Finite Element Analysis of Thermal Deformation of Worm Gears’ Meta-action in Numerical Control Turntables

doi:10.12068/j.issn.1005-3026.2024.01.010

Abstract

Abstract:

In order to study the thermal error modeling method of the transmission system meta-action in numerical control turntable. Firstly， the meta-action theory and the temperature field model assumptions and boundary condition of finite element simulation are introduced. Then， the mechanism of the friction temperature field simulation and the parameter calculation method of the ANSYS simulation are analyzed the rotation meta-action is numerically simulated from the two aspects of steady state and transient state， and the temperature rising curve of key meta-action is given. The temperature distribution and temperature rising and thermal deformation are obtained by coupling the temperature field and deformation field of meta-action transient state, steady state and thermal-structure. The thermal error model of the transmission system is obtained by the theoretical calculation and finite element analysis of thermal deformation.

Key words: transmission system, meta-action, finite element simulation, thermal deformation, thermal error

CLC Number:

TP 301.6

Xin YANG, Zhen-gang FAN, Gen-bao ZHANG, Yan RAN. Finite Element Analysis of Thermal Deformation of Worm Gears’ Meta-action in Numerical Control Turntables[J]. Journal of Northeastern University(Natural Science), 2024, 45(1): 76-84.

Figures/Tables 19

Fig. 1 FMA decomposition mapping of meta‐action

Fig. 2 Gears' rotation meta‐action of the single tooth calculation area

Table 1 Boundary conditions of various surfaces of gears' rotation meta‐action

工作部位	表面区域	所属边界/边界条件
齿面啮合区	m区	既有摩擦热流量的输入，又有对流换热边界条件，同时符合第2、3类边界条件：
齿面啮合区	m区	$- λ ? T ? n = h m (T - T m) + q m$ .
齿顶面、齿根面及啮合面非工作区	t区	只有对流换热边界条件，属于第3类边界条件：
齿顶面、齿根面及啮合面非工作区	t区	$- λ ? T ? n = h t (T - T t)$ .
齿轮端面	s区	和齿顶面一样边界条件为对流换热.
齿轮端面	s区	$- λ ? T ? n = h s (T - T s)$ .
齿轮底面	d区	因为离啮合面较远，温度梯度变化小，可以假设为绝热表面.
齿轮底面	d区	$? T ? n = 0$ .

Table 1 Boundary conditions of various surfaces of gears' rotation meta‐action

工作部位	表面区域	所属边界/边界条件
齿面啮合区	m区	既有摩擦热流量的输入，又有对流换热边界条件，同时符合第2、3类边界条件：
齿面啮合区	m区	$- λ ? T ? n = h m (T - T m) + q m$ .
齿顶面、齿根面及啮合面非工作区	t区	只有对流换热边界条件，属于第3类边界条件：
齿顶面、齿根面及啮合面非工作区	t区	$- λ ? T ? n = h t (T - T t)$ .
齿轮端面	s区	和齿顶面一样边界条件为对流换热.
齿轮端面	s区	$- λ ? T ? n = h s (T - T s)$ .
齿轮底面	d区	因为离啮合面较远，温度梯度变化小，可以假设为绝热表面.
齿轮底面	d区	$? T ? n = 0$ .

Fig. 3 Relative sliding speed of worm gears and worms

Table 2 Simulation parameters of the

参数	蜗轮	蜗杆	齿轮轴	电机花键
$λ / W · m · K - 1$	84	46.4	32.6	32.6
$ρ / k g · m - 3$	8 760	7 830	7 850	7 850
$v i / m · s - 1$	0.184	3.78	6.75	7.005
$c / J · k g ? K - 1$	380	502.4	460	460
$β$	0.22	0.779	0.662	0.338
$p m *$	0.892	0.892	—	—
$ω / r a d · s - 1$	2	180	180	360
$E r e d / N · m m - 2$	110 000	210 000	211 000	211 000
$a / m m$	109		53.23
$f$	0.145		0.075

Table 2 Simulation parameters of the

参数	蜗轮	蜗杆	齿轮轴	电机花键
$λ / W · m · K - 1$	84	46.4	32.6	32.6
$ρ / k g · m - 3$	8 760	7 830	7 850	7 850
$v i / m · s - 1$	0.184	3.78	6.75	7.005
$c / J · k g ? K - 1$	380	502.4	460	460
$β$	0.22	0.779	0.662	0.338
$p m *$	0.892	0.892	—	—
$ω / r a d · s - 1$	2	180	180	360
$E r e d / N · m m - 2$	110 000	210 000	211 000	211 000
$a / m m$	109		53.23
$f$	0.145		0.075

Fig. 4 Gear meshing line and its calculation points

Fig. 5 Three dimensional geometric model of the platform gearing system

Fig. 6 Worm gears rotation unit output

Table 3 Numerical simulation data of the temperature field of worm gears’ rotating unit output

序号	时间/s	t_min/°C	t_max/°C	t_ave/°C
1	4	23.427	25.404	24.669
2	8	25.085	27.324	26.569
3	20	28.440	30.673	29.928
4	56	31.939	34.090	33.355
5	96	33.180	35.300	34.568
6	136	33.576	35.686	34.954
7	176	33.702	35.810	35.078
8	216	33.743	35.849	35.117
9	256	33.755	35.861	35.130
10	296	33.760	35.865	35.134
11	336	33.761	35.867	35.135
12	376	33.761	35.867	35.135
13	400	33.761	35.867	35.135

Fig. 7 Temperature rising curves of output part in worm gears' rotating meta‐action unit

Fig. 8 Thermal‐structure coupling analysis flow of the rotary meta‐action chain of turntables

Fig. 9 Meshing and local encryption of the rotary meta‐action chain model

Fig. 10 Meshing and local encryption of the rotary meta‐action chain model

Fig. 11 Equivalent elastic strain diagram and local enlargement diagram

Fig. 12 Isoeffect strain diagram and local enlargement

Fig. 13 Cloud diagram of thermal deformation in various directions of meta‐action chain

Fig. 14 Tooth thickness error of the worm wheel

Fig. 15 One‐tooth meshing error of the worm wheel

Fig. 16 Cloud diagram of thermal deformation in various directions of worm gear

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