機(jī)械設(shè)計(jì)外文翻譯-凸輪速度對(duì)凸輪系統(tǒng)影響的實(shí)驗(yàn)研究【中文4500字】【PDF+中文WORD】

機(jī)械設(shè)計(jì)外文翻譯-凸輪速度對(duì)凸輪系統(tǒng)影響的實(shí)驗(yàn)研究【中文4500字】【PDF+中文WORD】,中文4500字,PDF+中文WORD,機(jī)械設(shè)計(jì),外文,翻譯,凸輪,速度,系統(tǒng),影響,實(shí)驗(yàn),研究,中文,4500,PDF,WORD 凸輪速度對(duì)凸輪系統(tǒng)影響的實(shí)驗(yàn)研究摘要：傳統(tǒng)上，在一個(gè)凸輪系統(tǒng)，一旦確定凸輪位移曲線的設(shè)計(jì),從動(dòng)件是以恒定的速度和運(yùn)動(dòng)特性運(yùn)動(dòng)的。從運(yùn)動(dòng)學(xué)角度看，通過改變輸入速度是一個(gè)改善從動(dòng)件運(yùn)動(dòng)特征的可行方法。本文中，我們說明如何找到一個(gè)多項(xiàng)式的速度軌跡來減少運(yùn)動(dòng)特性的峰值。此外，通過約束和系統(tǒng)設(shè)計(jì)程序產(chǎn)生一個(gè)適當(dāng)?shù)耐馆喗撬俣溶壽E的方法正在開發(fā)。設(shè)計(jì)實(shí)例說明了這個(gè)程序能為變速凸輪系統(tǒng)的速度得到適當(dāng)?shù)乃俣溶壽E。此外，一個(gè)帶有伺服控制器的實(shí)驗(yàn)裝置正在開發(fā)用來研究這種方法的可行性。實(shí)驗(yàn)數(shù)據(jù)表明，結(jié)果是非常接近那些理論。術(shù)語a-從動(dòng)件的加速度A, At-從動(dòng)件的標(biāo)準(zhǔn)加速度c, d, e, n, Ta, Tb, x, y-常參數(shù)h-從動(dòng)件的最大位移j-從動(dòng)件的急動(dòng)值J, Jc-從動(dòng)件的標(biāo)準(zhǔn)急動(dòng)值s-從動(dòng)件的位移S-從動(dòng)件的標(biāo)準(zhǔn)位移t-凸輪轉(zhuǎn)動(dòng)過的時(shí)間T, Tpa, Tpj, pv-標(biāo)準(zhǔn)時(shí)間v-從動(dòng)件的速度V, Vc-從動(dòng)件的標(biāo)準(zhǔn)速度-凸輪上升h高度是轉(zhuǎn)過的角度1234-凸輪轉(zhuǎn)過的角度-標(biāo)準(zhǔn)凸輪轉(zhuǎn)過的角度-凸輪轉(zhuǎn)過的角度-凸輪上升h高度用的時(shí)間1234-凸輪旋轉(zhuǎn)的-凸輪角速度ave-一個(gè)完整循環(huán)中凸輪的平均角速度s1s2s3s4-在從動(dòng)件一個(gè)運(yùn)動(dòng)周期中凸輪平均角速度-的一階導(dǎo)-的二階導(dǎo)-標(biāo)準(zhǔn)凸輪角速度-的一階導(dǎo)-的二階導(dǎo)引言在一個(gè)凸輪系統(tǒng)中，慣性力所產(chǎn)生的負(fù)載是容易變形和產(chǎn)生振動(dòng)的。而且急動(dòng)所產(chǎn)生的負(fù)載也可能造成振動(dòng)，這些都會(huì)影響凸輪的工作。因此，設(shè)計(jì)的運(yùn)動(dòng)曲線來盡量減少動(dòng)態(tài)加載對(duì)高速凸輪機(jī)構(gòu)很重要。眾所周知，速度和加速度曲線需要是連續(xù)的且有較小的峰值。此外，急動(dòng)曲線應(yīng)該是有限性的。在設(shè)計(jì)一個(gè)凸輪機(jī)構(gòu)是凸輪速度往往被假定是不變的。然而，從動(dòng)件的運(yùn)動(dòng)特性是隨凸輪速度變化而變化的。想要達(dá)到理想的運(yùn)動(dòng)狀態(tài)是一個(gè)合成有較好動(dòng)態(tài)特性的新曲線的應(yīng)用。在本文中，我們提出一個(gè)通過改變轉(zhuǎn)速的方法。在凸輪系統(tǒng)設(shè)計(jì)中用變速的觀念很少在文獻(xiàn)中有研究。羅特巴特 1 設(shè)計(jì)了一個(gè)變速凸輪機(jī)構(gòu)，在其中凸輪的輸入是輸出一個(gè)急回機(jī)構(gòu)。特薩和馬太福音 2 通過考慮變速凸輪的案例導(dǎo)出了從動(dòng)件的運(yùn)動(dòng)方程。選擇消除從動(dòng)件不連續(xù)運(yùn)動(dòng)特性的適當(dāng)角度是Yan等人設(shè)計(jì)的 3 。從運(yùn)動(dòng)學(xué)角度講，這項(xiàng)工作的任務(wù)是找到減少從動(dòng)件運(yùn)動(dòng)峰值的凸輪速度。此外，通過約束和系統(tǒng)設(shè)計(jì)程序產(chǎn)生一個(gè)適當(dāng)?shù)耐馆喗撬俣溶壽E的方法正在開發(fā)。設(shè)計(jì)實(shí)例說明，對(duì)于一個(gè)給定的從動(dòng)系統(tǒng)程序設(shè)計(jì)適合的角速度。一個(gè)實(shí)驗(yàn)凸輪系統(tǒng)是建立在其中一個(gè)伺服電機(jī)控制生成所需的速度軌跡來進(jìn)行性能評(píng)估上的。運(yùn)動(dòng)方程對(duì)于一個(gè)凸輪系統(tǒng)，從動(dòng)件的位移s（t）是凸輪旋轉(zhuǎn)角度（t）的應(yīng)變量。在算術(shù)上，他們可以這樣表達(dá)： 旋轉(zhuǎn)角度（t）在t時(shí)間時(shí)，從動(dòng)件的速度v（t）是： f（）=df（）/d，而且加速度（t）=d/dt。而且對(duì)應(yīng)的從動(dòng)件加速度a（t）和jerk，j（t）是： 方程14呈現(xiàn)了凸輪輸入角速度（t）和從動(dòng)件運(yùn)動(dòng)參數(shù)s（t），v(t) ，a（t）的關(guān)系。很明顯，如果（t）是連續(xù)的，它們就很簡單。令h是凸輪在時(shí)間t內(nèi)轉(zhuǎn)過角時(shí)從動(dòng)件的位移。讓T=t/，=/，S=s/h。我們有。那么方程14可以寫成如下的標(biāo)準(zhǔn)形式：s(t) 是標(biāo)準(zhǔn)的凸輪角速度，V（T），A（T），和J（T）分別是標(biāo)準(zhǔn)的速度，加速度和從動(dòng)件的急動(dòng)值方程18關(guān)系可以表示成： 當(dāng)凸輪以連續(xù)速度工作時(shí)，（T）=1，從動(dòng)件的標(biāo)準(zhǔn)的速度Vc（T），加速度Ac（T），和jerk，Jc（T）可以寫成： 則（T）=T。（T）的設(shè)計(jì)準(zhǔn)則對(duì)于一個(gè)給定的凸輪從動(dòng)件系統(tǒng)，如果我們正確的控制輸入速度軌跡，由恒定速度導(dǎo)致的標(biāo)準(zhǔn)速度，加速度，急動(dòng)值的峰值可能減小。例如，為了減小標(biāo)準(zhǔn)速度的峰值，（T）能能夠改變，那么，則Vc在標(biāo)準(zhǔn)時(shí)間Tpv時(shí)有峰值Vc。那么，從方程6，13我們知道（T）必須滿足下面條件： 為了減小標(biāo)準(zhǔn)加速度的峰值，是，當(dāng)在標(biāo)準(zhǔn)時(shí)間Tpa時(shí)Ac取峰值。在方程7和14的基礎(chǔ)上，（T）應(yīng)該這樣選： 請(qǐng)注意必須為非零。相似的，如果要求，則Jc在標(biāo)準(zhǔn)時(shí)間Tpj時(shí)有峰值。那么從方程（8）和（15）知，（T）需要滿足： 當(dāng)為了避免從動(dòng)件的過度振動(dòng)，（T）的諧波應(yīng)該越小越好。這里，我們選擇一個(gè)合適的速度軌跡。由于速度和加速度曲線，方程（6）和（7），要求是連續(xù)的，且急動(dòng)值曲線，方程（9），也需要有限的，那么（T）必須至少是二階可微?？紤]到（T）的連續(xù)性，（T）的斜率在T為0和1是，可能為了0，即（0）=0，（1）=0。而且，由于標(biāo)準(zhǔn)凸輪旋轉(zhuǎn)角度的邊界限制，（0）=0和（1）=1，（T）整合必須滿足下列條件： C是連續(xù)的。在一個(gè)變速凸輪從動(dòng)件系統(tǒng)中，凸輪在時(shí)間周期中以角速度（t）運(yùn)轉(zhuǎn)，轉(zhuǎn)過角度為（t），我們得到： 由于，那么方程（20）實(shí)際等同于： 這里，我們只考慮（T）0的情況，凸輪速度方向不改變。因此選擇（T）來減小從動(dòng)件峰值的標(biāo)準(zhǔn)是：(a) (I)為了減小標(biāo)準(zhǔn)速度的峰值： (II)為了減小標(biāo)準(zhǔn)加速度的峰值： (III)為了減小標(biāo)準(zhǔn)急動(dòng)值的峰值：(b) （T）至少二階可微(c) （0）=（1）=0(d)根據(jù)邊界條件（0）=0和（1）=1，連續(xù)的c滿足：(e)（T）有盡可能低的諧波(f) (g)（T）0讓方程（5）方程（8），在從動(dòng)件的上升期，代表標(biāo)準(zhǔn)運(yùn)動(dòng)特性。那么，下降期的運(yùn)動(dòng)特性為： 很容易可以發(fā)現(xiàn)，標(biāo)準(zhǔn)速度，加速度，急動(dòng)值在上升期和下降期是分別相等的。所以，我們有以下事實(shí)：如果同樣的位移曲線用在從動(dòng)件的上升期和下降期，函數(shù)（T）在兩個(gè)階段是相同的。角速度（T）考慮到一個(gè)有凸輪提供擺線運(yùn)動(dòng)的凸輪從動(dòng)件系統(tǒng)，并且凸輪輸入（T）是多項(xiàng)式。要在上升（或下降）時(shí)間，用標(biāo)準(zhǔn)（a）和標(biāo)準(zhǔn)（g）來減小運(yùn)動(dòng)曲線的峰值，我們選擇如下多項(xiàng)式（T），F(xiàn)ig. 1:圖. 1上升或下降時(shí)期的多項(xiàng)式角速度圖. 2休止時(shí)間的多項(xiàng)式角速度變速 定速圖. 3.擺線運(yùn)動(dòng)表1擺線運(yùn)動(dòng) 定角速度 變角速度 相差% V的峰值 2.00 1.83 -8.5 A的峰值 6.28 5.97 -4.9 J的峰值 39.48 52.55 33.1 當(dāng) 當(dāng) 當(dāng) 恒定參數(shù)d，e，x，y，Ta和Tb是要確定的。參數(shù)d呈現(xiàn)了函數(shù)（T）的波動(dòng)，根據(jù)準(zhǔn)則（g）-1d1。為了滿足設(shè)計(jì)準(zhǔn)則（b）（d），要： 參數(shù)x，y是根據(jù)凸輪位移曲線決定于凸輪位移曲線和設(shè)計(jì)準(zhǔn)則。參數(shù)e服從準(zhǔn)則f如下： 顯然我們可以在多項(xiàng)式（T）圖中，選擇合適的d，Ta，和Tb來獲得最小的運(yùn)動(dòng)特性峰值。由于擺線運(yùn)動(dòng)特性具有對(duì)稱性，為了簡單和對(duì)稱我們讓Tb=1-Ta。另外，當(dāng)從動(dòng)件在休止階段是，根據(jù)設(shè)計(jì)準(zhǔn)則（c），（g）和方程（26），（29），我們得到（T）： 在設(shè)計(jì)準(zhǔn)則（d）和方程（30）條件下，我們得到： 且從設(shè)計(jì)準(zhǔn)則（c）和（g），我們得到： 圖4.變速凸輪從動(dòng)件系統(tǒng)角速度角速度位移速度加速度急動(dòng)值-恒速 變速 圖5：擺線凸輪運(yùn)動(dòng)（n=0，d=0，1，ave=100rpm） -1n1假設(shè)凸輪具有如下擺線運(yùn)動(dòng)： 因此 為了減小給定擺線凸輪運(yùn)動(dòng)的標(biāo)準(zhǔn)速度與加速度的峰值，我們認(rèn)為要選取，Ta=0，Tb=1，d=0.1，且x=y=2 把方程（34）（38）代用到方程（5）（8），我們的到像圖3所示標(biāo)準(zhǔn)的位移，速度，加速度和急動(dòng)值。在圖3中，我們可以看到連續(xù)的標(biāo)準(zhǔn)的速度，加速度曲線和急動(dòng)值曲線是有限的。表1中的V，A，和J的峰值表明標(biāo)準(zhǔn)速度和加速度的峰值減小了。設(shè)計(jì)實(shí)例設(shè)計(jì)一個(gè)滿足如下條件的凸輪從動(dòng)件系統(tǒng)：當(dāng)凸輪裝過60度時(shí)，徑向滾子停止，且再下一個(gè)120度做擺線運(yùn)動(dòng)，總共上升30mm。從動(dòng)件停止當(dāng)凸輪在轉(zhuǎn)60度過程中，然后在最后的120度做擺線運(yùn)動(dòng)退回30mm到遠(yuǎn)處。讓s1，s2，s3，s4分別代表從動(dòng)件在第一個(gè)休止階段，上升階段，第二個(gè)休止階段，下降階段的平均角速度。相似的讓1，2，3，4凸輪旋轉(zhuǎn)角度，1，2，3，4分別是以上階段的時(shí)間。然后我們得到以下事實(shí)： 其中ave是整個(gè)循環(huán)中的平均角速度。如圖4所示的（t）必須是連續(xù)的。根據(jù)方程（26）（29）和（30）（33），我們得到： 角速度位移速度加速度急動(dòng)值圖6.擺線運(yùn)動(dòng)凸輪從動(dòng)件系統(tǒng)（n=2，d=0.1，ave=100rpm）（基圓與標(biāo)準(zhǔn)圓）圖7.擺線凸輪 其中1=/3，2=2/3，3=/3，4=2/3，d=0.1，且s1=s2。令ave為100rpm且為0。凸輪的角速度，速度，加速度和從動(dòng)件的急動(dòng)值就能計(jì)算出來。圖5可以看出凸輪從動(dòng)件在定速100rpm是的運(yùn)轉(zhuǎn)結(jié)果。相似的，圖6中ave為100rpm，n為0.2。而且，圖5和圖6表面在變速和定速凸輪從動(dòng)件系統(tǒng)中峰值出現(xiàn)的時(shí)間是不同的。這意味著，可以設(shè)計(jì)一個(gè)合理的速度來改變峰值出現(xiàn)的時(shí)間，以使凸輪從動(dòng)件系統(tǒng)具有的更好的運(yùn)動(dòng)特性。圖8.實(shí)驗(yàn)裝置示意圖和使用儀器圖9.基于DSP控制的實(shí)驗(yàn)系統(tǒng)實(shí)驗(yàn)計(jì)劃和步驟這項(xiàng)研究中用到了變速凸輪從動(dòng)件實(shí)驗(yàn)系統(tǒng)和設(shè)備。實(shí)驗(yàn)用的盤形凸輪的直徑，有$50構(gòu)成，有60mm長?？紤]到靜態(tài)平衡，如圖7所示盤形凸輪的質(zhì)心是（-0.478mm，0mm）。凸輪的厚度、質(zhì)量、面積和慣性矩分別是13mm，9777.7mm，1082g和0.0031kg.m2。最大的凸輪壓力角是18度，那樣設(shè)計(jì)可以正確承受側(cè)向推力。如圖8所示這個(gè)凸輪從動(dòng)件系統(tǒng)是安裝在一個(gè)與地基固定的框架上的。滾子從動(dòng)件可在固定在導(dǎo)軌上的支架上水平移動(dòng)，且可以由旋轉(zhuǎn)的凸輪驅(qū)動(dòng)。從動(dòng)件是一個(gè)長為495mm，直徑為20mm的圓柱棒。滾子固定在靠近凸輪端的棒子，直徑為22mm厚度為10ram，可以繞著直徑為8長度為36mm的滾子銷旋轉(zhuǎn)。由于凸輪從動(dòng)件系統(tǒng)截面尺寸大且是由碳鋼做的，所以被認(rèn)為是剛性的。預(yù)設(shè)彈簧為規(guī)格d13160，剛度為0.146公斤每毫米且長36mm，那樣就可以保證滾子從動(dòng)件可以和凸輪始終保持接觸。因?yàn)橥馆嗇S和電機(jī)軸由剛性聯(lián)軸器（剛性，MJC50）連接的，所以凸輪角速度和點(diǎn)擊轉(zhuǎn)速要相同。因此要利用直流私服電機(jī)（三洋，點(diǎn)擊，cn-800-10，850w，1000轉(zhuǎn)每分）和圖8所示的驅(qū)動(dòng)馬達(dá)，它可以很容易的控制凸輪輸入角速度來驅(qū)動(dòng)凸輪從動(dòng)件系統(tǒng)。圖10a采用速度控制系統(tǒng)（4），可以最容易的通過電機(jī)來改變角速度的變化。在插件板上的IBM電腦，TMS320C30系統(tǒng)板，是用在實(shí)時(shí)實(shí)驗(yàn)安裝的。圖9描述了實(shí)驗(yàn)系統(tǒng)的硬件配置。除了通過AT總線來數(shù)字通信外，輸入/輸出模擬信號(hào)可以通過車載模數(shù)轉(zhuǎn)換器（模數(shù)轉(zhuǎn)換器）和數(shù)模轉(zhuǎn)換器（數(shù)模轉(zhuǎn)換器）。這些輸入和輸出通道的分別是為了對(duì)DSP的信號(hào)反饋和對(duì)控制裝置的控制信號(hào)。在實(shí)時(shí)控制中，采樣頻率60每秒是合適的，那樣遙控器可以連續(xù)的控制。圖10b圖10c控制的輸出響應(yīng)通過機(jī)載模數(shù)轉(zhuǎn)換器測量，并存儲(chǔ)在記憶板中。驅(qū)動(dòng)電機(jī)的轉(zhuǎn)速是從電機(jī)電壓信號(hào)，即內(nèi)置的轉(zhuǎn)速，并輸入到個(gè)人計(jì)算機(jī)486來完成。加速度和位移可以通過如圖8所示的加速度測量工具（PCB，353A34），和線性編碼器（HEIDENHAIN，LS404）。圖10d圖10e來自加速度儀的信號(hào)取決于電源單元(PCB, mode 480E09 ICP).利用最小二乘擬合方法 7 ，從位移信號(hào)獲得從動(dòng)件加速度，從加速度信號(hào)獲得急動(dòng)值。測量的數(shù)據(jù)傳回電腦來進(jìn)行性能評(píng)價(jià)。分別能從圖10，11中得到凸輪在轉(zhuǎn)速為150和200rpm是n=0，d=0.1的實(shí)驗(yàn)和理論結(jié)果。雖然運(yùn)轉(zhuǎn)速度波動(dòng)，但圖10，11表明在每個(gè)中期中實(shí)驗(yàn)與理論結(jié)果一致。本實(shí)驗(yàn)結(jié)果表面該方法是可行的。圖11a圖11b圖11c圖11d圖11e結(jié)論在這項(xiàng)工作中，從運(yùn)動(dòng)學(xué)角度，根據(jù)凸輪輸入速度的控制，我們提出了一個(gè)來提高凸輪從動(dòng)件系統(tǒng)的運(yùn)動(dòng)特性可行的方法。通過約束和系統(tǒng)設(shè)計(jì)程序產(chǎn)生一個(gè)適當(dāng)?shù)耐馆喗撬俣溶壽E的方法正在開發(fā)。實(shí)例表明，當(dāng)速度加速度是連續(xù)，急動(dòng)值是有限時(shí)，這個(gè)變速凸輪設(shè)計(jì)方法是有效的。另外，一個(gè)用來實(shí)驗(yàn)的變速凸輪從動(dòng)件系統(tǒng)已建立用來研究變速凸輪從動(dòng)件系統(tǒng)。在每個(gè)周期的實(shí)驗(yàn)于理論結(jié)果都高度一致。實(shí)驗(yàn)結(jié)果表明，該方法是可行的。文獻(xiàn)1. H. A. Rothbart, Cams: Design, Dynamics, and Accuracy. Wiley, New York (1956).2. D. Tesar and G. K. Matthew, The Dynamics Synthesis, Analysis, and Design of Modeled Cam Systems. Lexington Books(1976).3. H. S. Yah, M. H. Hsu, M. K. Fong and W. H. Hsieh, 3rd National Applied Mechanisms & Robotics Conf., Vol. 2,AMR-93-076-81 (1993).4. C. K. Benjamin and T. Jacob, Incremental Motion Control: DC Motors and Control System.5. Loughborough Sound Images, TMS320C30 System Board User Manual, Loughborough, U.K. 0990).6. Texas Instruments, TMS320C30x Users Guide. Texas Instruments Inc. (1991).7. R. H. Brown, S. C. Schneider and M. C. Mulligan, IEEE Trans. on Industrial Electronic 39, No. I (1992).陽酬。因侃”，4-114x份制刷7-9Mech. Mach. Theory Vol 31, No. 4, pp. 397-412, I”6 Copyright 1996 El回vier Science LtdPrinted in Great Britain. All rights re甜”“0094-l 14X/96 $15.00 + 0.00AN EXPERIMENTAL STUDY OF THE EFFECTS OF CAM SPEEDS ON CAM-FOLLOWER SYSTEMSH. S. YAN and M. C. TSAIDepartment of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, Republic of ChinaM. H. HSUDepartment of Mechanical Engineering, Kung Shan Institute of Technology and Commerce, Yungkang,Tainan 71016, Taiwan, Republic of China(Received 9 September 1994; received for publication 26 October 1995)Abstract-Traditionally, in a cam-follower system, the cam is often operated at a constant sp臼d and the motion characteristics of the follower are determined once the cam displacement curve is designed. From the kinematic point of view, the approach by varying cam input driving speed is an alternative way forimproving the follower motion characteristics. Here we show how to find a polynomial speed tr句“toryfor reducing the peak values of the motion characteristics. Furthermore, constraints and systematic design procedures for generating an appropriate tr句“tory of the cam angular velocities are developed. Design examples are given to illustrate the procedure for getting an appropriate speed trajectoas variable speedcam-follower systems. Furthermore, an experimental setup 叨th a servo controller is developed to study the feasibility of this approach. Experimental data show that the results are very close to those of theory.NOME NCLATUR Ea-acceleration of the followerA, A0-normalized acceleration of the followerc,d,e爪，孔，孔，x, y-constant parametersh-maximum displacement of the followerJ一jerk of the followerJ, J0-normalized jerk of the followers-displacement of the followerS-normalized displacement of the follower t一time for the cam to rotate through angle (T，凡，T 凡，normalized timev-velocity of the followerV, V0-normalized velocity of the follower：F am angle rotation for total rise hA ，品，島，Aam rotation angle1扣一normalized cam angle of rotation 0一切m an阱 of rotation time of cam rotation for total rise hh，勺，勺，馬一time of cam rotation 護(hù)-cam angluar velocityw.,.-average cam angular velocity of a complete cycle時(shí) w,2, w,3 , w54-average cam angular velocity in a follower motion period the 1st derivative of wd頭 the 2nd derivative of w。一1-normalized cam angular velocity 0-the 1st derivative of nthe 2nd derivative of nINTR ODUCTIONIn a cam-follower system, the load produced by inertia forces is prone to deflection and creates vibrations; and the load introdu臼d by jerks may cause vibrations as well. These will d創(chuàng) the operating life of the cam. Therefore, the design of motion cu凹臼 to minimize dynamic loading is of importan四 for high sp臼d cam mechanisms. It is well known that the velocity and acceleration cu凹es are required to be continuous and to have smaller peak values. In addition, the jerk curve should be finite.397398H. S. Yan et al.A cam is often assumed to be operated at a constant speed in designing a cam-follower system. However, the motion characteristics of the follower are changed as the cam speed varies. Traditionally, to achieve the desired motion is an application of synthesis for obtaining new displacement curves which have better dynamic characteristics. In this paper, we propose an alternative method by varying the cam speeds. The concept of using variable speeds in a cam-follower system design was seldom studied in the literature. Rothbart 1 designed a variable speed cam mechanism in which the input to the cam is the output of a Withworth quick-return mechanism. Tesar and Matthew 2 derived the motion equations of the follower by considering the case of variable speed cams. The criteria for selecting proper angular velocities which will eliminate the discontinuity in motion characteristics of the follower are investigated by Yan et al. 3. From the kinematic point of view, the objective of this work is to find cam speed tr哉jecto for reducing the peak values of the follower-output motion. Furthermore, constraints and system design procedures for generating a proper trajecto of the cam angular velocities are deveoped. Design examples are given to illustrate the procedure for getting a proper angular speed for a given follower system. An experimental cam-follower system is set up in which a servo motor is controlled to generate the desired speed trajectory for performance evaluations.MOTIO N EQU ATIO NSFor a cam-follower system, the follower displacement, s( t ), is a function of cam rotation angleO( t ). Mathematically, it can be expressed as:、且，a，飛s(t ) =f (O( t )where O ( t ) is the cam rotation angle at time t. The follower velocity, v( t ), of the follower is then given by:v( t ) =f （O)ro(t)(2)where f (O ) = df 刷刷，and ro(t) = dO( t )/d t is the cam angular velocity. Furthermore, the corresponding follower acceleration, a( t ), and jerk, j( t ), are:a( t ) =f j( t ) = f (O)ro3(t) ，（O)ro(t ).iJ(t ) +f (O ）（t ）(3)(4)where f (O) = df 2() )/d0 2, f ”（0) = df 3(0)/d03，（t ) = dw( t )/d t, and co( t ) = dw 2( t )/d t 2. Equations (1)-(4) present the relationship between cam input angular velocity w(t) and follower-output motions s( t), v( t ), a( t ), and j(t ). Obviously if w( t ) is a constant, they can be greatly simplified.Let h be the total displacement of the follower as the cam rotates an angle P in time period Furthermore, denote T = t /t, y 咐，and S = s/h. Now we have T eO, 1, vt e O， ，y e O, 1, vO eO, P and SeO, l, vs eO, h. Then, equations (1)-(4) can be rewritten in terms of their normalized forms as follows:= hs( t )S(T) = g( y )V(T) = g ）Q(T)A(T ) = g 氣y )Oi(T ) +g ）Q(T)J(T) = g ”（y )03(T) + 3g 氣y )0(T)Q( T) +g ）凸（T)(5)(6)(7)(8)where O(T) = dy (T)/dT is the normalized cam angular velocity and V(T), A(T ), and J(T) are the normalized velocity, acceleration, and jerk of the follower, respectively. The relationship between equations (1) to (8) can be found as:s = hS(9)V雪 Vt(10)、l且，SE飛Effects of cam speeds on cam-follower systems399 Ari與EF(12)When the cam operates at a constant speed, i.e. Q(T) = 1, the normalized velocity, Vc (T),acceleration, Ac(T ), and jerk, Jc (T), of the follower can be expressed as:where y (T) = T.F二 （ T) = g(y )Ac (T) = g( y )Jc (T) = g （y )(13)(14)(15)CR ITER IA FOR DESIG N ll( T)For a given cam-follower system, the peak values of the normalized velocity, acceleration, and jerk resulting from a constant driving speed may possibly be reduced if we properly control its input speed trajectory !l(T). For example, to redu臼the peak values of normalized velocity, !l(T) canbe chosen so that IV(Tpv )I 副Vc (Tpv )I where Ve has peak values at normalized time Tpv Then, from equations (6) and (13), we select that !l(T) must satisfy the condition:一 1 Q(Tpv) l.(16)“For the case that we want to reduce the peak values of the normalized acceleration, i.e.IA (Tpa )I 運(yùn) Ac(T pa)I were A0 has peak values at normalizedme Tpa . Based on equations (7) and(14），（T) should be chosen such that一 l + 02(Tp，） g(y (1盧）!l(T歸. ） I -2Q (T ).(lg(y (T陽. ）陽Note that g氣y (Tpa ) must be nonzero. Similarly, if it requires /J （馬） 運(yùn) 孔（馬）I where J0 has peak values at normalized time TPi then from equations (8) and (15), we need some Q(T) which satisfies一 l + Q3(Tpi ) as g( y (Tpi ) # 0.3g 氣y （馬）Q(Tpi)!l(Tpi ) +g(y (Tpi )fi(T.陽） 1 -!l3(T凹）(18)g叨叨（y (Tp)For avoiding excessive vibration in the follower, the harmonics of !l(T) should be chosen as low as possible. Here, we choose an appropriate speed trajectory.Since the velocity and acceleration curves, equations (6) and (7), are required to be continuous and the jerk curve, equation （例，should be finite as well, O(T) must be at least second orderdifferentiable.Considering the the continuity of !l(T), the slope of !l(T) may be zero at T = 0 and I , i.e.0(0) = 0 and 0(1) = 0.Furthern ore, due to the boundary conditions of the normalized cam rotation angle, y(O ) = 0 andy(l ) = l, the integration of !l(T) must satisfy the following condition:f !l(T) dT + c = y (T)(19)for some constant c.；叫小In a variable-speed cam-follower system, the cam operates at an angular velocity w (t ) and rotates an angle f3 in a time period ，we have:FSinc晴 w(t) = p IQ(T), equation (20) is in fact equal to:f O(T) dT = I.(21)400H. S. Yan et al.Here, we only consider the case of !l(T） 0, i.e. the direction of cam speed is not changed. As a result, design criteria for selecting !l(T) t reduce the peak values of follower-outputs are:(a) (I) for the case of reducing the peak value of the normalized velocity:-1 :; !l(Tp.） 運(yùn) l(II) for the case of reducing the peak value of the normalized acceleration:一 l + T0.) :; g （匯.）O(Toa ) :; l -, Toa )JQl(Ppaff(四g( y( T歸. ）陽(III) for the case of reducing the peak value of the normalized jerk:一l +Q3(T.陽） :,;Pl l -!l3(T四）3g ”（y （乓，j)!l(Tpi)O(Tpi ) +g(y(T.回）nc r.;)g” （幾）(b) !l(T) is at least second order differentiable.(c) 0(0) = 0(1) = 0.(d) Constant c satisfies J !l(T) d T + c = y (T) subject to the boundary conditions y (O) = 0, y(l) = I.(e) !l(T) has the harmonics as low as possible.( f ) JA !l(T) d T = I.(g) !l(T) 注 0.Let equations (5問8) represent the normalized motion characteristics of the follower in the rising period. Then, the motion characteristics in the falling period are:S(T) = 1 -g(y)(22)V(T) = -g ）!l(T)(23)A(T ) = -g飛y )!l2(T) -g ）O(T)(24)J（ -g”（y )!l3(T) -3g ”（y )!l(T)O(T) -g ） （T).(25)It is easy to find that the absolute values of the normalized velocity, acceleration, and jerk in the falling period are equal to those in the rising period, respectively. Hence, we have the following fact:Ifthe same displacement cu凹e is used in the rising and falling periods of a follower, the functions of !l（ in these two periods are identical.A NGULA R VELOCITY O( T)Consider a cam-follower system which has a cam providing a cycloidal motion cu凹e where cam-input !l(t) is a polynomial.In the rising (or falling) period and applying criteria (a) and (g) to reduce the peak values of motion curves, we choose the following polynomial !l(T), Fig. 1:.Q(T)0TaTbFig. 1. Polynomial angular velocity in rising or falling period.Effects of cam speeds on cam-follower systemsQ）401。T0.5Fig. 2. Polynomial angular velocity in dwell period.Q(f)1.151.101.051.00 .,_ - - ，甲， ， F ， 一 - - - -。0.250.50.750.950.90T。0.250.50.75s）1.000.750.500.250.00TV口J2.50丘2.001.50t ”，”。. T0.250.50.75A(T)10。-5。-10T0.250.50.75505。25J(T)752- 咽h ”-50 。T0.250.50.75一一一一一 Vanable speed- Constant speed Fig. 3. Cycloidal motion.402H. S. Yan et al.Peak value of V Peak value of A Peak value of JTable l. Cycloidal motion ConstantVariableanular 飛 elocityangular velocityDifference %2.001.83-8.56.285.97-4.939.4852.5533.1for O 運(yùn) TT.,for T.T Tb,for Tb T 1,Q(T) = 1 + d(26a)Q(T) = 1 + dl -e(T - T.Y(T - Tb川(26b) Q(T) = 1 + d(26c)where constant parameters d，鳥 兒 y，丸，and Tb are to be determined. Parameter d presents the fluctuation of Q(T), where 一 1 d 1 by criterion (g).To satisfy design criteria (b）一（喲，imply:X +y 注 2X = y = 0, 2, 4, . . .(27)(28)Parameters x and y are determined based on the type of cam displacement curves and the design criteria (e). Furthermore, parameter e subject to desi伊 criteria (t) is given by:30e一 （衛(wèi)一一一一 一Tb)5 (29)Apparently, we can properly select d，叉，and Tb to obtain the lowest peak values of motion characteristic under the polynomial Q( T) of Fig. 1. Since the cycloidal motion curve is of symmetry, we let Tb = 1 - T. for simplicity and symmetry.In addition, when the follower is in the dwell period, from design criteria (c) and (g) and equations (26)-(29), we obtain Q(T), Fig. 2, as follows:Q(T) = 2n (2T3 - 3T2 ) + 1 + n.(30)Under design criterion (d) and equation (30), we have:y (T) = n(T4 - 2T3 ) +(1 + n)T(31)and from design criteria (c) and （剖，we imply:O(T) = 12n(T2 - T)(32)(Os4Time (t)0tiI 2I 23I234Fig. 4. Angular velocity of a variable-speed cam-follower system.Effects of cam speeds on cam-follower systems403g（ gbE苦HS昆寫1101051009590，、dh85 。0.120.240.360.480.6Time(s)JAUaJAUJAJ句3句3肉，噸，且且nu hn321g（ g）Hgg82房擊。0.120.240.360.480.6Time(s)nunununUAU Anuhu nunqd Jm（ BE）hM忘。古。mEE）gzEU38（N5,0002,500。-2,5000.120.240.360.480.6Time(s)-5,000 。0.120.240.360.480.6Time(s)300,000回白g 捋 同（mu150,000。） HU-150,000-300,000 。0.120.240.360.480.6Time(s)- - - - Constant speed一一一Variable speedFig. 5. Cam-follower system with cycloidal motion (n = 0, d = 0.1, w = 100 rpm).MMT 31陣 D404H. S. Yan et al.。（T) = 12n(2T 一 1)(33)where -1 n 1.Assume that the cam has cyclo1dal motion which is given by:Therefore:g(y ) = y 一 siL冗g(y ) = I -cos (2y )g”（y ) = 2n sin (2ny ) g” ）42 cos (2y ).(34)(35)(36)(37)We c。nsider to reduce the peak values of both normalized velocity and acceleration for the givencycloidal motion curves, and then choose T.= 0, Tb = 1, d = 0.1, and x = y = 2, i.e.。（T) = I + 0.11 - 30T2( T - 1)2.(38)Substituting equations (34）一（38) into equations (5問8), we have normalized displacement, velocity, acceleration, and jerk as shown in Fig. 3, in that we can see the normalized velocity and acceleration curves being continuous and the normalized jerk curve is being 缸1ite. The peak values of V, A, and J as demonstrated in Table 1 shows the peak values of the normalized velocity and acceleration are reduced.DESIG N EX AMPLEDesign a variable-speed cam-follower system to satisfy the following condition:A radial roller follower dwells while the cam rotates 60, and rises through a total displacement of 30 mm with cycloidal motion for the next 120。.The follower dwells again in 60。of cam rotation,and then returns 30 mm with cycloidal motion for the final 120.Let 時(shí) ，ro,2 , Ws3 , and ro84 denote the average angular velocities while the follower is in the first dwell, rising, second dwell, and falling period, respectively. Similarly, let 31 , /32 ，比，/34 be the camrotation angles; and h，勺，勺，r4 be the times in the above period. Then, we have the following facts:自11(39)W,1一/3一22s2/333W,3(40)(41)(42)盧4一一4）s4一一21 2 3 4(43)wwhere w is the average cam angular velocity of a complete cycle.The polynomial ro(t), Fig. 4, must be continuous. Based on equations (26)-(29) and (30)-(33), we have:(1 -n )w,1 = (1 + d)w,2(1 + d)w,2 = (1 - n)w 。(1 + n)w,3 = (1 + d)w 抖。d)ws4 = (1 + n)ro,1(44)(45)(46)(47)Effects of cam speeds on cam-follower systems405J宮幸1401201008060 。0.120.240.360.480.6 Time(s)3530、，、5、20、15、自.日s10。s。0.12-5、0.240.360.480.6 Time(s)400200。號(hào)-200。-4000.120.240.360.480.6 Time(s)M 氣8,0004,000。-4,000-8,000 。， ，0.120.240.360.480.6 Time(s)。300,000”、M-300,000曾-600,000 。、_ ,0.120.240.360.480.6 Time(s)- - - C。nstant speedVariable speedFig. 6. Cam-follower system with cycloidal motion (n = 0.2, d = 0.1, w.= 1傭rpm).406日.S. Yan et al.y盧60 X、base circle Fig. 7. Profile of a cycloidal cam.I II + dl + dlI -一2 II P1 一n P2 品 I + n P4 lIroave(48)where p13，品 23, /J3 3, p4 = 2 3, d = 0., and W,1 = W,3. Let Wave be ）（） 叩m and n be equal to 0. The angular velocity of the cam and the displacement, velocity, acceleration, and the jerk of follover can be calculated. Figure 5 shows the result where the dashed lines are the corresponding results when a cam-follower system operates at a constant speed of 100 rpm. Similarly, the results of the case, Wave is 100 甲m and n is equal to 0.2, are demonstrated in Fig. 6. Furthermore, Figs 5 and 6 show that the time at the peak values occurs in a variable speed cam-follower system is not the same as the constant-speed case. This implies that a proper variable cam speed design can provide a method to change the times of various period to get desired orbetter motion characteristics in a cam-follower system.Fig. 8. Schematic of experimental apparatus and instrumentation.Effects of cam speeds on cam-follower systemsIBM-PC BUSFig. 9. Experimental system for DSP-based control.407EX PER IMENT A L SETUP A N D R ESU LTSThe experimental variable-speed cam-follower system and apparatus are used in this study. The base circle diameter of this experimental disk cam, made of S50C, is 60 mm. Consider the condition of static balance, the centroid of the disk cam as shown in Fig. 7 is (-0.478 mm, 0 mm). The thickness, area, mass, and mass moment of inertia of the cam are 13 mm, 9777.7 mm2, 1080 g, and 0.0031 kg-m2, respectively. The maximum cam pressure angle is 18 that the guide is properly designed to withstand the side thrust. The cam-follower system is mounted on a frame which is rigidly fixed to the foundation and shown in Fig. 8. 1e roller follower moves horizontally in the guide fixed to the frame, and the follower is actuated by the rotation of the ca噸The follower is made of a bar (THK, LBH25) of length 495 mm and diameter 20 mm at both ends. The roller is fixed at the cam-end of the bar and is of cylindrical type, 22 mm in diameter and IO mm thick, which rotates on a roller-pin of diameter 8 mm and length 36 mm. The cam-follower system is assumed to be rigid because it processes the large cross-section dimensions and is made of carbon steel. The preset of the spring (SPEC, D13160) with 0.146 kg/mm spring stiffness is 36 mm so that the roller follower can keep contact with the cam.The angular velocities of the cam and the driving motor are assumed to be identical because a rigid Oldham coupling (MIGHTY, MJC50) is used to connect the cam shaft to motor shaft. Hence, using a d.c. servo motor (SANYO, DENKI, CN-800-10, 850W, 1000 rpm) and a driver of motor as shown in Fig. 8, it can be