漸開(kāi)線(xiàn)齒輪嚙合碰撞力仿真

黃中華，張曉建，周玉軍

(1. 湖南工程學(xué)院 機(jī)械工程學(xué)院，湖南 湘潭，411101；2. 中南大學(xué) 機(jī)電工程學(xué)院，湖南 長(zhǎng)沙，410083)

漸開(kāi)線(xiàn)齒輪嚙合碰撞力仿真

黃中華1,2，張曉建2，周玉軍2

(1. 湖南工程學(xué)院 機(jī)械工程學(xué)院，湖南 湘潭，411101；2. 中南大學(xué) 機(jī)電工程學(xué)院，湖南 長(zhǎng)沙，410083)

為獲得漸開(kāi)線(xiàn)齒輪嚙合傳動(dòng)時(shí)輪齒碰撞力的變化規(guī)律，提出基于動(dòng)力學(xué)仿真的漸開(kāi)線(xiàn)輪齒碰撞力計(jì)算方法。建立一對(duì)漸開(kāi)線(xiàn)齒輪嚙合傳動(dòng)的動(dòng)力學(xué)模型，給出基于Hertz接觸理論的齒輪嚙合傳動(dòng)時(shí)輪齒碰撞力的計(jì)算方法。對(duì)齒輪嚙合傳動(dòng)時(shí)的輪齒碰撞力、x向碰撞力和y向碰撞力的變化規(guī)律及其頻譜特征進(jìn)行仿真研究。仿真結(jié)果表明：齒輪嚙合傳動(dòng)時(shí)碰撞力的幅值波動(dòng)顯著，輪齒從嚙入到嚙出，碰撞力從0 kN增加到最大碰撞力后又減小至0 kN，具有明顯的周期性；碰撞力頻譜中會(huì)出現(xiàn)齒輪嚙合頻率的1倍頻和2倍頻；x向碰撞力和y向碰撞力幅值波動(dòng)顯著，具有相同的頻譜特征，相位相差約90°；頻譜中出現(xiàn)齒輪的旋轉(zhuǎn)頻率和嚙合頻率，存在明顯的調(diào)制現(xiàn)象，其中載波為齒輪的嚙合頻率，調(diào)制波為齒輪的旋轉(zhuǎn)頻率。

碰撞力；齒輪嚙合；仿真

漸開(kāi)線(xiàn)齒輪是機(jī)械設(shè)備中廣泛應(yīng)用的一種傳動(dòng)裝置。齒輪的強(qiáng)度、剛度和疲勞壽命與齒輪嚙合傳動(dòng)時(shí)的碰撞力密切相關(guān)[1?3]。傳統(tǒng)的機(jī)械設(shè)計(jì)通常把齒輪嚙合傳動(dòng)過(guò)程中輪齒產(chǎn)生的碰撞力視為恒力。由于結(jié)構(gòu)的原因，齒輪在嚙合傳動(dòng)過(guò)程中嚙合剛度會(huì)發(fā)生周期性變化，導(dǎo)致輪齒碰撞力發(fā)生相應(yīng)變化，進(jìn)而引起齒輪的沖擊和振動(dòng)[4?7]。為了掌握漸開(kāi)線(xiàn)齒輪嚙合傳動(dòng)時(shí)的動(dòng)力學(xué)行為，有必要對(duì)齒輪嚙合傳動(dòng)時(shí)輪齒產(chǎn)生的碰撞力進(jìn)行研究，以獲取輪齒碰撞力的變化規(guī)律，為齒輪系統(tǒng)的設(shè)計(jì)提供相關(guān)的技術(shù)參數(shù)。由于齒輪嚙合時(shí)輪齒碰撞力波動(dòng)幅度大、齒輪處于旋轉(zhuǎn)運(yùn)動(dòng)狀態(tài)，導(dǎo)致齒輪嚙合傳動(dòng)時(shí)輪齒碰撞力的實(shí)驗(yàn)測(cè)取存在許多困難。隨著計(jì)算機(jī)仿真技術(shù)和虛擬樣機(jī)技術(shù)的發(fā)展，通過(guò)建立齒輪嚙合傳動(dòng)的虛擬樣機(jī)模型，運(yùn)用動(dòng)力學(xué)仿真方法獲取齒輪嚙合傳動(dòng)時(shí)碰撞力已成為現(xiàn)實(shí)[2,8?11]。本文作者擬建立齒輪嚙合傳動(dòng)的虛擬樣機(jī)模型，通過(guò)仿真研究獲取齒輪嚙合傳動(dòng)時(shí)輪齒碰撞力的變化規(guī)律。

1 仿真模型

圖1所示為1對(duì)漸開(kāi)線(xiàn)直齒齒輪嚙合傳動(dòng)的動(dòng)力學(xué)仿真模型。在2個(gè)齒輪的幾何中心施加了旋轉(zhuǎn)運(yùn)動(dòng)副，確保2個(gè)齒輪在運(yùn)動(dòng)時(shí)繞軸心旋轉(zhuǎn)，2個(gè)齒輪的中心距等于2個(gè)齒輪分度圓直徑之和的一半，兩齒輪的輪齒之間定義了實(shí)體對(duì)實(shí)體的接觸力。齒輪材料均為45鋼，具體結(jié)構(gòu)參數(shù)如表1所示。

圖1 齒輪嚙合傳動(dòng)動(dòng)力學(xué)仿真模型Fig.1 Dynamic simulation model of gear meshing

表1 齒輪結(jié)構(gòu)參數(shù)Table 1 Structural parameters of gears

2 碰撞力計(jì)算方法

齒輪在嚙合過(guò)程中，輪齒因接觸產(chǎn)生碰撞力。對(duì)于相互接觸的2個(gè)輪齒，設(shè)輪齒間的距離為x，忽略齒輪的彈性波動(dòng)和運(yùn)動(dòng)副的間隙，當(dāng)x≥0時(shí)，兩齒輪不發(fā)生接觸，碰撞力為0 kN；當(dāng)x＜0時(shí)，輪齒發(fā)生接觸，接觸力f與輪齒的嚙合剛度、接觸變形量、非線(xiàn)性指數(shù)、阻尼系數(shù)和穿入深度有關(guān)，具體計(jì)算方法如下：

式中：p為施加在物體上的載荷。

由式(1)～(5)可知：輪齒碰撞力的計(jì)算需要確定輪齒的嚙合剛度、非線(xiàn)性指數(shù)、阻尼系數(shù)和最大阻尼時(shí)的穿入深度等參數(shù)。

3 仿真分析

仿真條件：給小齒輪施加1 r/min的轉(zhuǎn)速。齒輪的彈性模量E1=E2=2×105N/mm2，泊松比ν1=ν2=0.285，根據(jù)式(2)可得K=7.5×105N/mm2，根據(jù)文獻(xiàn)[15?17]可知，碰撞力指數(shù)e=1.5，阻尼系數(shù)C=10 N·s/mm。

圖2所示為齒輪嚙合傳動(dòng)時(shí)輪齒碰撞力的變化，共給出了5個(gè)齒輪接觸力的變化曲線(xiàn)。從圖2可以看出：輪齒碰撞力變化顯著，具有明顯的周期性。從輪齒的嚙入到嚙出，碰撞力經(jīng)歷了一個(gè)從0 kN到最大碰撞力再回到 0 kN的過(guò)程。碰撞力的變化周期大約為0.04 s，與設(shè)定的小齒輪轉(zhuǎn)速相對(duì)應(yīng)。

圖2 輪齒碰撞力隨時(shí)間的變化Fig.2 Variation of gear teeth contact force with time

圖3所示為齒輪嚙合傳動(dòng)時(shí)輪齒x向碰撞力的變化。從圖3可以看出：x向碰撞力變化顯著，波動(dòng)幅度為0～9.2 kN，碰撞力存在明顯的調(diào)制現(xiàn)象。圖4所示為x向碰撞力的頻譜。從圖4可以看出有2條幅值突出的譜線(xiàn)：一條是頻率為1 Hz的譜線(xiàn)，它對(duì)應(yīng)于小齒輪的旋轉(zhuǎn)頻率，另一條是頻率為24 Hz的譜線(xiàn)，它對(duì)應(yīng)于小齒輪的嚙合頻率。由此可見(jiàn)：x向碰撞力的載波頻率為齒輪的嚙合頻率，調(diào)制頻率為齒輪的旋轉(zhuǎn)頻率。

圖5所示為齒輪嚙合傳動(dòng)時(shí)輪齒y向碰撞力的變化。從圖5可以看出：y向碰撞力變化顯著，波動(dòng)幅度為0～9.6 kN，碰撞力存在明顯的調(diào)制現(xiàn)象。圖6所示為y向碰撞力的頻譜。從圖6可以看出有2條幅值突出的譜線(xiàn)：一條是頻率為1 Hz的譜線(xiàn)，它對(duì)應(yīng)于小齒輪的旋轉(zhuǎn)頻率；另一條是頻率為24 Hz的譜線(xiàn)，它對(duì)應(yīng)于小齒輪的嚙合頻率。對(duì)比圖3和圖5可知：x向碰撞力和y向碰撞力的幅值基本相當(dāng)，頻譜特性相同，區(qū)別在于兩者存在大約90°的相位差。

圖7所示為齒輪嚙合傳動(dòng)時(shí)齒輪碰撞力的變化。從圖 7中可以看出：碰撞力變化顯著，波動(dòng)幅度為0～9.6 kN。圖8所示為齒輪碰撞力的頻譜。從圖8可以看出：在24 Hz處有1條幅值突出的譜線(xiàn)，它對(duì)應(yīng)于齒輪對(duì)的嚙合頻率，在48 Hz處還出現(xiàn)了嚙合頻率的2倍頻。由此可見(jiàn)：齒輪嚙合傳動(dòng)時(shí)，輪齒碰撞力的頻譜特征為頻譜圖中出現(xiàn)的齒輪嚙合頻率的1倍頻和2倍頻。

圖3 齒輪x向碰撞力隨時(shí)間的變化Fig.3 Variation of gear contact force in x direction with time

圖4 齒輪x向碰撞力頻譜Fig.4 Frequency spectrum of gear contact force in x direction

圖5 齒輪y向碰撞力隨時(shí)間的變化Fig.5 Variation of gear contact force in y direction with time

圖6 齒輪y向碰撞力頻譜Fig.6 Frequency spectrum of gear contact force in y direction

圖7 齒輪碰撞力隨時(shí)間的變化Fig.7 Variation of gear contact force with time

圖8 齒輪碰撞力頻譜Fig.8 Frequency spectrum of gear contact force

4 結(jié)論

(1) 建立了齒輪嚙合傳動(dòng)動(dòng)力學(xué)模型，給出了輪齒碰撞力的計(jì)算方法，并對(duì)齒輪嚙合傳動(dòng)時(shí)的碰撞力進(jìn)行了仿真研究。

(2)齒輪嚙合傳動(dòng)時(shí)碰撞力的幅值波動(dòng)顯著，頻譜中會(huì)出現(xiàn)齒輪嚙合頻率的1倍頻和2倍頻；x向碰撞力和y向碰撞力存在明顯的調(diào)制現(xiàn)象，具有相同的頻譜特征，相位相差約90°。

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(編輯 陳愛(ài)華)

Simulation of contact force of involute gear meshing

HUANG Zhong-hua1,2, ZHANG Xiao-jian2, ZHOU Yu-jun2

(1. College of Mechanical Engineering, Hunan Institute of Engineering, Xiangtan 411101, China;2. School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China)

In order to obtain the changing pattern of contact force of involute gear meshing, a computation method based on dynamic simulation was proposed. The dynamic model of involute gear meshing was established and the computation method for contact force of gear meshing based on Hertz theory was also introduced. With the dynamic model, the contact force and the frequency spectrum of gear meshing,xdirection andydirection were simulated. The simulation results show that the amplitude of the contact force of gear meshing changes obviously. From the moment of meshing approach and meshing recess, the experience of contact force is from 0 kN to maximum and from maximum to 0 kN.Frequency multiplication of 1 and 2 of meshing frequency are in the frequency spectrum of contact force. The contact force of gear meshing ofxdirection andydirection changes obviously with explicit frequency modulation, and the frequency of rotation and frequency of meshing are in the frequency spectrum. The carrier wave is the meshing frequency of gear and the modulation wave is the rotary frequency of gear. The contact force of gear meshing ofxdirection andydirection has the same frequency spectrum and phase difference of 90°.

contact force; gear meshing; simulation

TH132.41；TP391.9

A

1672?7207(2011)02?0379?05

2009?12?07；

2010?03?05

國(guó)家自然科學(xué)基金資助項(xiàng)目(50804054)；湖南省自然科學(xué)基金資助項(xiàng)目(08JJ4011)

黃中華(1979?)，男，湖南婁底人，博士，副教授，從事機(jī)電一體化研究；電話(huà)：15616273368；E-mail：csu707@163.com