Influence of thermal-mechanical coupling effect on vibration of double-drive feed system

2.1 Heat source analysis of double-drive feed system

The main heat sources of the double-drive feed system are as follows: (1) motor heating power, (2) rolling guide rail and the slider friction heat, (3) ball screw and nut friction heat, (4) bearing friction heat. As is shown in Fig.2, the heat transfer model is complex due to the friction and combination of the components in a double drive feed system. In this model, the sources of heat are composed of four parts: two servo-motors, four guide rail-sliders, two screw-nuts, and four ball bearings, followed by Q₁, Q₂, Q₃ and Q₄. It is necessary to study the capacity of heat sources and the temperature, which is a vital basis for investigating the thermal force coupling mechanism of the double-drive feed system.

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Figure 2. Thermal source of double-drive feed system

2.2 Temperature field analysis theory

Generally, the finite element method and finite volume method were used to simulate the temperature field of a CNC machine tool feed system [10]. In this paper, the finite element method is used. The temperature field of double-drive feed system analysis of the finite element and the equation are:

$[{{C}_{t}}]\{\overset{.}{\mathop{T}}\,\}+[{{K}_{t}}]\{T\}=\{Q\}$          (1)

For the temperature field of the double-drive feed system, the elements of [Kt], [Ct] and {Q} are obtained by the corresponding matrix elements.

${{K}_{ij}}=\sum\limits_{e}{K_{ij}^{e}}+\sum\limits_{e}{H_{ij}^{e}}$           (2)

where

$K_{ij}^{e}=\int_{{{\Omega }^{e}}}{({{\lambda }_{x}}}\frac{\partial {{N}_{i}}}{\partial x}\frac{\partial {{N}_{j}}}{\partial x}+{{\partial }_{y}}\frac{\partial {{N}_{i}}}{\partial y}\frac{\partial {{N}_{j}}}{\partial y})d\Omega $,

$H_{ij}^{e}=\int\limits_{S_{s}^{e}}{\alpha {{N}_{i}}{{N}_{j}}dS}$

${{C}_{ij}}=\sum\limits_{e}{C_{ij}^{e}}$          (3)

where,

$Pi=\sum\limits_{e}{p_{qi}^{e}}+\sum\limits_{e}{p_{Hi}^{e}}+\sum\limits_{e}{p_{Qi}^{e}}$          (4)

where

$p_{qi}^{e}=\int\limits_{s_{2}^{e}}{{{N}_{i}}}qdS$, $p_{Hi}^{e}=\int\limits_{s_{3}^{e}}{{{N}_{\alpha }}}{{T}_{\alpha }}dS$, $p_{Qi}^{e}=\int\limits_{\Omega }{{{N}_{i}}}{{q}_{\upsilon }}dS$

Dynamic performance analysis of the thermal-mechanical coupling of the machine tool’s feed system belongs to the research field of multi-discipline integration, which is related to the structure, temperature field and vibration, so it is necessary to analyze the relationship among them. Figure 3 shows the effects of thermal-mechanical coupling for the machine tool’s feed system. In processing the feed system, the stress produced by the structure field is coupled with the temperature field generated by the heat, which affects the mechanical properties of the feed system, causing the influence of the vibration [11].

2.3 Effect of thermal-mechanical coupling on dynamic performance of feed system

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Figure 3. Vibration mechanism of thermal-mechanical coupling of the double-drive feed system

In order to facilitate the analysis, the coupling problem of the structure field, temperature field and mechanical vibration for the machine tool’s feed system is divided into three sub-modules [12], which include the dynamic performance problem, the thermal-mechanical coupling effect problem and the thermo-mechanical-coupled vibration problem and its direct or indirect coupling between the sub-modules [13] [14].

Analysis for thermal elasticity and the coupling relationship between temperature field and strain of the feed system are determined by the Eq (5):

$\vartheta _{{}}^{2}T-\frac{1}{k}T-\upsilon e=0$      (5)

where

$\vartheta =\frac{\partial _{{}}^{2}}{\partial x_{{}}^{2}}+\frac{\partial _{{}}^{2}}{\partial y_{{}}^{2}}+\frac{\partial _{{}}^{2}}{\partial z_{{}}^{2}}$

The intensity of the coupling relationship between the temperature field and the strain in the structure field is expressed in the double-drive feed system [15] [16].

$e={{\varepsilon }_{x}}+{{\varepsilon }_{y}}+{{\varepsilon }_{z}}$       (6)

Based on the theory analysis of the thermal elastic mechanics, the influence of the thermal-mechanical coupling effect is considered, and thus the relationship between the structure field and temperature field of machine tool’s feed system can be expressed by Eq (7) and Eq (8).

$\begin{align}  & \vartheta _{{}}^{4}q+\frac{\rho h}{D}\cdot \frac{\partial _{{}}^{2}q}{\partial t_{{}}^{2}}+(1+\upsilon )\vartheta _{{}}^{2}{{M}_{T}}- \\ & \frac{h}{D}\left( \frac{\partial _{{}}^{2}\varphi }{\partial y_{{}}^{2}}\cdot \frac{\partial _{{}}^{2}q}{\partial x_{{}}^{2}}-2\frac{\partial _{{}}^{2}\varphi }{{{\partial }_{x}}{{\partial }_{y}}}\cdot \frac{\partial _{{}}^{2}q}{{{\partial }_{x}}{{\partial }_{y}}}+\frac{\partial _{{}}^{2}\varphi }{\partial x_{{}}^{2}}\cdot \frac{\partial _{{}}^{2}q}{\partial y_{{}}^{2}} \right)=0 \\\end{align}$         (7)

$\vartheta _{{}}^{4}\varphi +E\alpha \vartheta _{{}}^{2}{{N}_{T}}=E\left[ \left( \frac{\partial _{{}}^{2}q}{{{\partial }_{x}}{{\partial }_{y}}} \right)_{{}}^{2}-\frac{\partial _{{}}^{2}q}{\partial x_{{}}^{2}}\cdot \frac{\partial _{{}}^{2}q}{\partial y_{{}}^{2}} \right]$      (8)

FT and MT are the additional internal force and additional torque of internal structure of the feed system caused by the increasing temperature respectively.

$\left\{ \begin{align}  & {{F}_{T}}=\frac{1}{h}\int_{-{h}/{2}\;}^{{h}/{2}\;}{T(x,y,z,t)dt} \\ & {{M}_{T}}=\frac{12}{h_{{}}^{3}}\int_{-{h}/{2}\;}^{{h}/{2}\;}{T(x,y,z,t)dt} \\\end{align} \right.$         (9)

The Eq (9) shows the coupling relationship between the temperature field and structure field. Mechanical vibration in the process of the machine tool’s feed system is as follows:

1) Temperature field results: the temperature will affect the structure deformation, stress and the vibration amplitude of the feed system.

2) Structural field results: Deformation of feed system’s structure will lead to a change of the temperature field and temperature load, and the vibration mode shapes are also affected. They can be obtained the thermal-mechanical coupling effect between the temperature field and structure field. Herein, the influence of the thermal-mechanical coupling effect must be considered in the dynamic performance of the machine tool’s feed system.

4.1 Steady analysis of thermal-mechanical coupling vibration of feed system

Table 4 shows that the natural frequency of feed system is reduced with the heat impact of the screw. With the thermal deformation of the screw, the tensile force is reduced, but the stiffness of the screw and the frequency are decreased.

Table 4. Comparison of 1 to 6 Frequency

The dynamic experimental amplitude of the feed system under the initial condition (a1, b1, c1) and thermal equilibrium (a2, b2, c2) are shown in Figure.6. Compared with the results of the finite element calculation and experiments, the errors are not great, so the accuracy of the analysis is reasonable. It can also be clearly seen that the amplitude of the thermal equilibrium is significantly higher than that of the initial state in the x, y, z directions for the feed system.

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Figure 6. Amplitude comparison between static and thermal balance of double-drive feed system

4.2 Transient analysis of the thermal coupling vibration of the feed system

As shown in Figure 7, the transient analysis of the feed system coupled with the thermal-mechanical vibration includes the formulation of transient analysis time, the simulation of the transient process and the result of thermal mechanical coupling. Four coupling steps as shown in Table 5 have been selected for dynamic analysis of transient thermal coupling, and the results are shown in Figure 8.

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Figure 7. Highest temperature of double-drive feed system rising process

Table 5. Thermal-mechanical coupling time steps

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Figure 8. Transient dynamic amplitude contrast

As shown in Figure. 8, the vibration amplitude increased with the increase in temperature. When the frequency is higher, this phenomenon is more obvious. The heat source is mainly caused by the ball screw rotation of the bearing before the thermal balance. The above analysis results show that before the double-drive feed system reaches a thermal steady state, as its thermal energy continually transforms into mechanical energy, the vibration amplitude increases, with the maximum rate of increase up to 14.06%. As a consequence, the effect of thermal-mechanical coupling must be considered in the process of optimizing the structure of the feed system.

This project is supported by the natural science foundation of Jiangsu Province (BK20141400), and Key Special Project of Numerical Control Machine Tool (2013ZX04002-001 and 2015ZX04014021).