基于布尔求差的对数螺旋锥齿轮精确建模

Accurate Modeling for Logarithmic Spiral Bevel Gear Based on Boolean Subtraction Operation

为了获得新型螺旋锥齿轮——对数螺旋锥齿轮的小齿轮的精确三维模型,提出一种新的建模方法,即基于对数螺旋锥齿轮的形成机理,建立精确的等螺旋角圆锥对数螺旋线,并作为对数螺旋锥齿轮的齿向线.对数螺旋锥齿轮大小两端的齿廓线采用精确渐开线及圆弧过渡,通过沿引导线精确扫掠的方式建立第一个齿槽.对齿槽进行阵列,用面锥与齿槽进行布尔求差的方式实现对数螺旋锥齿轮三维模型的精确建模,建模理论误差为零,实际距离误差≤0.1μm.以齿数为9,模数为4.5mm,压力角为20°,螺旋角为35°的对数螺旋锥齿轮小齿轮为例进行建模,在德玛吉DMU 40五轴联动铣床进行数控加工,证明方法的有效性和实用性.

The logarithmic spiral bevel gear is a new type of spiral bevel gear.In order to obtain the accurate three dimension model of the pinion,a new modeling method is presented in this paper.It can be described as follows.Based on the formation mechanism of logarithmic spiral bevel gear,an equal spiral angle conical logarithmic spiral line as the logarithmic spiral bevel gear is constructed.The both transverse tooth profile lines of logarithmic spiral bevel gear are constructed by precise involutes and arcs.The first tooth space is established by sweeping the precise transverse tooth profile lines along the logarithmic spiral line.All tooth spaces can be formed by arraying the first tooth space.The high precision logarithmic spiral bevel gear three dimension model can be built by Boolean subtraction operation between face cone and all tooth spaces.There is no theory error of this three dimension model while the actual distance error is less than 0.1μm.Taking apinion as an example for modeling,which the number of the gear′s teeth are 9,module is 4.5mm,the pressure angle is 20 degrees and spiral angle is 35 degrees.The numerical control processing on the DMG DMU 40 five axis NC milling machine has been done to demonstrate the validity and practicability of this new modeling method.