frequency is proportional to the square root of stiffness, K, and inversely proportional to the square root of weight, W, or mass, M. Therefore, increasing the stiffness of the spring or decreasing the weight of the mass increases natural frequency.DampingDamping is any effect that removes kinetic and/or potential energy from the spring mass system. It is usually the result of viscous (fluid) or frictional effects. All materials and structures have some degree of internal damping. In addition, movement through air, water, or other fluids absorbs energy and converts it to heat. Internal intermolecular or intercrystalline friction also converts material strain to heat. And, of course, external friction provides damping. Damping causes the amplitude of free vibration to decrease over time, and also limits the peak transmissibility in forced vibration. It is normally characterized by the Greek letter zeta () , or by the ratio C/Cc, where c is the amount of damping in the structure or material and Cc is critical damping. Mathematically, critical damping is expressed as Cc = 2(KM)1/2. Conceptually, critical damping is that amount of damping which allows the deflected spring mass system to just return to its equilibrium position with no overshoot and no oscillation. An underdamped system will overshoot and oscillate when deflected and released. An overdamped system will never return to its equilibrium position; it approaches equilibrium asymptotically.Displacement, Velocity, and AccelerationSince vibration is defined as oscillatory motion, it involves a change of position, or displacement (see Figure 1).Figure 1. Phase relationships among displacement, velocity, and acceleration are shown on these time history plots.Velocity is defined as the time rate of change of displacement; acceleration is the time rate of change of velocity. Some technical disciplines use the term jerk to denote the time rate of change of acceleration.Sinusoidal Motion Equation. The single-degree-of-freedom spring mass system, in forced vibration, maintained at a constant displacement amplitude, exhibits simple harmonic motion, or sinusoidal motion. That is, its displacement amplitude vs. time traces out a sinusoidal curve. Given a peak displacement of X, frequency f, and instantaneous displacement x: (2) at any time, t. Velocity Equation. Velocity is the time rate of change of displacement, which is the derivative of the time function of displacement. For instantaneous velocity, v: (3) Since vibratory displacement is most often measured in terms of peak-to-peak, double amplitude, displacement D = 2X: (4) If we limit our interest to the peak amplitudes and ignore the time variation and phase relationships: (5) where: V = peak velocity Acceleration
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