1 Introduction

With the development of high-speed and high-precision machine tools, preloaded ball screw feed drive systems are widely used [1, 2], which can increase the axial rigidity, modify the nature frequency in order to avoid the vibration, and improve the manufacturing quality [3, 4]. Hence, the dynamic characteristics of high-speed ball screw systems have attracted the attention of many researchers [5,6,7,8,9,10,11,12,13,14,15,16,17,18]. Erkorkmaz et al. [8] proposed a new method of identifying the dynamic parameters and the friction characteristics of feed drive systems used in machine tools. Jerzy et al. [9] investigated a resonance frequency of ball screw feed drive systems of CNC machine tools. Li et al. [10] presented a modal identification method to investigate the dynamics of machine tool feed drive systems under different feed rates. Feng et al. [11] proposed a lumped dynamic model of the screw system to study preload influences on the dynamic response. Zhou et al. [12] found an accurate relationship between the preload and no-load drag torque. Frey et al. [13] studied the behavior of ball screw feed drives and its dominant influence. Frey et al. [14] found that the preload is one of main parameters of the working performance of ball screws. Vicente et al. [15] proposed a new high-frequency dynamic model of ball screw systems, using this model, the frequency variation of each mode of the systems was investigated under different carriage positions and masses. Wei et al. [16] proposed the kinematics model of the preloaded single-nut high-speed screw. Varanasi et al. [17] proposed a new dynamics model of screw systems that accounts for the screw inertia, compliance, and damping of the main elements. Kamalzadeh et al. [18] proposed a precision control method for ball screw systems and axial vibrations were compensated using the method.

However, preloaded high-speed screw systems generate excessive friction heat; the thermal effects were widely investigated by previous literatures [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. On one hand, with the increase of feed rates, the positioning precision is affected by the thermally induced characteristics [19,20,21,22]. In the previous researches, the thermal error accounts for about 40–70% of the whole error of machine tools [23,24,25,26,27]. The methods of reducing thermal error in machine tools are usually the error control and error compensation [22,23,24,25,26,27,28,29,30,31,32,33,34,35]. On the other hand, due to the thermally induced deformation of the screw, the initial preload of the system is changed, which modify the dynamic characteristics, such as the axial rigidity and natural frequency. The preload variation with temperature rise of the double nut preloaded ball screw was analyzed by experiments [36]. A finite element model was proposed to determine the remarkable influence of the preload of a bearing on thermal stabilization of ball screw systems [37]. In literature 38, the remarkable decrease of the preload in preloaded ball screw systems was found by experiments. When the temperature rises by 12–16 K, the preload decreases by 32%.

In summary, many researches focused on the modeling of the thermo-mechanical behavior of ball screw systems. However, most of these researches were carried out to predict and compensate thermal errors [22,23,24,25,26,27,28,29,30,31,32,33,34,35]. Though some of these researchers studied the thermal-preload-mechanical dynamic characteristics by experiments and FEM [33, 36,37,38], the studies did not consider real-time characteristics of preload ball screw systems with temperature rise.

To better investigate the thermally induced dynamic behavior of the preload ball screw systems and further control the thermal-dynamic behavior, a real-time dynamic model of the preload ball screw systems is necessary, which, however, has been rarely studied by now.

In this paper, the real-time thermal-dynamic model to predict the variation of the preload, stiffness, and natural frequency of preloaded ball screw systems due to temperature increase was proposed by using a heat transfer method. The solution of this model was validated by experiments. It provided the theoretical and practical foundation for real-time monitoring and controlling of dynamic characteristics of preloaded ball screw systems.

2 Thermal dynamic model

The main heat sources in ball screw feed drive systems used for CNC machine tools include the two supporting bearings and moving nut, and the heat rates result from the friction among the components, which are dependent on the feed rate, assembly conditions, and lubricant conditions. Although the system working under the same feed rate, assembly conditions, and lubricant conditions, the thermal deformation also changes the coherence among the moving nut, the screw, and the supporting bearings.

The thermally induced axial deformation of the screw affects the supporting system stiffness and the friction torque, which changes the dynamic performance of the ball screw system [20].

2.1 Stiffness model of the supporting system

The stiffness of the supporting system of the ball screw feed drive system is composed of the stiffness of the supporting bearings and the stiffness of the screw.

2.1.1 Stiffness of the support bearing

When the ball screw feed drive system is fixed by one end, the stiffness of the support bearing is given by [21]

$$ {K}_{\mathrm{b}}=\frac{3{F}_{\mathrm{b}0}}{\delta_{\mathrm{b}0}} $$
(1)

The axial deformation of the supporting bearing δb0 in Eq. 1 can be expressed as

$$ {\updelta}_{\mathrm{b}0}=\frac{0.45}{{\mathrm{sin}\upbeta}_{\mathrm{b}}}{\left(\frac{{\mathrm{Q}}^2}{{\mathrm{d}}_{\mathrm{b}}}\right)}^{1/3} $$
(2)

where Q is expressed as

$$ \mathrm{Q}=\frac{{\mathrm{F}}_{\mathrm{b}0}}{\mathrm{nsin}{\upbeta}_{\mathrm{b}}} $$
(3)

2.1.2 Stiffness of the screw shaft

One end of the ball screw system is fixed, and the other end is free. According to material mechanics, the elongation of the screw caused by the temperature arising makes the tensile preload of the supporting bearing weak. The thermal deformation of the screw changes the coherence of bearings [20], and the tensile preload variation influences the screw stiffness.

According to material mechanics, the lessened tensile preload ΔF0 due to the temperature increasing is defined as

$$ {\Delta F}_0={\mathrm{S}}_{\mathrm{s}}\mathrm{E}\Delta \mathrm{L} $$
(4)

where ΔL denotes the thermal elongation value of the screw shaft.

In order to obtain the thermal elongation value ΔL of the screw shaft, the determination of temperature arising of the screw shaft is necessary.

Real-time modeling of the temperature arising of the screw shaft

In this paper, a new real-time thermal model of multiple varying and moving heat sources was proposed and used for the screw shaft.

Figure 1 shows the ball screw feed drive system is fastened at the left end, and it is stretched at the other end.

Fig. 1
figure 1

Heat transfer of the screw

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The expression used for calculating heat transfer of the screw can be described as

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial^2T\left(x,t\right)}{\partial {x}^2}-\frac{\rho c}{k}\frac{\partial T\left(x,t\right)}{\partial t}+\frac{Ph}{k{A}_c}\left(T\left(x,t\right)-{T}_{\mathrm{air}}(t)\right)=0x\in \left[0,L\right]\\ {}T\left(x,0\right)={T}_{\mathrm{air}}(0)x\in \left[0,L\right]\\ {}T\left(0,t\right)={T}_{cb1}\left({t}_j\right)\\ {}T\left(L,t\right)={T}_{cb2}\left({t}_j\right)\\ {}T\left(\mathrm{x},t\right)={T}_{cn}\left({x}_i,{t}_j\right)x\in \left[0,S\right]\end{array}} $$
(5)

The three key point temperatures Tcb1(tj), Tcb2(tj), and Tcn(tj) of the ball screw connecting with the bearing 1, bearing 2, and nut are important for investigating thermal deformation, preloads, and thermal characteristics. However, these key point temperatures of the system are difficult to be measured, for the temperature sensors could not be put into the prescribed position.

In order to obtain the key point temperatures Tcb1(tj), Tcb2(tj), and Tcn(tj) of the ball screw connecting with the bearing 1, bearing 2, and nut, the model of the temperature arising of the bearing and nut is required.

In this paper, the bearing and nut are simplified as hollow cylinders, and a new real-time temperature model of hollow cylinders is proposed. This model was used for obtaining the key point temperatures Tcb1(tj), Tcb2(tj), and Tcn(tj) of the ball screw contacting with the bearings and moving-nut.

Modeling of real-time temperature field for a hollow cylinder

In order to establish a real-time temperature model of the supporting bearings and moving-nut, the assumptions were as follows [5, 7]:

  1. 1.

    The bearings and screw-nut are treated as hollow cylinders.

  2. 2.

    The friction heat generated by the bearings and screw-nut is uniform.

In this work, a real-time temperature distribution model of a hollow cylinder is proposed according to the heat transfer theory.

Figure 2 shows that the bearings and moving-nut are simplified to be the hollow cylinders with a uniform heat generation rate.

Fig. 2
figure 2

Heat transfer of the hollow cylinder

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With regard to the convection heat dissipation of the cylinder surfaces, the convection heat loss is treated as a virtual negative heat source.

The transient heat transfer equation for the hollow cylinder with a negative heat source can be expressed as

$$ \frac{1}{r}\frac{\partial }{\partial r}\left( kr\frac{\partial T\left(r,t\right)}{\partial r}\right)=\rho c\frac{\partial T\left(r,t\right)}{\partial t}+\dot{\phi} $$
(6)

where \( \dot{\phi} \) denotes the heat flow of the negative heat source of the cylinder surface convection.

The total heat for the surface convectionϕscan be written as

$$ {\phi}_s={P}_c dr{h}_s\left(T-{T}_f\right) $$
(7)

where Pc is the circumference of the heat transfer section.

Considering the convection heat transfer of the up, down, and top surfaces, the circumference of the heat transfer section is written as

$$ {P}_c=2\left(2\pi r+\frac{W}{2}\right) $$
(8)

The negative heat source \( \dot{\phi} \)can be rewritten as

$$ \dot{\phi}=-\frac{\phi_s}{A_c dr}=-\frac{h_s{P}_c\left(T-{T}_f\right)}{A_c} $$
(9)

where Ac is the area of the heat transfer section.

The area of the heat transfer section Ac is expressed as

$$ {A}_c=2\pi rW $$
(10)

where W denotes the width of the hollow cylinder.

Substituting Eq. 9 into Eq. 6, the temperature model of the hollow cylinder can be rewritten as

$$ \frac{1}{r}\frac{\partial }{\partial r}\left( kr\frac{\partial T\left(r,t\right)}{\partial r}\right)=\rho c\frac{\partial T\left(r,t\right)}{\partial t}-\frac{P_c{h}_s}{A_c}\left(T-{T}_f(t)\right) $$
(11)

where T(r, t) is the temperature distribution of the hollow cylinder, which is the function of time t and the radius r, and hs is the surface convection coefficient of the hollow cylinder.

According to the variable separation theory, the temperature distribution of the hollow cylinder T(r, t) is written as

$$ T\left(r,t\right)={T}_r(r)\cdot {T}_t(t) $$
(12)

Temperature of a hollow cylinder always shows a logarithm decrease with the radius [39]. In this paper, the function Tr(r) of the position can be written as

$$ {T}_r(r)=\frac{1}{r\ln \left({R}_{out}/{R}_{in}\right)} $$
(13)

The temperature of the cylinder always shows an exponential increase with time [39, 40]. In this work, the function Tt(t) of the time is expressed as

$$ {T}_{\mathrm{t}}(t)=A\left(1-{e}^{-\frac{t}{B}}\right) $$
(14)

where A and B are constants.

Therefore, the transient temperature distribution of the hollow cylinder T(r, t) can be rewritten as

$$ T\left(r,t\right)=A\frac{1}{r\ln \left({R}_{out}/{R}_{in}\right)}\left(1-{e}^{-\frac{t}{B}}\right) $$
(15)

The \( \frac{\partial T}{\partial t} \) expression can be derived as

$$ \frac{\partial T}{\partial t}=\frac{A{e}^{-\frac{t}{B}}}{B}\frac{1}{r\ln \left({R}_{out}/{R}_{in}\right)} $$
(16)

The \( \frac{1}{r}\frac{\partial }{\partial r}\left( kr\frac{\partial T\left(r,t\right)}{\partial r}\right) \)expression can be derived as

$$ \frac{1}{r}\frac{\partial }{\partial r}\left( kr\frac{\partial T\left(r,t\right)}{\partial r}\right)=\frac{kA\left(1-{e}^{-\frac{t}{B}}\right)}{r^3\ln \left({R}_{out}/{R}_{in}\right)} $$
(17)

Substituting Eqs. 16 and 17 into Eq. 11, Eq. 11 can be rewritten as

$$ \frac{kA\left(1-{e}^{-\frac{t}{B}}\right)}{r^3\ln \left({R}_{out}/{R}_{in}\right)}-\rho c\frac{A{e}^{-\frac{t}{B}}}{B}\frac{1}{r\ln \left({R}_{out}/{R}_{in}\right)}+\frac{P_c{h}_s\left(T-{T}_f\right)}{A_c}=0 $$
(18)

The analytical solution of Eq. 18 is

$$ T=\frac{\rho cA{A}_c{r}^2{\mathrm{e}}^{-\frac{t}{B}}+ AB{A}_ck{\mathrm{e}}^{-\frac{t}{B}}- AB{A}_ck}{B{h}_s{P}_c{r}^3\ln \left({R}_{out}/{R}_{in}\right)}+{T}_f $$
(19)

Equation 19 is adaptive to not only the bearing temperature field but also a nut temperature field.

The real-time surface temperature T(Rout, ti) used for the boundary can be obtained by the thermocouple. The thermocouples on the bearing seat surface and the moving-nut flank surface can be used to collect the real-time temperature and time information. Two sets of data from the measured point of the heat source surface are enough to solve Eq. 19, and then A and B are obtained.

Using the thermocouples to read the surface temperature T(Rout, ti) of the heat source, this real-time model can be used to obtain the real-time temperature Tcb1(tj), Tcb2(tj), and Tcn(tj) of the screw contact surface at the position with the radius Rin of the bearings and nut.

The T(Rout, ti) is the real-time temperature for the point measured from the surface of the kinematics pair collected by thermocouples. The Tcb1(tj), Tcb2(tj), and Tcn(tj) are the contacting surface temperatures between the screw shaft and the kinematics pairs.

Hence, this modeling of real-time temperature field for a hollow cylinder can be used to obtain the Tcb1(tj), Tcb2(tj), and Tcn(tj)of the contacting surface temperature between the screw shaft and the bearings and nut.

Numerical solution method of the real-time modeling of the temperature arising of the screw shaft

When the contact surface temperatures are obtained, Eq. 5 can be used to calculate the temperature distribution of the screw shaft.

Equation 5 can be solved by the finite difference method. Partition the setting region into mesh by step size of space s and step size of time t. xi = i ⋅ s, tj = j ⋅ τ. Node (xi, tj) is simplified to (i, j). i[1, M − 1], j[1, N − 1], M, and N are positive integers, and \( {T}_j^i=T\left(i,j\right) \).

The following iterative form of solving Eq. 5 is rewritten as

$$ \frac{1}{s^2}{T}_{j+1}^{k+1}-\left(\frac{2}{s^2}+\frac{\rho c}{k}\frac{1}{\tau}\right){T}_j^{k+1}+\frac{1}{s^2}{T}_{j-1}^{k+1}=-\frac{\rho c}{k}\frac{1}{\tau }{T}_j^k+\frac{Ph}{k{A}_c}{T}_j^k-\frac{Ph}{k{A}_c}{T}_{air} $$
(20)

The forced convective coefficient h is rewritten as [40]

$$ h=0.133{\left(\frac{\pi n}{60\upsilon}\right)}^{\frac{2}{3}}{d}_s^{\frac{1}{3}}{{\mathrm{P}}_r}^{\frac{1}{3}} $$
(21)

Determining thermal elongation of the screw shaft

During the feed working, the axial thermal elongation of a screw shaft is always produced, for the temperature of the screw caused by frictional heat increases.

Hence, based on the result of real-time modeling of the temperature arising of the screw shaft, the expression of thermal elongation of the screw shaft can be obtained as:

$$ \varDelta L={\alpha}_s{\int}_0^L\varDelta T\left(x,t\right) dx $$
(22)

where ΔT(x, t) denotes the temperature rise of the screw shaft, and αs denotes the thermal expansion coefficient.

The axial deformation caused by the friction heat mainly affects the preloads and of the stiffness of the systems.

Calculating the rest tensile preload

Because of the thermal elongation of the screw shaft, the rest tensile preload is described as,

$$ {F}_{ap}={F}_0-\varDelta {F}_0 $$
(23)

Determining the axial stiffness of the circle bar

Considering that the tensile preload variation influences the screw stiffness, the axial stiffness of the circle bar is defined as

$$ {k}_s=\frac{P{S}_sE}{\left(P-{F}_{ap}\right)x} $$
(24)

where P is the external axial load.

2.1.3 Stiffness of the supporting system

The stiffness of the screw supporting system consists of the stiffness of support bearing and the axial stiffness of the screw shaft, which are connected in series. Hence, the stiffness of the supporting system is written as

$$ {k}_e=\frac{k_s{k}_b}{k_s+{k}_b} $$
(25)

2.2 Screw-nut stiffness model

The axial contact stiffness between the ball screw shaft and the nut is given by [41]

$$ {k}_{\mathrm{n}}=0.8{\left(\frac{F_{\mathrm{n}0}}{\upvarepsilon {C}_{\mathrm{a}}}\right)}^{1/3}K $$
(26)

2.3 Ball screw system governing equation

In order to investigate the dynamic characteristics of the ball screw system varying with the different increment of temperature, the entire system is modeled, as shown in Fig. 3. The system is composed of the screw shaft, working table, and supporting bearings.

Fig. 3
figure 3

Modeling of the ball screw system with a lumped parameter system

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To consider the effects of the different increment of temperature, the supporting system stiffness and the screw-nut stiffness are treated as variables.

The matrix[M] is written as

$$ \left[\mathrm{M}\right]=\left[\begin{array}{c}{\mathrm{M}}_{\mathrm{w}}\kern0.75em 0\kern0.5em \\ {}0\kern0.75em {\mathrm{M}}_{\mathrm{s}}\end{array}\right] $$
(27)

The matrix[C] is written as

$$ \left[\mathrm{C}\right]=\left[\begin{array}{c}{\mathrm{C}}_{\mathrm{w}}\kern0.75em 0\kern0.75em \\ {}0\kern0.75em {\mathrm{C}}_{\mathrm{s}}\end{array}\right] $$
(28)

The matrix[K] is written as

$$ \left[\mathrm{K}\right]=\left[\begin{array}{c}{k}_{\mathrm{n}}\kern2.5em -{k}_{\mathrm{n}}\\ {}-{k}_{\mathrm{n}}\kern1.25em {k}_{\mathrm{n}}+{k}_{\mathrm{e}}\end{array}\right] $$
(29)

The vector X is written as

$$ \mathrm{X}=\left[\begin{array}{c}{\mathrm{x}}_{\mathrm{w}}\\ {}{\mathrm{x}}_{\mathrm{s}}\end{array}\right] $$
(30)

The vector F is written as

$$ \mathrm{F}=\left[\begin{array}{c}0\\ {}0\end{array}\right] $$
(31)

The matrix [M], [C],  and [K] are the system mass, the system damping, and the system stiffness matrix, respectively. [F] is the excitation force vector, and [X] is the displacement vector.

Combining Eqs. 27–31, the system governing equation can be presented as the matrix form

$$ \left[\mathrm{M}\right]\left\{\ddot{\mathrm{X}}\right\}+\left[\mathrm{C}\right]\left\{\dot{\mathrm{X}}\right\}+\left[\mathrm{K}\right]\left\{\mathrm{X}\right\}=\left\{\mathrm{F}\left(\mathrm{t}\right)\right\} $$
(32)

2.4 Solving dynamic model

This model belongs to a SDOF system. The parameters of the feed drive system were listed in Table 1. Through taking the Laplace transform of all the equations and summarizing the results in matrix form, the dynamic characteristics were obtained.

Table 1 Parameters of the feed drive system
Full size table

2.4.1 Axial natural frequency of the working table

The axial natural frequency of the working table is obtained as

$$ {\displaystyle \begin{array}{l}{\omega}_W=\frac{1}{2{M}_1{M}_2}\left\{\sqrt[]{2}\right[{M}_1{M}_2\Big({k}_e{M}_1+{k}_n{M}_1+{k}_n{M}_2+\\ {}\sqrt{k_e^2{M}_1^2+2{k}_e{k}_n{M}_1^2-2{k}_e{k}_n{M}_1{M}_2+{k}_n^2{M}_1^2+2{k}_n^2{M}_1{M}_2+{k}_n^2{M}_2^2}\left)\right]{}^{1/2}\Big\}\end{array}} $$
(33)

2.4.2 Axial natural frequency of the screw

The axial natural frequency of the screw is obtained as

$$ {\displaystyle \begin{array}{l}{\omega}_s=\frac{1}{2{M}_1{M}_2}\left\{\sqrt[]{2}\right[{M}_1{M}_2\Big({k}_e{M}_1+{k}_n{M}_1+{k}_n{M}_2-\\ {}\sqrt{k_e^2{M}_1^2+2{k}_e{k}_n{M}_1^2-2{k}_e{k}_n{M}_1{M}_2+{k}_n^2{M}_1^2+2{k}_n^2{M}_1{M}_2+{k}_n^2{M}_2^2}\left)\right]{}^{1/2}\Big\}\end{array}} $$
(34)

3 Experimental verification and dynamics simulation for ball screw feed drives

3.1 Experimental verification

3.1.1 Experimental setup

In order to investigate the thermal characteristics of the ball screw feed drive system, experiments were performed on CNC machine tools; the z-axis feed system was tested under different feed rates. Figure 4 shows the experimental setup of testing the screw system.

Fig. 4
figure 4

Experimental setup

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The experimental setup of the lathe consists of an infrared thermal imaging instrument, a temperature collecting instrument with four thermocouples, a computer with the Visual C++ FOCAS application, a computer with LAMS software, and a CNC lathe.

3.1.2 Sensor locations

In order to investigate the temperature characteristics of the system, experiments were carried out using the measuring location arrangement shown in Fig. 5. The 14 testing points of temperature were shown in Fig. 5. The 1st, 13th, and 14th testing points were respectively located on the surfaces of bearing seat 1, the surfaces of bearing seat 2, and the flank surface of the nut, and the three points were measured by three thermocouples, respectively. The testing points (2–12) of temperature on the screw shaft (shown in Fig. 5) is evenly taken every 42 mm from the reference origin point of the machine tool up to 420 mm. The ball screw system is warmed up by the repeated movement of the nut along the screw shaft, between points 2 and 12. To measure the measuring point temperature in the stroke of the ball screw shaft, an infrared thermal imaging instrument was selected.

Fig. 5
figure 5

Measuring location arrangement

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3.1.3 Testing procedure

The z-axis feed drive of a lathe was tested using no load. First, the CNC lathe was initially begun, and the initial characteristics were collected. After this, by moving the worktable repeatedly in its stroke range, the ball screw feed system of the CNC lathe was heated. Meanwhile, the temperature of the bearings, and moving-nut was collected using the thermocouples at an interval of 0.048 s. Temperature of the screw shaft was taken using a thermal imaging instrument at an interval of 10 min. After the lathe was warmed up for 10, 20, 30, 40, and 50 min, the natural frequencies of the carriage were measured using LAMS. To obtain the position of the moving-nut, an application program based on the FOCAS was used to collect the real-time position of the carriage in sampling intervals of 0.048 s.

3.2 Inverse identification method of the heat excitation

In order to study the thermal dynamic performance deeply, this paper proposed an inverse identification method of the heat excitation, as shown in Fig. 6. To obtain the heat flow rates of the fixed bearing 1, the stretched bearing 2, and the moving-nut, the Monte Carlo method is used. Once the MC process is completed, the FEM process is preformed to obtain the temperature field of the ball system. In order to capture the three heat flow rates, experiments were performed to obtain the temperature rise of 14 designated points, which are denoted by M1, M2, ⋯, M14 in Fig. 5. The method to obtain the heat flow rates of three heat sources is that the temperature rises of the FEM calculation agrees with the data measured by experiments. The calculating process can search the heat flow rates, through minimizing the following expression:

$$ F\left({x}_1,{x}_2,{x}_3\right)=\sum \limits_i\sum \limits_j{\left(\Delta {M}_{ij}^{\mathrm{EM}}-\Delta {M}_{ij}^{\mathrm{MC}}\right)}^2 $$
(35)
Fig. 6
figure 6

Thermal dynamic identification model of the ball screw feed drive system

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where \( \Delta {M}_{ij}^{\mathrm{EM}} \) denotes the measured temperature for given point i at the jth sampling time step of tj, i = 1, 2, ⋯, 14, and j = 1, 2, ⋯, N, \( \Delta {M}_{ij}^{\mathrm{MC}} \) denotes the simulated temperature for given point i at the jth sampling time step of tj, i = 1, 2, ⋯, 14, and j = 1, 2, ⋯, N. Figure 7 shows the flow diagram of the thermal dynamic identification model of the ball screw feed drive system.

Fig. 7
figure 7

Flow diagram of the thermal dynamic identification model of the ball screw feed drive system

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The thermal dynamic identification process of the ball screw feed drive system is as follows:

  1. 1.

    Generating the thermal dynamic finite element model of the system according to the geometric dimension of the ball screw feed drive system.

  2. 2.

    Inputting the initial temperature, convention coefficient, and the thermal flux regions of the fixed bearing, stretched bearing, and nut xh ∈ [xLh, xUh], δh = xUh − xLh, h = 1, 2, 3.

  3. 3.

    Using the Monte Carlo method, the friction heat fluxes xh = xLi + random(0, δh), h = 1, 2, 3. are generated.

  4. 4.

    Executing the simulation of the model and collect the 14 key temperatures of the simulation.

  5. 5.

    Calculating the objective value of the expression (35): F and search the minimum objective value of the expression (35): F*

  6. 6.

    If F* is less than the setting small value Ɛ, the friction heat fluxes are obtained. If F* is bigger than the setting small value Ɛ, the regions of the friction heat fluxes are updated, and repeated the MC process until F* is less than the setting small value Ɛ.

  7. 7.

    Extracting the temperatures of elements.

  8. 8.

    Executing thermal-force simulation and extract the tensile preloads of the screw.

  9. 9.

    Executing thermal-modal simulation and extract the natural frequencies of the system.

4 Results and discuss

In order to verify the thermal model, the tests of machine tools were conducted at Z axis positions, under the different increment of temperature of the screw using the feed rate 5 m/min.

4.1 Temperature distribution of the screw system

In order to investigate the thermal dynamic performance of ball screws, the temperature distribution of the screw system is necessary.

The comparison of key point temperatures of the thermal dynamic model, experiments, and inverse identification by experiments of the ball screw feed drive system was shown in Fig. 8.

Fig. 8
figure 8

Comparison of key point temperatures of the thermal dynamic model, experiments, and inverse identification by experiments of the ball screw feed drive system

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The point line denotes the measured temperature of testing points of the bearings and nut, the solid line denotes temperature from the inverse identification of the testing points and the centers of the bearings and nut, and the dash line denotes the temperature obtained by the thermal dynamic model of the centers of the bearings and nut. The results shows that the temperature rise of the inverse identification by experiments agrees with the data of the experiments, and the temperature rise of the thermal dynamic model agrees with the data of the inverse identification by experiments under the feed rate 5 m/min. The effectiveness of the proposed thermal dynamic model was proved through the above results.

4.2 Preload variation of the screw due to temperature increase

The thermally induced axial deformation of the screw certainly affects the preload, which changes the dynamic performance of the ball screw system.

Figure 9 shows that comparison of tensile preloads of the dynamic model and inverse identification method. From the results, it is can be seen that the tensile preload of system sharply decreases with the increase of temperature. The calculated results of the model could match quite well with the inverse identification results. During ball screw systems working, due to the increase of temperature, the thermal deformation increases, which causes the decrease of the tensile preload of the screw shaft.

Fig. 9
figure 9

Comparison of tensile preloads of the dynamic model and inverse identification

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4.3 Variation of the stiffness and natural frequency due to temperature increase

The thermally induced axial deformation of the screw certainly affects the supporting system stiffness and the natural frequency, which changes the dynamic performance of the ball screw system.

According to the dynamic model, the axial supporting stiffness of the working table at Z axis position of 336 mm is obtained under the feed rate 5 m/min. Figure 10 shows that the axial supporting stiffness of the working table nonlinearly decreases with the increase of temperature. This is because the decrease of the tensile preload of the screw shaft causes the system axial stiffness nonlinearly changes.

Fig. 10
figure 10

Axial supporting stiffness of the screw system sharply decreases at the position of 336 mm (obtained by the thermal dynamic model)

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Using the above solutions Eqs. 33 and 34 of the dynamic model, the dynamic characteristics varying with the temperature of the screw system was exposed. And the natural frequencies of the working table at Z axis position of 336 mm was measured under the feed rate 5 m/min.

Figure 11 shows the comparison of natural frequencies of the dynamic model, inverse identification, and experiments. From the results, it can be seen that the natural frequency of system slowly decreases. The calculated results of the model could match quite well with the actual measured results and inverse identification.

Fig. 11
figure 11

Comparison of the natural frequencies of working table for the dynamic model, inverse identification, and experiments at the position of 336 mm

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This is because in the working stage, the tensile preload of system sharply decreases, the axial stiffness of system sharply decreases, and the natural frequency of system slowly decreases.

The experimental results and analytical solution results showed the effective of the proposed dynamic model. The proposed dynamic model has been validated in the testing machine tool.

5 Conclusions

Based on the analysis of the preloads and stiffness of the feed drive system varying with temperature rise, a new lumped, thermally induced dynamic model has been successfully presented for studying the dynamic performance due to the temperature variation of the ball screw feed drive system.

  1. 1.

    A new analytical model of the real-time temperature field of a hollow cylinder with the varied heat flow was proposed, based on the heat theory. By using the thermocouples to read the surface temperatures of heat sources, this model can be used to obtain the real-time temperature field of the bearings and moving-nut.

  2. 2.

    A new finite difference heat transfer model with real-time moving heat sources was proposed. By using the FOCAS function to read the real-time position of moving heat sources, this model can be used to obtain the temperature distribution of the screw. Based on this model, the real-time thermally induced preloads and stiffness models of the ball screw system were obtained. Furthermore, an analytical real-time thermally induced dynamic model of the ball screw system has been successfully presented, and the analytical solutions of this model were obtained.

  3. 3.

    An inverse identification method of the heat excitation was proposed. Using this method and experiments, the relationships between the preload, axial stiffness, and natural frequency and the temperature variation of the ball screw system were investigated. The extension preload and the axial stiffness markedly decrease with the increase of temperature. However, the natural frequency slowly decreases with the increase of temperature. The results proved that the effectiveness of the presented thermal dynamic models. These models provided the theoretical and practical foundation of the real-time monitoring and controlling of dynamic characteristics of preloaded ball screw systems. These methods focus on real-time thermal dynamic characteristics of preloaded ball screw systems, which can be used for other machines by suitable modification.