1 Introduction

Because of its simplicity and stability in assembly and manufacturing, the bolted joint is an important mechanical connection used in the crucial equipment of multiple industrial areas, including aerospace, automobiles, energy, and shipping [1, 2]. The contact stiffness of the bolted joint is an important performance of machine tools, representing the safety and reliability of the fixed joint [3,4,5]. Considering the complexity and nonlinearity resulting from factors such as material properties, contact pressure, and surface roughness, a simplified contact stiffness model of the bolted joint is required at the initial conceptual design stage to accurately and efficiently predict the characteristics of the fixed joint in machine tools.

The available methods for contact stiffness modeling of bolted joints can be roughly categorized into three types, i.e., parameter identification method (PIM) [6,7,8,9,10,11,12,13], fractal theory (FT) [14,15,16,17,18,19,20,21,22,23,24,25,26], and equivalent material method (EMM) [27,28,29,30,31,32,33,34]. The contact elasticity of bolted joints in the PIM is treated as a system comprising several springs and damping, which are adjusted iteratively from the frequency response function (FRF) comparison studies of modal analysis tests and identified parameters. For instance, Mao et al. proposed a general dynamic model of fixed joints considering the coupling effects between substructures and used the model to identify the stiffness parameters of fixed joints according to the inverse relationship between the dynamic stiffness matrix and FRF matrix [6]. The maximum error in the proposed model relative to the experimental results was < 10%. Based on the established identification model, they illustrated that the contact stiffnesses of bolted joints under different material combinations were proportional to the equivalent elastic modulus, consistent with the Hertz contact theory [8]. The equivalent stiffness and Valanis parameters of a metric M12 bolted joint were determined from quasistatic experiments [9]. Li et al. proposed an improved nonlinear approach for the microslip modeling of bolted joints considering residual shear deformation and identified the parameters in the hysteresis loop through quasistatic experiments [13]. According to the experimental results, the identified stiffness parameters in the PIM are relatively accurate. However, this method is typically appropriate for model verification or database generation under different conditions owing to repeatedly time-consuming measurements.

To investigate the influence of various factors on the contact stiffness of rough surfaces, a simplified theoretical calculation model is required. During the development of microscopy technology, scholars discovered that metal surfaces exhibit statistical self-similarity and self-affinity at different scales. Thus, the FT was proposed [14, 15]. In FT, the characteristics of bolted joints are described from a single asperity and then extended to the entire joint surface according to the distribution of asperities, including the Greenwood and Williamson (GW) model, Persson’s model, and Majumdar and Bhushan (MB) model. To calculate the normal contact stiffness accurately and effectively, Liu et al. proposed an improved fractal model by introducing a specific coefficient related to friction, which helped reveal the influence of friction on the dynamic performance of the contact area [16]. Considering the differences in deformation regimes, Xiao et al. formulated a novel expression for normal contact stiffness between rough surfaces [20]. The deformation modes of a single asperity were determined as fully elastic, elastoplastic, and fully plastic, which changed smoothly at critical points between different deformation regimes. Subsequently, Zhang et al. introduced a novel parameter associated with the contact angle between the upper and lower asperities of rough surfaces, establishing a normal contact stiffness model considering the oblique contact mode [24]. Experiments demonstrated that the modified model was more accurate and appropriate than the previous models in predicting the contact stiffness of surfaces with larger fractal dimensions. Considering the inclination angle, Liu et al. proposed normal and tangential contact stiffness models of the bolted joint surface of a heavy-duty machine tool [25]. Combining the contact mechanism at macro- and microscales, Chang et al. established a hybrid contact stiffness model for bolted joints [26]. By replacing M10 bolts with M8 bolts, a 2.283% reduction was achieved in the dynamic response of the workpiece in a machine tool by optimizing the bolt arrangement. Although the FT provides valuable information concerning the contact characteristics on a microscale, it neglects the stress distribution on a rough surface.

The joint surface in the EMM is treated as a virtual material layer whose material properties are obtained from the above theoretical calculation model. This approach is advantageous because it can be easily integrated into an FEA (finite element analysis) program. For instance, Tian et al. assumed the microcontact area of contact surfaces in the fixed joint as a virtual isotropic material and then derived analytical expressions for the elastic modulus, Poisson ratio, shear modulus, and density by considering normal and tangential characteristics [27]. The experimental results illustrated that the mode shapes of the specimen estimated by the theoretical model agreed very well with those obtained from the modal analysis tests, and the differences between the natural frequencies of the two models were less than 9%. Liao et al. employed an orthogonal virtual material model to describe the properties of a joint contact surface, which was divided into several sublayers, thereby deducing a novel equivalent gradient material model [29]. Assuming an uneven stress distribution on a contact surface, Zhao et al. presented a nonlinear equivalent material model to accurately predict the performances of a bolted joint assembly [31]. Zhang et al. derived the expressions for the equivalent material properties based on the oblique asperity contact FT and demonstrated that a bolted joint with small size, large roughness, and tightening torque displayed a more pronounced effect of EM (equivalent material) [33]. A novel EM model was proposed by Yang et al. to predict the lower-order natural frequencies of composite bolted joints with carbon fiber-reinforced plastic, which was crucial in the optimization design of the composite material [34]. However, the coupling effects between the adjacent bolted joints were ignored in the existing FT and EMM, which is an important issue when determining a bolt arrangement.

To sufficiently guide the optimization design of bolt arrangements, this paper introduces a semi-analytical stiffness model for bolted joints in a machine tool system. This work provides accurate and efficient modeling of bolted joints considering the nonlinear stress distribution and coupling effects between adjacent bolts. Following a brief introduction to stiffness modeling methodologies, an enhanced contact stiffness model for bolted joints combining FT and FEA is presented. This model simulates the nonlinear pressure distribution inside the contact area of a bolted joint using FEA, affording fractal-based analytical expressions for normal and tangential contact stiffnesses. Then, two groups of test specimens are considered as an illustration and comparison studies between the proposed model and modal analysis tests are conducted to demonstrate the accuracy and validity of the proposed model. Further, a box-in-box precision horizontal machine tool is used to verify the developed contact stiffness model. Finally, several conclusions of this paper are summarized.

2 Contact Stiffness Modeling of a Bolted Joint

In this section, we develop semi-analytical expressions for the normal and tangential contact stiffnesses of a single-bolted joint considering nonlinear contact stress distribution and coupling effects. Figure 1 shows the schematic diagram of a single-bolted joint in a machine tool system, and the equivalent elastic model of a single-bolted joint is considered to be a three-dimensional spring.

Fig. 1
figure 1

Schematic of the equivalent elastic model of a single-bolted joint

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2.1 Fractal Model of Rough Surfaces

In this subsection, we introduce the fractal model of the rough surface. The micromodel of two contact surfaces can be simplified as the contact of a rigid plane with a rough surface, whose two-dimensional topography is illustrated in Fig. 2. Based on the Weierstrass and Mandelbrot (WM) fractal function, the height of surface topography can be described by the superposition of several cosine waves as [35]

$$z\left( x \right) = G^{D - 1} \sum\limits_{n = 0}^{\infty } {\frac{{\cos 2\uppi \gamma^{n} x}}{{\gamma^{{\left( {2 - D} \right)n}} }}}$$
(1)

where \(D \in \left( {1,2} \right)\) and G denote the fractal dimension and roughness parameters of the surface topology, which represent the irregularity and amplitude of z(x) at the sample length, respectively. x represents the coordinate of the measurement point, and γ > 1 is the spectrum of the surface topography. For most rough surfaces, γ = 1.5. n denotes the number of cosine waves.

Fig. 2
figure 2

Contact model of a rigid plane with a rough surface

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Based on the assumptions of the WM model, the deformed asperities between two rough surfaces are treated as hemispheres. The contact model is shown in Fig. 3. Considering the sample length on the rough surface, the topography can be described as

$$z\left( x \right) = G^{D - 1} l^{2 - D} \cos \frac{\uppi x}{l},\quad x \in \left( { - \frac{l}{2},\frac{l}{2}} \right)$$
(2)

where l = a0.5 denotes the sample length topography, and a = πr2 is the microcontact area. r represents the radius of the microcontact area. Therefore, radius R and interference δ of a single asperity are derived as

$$R = \left| {{1 \mathord{\left/ {\vphantom {1 {\left| {\frac{{{\text{d}}^{2} z\left( x \right)}}{{{\text{d}}x^{2} }}} \right|_{x = 0} }}} \right. \kern-0pt} {\left| {\frac{{{\text{d}}^{2} z\left( x \right)}}{{{\text{d}}x^{2} }}} \right|_{x = 0} }}} \right| = \frac{{a^{0.5D} }}{{\uppi^{2} G^{D - 1} }},\quad \delta = G^{D - 1} l^{2 - D} = G^{D - 1} a^{1 - 0.5D}$$
(3)

where δ represents the deformation height of the top of a single asperity, as shown in Fig. 3.

Fig. 3
figure 3

Normal contact of a single asperity condensed by a rigid plane

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2.2 Simulation of the Contact Stress Distribution

According to the FEA simulation, normal contact pressure undergoes exponential decay along the radial axis within the contact area of a single-bolted joint, as shown in Fig. 4. The normal contact stress in the contact area is described by a negative exponential function as

$$p_{{\text{n}}} \left( r \right) = \alpha \text{e}^{ - \beta r} \,\left( {{\text{MPa}}} \right)$$
(4)

where α and β are coefficients, which are both related to the geometry, preload, and material properties of a single-bolted joint’s contact area and can be calculated from FE (finite element) software. The contact area of a single-bolted joint is then described as a hollow ring with a radius \(r \in \left( {r_{0} ,R_{0}} \right)\), where R0 is defined by

$$p_{{\text{n}}} \left( {R_{0} } \right) = \eta p_{{{\text{n}},\max }}$$
(5)

where \(\eta \in \left( {0,1} \right)\) is the scaling factor. Subsequently, we consider the distributions of the normal contact stress of two adjacent bolted joints, as shown in Fig. 5. Each single-bolted joint has the same distribution of contact stress, as shown in Fig. 4. The overlapping area of two contact areas is the coupling area of two adjacent bolted joints, where the normal contact stress is cumulative in this area, as shown in Fig. 6. Therefore, the equivalent normal resultant force in the contact area of a single-bolted joint is described as

$$\begin{aligned} P & = \int_{ - \uppi }^{\uppi } {\int_{0}^{{{R_{0}} }} {p_{{\text{n}}} \left( r \right){\text{d}}r{\text{d}}\theta } } + \int_{ - \uppi }^{{{ - \uppi + \theta_{0}} }} {\int_{{r_{\theta 1} }}^{{{R_{0}} }} {p_{{\text{n}}} \left( r \right){\text{d}}r{\text{d}}\theta } } \\ & \quad + \int_{{\uppi - \theta_{0} }}^{\uppi } {\int_{{r_{\theta 1} }}^{{{R_{0}} }} {p_{{\text{n}}} \left( r \right){\text{d}}r{\text{d}}\theta } } \\ \end{aligned}$$
(6)

where θ0 is the contact angle related to the center distance d0 between two adjacent bolts and R0, as shown in Fig. 6. rθ1 is a function of θ1 and is solved by

$$r_{\theta 1} = d_{0} \cos \theta_{1} - \sqrt {R_{0}^{2} - d_{0}^{2} \left( {1 - \cos^{2} \theta_{1} } \right)}$$
(7)
Fig. 4
figure 4

Distribution of normal contact stress in a single-bolted joint

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Fig. 5
figure 5

Distributions of normal contact stress in two adjacent bolted joints

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Fig. 6
figure 6

Geometrical parameters of two adjacent bolted joints

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2.3 Model of a Single Asperity and Multi-asperities

The critical deformation height determining the elastic or plastic deformation of the spherical asperities is

$$\begin{aligned} \delta_{{\text{c}}} & = \left( {\frac{\uppi k\phi }{2}} \right)^{2} R = \left( {\frac{k\phi }{2}} \right)^{2} \frac{{a^{0.5D} }}{{G^{D - 1} }} \\ k & = {H \mathord{\left/ {\vphantom {H {\sigma_{{\text{y}}} }}} \right. \kern-0pt} {\sigma_{{\text{y}}} }},\quad \phi = {{\sigma_{{\text{y}}} } \mathord{\left/ {\vphantom {{\sigma_{{\text{y}}} } {E^{{\prime }} }}} \right. \kern-0pt} {E^{{\prime }} }},\quad E^{{\prime }} = \left( {\frac{{1 - v_{1}^{2} }}{{E_{1} }} + \frac{{1 - v_{2}^{2} }}{{E_{2} }}} \right)^{ - 1} \\ \end{aligned}$$
(8)

where σy and H denote the yield strength and hardness of the softer material between the lower and upper rough surfaces, respectively. E′ represents the equivalent elastic modulus. v1(E1) and v2(E2) represent the Poisson’s ratio (elastic modulus) of two contact surfaces. If δ > δc, then the asperity is in plastic deformation; otherwise, it is in elastic deformation. Obviously, δc is a constant value, and it only corresponds to the material properties. Substituting Eq. (3) into Eq. (8) leads to

$$a_{{\text{c}}} = G^{2} \left( {\frac{k\phi }{2}} \right)^{{\frac{2}{1 - D}}}$$
(9)

where ac represents the critical contact area of the asperities, which is associated only with fractal parameters and material properties. If a > ac, then the asperity is in elastic deformation; otherwise, it is in plastic deformation.

Based on the Hertz contact theory, the plastic or elastic loads on a spherical asperity are described as

$$p = \left\{ {\begin{array}{*{20}l} {p_{{\text{e}}} = \frac{4}{3}E^{{\prime }} R^{0.5} \delta^{1.5} = \frac{4}{3\uppi }E^{{\prime }} G^{D - 1} a^{1.5 - 0.5D} ,} \hfill & {\quad a > a_{{\text{c}}} } \hfill \\ {p_{{\text{p}}} = k\sigma_{{\text{y}}} a,} \hfill & {\quad a < a_{{\text{c}}} } \hfill \\ \end{array} } \right.$$
(10)

The multi-asperities during the contact area satisfy the following distribution density function [36]

$$n\left( a \right) = 0.5D\psi^{1 - 0.5D} a_{{\text{l}}}^{0.5D} a^{ - 1 - 0.5D} ,\quad 0 \le a \le a_{{\text{l}}}$$
(11)

where ψ > 1 denotes the domain extension parameter of the microcontact area, and al represents the largest microcontact area. ψ is obtained as

$$\psi^{1 - 0.5D} - \left( {1 + \psi^{ - 0.5D} } \right)^{0.5D - 1} = {2 \mathord{\left/ {\vphantom {2 D}} \right. \kern-0pt} D} - 1$$
(12)

Then, the real contact area A, the elastic contact area Ae, and the plastic contact area Ap are derived as

$$\begin{aligned} A & = \int_{0}^{{a_{{\text{l}}} }} {n\left( a \right)a{\text{d}}a} = \frac{{D\psi^{1 - 0.5D} }}{2 - D}a_{{\text{l}}} \\ A_{{\text{e}}} & = \int_{{a_{{\text{c}}} }}^{{a_{{\text{l}}} }} {n\left( a \right)a{\text{d}}a} = \frac{{D\psi^{1 - 0.5D} }}{2 - D}\left( {a_{{\text{l}}} - a_{{\text{l}}}^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0pt} 2}}} a_{{\text{c}}}^{{{{\left( {2 - D} \right)} \mathord{\left/ {\vphantom {{\left( {2 - D} \right)} 2}} \right. \kern-0pt} 2}}} } \right) \\ A_{{\text{p}}} & = \int_{0}^{{a_{{\text{c}}} }} {n\left( a \right)a{\text{d}}a} = \frac{{D\psi^{1 - 0.5D} }}{2 - D}a_{{\text{l}}}^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0pt} 2}}} a_{{\text{c}}}^{{{{\left( {2 - D} \right)} \mathord{\left/ {\vphantom {{\left( {2 - D} \right)} 2}} \right. \kern-0pt} 2}}} \\ \end{aligned}$$
(13)

Integrating the plastic and elastic loadings inside the contact area obtains the resultant normal force:

$$\begin{aligned} P & = P_{{\text{e}}} + P_{{\text{p}}} \\ & = \frac{4}{3\uppi }E^{{\prime }} G^{D - 1} \int_{{a_{{\text{c}}} }}^{{a_{{\text{l}}} }} {n\left( a \right)a^{1.5 - 0.5D} {\text{d}}a} + k\sigma_{{\text{y}}} \int_{0}^{{a_{{\text{c}}} }} {n\left( a \right)a{\text{d}}a} \\ \end{aligned}$$
(14)

Substituting Eq. (11) into Eq. (14) leads to

$$P = \left\{ {\begin{array}{*{20}l} {\psi^{0.25} \left( {\frac{{E^{{\prime }} \sqrt G a_{{\text{l}}}^{0.75} }}{\uppi }\ln \frac{{a_{{\text{l}}} }}{{a_{{\text{c}}} }} + 1.5k\sigma_{{\text{y}}} a_{{\text{l}}}^{0.75} a_{{\text{c}}}^{0.25} } \right)} \hfill & {\quad D = 1.5} \hfill \\ {\psi^{1 - 0.5D} D\left( \begin{gathered} \frac{{4E^{{\prime }} G^{D - 1} a_{{\text{l}}}^{0.5D} }}{{3\uppi \left( {3 - 2D} \right)}}\left( {a_{{\text{l}}}^{1.5 - D} - a_{{\text{c}}}^{1.5 - D} } \right) \hfill \\ + \frac{{k\sigma_{{\text{y}}} a_{{\text{c}}}^{1 - 0.5D} a_{{\text{l}}}^{0.5D} }}{{2\left( {2 - D} \right)}} \hfill \\ \end{gathered} \right)} \hfill & {\quad D \ne 1.5} \hfill \\ \end{array} } \right.$$
(15)

Equation (15) clearly shows that P is associated with al, which can be resolved by combing Eqs. (6) and (15).

2.4 Normal and Tangential Stiffness Model of a Bolted Joint

Following the Hertz contact theory, the interference δ and the radius of the contact area of a single asperity can be described as

$$\delta = \frac{{r^{2} }}{R} = \left( {\frac{9}{16}\frac{{p^{2} }}{{RE^{{{\prime }2}} }}} \right)^{\frac{1}{3}} ,\quad r = \left( {\frac{3pR}{{4E^{{\prime }} }}} \right)^{\frac{1}{3}}$$
(16)

The normal contact stiffness of a single asperity is then derived as

$$k_{{\text{n}}} = \frac{{{\text{d}}p}}{{{\text{d}}\delta }} = 2E^{{\prime }} \sqrt {R\delta } = 2E^{{\prime }} r = 2E^{{\prime }} \sqrt {\frac{a}{\uppi }}$$
(17)

Integrating kn within the contact area leads to the normal contact stiffness of a single-bolted joint

$$K_{{\text{n}}} = \int_{{a_{{\text{c}}} }}^{{a_{{\text{l}}} }} {k_{{\text{n}}} n\left( a \right){\text{d}}a}$$
(18)

Substituting Eqs. (11) and Eq. (17) into Eq. (18) leads to

$$K_{{\text{n}}} = \frac{{2E^{{\prime }} \psi^{1 - 0.5D} Da_{{\text{l}}}^{0.5D} }}{{\sqrt \uppi \left( {1 - D} \right)}}\left( {a_{{\text{l}}}^{0.5 - 0.5D} - a_{{\text{c}}}^{0.5 - 0.5D} } \right)$$
(19)

As shown in Fig. 7, the shear deformation ε produced by the tangential load \(\overline{p}\) on a single asperity is described as [37]

$$\varepsilon = \frac{{3\left( {2 - \nu } \right)\mu p}}{{16G^{{\prime }} r}}\left( {1 - \left( {1 - \frac{{\overline{p}}}{\mu p}} \right)^{\frac{2}{3}} } \right),\quad G^{{\prime }} = \frac{{E^{{\prime }} }}{{2\left( {1 + \nu } \right)}}$$
(20)

where G′ represents the equivalent shear modulus of two contacting rough surfaces. μ denotes the coefficient of sliding friction. Then, the tangential compliance ct of a single asperity is determined as

$$c_{{\text{t}}} = \frac{{{\text{d}}\varepsilon }}{{{\text{d}}\overline{p}}} = \frac{2 - \nu }{{8G^{{\prime }} r}}\left( {1 - \frac{{\overline{p}}}{\mu p}} \right)^{{ - \frac{1}{3}}}$$
(21)

Assuming that the tangential and normal loads on the asperities are proportional to their contact area, the tangential contact stiffness of a single asperity is derived as

$$k_{{\text{t}}} = \frac{{8G^{{\prime }} r}}{2 - \nu }\left( {1 - \frac{{\overline{p}}}{\mu p}} \right)^{\frac{1}{3}} = \frac{{8G^{{\prime }} \sqrt a }}{{\left( {2 - \nu } \right)\sqrt \uppi }}\left( {1 - \frac{{\overline{P}}}{\mu P}} \right)^{\frac{1}{3}}$$
(22)

where \(\overline{P}\) denotes the resultant tangential force within a single-bolted joint’s contact area. Obviously, the single asperity slides tangentially relative to the rigid plane when \(\overline{P} > \mu P\). Integrating kt within the contact area leads to the tangential contact stiffness of a single-bolted joint:

$$K_{{\text{t}}} = \int_{{a_{{\text{c}}} }}^{{a_{{\text{l}}} }} {k_{{\text{t}}} n\left( a \right){\text{d}}a}$$
(23)

Substituting Eqs. (11) and (22) into Eq. (23) leads to

$$K_{{\text{t}}} = \frac{{8\psi^{1 - 0.5D} G^{{\prime }} Da_{{\text{l}}}^{0.5D} }}{{\left( {1 - D} \right)\left( {2 - \nu } \right)\sqrt \uppi }}\left( {1 - \frac{{\overline{P}}}{\mu P}} \right)^{\frac{1}{3}} \left( {a_{{\text{l}}}^{0.5 - 0.5D} - a_{{\text{c}}}^{0.5 - 0.5D} } \right)$$
(24)

According to Eqs. (19) and (24), the equivalent normal and tangential stiffnesses of a single-bolted joint are related to the elastic modulus, the tangential modulus, the fractal parameters, and the preload of the contact rough surfaces.

Fig. 7
figure 7

Tangential contact of a single asperity

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3 Experimental Verifications

In this section, we conducted modal analysis tests on two designed specimens and a box-in-box precision horizontal machine tool to demonstrate the accuracy and efficiency of the proposed contact stiffness model.

To illustrate the coupling effect, two test specimens made of HT300 were designed and assembled with different center distances between two adjacent bolted joints (70 mm and 100 mm), as shown in Fig. 8a. The bolts connecting the upper and lower surfaces in the two test specimens were identically M24 × 110, and the contact surfaces were machined using an accurate milling process. Assume that the machined surfaces have the same fractal dimension and roughness. The bolts in both test specimens were applied to identical preload torques of 340 Nm using a digital torque wrench (Stanley SD-340-22), as shown in Fig. 8b. Table 1 provides the material properties of the two contact surfaces, and Table 2 shows the mechanical parameters of a single-bolted joint.

Fig. 8
figure 8

Schematic of test specimens and preload torque loading equipment

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Table 1 Material properties of HT300 gray cast iron
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Table 2 Mechanical parameters of a single-bolted joint (preload: 340 Nm)
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3.1 Fractal Dimensions and Roughness Calculation

This subsection identifies machined rough surfaces’ fractal dimension and roughness parameters.

Considering a 20 × 2.4-mm rectangular area in a rough surface as a sample surface, the topography height was measured by the surface roughness measuring instrument (MarSurf PS 10) illustrated in Fig. 9. The sensor precision of the instrument was 2 μm. The sample surface was composed of 21 measurement lines, and the probe moved along each measurement line for 2.4 mm. The measuring point interval Δx was 0.5 μm, leading to 4801 points for each measurement line. Figure 10 illustrates the three-dimensional topography of the milling surface.

Fig. 9
figure 9

Measurement of three-dimensional surface topography

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Fig. 10
figure 10

Three-dimensional surface topography of the sample surface

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Based on the structure function method, the function of the rough surface profile S(τ) is described as [38]

$$S\left( \tau \right) = \frac{1}{N - n}\sum\limits_{i = 0}^{N - n} {( {z( {x_{i} + n} ) - z( {x_{i} } )} )^{2} }$$
(25)

where N − n represents the number of calculation data separated by an arbitrary interval \(\tau = n \times \Delta x\) between the recorded data. The fractal dimension and roughness parameters are then identified from double logarithmic coordinates as

$$\begin{aligned} D & = \frac{4 - k}{2},\quad \log G = \frac{b - \log C}{{2\left( {D - 1} \right)}} \\ C & = \frac{{\Gamma \left( {2D - 3} \right)\sin \left( {{{\uppi \left( {2D - 3} \right)} \mathord{\left/ {\vphantom {{\pi \left( {2D - 3} \right)} 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {4 - 2D} \right)\ln \gamma }} \\ \end{aligned}$$
(26)

where k and b represent the slope and intercept on the y-axis of the fitting line between log(S(τ)) and log(τ). Figure 11 illustrates the fit line of the middle line (11th) in the sample surface when n = 25, k = 0.7938, and b = – 2.1391. Thus, D = 1.6031 and G = 1.90 × 10−9 m. The fractal dimension and roughness parameters of the machined rough surfaces are calculated as the average value of 21 measurement lines, as shown in Table 3. Based on the proposed contact stiffness model, Table 4 provides the equivalent normal and tangential stiffnesses of a single-bolted joint in test specimens 1 and 2 without considering the coupling effect.

Fig. 11
figure 11

Fitting line of the double logarithmic function of the measuring data in the middle line

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Table 3 Average fractal parameters of the machined surface of a test specimen
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Table 4 Equivalent normal and tangential stiffnesses of a single-bolted joint under different conditions (× 1010 N/m)
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3.2 Modal Analysis Tests of the Test Specimens

This subsection conducts a modal test of the designed specimens to illustrate the accuracy of the proposed contact stiffness model.

Figure 12 illustrates the experimental setup for modal analysis tests on the designed specimens. During the experiment, the test specimens were suspended by steel wires to measure the free vibration modes. An impact hammer (ICP 086D20) was used to generate a transient excitation force on the test specimens, and an acceleration sensor (PCB 356A26) was employed to record vibration response signals from each measuring point. Then, using a dynamic testing and analysis system (SCADAS III), the data acquisition and post-processing of the excitation and response signals, as well as the identification of modal parameters, were accomplished, resulting in a detailed analysis of the results of the mode shapes of the test specimens.

Fig. 12
figure 12

Experimental setup for the modal test of the specimen 1

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Using the stiffness data in Table 4, the FEA models of the test specimens were established in SAMCEF commercial software. Each single-bolted joint was connected using a bushing unit, which was considered a virtual spring connecting the upper and lower specimens. The mesh type and average length of the upper and lower specimens were set as tetrahedral and 40 mm, respectively. The FEA software’s corresponding boundary conditions and material properties were set consistent with those parameters of the test specimens, leading to an FEA model with 3.15 × 103 degrees of freedom. Figures 13 and 14 show the first six mode shapes of test specimen 1 (d0 = 70 mm) obtained by the FEA and modal analysis tests, respectively. The first- and third-order mode shapes are the bending vibrations of the entire test specimen along the x- and z-directions of \({\mathcal{K}}^{{\prime }}\), respectively. The second-order mode shape is a torsional vibration of the entire test specimen along the y-direction. The fourth-order mode shape is a translational vibration of the upper and lower specimens along the y-direction, and the sixth-order mode shape is a local torsional vibration of the upper specimen. Obviously, the first six mode shape orders estimated by the FEA model agree very well with those obtained from modal tests. In addition, the MAC (modal assurance criterion) is employed to verify the above two models’ consistency as

$${\text{MAC}}_{ij} = \frac{{\left( {{\varvec{\varPhi}}_{{{\text{FE}},i}}^{\text{T}}{\varvec{\varPhi}}_{{{\text{MT}},j}} } \right)^{2} }}{{\left( {{\varvec{\varPhi}}_{{{\text{FE}},i}}^{\text{T}}{\varvec{\varPhi}}_{{{\text{FE}},i}} } \right)\left( {{\varvec{\varPhi}}_{{{\text{MT}},j}}^{\text{T}}{\varvec{\varPhi}}_{{{\text{MT}},j}} } \right)}},\quad i,j = 1,2, \ldots ,6$$
(27)

where ΦFE,i and ΦMT,j are the ith and jth order mode shapes from the FEA and the developed modal test, respectively. Figure 15 shows the MAC of the first six mode shape orders of the two models. The diagonal elements’ values are much higher than the remaining nondiagonal elements’ values (0.95 against 0.23), demonstrating the accuracy of the developed contact stiffness model for calculating the mode shapes of the test specimen. In addition, the mode shapes of test specimen 2 (d0 = 100 mm) obtained by the FEA and modal analysis tests are similar to those of specimen 1.

Fig. 13
figure 13

First six mode shapes of test specimen 1 predicted by the FEA

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Fig. 14
figure 14

First six mode shapes of test specimen 1 obtained from the modal test

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Fig. 15
figure 15

MAC of the first six mode shape orders of test specimen 1 from the FEA and modal tests

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Tables 5 and 6 compare the errors in the natural frequencies of the first six orders of test specimens 1 and 2, respectively, for the FEA models and the modal test. The contact stiffnesses of bolted joints in FEA model 1 were calculated using the proposed model considering the coupling effect, which was ignored in FEA model 2. For test specimen 1, the natural frequencies predicted by model 1 are more accurate than those of model 2 (maximum error: 2.73% against − 7.72%), illustrating the considerable impact of the coupling effect on a single-bolted joint’s contact stiffness when d0 is much smaller than 2R0. In contrast, for test specimen 2, the coupling effect can be negligible when d0 is near 2R0. The above results illustrate the accuracy and efficiency of the proposed semi-analytical model for calculating the natural frequencies of the test specimen.

Table 5 Natural frequencies of the first six mode orders of test specimen 1 obtained from an FEA and the modal test
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Table 6 Natural frequencies of the first six mode orders of test specimen 2 obtained from an FEA and the modal test
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3.3 Modal Analysis Tests of the Entire Machine Tool

This subsection conducts a modal test of a box-in-box precision horizontal machine tool to illustrate the application of the proposed semi-analytical model in the dynamic predictions of an entire machine tool.

To increase the contact stiffness of a single-bolted joint while reducing the zero-pressure contact area in the fixed joint, the center distance d0 between two adjacent bolts (M24 × 160) was set to 80 mm. The preload torque of each bolt was set to 340 Nm. Based on the developed model considering the coupling effect, the normal and tangential contact stiffnesses of a single-bolted joint are 3.59 × 1010 N/m and 3.43 × 1010 N/m, respectively. Employing the dynamic model presented in [39], the dynamic responses of the tool and workpiece of the entire machine tool can be predicted. Meanwhile, a modal analysis test was performed on a prototype, as shown in Fig. 16. Figure 17 compares FRFs along the x-, y-, and z-axes of \({\mathcal{K}}^{\prime }\) of the modal test and the employed dynamic model. Obviously, the FRFs of the employed model agree well with those of the modal test. Table 7 provides the natural frequencies of the experiments and the proposed model at the reference configuration. The differences between these two methods were less than 15.61%, demonstrating the accuracy and efficiency of the developed contact stiffness model of bolted joints and the employed dynamic model.

Fig. 16
figure 16

Experimental setup for the modal test of a box-in-box precision horizontal machine tool

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Fig. 17
figure 17

FRF comparisons between the modal analysis test and the employed dynamic model

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Table 7 Experimental and employed dynamic model natural frequencies of the entire machine tool at the reference configuration
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4 Conclusions

This article presents a semi-analytical approach for the normal and tangential contact stiffness modeling of bolted joints considering the coupling effect. The following inferences were drawn from the results obtained using the proposed method.

  1. 1.

    According to FEA simulation results, a nonlinear distribution model of contact stress inside a single-bolted joint is proposed using a negative exponential function. Subsequently, the analytical normal resultant force of the contact area of a single-bolted joint considering the coupling effect is derived based on the Hertz contact theory and FT, solving the largest microcontact area. Finally, the semi-analytical expressions for the normal and tangential contact stiffnesses of a single-bolted joint are formulated by integrating the stiffnesses of multi-asperities. The developed model indicates the influence of preload torque, material properties, center distance, and fractal parameters on the contact stiffnesses of bolted joints.

  2. 2.

    Considering two designed test specimens as illustrations, the effectiveness and accuracy of the presented model are demonstrated. The mode shapes of the two test specimens from a modal analysis test and the proposed model are highly consistent. The proposed model considering the coupling effect is more accurate than the traditional method in predicting the contact stiffnesses of bolted joints. The maximum error in the natural frequencies of the proposed approach relative to experimental results is < 2.73%, illustrating the accuracy and efficiency of the developed contact stiffness model. In addition, a modal test on a box-in-box precision horizontal machine tool prototype illustrates the application of the proposed model in dynamic predictions of an entire machine tool. The developed approach is instrumental in revealing the influence of microcontact factors on the contact stiffness of bolted joints and in guiding the optimization of bolt arrangements.