【正文】 ()( ???? jjj trrjtt FKFahKF ?? Where a is the axial depth of cut, and tK and rK are the cutting force coefficients. After substituting jh from equation, the dynamic milling forces can be resolved in x and y directions as follows: ?????????????????????? yxaaaaaKFFyyxyyxxxtyx 21 Where xya are the directional coefficients[9,10]. The directional coefficients depend on the angular position of the cutter which makes Eq.(6) time varying: ? ? ? ?? ?)()(21)( ttAaKtF t ?? ? ?)(tA is periodic at the tooth passing frequency ??N? .In general, the Fourier series expansion of the periodic term is used for the solution of the periodic systems. However, in chatter stability analysis the inclusion of the higher harmonics in the solution may not be required for most cases as the response at the chatter limit is usually dominated with a single chatter frequency. Starting from this idea, Budak and Altintas [810] and later Merdol and Altintas [16] have shown that the higher harmonics do not affect the accuracy of the predictions unless the radial depth of cut is extremely small pared to the tool diameter .Thus , it is sufficient to include only the average term in the Fourier series expansion of ? ?)(tA in which case the directional coefficients take the following form [810]: ? ?? ?? ?? ? exstexstexstexstrryyryxrxyrrxxKKKKKK????????????????????????2s i n22c o s212c o s22s i n212c o s22s i n212s i n22c o s21??????????????? Then, Eq.(7)reduces to the following form : {F(t)= ? ?? ?)(210 tAaK t ? (9) Chatter stability limit The dynamic displacement vector in Eq.(9) can be determined using the synamic properties of the structures, transfer function or frequency response functions , and the dynamic forces .By substituting the response and the delay terms in Eq.(9), the following expression is obtained [9,10]: ? ? ? ?? ? ticTitti ccc eFiGAeaKeF ??? ? }{)()1(21 0??? (10) where {F}represents the amplitude of the dynamic milling force {f(t)}, and the transfer function matrix is given as: ),(][ wcpGGGGGpy ypx ypy xpx xp ?????????? (11) Where the total transfer function can be obtained form the summation of the cutter and workpiece transfer function, .,[].Eq.(10) has a nontrivial solution only if its determinant is zero, Det[[I]+A[G0(iwc)]]=0, (12) Where [I] is the unit matrix, and the oriented transfer function matrix is defined as ]][[][ 00 GAG ? (13) And the eigenvalue (? ) in Eq. (12) is given as ),1(4 Tit ceaKN ?? ????? (14) ? can easily be puted from Eq. (12) numerically. However, an analytical solution is possible if the cross transfer functions, xyG and yxG , are neglected: ),4(2 1 02110 aaaa ???? ?? (15) where ).()( ))(()(10 cyyyycxxxx yxxyyyxxcyycxx iGiGa iGiGa ???? ?????? ?? ?? (16) Since the transfer functions are plex, ? will have plex and real parts .However , the axial depth of cut (a) is a real number .Therefore , when IR i????? and TiTe ccTi c ??? s inc o s ??? is substituted in Eq.(14), the plex part of the equation has to vanish yielding TTccR ??? c os1 sin1 ????? (17) The above can be solved to obtain a relation between the chatter frequency and the spindle [9, 10]: ,60,tan,2,21NTnkTc??????? ???????? (18) where? is the phase difference between the inner and outer modulations, ? is an integer corresponding to the number of vibration waves within a tooth period, and n is the spindle speed(rpm). After the imaginary part in Eq. (14) is vanished, the following is obtained for the stability limit: )1(2 2l i m ?? ???? tNK Ra (19) Therefore , for given cutting geometry ,cutting force coefficients, tool and work piece transfer functions , and chatter frequency 1,?c? and R? can be determined from Eq.(15), and can be used in Eqs. (18) and (19) to determine the corresponding spindle speed and stability limit .Eq.(15)provides two s39。 revised 17 September 2021。 第 II 部 分：在共同的碾碎的系統(tǒng)， ASME， Jouranal的交易的應(yīng)用動力系統(tǒng)、測量和控制 120 (1998) 2236。 [9] ， E. Budak，穩(wěn)定的分析預(yù)言在碾碎， CIRP 44 (1) (1995) 357362 的史冊的。 [8] E. Budak、碾碎薄壁結(jié)構(gòu)技工和動力學(xué) ， 。 [6] R. Sridgar， . Hohn，長期 .， A碾碎的過程的穩(wěn)定算法， ASME，設(shè)計學(xué)報的交 易產(chǎn)業(yè)的 90 (1968) 330334。 [4] H. Opitz， F. Bernardi，車床聊天 begavior 的調(diào)查和演算和銑床，史冊 thr CIRP 18 (1970) 335343。 [2] . 托拜厄斯，機(jī)械工具 Vibration， Blackie，倫敦 1969 年。 這些方法可以用于工業(yè)生產(chǎn)方法為增加的碾碎的表現(xiàn)。 一個分析方法為塑造消除對傳遞函數(shù)測量的需要，每個工具匯編的立銑床動力學(xué)被提出。 方法是非?？焖俸蛯嵱玫脑谝鸱€(wěn)定可以被用于任意最大化振動物質(zhì)撤除率的葉片圖。 磨床的穩(wěn)定復(fù)雜歸結(jié)于旋轉(zhuǎn)的切割工 具造成時間變化的動力學(xué)。 6. 結(jié)論 振動是其中一個在用機(jī)器制造的主要局限造成質(zhì)量差和低生產(chǎn)力。 使用分析組分 FRFs 和 RCSA的被測量的和被預(yù)言的 FRFs為最短和最長的工具圖 12 被測量。 同一刀具柄使用用不同的立銑床，并且同一 FRF (H33)用于 RCSA。 工具中徑和阻止系數(shù)被確定了作為 mm 和 5 Ns/m，分別。 . 例子應(yīng)用 與 4支長笛 8 mm 直徑和 100 mm 長度的碳化物立銑床為測試使用。 從分析組分方式被提出的和實驗性數(shù)據(jù)夾緊的僵硬和阻止可以從工具點(diǎn) FRF 被辨認(rèn)。mn=iwNmn, P39。mn=iwHmn, L39。33+cqN39。33+cqP39。33+cqN39。21)E13(kqN21+cqN39。33+cxL39。33+cxH39。 終于，在完全的 RCSA 以后，對于穩(wěn)定和聊天退避是必需的， Schmizt 等給在工具套子 (G11)的分析位移或力量關(guān)系 [25] ： G11=X1/F1=H11H12E2E11E31L12((kx+cqN2139。對于刀具柄或紡錘組分直接偏折 e 期限 (H33)在交叉點(diǎn)地點(diǎn)被測量由沖擊試驗。 ζ= 和 ζ= 的平 均值從實驗性數(shù)據(jù)得到了為 HSS和碳化物刀具，分別。 工具的中徑和第二個轉(zhuǎn)動慣量可以被計算使用在本文 [30 的 ]提出的分析等式第 (1)部分射線的。 對于演算， (i)需要密度 (ρ)，彈性模數(shù) (e)，黏阻止的系數(shù) (c)和第二個轉(zhuǎn)動慣量。 在公式化的， 計算機(jī)方式由 G 是為匯編 FRFs 的 H代表。 RCSA 是預(yù)言的工具動力特性一個非常高效率的方法沒有測量為每個工具、刀具柄和紡錘組合。 必須確定四個連接參量預(yù)言工具點(diǎn)頻率特性作用 (FRF)。 . 工具動力學(xué)包括機(jī)器靈活性 . 聯(lián)結(jié)工具動力學(xué)的亞結(jié)構(gòu)分析 在這個模型的，完全機(jī)器結(jié)構(gòu)被 劃分成二部分： 工具和刀具柄或者紡錘。屈服介入這些常數(shù)的等式 [26]。 在 x=L1 和 y=0， ， 空間和剪切力的連