Dynamic modeling and frequency response characteristic analysis of a heavy boring machine tool post system based on 5-DOF

1 Introduction

As a primary machine tool for specialized operations such as inner hole boring and cylindrical turning, heavy boring machines are extensively employed in fields including defense, aerospace, automotive manufacturing, and paper machinery. Tool-post vibration in heavy boring machines is a predominant constraint on achievable workpiece machining accuracy. Therefore, investigation of the vibration characteristics of the tool post system carries significant theoretical and practical value. These vibrations primarily arise from loss of structural dynamic stability and the onset of self-excited chatter during the boring process. The dynamic interaction between the workpiece and the machine tool leads to degraded overall machining performance, impaired surface integrity, and accelerated tool wear (Yigit et al., 2017).

(1) Theoretical modeling. Researchers have mathematically modeled the dynamic characteristics of various boring systems. Sherbakov et al. (2018) and Shi et al. (2017) developed a dynamic model by integrating the continuity equation, mass balance equation, displacement-strain relationship, and Hooke’s law. By investigating the dynamic behavior under different boring radii, 24 distinct mathematical models were summarized and subsequently validated through comparison with finite element analysis results. Ekrem and Kubilay (2025) employed a discretization approach for the equations of motion, simplifying the cutting portion of the boring bar into a spring-mass model, and utilized the MCK equation to theoretically model a proposed vibration reduction technique for boring bars. The classical analytical method, frequently applied to study boring machine dynamics, typically simplifies the cutting portion to a single degree of freedom model with a spring–damper system at the free end. This simplification, however, can lead to an incomplete representation of the vibration response. To overcome this limitation, Yadav et al. (2020) introduced an admittance coupling method, representing the boring bar’s substructure admittance via a coupled system of a cantilever Euler beam and a damper. Their work highlighted the difficulty of fully capturing boring vibrations using a single-degree-of-freedom model. Based on Bernoulli beam theory and the Hamiltonian principle, Ma et al. (2022) established the governing equation for the cutting stability of a composite variable cross-section boring bar. Considering the variable cross-section’s motion, they analyzed the relationship between damping ratio, length, and diameter by combining the Galerkin and semi-discrete methods, thereby accurately extracting the dynamic characteristics of variable cross-section boring. Zhang et al. (2020) modeled the workpiece in the boring process as a rotating, cantilevered Euler–Bernoulli axis. Using Hamilton’s principle, they formulated the system’s governing equation and discretized the differential equations of motion via the Galerkin method. Their results indicate that the instability induced by internal damping is directly correlated with cutting chatter as the material damping increases. Sentyakov et al. (2020) modeled the boring machine’s inertial body, tool bracket, and boring bar as a four degrees-of-freedom (4-DOF) spring–damper system. They derived the differential equations of motion for this configuration using the finite difference method and Laplace transforms. By substituting designed damping values into the model for response analysis, their research demonstrated a significant reduction in the vibration response of boring bars equipped with damping devices. Takaahashi et al. (2021) applied the theoretical relationship between mechanical structural dynamics and boring chatter stability to practical tool design. They proposed an anisotropic structural design method to provide theoretically infinite dynamic stiffness for the boring bar, markedly improving its dynamic characteristics. Marzur et al. (2021) proposed a linear dynamic equation based on modal coupling technology to estimate the stiffness of complex support structures in discrete models. They combined the particle swarm optimization algorithm with finite element analysis to identify and predict different vibration modes.

(2) Finite element method. With advances in computer technology, the finite element method has become an indispensable tool for investigating the dynamic characteristics of machine tools. By analyzing the stress and deformation of individual machine tool components under actual operating loads, structural weaknesses can be identified and optimized, thereby enhancing machining accuracy and extending service life. Soori and Arezoo (2023) utilized simulation software to perform a finite element analysis, comparing the simulated static and dynamic results with experimental data to validate the reliability of the method. Aiming to improve the natural frequency and damping performance of boring bars, Athifah (2025) conducted finite element-based modal and harmonic response analyses on a boring bar structure incorporating damping materials. Their study found a significant reduction in chatter for the damped structure compared to the original design. Kri et al. (2021) investigated chatter near the boring tool, a form of undesirable self-excited vibration during deep-hole machining, that is often difficult to avoid (Patel and Rai, 2025). To analyze this dynamic behavior, many researchers use damping-based vibration suppression. By varying material and structural damping to examine different damping ratios, they seek to identify optimal parameters for vibration control. Ramesh et al. (2023) noted that the boring bar operates as a cantilevered structure, making the cutter head particularly prone to vibration. To suppress this, they explored different internal materials within the boring bar. By comparing the effects of various damping ratios, they adjusted the bar’s effective stiffness to achieve vibration suppression. Hayati et al. (2021) designed a boring bar with an internal friction damping structure, achieved by inserting pins that dissipate energy during bending vibrations. They established an analytical model to calculate the energy dissipated by the damping mechanism. Finally, by combining mathematical modeling with finite element simulation, they verified the optimal configuration for this frictional damping structure under different damping ratios, providing critical data for subsequent prototyping.

(3) Investigation of heavy machines. Heavy machines are primarily used to machine inner holes in super-large and heavy components. Characterized by their substantial model size, high manufacturing complexity, and considerable experimental costs, the analysis and research of these machines present significant challenges. In studying heavy-duty boring machines, Kalinski et al. (2021) proposed modeling the boring bar as a flexible beam. They established a finite element computational model for the boring process that incorporates a Bernoulli beam coupled with spring–damper units. This model can accurately capture multi-directional vibrations during heavy boring operations. Liu et al. (2018) noted that the stiffness of a boring bar in deep-hole machining decreases as its aspect ratio increases. To address this issue, they developed a dynamic model of the boring machine during cutting based on vibration mechanics theory. Utilizing this model, they designed a boring bar with variable stiffness. Targeting a similar problem, Feng et al. (2020) established a dynamic model for boring bar vibration by integrating the principles of self-excited vibration-coupled flutter and forced vibration dynamic damping. They further optimized the conventional structure by proposing a double-layer design for the boring bar.

Conventionally, heavy machine design and manufacturing have relied on empirical and analogical methods, resulting in a lack of reported data on dynamic characteristics. This study focuses on the dynamic behavior of the CNCT250 heavy boring machine. A mathematical model is developed through theoretical analysis to obtain its frequency response characteristics. The actual frequency response is investigated using finite element simulation, and the results are compared with theoretical predictions. The system’s response under different cutting frequencies is analyzed. By modeling and calculating the vibration characteristics under varying damping conditions, the optimal damping parameters are identified, providing theoretical support for industrial application.

2 Theoretical modeling

2.1 Five-degrees-of-freedom tool post modeling

Figure 1 shows the CNCT250 heavy boring machine structure. It is equipped with a 17 m boring bar. It can process workpieces with an outer diameter of 2.5 m, an inner diameter of 1.8 m, and a length of 12 m. This capacity enables the concurrent machining of both the inner and outer diameters of large pipe components. When in working mode, the workpiece is fixed by the chuck mechanism of the spindle box of the machine tool. The spindle provides the rotating power to realize the rotating motion of the workpiece. A floating support structure is installed at a certain distance to assist the rotating motion of the workpiece. When the machine tool is boring, the workpiece is in a rotating motion state. The tool post system moves linearly along the boring bar inside the workpiece to accomplish the boring.

The tool post system is a critical component of heavy boring machines, directly influencing machining stability and precision. Vibration can significantly influence the machine’s performance stability and achievable machining accuracy. Hence, investigating the dynamic response characteristics of the tool post structure in heavy boring machines holds considerable theoretical and practical importance. As illustrated in Figure 2a, the primary system components are the guiding mechanism, the support and drive mechanism, the left and right tool holders, and the boring bar.

Surface contacts exist between the boring bar and both the guiding mechanism and the support and drive mechanism. A bolt connection is used between the left and right tool holders and the guiding mechanism, the support and drive mechanism, the guiding mechanism, and the support and drive mechanism, so the joint between them can also be equivalent to a spring and damping. These interfaces can be dynamically modeled as equivalent spring–damper pairs. The guiding mechanisms are also represented as equivalent springs and dampers. The tool post structure is simplified into a 5-DOF spring–damper system, as shown in Figure 2b. The interfacial contacts between these components are simplified as equivalent damping and spring elements.

The 5-DOF dynamic equation can be expressed as (Meirovitch, 2001)

M x ¨ + C x ˙ + K x = F t . (1)

M 1 and M 2 are the masses of the left and right tool holders, with a mass of 409 Kg each. M 3 is the mass of the guiding mechanism, which is 285 kg. M 4 is the mass of the boring bar, which is 21,115 kg. M 5 is the mass of the support and drive mechanism, which is 1515 kg. The maximum M can be expressed as

M = M 1 0 0 0 0 0 M 2 0 0 0 0 0 M 3 0 0 0 0 0 M 4 0 0 0 0 0 M 5 .

The stiffness matrix K is

K = K 13 + K 15 0 − K 13 0 − K 15 0 K 23 + K 25 − K 23 0 − K 25 − K 13 − K 23 K 13 + K 23 + K 34 + K 35 0 − K 35 0 0 0 K 34 + K 45 − K 45 − K 15 − K 25 − K 35 − K 45 K 15 + K 25 + K 35 + K 45 .

The damping matrix C is

C = C 13 + C 15 0 − C 13 0 − C 15 0 C 23 + C 25 − C 23 0 − C 25 − C 13 − C 23 C 13 + C 23 + C 34 + C 35 0 − C 35 0 0 0 C 34 + C 45 − C 45 − C 15 − C 25 − C 35 − C 45 C 15 + C 25 + C 35 + C 45 .

When determining the natural frequency of the system, it is modeled as an undamped system undergoing free vibration. Accordingly, the solution for free vibration can be assumed as

x = X e i ω t . (2)

Substituting Equation 2 into Equation 1, the system is expressed as Equation 3:

K − ω 2 M X = 0 . (3)

The cutting is simplified into two harmonic excitations acting on the mass blocks in opposite directions. The excitation matrix F is expressed by Equation 4:

F = F t − F t 0 0 0 T , (4)

where F(t) is the unilateral cutting force.

Using the modal superposition method, the displacement vector is expanded into a linear combination:

x t = ∑ k = 1 5 q k t φ k , (5)

where q k t is the modal coordinate and k is the kth modal of the tool post system.

Substituting Equation 5 into Equation 1, the complex multiple degree of freedom equation is decoupled into an ordinary differential equation with a single generalized coordinate q k t by using the orthogonality of the vibration mode. The kth modal equation of the tool post system can be expressed as

M k q ¨ k t + C k q ˙ k t + K k q k t = F k t . (6)

2.2 Dynamic modeling of the tool post joints

Figure 2 shows that most of the stiffness and damping of the tool post come from the joint between the parts, including the bolt joint, cone joint, movable joint, and so on. The contact stiffness K and damping C are derived from the theoretical empirical formula and cannot accurately obtain the response. According to the principle of the joint, the joint parameters of the tool post system are identified and extracted by combining the finite element simulation and modal experiment.

2.2.1 Parameter identification of movable joint

The guiding mechanism and support and drive mechanism of the boring machine are driven by gears to move linearly along the boring bar, which can be simplified to identify the parameters of the movable joint of the rolling linear guide. According to the finite element theory, each node of the three-dimensional solid element only needs 4-DOF to reflect the movement of the entity. The eight-node hexahedron in the finite element is used to simulate the feed movement of the boring machine tool holder on the boring bar.

For the ith node, the jth force node displacement is X ij, the node force is F ij, and K ij is the stiffness of the joint. In the tool examined in this article, the force that constitutes the vibration mainly comes from the feed directions and cutting directions. According to the stiffness coefficient method, the motion equation can be obtained using Equation 7 (Mao et al., 2017).

∑ n = 1 2 k 1 n i j x 1 n − x 5 n + ∑ n = 1 2 k 2 n i j x 2 n − x 6 n + ∑ n = 1 2 k 3 n i j x 3 n − x 7 n + ∑ n = 1 2 k 4 n i j x 4 n − x 8 n = F i j . (7)

For the stiffness matrix, because the hexahedral structure has eight nodes and four combinations, each sub-stiffness matrix is 8 × 8. By symmetry and block ability, the overall joint stiffness K of the joint is 16 × 16.

The values of M s, K s, and C s can be obtained by modal experiment, and the values of M j , K j, and C j can be obtained by theoretical calculation. According to the elastic damping analysis, c j = β K j. The kinematic equation is (Mao et al., 2017)

M s x ¨ t + β K j x ˙ t t + K s + K j x t = F t . (8)

Fourier transform on both sides of Equation 8 obtains Equation 9:

− ω 2 M s + j ω β K j + K s + K j X ω = F ω . (9)

By using the method of single-point excitation and multi-point pickup, a column matrix H j ω of the frequency response function can be obtained. From the relationship between the frequency response function and the stiffness matrix, the response function of the movable joint is given as Equation 10.

− ω 2 M s + j ω β K j + K s + K j H j ω = I j . (10)

Rearranging yields Equation 11:

1 + j ω β K j H 1 j H 2 j ⋮ H n j = 0 ⋮ 0 1 0 ⋮ 0 − K s + ω 2 M s H 1 s H 2 s ⋮ H n s . (11)

2.2.2 Parameter identification of the bolt joint

For the bolted joint on the boring machine tool holder, there are mainly two kinds of joint surface units: linear and array. However, regardless of the joint form, its vibration mode can be modeled using the eight-node hexahedron unit. The kinematic equation obtained by the stiffness coefficient method is Equation 12 (Mao et al., 2017).

∑ n = 1 3 k 1 n i j x 1 n − x 5 n + ∑ n = 1 3 k 2 n i j x 2 n − x 6 n + ∑ n = 1 3 k 3 n i j x 3 n − x 7 n + ∑ n = 1 3 k 4 n i j x 4 n − x 8 n = F i j . (12)

Compared with the movable joint, the theoretical modeling of the bolted joint adds a bolt preload stiffness in the z-direction; that is, its sub-stiffness matrix K is a 12 × 12 matrix, and its overall stiffness matrix K is a 24 × 24 matrix. Other mathematical modeling is consistent with the movable joint and is not repeated here.

2.3 The model of multimode coupling between the tool post system and the Bernoulli beam

The boring bar examined in this article is a heavy slender rod, and there is force transfer with the tool post system. If only the boring bar is equivalent to a mass block, it is difficult to describe the actual vibration of the boring bar. The boring bar and the tool holder system can be regarded as a simply supported beam structure, as shown in Figure 3. A coupled dynamic analysis integrating the 5-DOF spring–damper system with the continuous beam model is conducted. M A is the mass of the 5-DOF system, K A and C A are the equivalent stiffness and equivalent damping, respectively, and M B is the mass of the boring bar. The kinematic equation of the beam, which can be described by the Euler–Bernoulli beam theory, is described by Equation 13 (Meirovitch, 2001):

E I ∂ 4 y ∂ x 4 + ρ A ∂ 2 y ∂ t 2 = F x , t . (13)

The vibration mode of the boring bar can be expressed as

W n = sin ⁡ β n x = sin n π x l . (14)

The displacement field of the boring bar can be expressed as

w x , t = ∑ n = 1 + ∞ W n x q n t = ∑ n = 1 + ∞ sin n π l q n t . (15)

where q n t is the modal coordinate of the boring bar and n is the nth modal of the boring bar.

Combining with Equation 1 and Equation 6, the 5-DOF system is coupled with the modes of the boring bar, and according to modal orthogonality, the modal motion equation can be expressed as

M q ¨ A + C A W A 2 q ˙ A + K B − K A W A 2 q A + K A W A x A + C A W A x ˙ A = 0 . (16)

Substituting Equation 14 and Equation 15 into Equation 6, the excitation response of the coupled system under forced vibration is calculated using Equation 17.

M x ¨ + C A x ˙ + K A x − C A ∑ n = 1 + ∞ W n x q ˙ n t − K A ∑ n = 1 + ∞ W n x q n t = F t . (17)

According to the multi-modal coupling method, the r-order modal response motion equation can be expressed as Equation 18:

M r q ¨ r + K r q ˙ r + C r q r − C n ∑ n = 1 + ∞ W n x q ˙ n t − K A ∑ n = 1 + ∞ W n x q n t = F r t . (18)

where r is the rth modal of the coupling system, and the equivalent formula is expressed by Equations 19–21:

M r = x n 2 M A + q r τ 2 M B C 1 r = x n 2 C A + q r τ 2 C B C 2 r = x n 2 ξ r M B ω r K r = ω r 2 M r ξ n = C r 2 M r ω r ω d r = ω r 1 − ξ r 2 , (19)

q r t = 2 m r ω d r ∫ 0 t F r τ e − ξ r ω d r t − τ ⁡ sin ⁡ ω d r t − τ d τ , (20)

F r t = F 0 , 0 ≤ t ≤ T 0 , t > T . (21)

When 0 ≤ t ≤ T, then the response is given by Equation 22:

q r t = 2 F 0 x 1 m r ω d r 2 1 − e − ξ r ω d r t − T ⁡ cos ⁡ ω d r t − T − ξ r ω d r ω d r e − ξ r ω r t − T ⁡ sin ⁡ ω d r t − T . (22)

When t > T, then the response is given by Equation 23:

q r t = q r T e ‐ ξ r ω dr t ‐ T ⁡ cos ⁡ ω dr t ‐ T + q r T ‐ ξ r ω dr q r T ω dr e ‐ ξ r ω dr t ‐ T ⁡ sin ⁡ ω dr t ‐ T . (23)

3 Result analysis

3.1 Analysis of theoretical dynamic response characteristics

A hammer excitation test was simulated using the finite element method. An impact force was applied directly above the tool post to obtain the stiffness and damping matrices characterizing the different joints, where K 13, K 23, K 15, and K 25 are the stiffnesses of the bolted connection, and K 34, K 35, and K 45 are the stiffnesses of the moving joints. As the initially calculated stiffness and damping matrices are sparse, a matrix transformation function was employed to convert them into full matrices for subsequent computational operations. The resulting full matrices were then substituted into the right-hand side of Equation 16 to determine the actual stiffness and damping values. Substituting the parameters into the mathematical model, the theoretical natural frequency of the tool post system is obtained, shown in Table 1. The extracted stiffness parameters are K 13 = 5 × 10 5 N / m, K 23 = 1.5 × 10 6 N / m, K 34 = 5 × 10 7 N / m, K 35 = 2 × 10 6 N / m, K 45 = 8 × 10 8 N / m, K 15 = 5 × 10 5 N / m, and K 25 = 1.5 × 10 6 N / m. The extracted damping parameters are C 13 = 237.05 N . s / m, C 23 = 711.15 N . s / m, C 34 = 3873 N . s / m, C 35 = 948.2 N . s / m, C 45 = 3792 N . s / m, C 15 = 237.05 N . s / m, and C 25 = 711.15 N . s / m.

TABLE 1. Theoretical natural frequency.

The frequency response of the tool post system is shown in Figure 4. Under cutting forces, distinct response peaks occur at 7.7 Hz, 12.6 Hz, and 23.9 Hz, respectively. These frequencies correspond closely to the first three natural frequencies obtained from the theoretical calculation. From the theoretical results, the response of the left and right tool holders is very prominent at frequencies of 7.7 Hz and 12.6 Hz, while the response of other structures is relatively gentle. When the frequency of cutting excitation reaches low or medium frequency, the vibration is caused by the insufficient stiffness of the left and right tool holders. When the excitation reaches 23.9 Hz, the response of the guiding mechanism is the largest, and the responses of the left and right tool holders are basically gentle. The boring bar and the supporting mechanism also show response in this frequency band. The vibration mainly comes from the movement of the tool post system in this frequency band. Therefore, it can be concluded that the vibration response of the boring machine under cutting conditions is predominantly influenced by its first three natural frequencies.

Figure 5 shows that within the low-frequency range of 0–10 Hz, the phase of the right tool holder and the support and drive mechanism shifts from positive to negative, while the left tool holder transitions from approximately zero phase to a negative phase. The guiding mechanism remains in the negative phase region throughout this band. A negative phase indicates that the output lags behind the input, which reflects that the tool holder system is subjected to low-frequency excitation. Due to the insufficient stiffness of the tool holder structure and the high aspect ratio of the boring bar, the vibration is manifested as the bending vibration of the tool post system along the boring bar. This vibration is directly affected by the tool holder. The entire tool post system causes a sharp shift in the phase lag. Under such dynamic conditions, the tool tends to undergo significant radial jumps, thereby reducing the cylindricity of the bored hole.

In the mid-frequency band of 10–24 Hz, the right tool holder and the guiding mechanism shift directly from negative to positive phase. Similarly, the boring bar, left tool holder, and support and drive mechanism also transition from the negative to the positive phase around 20 Hz. A positive phase signifies that the output leads the input. Within this frequency range, the dominant dynamic behavior is modal coupling resonance between the boring bar and other components, primarily attributable to insufficient clamping stiffness of the tool holder, which fails to adequately suppress bending vibrations of the boring bar. This frequency band is particularly prone to the onset of cutting chatter.

In the high frequency band of 24–35 Hz, the left tool holder abruptly shifts from positive to negative phase, while the guiding mechanism remains in the negative phase. The right tool holder and support and drive mechanism stay near zero phase, and the boring bar undergoes a sharp transition from the positive to the negative phase around 35 Hz. In this band, the left tool holder forms a coupled system under cutting forces, and the vibration mode of the boring bar transitions from bending to torsional vibration. Through the analysis of the phase diagram, it can be concluded that low-frequency excitation significantly affects the machining accuracy of the heavy boring machine.

3.2 Analysis of dynamic response characteristics based on the finite element method

To ensure the results met engineering requirements, the model was configured such that the cumulative effective mass ratio exceeded 60%. The model comprised 4,568,790 nodes. The simulation natural frequencies are compared with the theory, as shown in Table 2. The frequencies are consistent. The maximum error value is 0.31%. The results show that the theory model is effective.

TABLE 2. Theoretical and simulation natural frequency contrast.

The first six vibration modes of the tool post system are extracted, as shown in Figure 6. The modal analysis reveals the following dominant motions: the first-order mode is primarily a translational movement along the x-axis. The second-order mode presents translational motion along the y-axis. The third-order mode is characterized by an alternating (or rocking) motion along the x-axis. Movement transitions to the z-axis in the fourth-order mode. The fifth-order mode features an alternating motion along the y-axis. Finally, the sixth-order mode shows a coupled motion, combining alternating translation along the x-axis with rotation about the z-axis.

To guarantee a cumulative effective mass ratio exceeding 60%, the upper limit of the sweep was set to 1.9 times the highest natural frequency of interest, and a frequency range of 0–100 Hz was selected prior to conducting the harmonic response analysis. To enhance computational efficiency, the modal superposition method was employed for the solution. The harmonic response results for the coupled 5-DOF system in the x-, y-, and z-directions are presented in Figure 7. The analysis indicates that the dominant vibration response of the system occurs along the x-axis, with pronounced peaks corresponding to the first three natural frequencies and a maximum amplitude of 4.6 mm. The results show that the vibration amplitudes in the y- and z-directions are significantly smaller than that in the x-direction.

Figure 7 shows that in the x-direction, the phase shifts repeatedly between positive and negative values, indicating that the predominant mode in this direction is bending vibration. In the y-direction, the phase transitions from negative to positive at low frequencies and remains positive thereafter, characterized by an output leading input dynamic. In the z-direction, the phase remains negative without abrupt shifts, which is consistent with a vibration mode dominated by torsional motion. The analysis of the harmonic response results indicates that the description of the spectrum and the phase curve is basically consistent with the results of the theoretical model.

3.3 Comparison of theoretical and simulation response results

The resulting peak frequencies are 5.8 Hz, 15.75 Hz, and 26.8 Hz, which correspond well with the peak frequencies from the theoretical dynamic response characteristics, with a maximum discrepancy not exceeding 3.5 Hz, as summarized in Table 3. The theoretical and simulated dynamic response of the tool post system are presented in Figure 8. The figure shows that the response curves exhibit general agreement in the locations of their peaks and troughs, thereby further validating the accuracy of the developed mathematical model.

TABLE 3. Comparison of harmonic response and theoretical peak frequency.

4 Structural response analysis of tool post with different frequencies

4.1 Transient response analysis under cutting force

The theoretical calculation and simulation results in the above analysis indicate that the tool post system is prone to resonance at the first three natural frequencies. Based on the theoretical calculation and simulation results, the transient analysis of the tool post system is carried out. The cutting force is used to study the specific deformation of the tool post system when it is subjected to the cutting force at the first three natural frequencies of 7 Hz, 12 Hz, and 24 Hz. Where the main cutting force F c is 20,000 N, the feed force F f is 5265 N, and the thrust force F r is 7020 N.

The deformation cloud of the extraction tool post under different frequency excitation is shown in Figure 9. When it is excited by 7 Hz low frequency, the maximum deformation displacement is 2.7 mm, which occurs on the tool holder. It belongs to the single large vibration problem and is chiefly affected by the vibration of the tool holder. The resonance of the tool post system is obvious. When excited by a 12 Hz intermediate frequency, the vibration began to concentrate on the tool holder, and the maximum deformation is 0.14 mm. When subjected to 24 Hz high-frequency excitation, the vibration of other mechanisms of the tool post is basically uniform, and the vibration is independent. The maximum deformation position is 2.87 mm at the tool holder. At this time, the maximum vibration area has been completely concentrated in the boring tool of the tool holder, which belongs to the normal vibration situation. The deformation cloud diagram indicates that when the tool post system is excited by a 7 Hz low frequency, the overall vibration is obvious, which has a great influence on the whole system.

4.2 The amplitude response results of the whole structure

The amplitude of the whole tool post system is shown in Figure 10. An obvious valley value appears under the excitation of the 7 Hz natural frequency, which indicates that the damping consumption ability is weak, the energy accumulates easily, and the time is long under the resonance excitation. Modal analysis revealed that the vibration mode corresponding to the first-order natural frequency is bending motion along the y-axis, which belongs to the vibration condition of low frequency and large amplitude. It can be seen in the figure that the maximum amplitude under 7 Hz excitation is 4.986 mm. When the machine works at this cutting frequency, it will directly affect the machining accuracy.

It can be seen from the figure that the attenuation rate under the action of 12 Hz excitation is faster than that of 7 Hz, which corresponds to the vibration between the tool holder and the slide rail. Because the joint structure easily dissipates the vibration energy, this vibration has little effect on the overall system. The curve waveform under 24 Hz cutting excitation is irregular and does not show a periodic amplitude. The reason for this result is that the transient response is jointly affected by free vibration and forced vibration. Due to the long action time, the free vibration process of 7 Hz and 12 Hz has been attenuated, which is dominated by forced vibration, so it shows the amplitude characteristics of a sine wave. The 24 Hz cutting excitation is a high-frequency excitation. The action time is short, and the free vibration process cannot be attenuated. The main amplitude characteristics are dominated by the natural frequency of each structure. At this cutting frequency, the structure is decoupled and vibrates according to its own natural frequency. The vibration is complex, so the amplitude curve presents multiple peaks and valleys.

From the perspective of vibration amplitude, the amplitude of 7 Hz is the largest, and the amplitude of 24 Hz is the smallest. Through the analysis of the overall amplitude response of the tool holder, the influence of 7 Hz cutting frequency on the boring machine is the largest.

4.3 The amplitude response results of each structure

The amplitude of each actuator in the tool post system under different frequency excitation is extracted as shown in Figure 11. In Figure 11a, when the low-frequency vibration is 7 Hz, the peak value appears when the tool holder is about 0.042 s, and the maximum amplitude is 4.8 mm. The figure shows that the amplitude curve of the tool holder is obviously isolated from the curves of other structures. Therefore, the vibration at this frequency is dominated by the tool holder, which belongs to the large vibration problem of a single structure. Due to the insufficient dynamic stiffness of the tool holder, the structure mainly affects the machining accuracy of the machine tool during the working process. Figure 11b shows that the amplitude of the support and drive mechanism gradually moves closer to the tool holder with a maximum amplitude of 4.78 mm under 12 Hz intermediate frequency excitation. The amplitude of the boring bar and the guiding mechanism is basically the same, and each component has a peak at 0.04 s at the same time. It can be concluded that under the excitation of this frequency, the vibration gradually spreads from the tool holder to other components. Figure 11c shows that when excited at high frequency, the phase amplitude of each component is obviously different. The maximum amplitude of the tool holder is 4.48 mm. Due to the different natural frequencies of each mechanism, decoupling and dispersion appear, and obvious peaks and valleys appear.

From the overall summary, with the continuous improvement of the cutting excitation frequency, the vibration of the heavy boring machine is transformed from the 7 Hz single large vibration problem to the 24 Hz overall decoupling decentralized vibration, but the tool holder is the key mechanism that affects the dynamic characteristics of the machine tool when the machine tool has a single large vibration.

4.4 Damping analysis results

The above analysis reveals that the maximum amplitude of the CNCT250 heavy boring machine occurs in the tool holder during the cutting process. In order to obtain a reasonable damping ratio to suppress the vibration amplitude of the tool holder, combined with the structural characteristics, resonance suppression theory, and engineering feasibility, etc., the optimal damping ratio interval for the vibration reduction optimization of the heavy boring machine is 0.01–0.05. In order to improve computational efficiency, the damping ratios used in this experiment are set to 0.01, 0.03, and 0.05 (Dassault, 2025). The amplitude of the tool holder under different damping effects is shown in Figure 12. When the damping ratio is 0.01, the vibration amplitude of 7 Hz cutting frequency in Figure 12a decreases from 4.8 mm to 4.5 mm, the vibration amplitude of 12 Hz cutting frequency in Figure 12b decreases from 4.78 mm to 4.75 mm, and the vibration amplitude of 24 Hz cutting frequency in Figure 12c decreases from 4.486 mm to 4.482 mm. When the damping ratio of the action is 0.03, the vibration amplitude of the cutting frequency of Figure 12a 7 Hz is reduced from 4.8 mm to 4.33 mm, the amplitude of the cutting frequency of Figure 12b 12H is reduced from 4.78 mm to 4.67 mm, and the amplitude of the cutting frequency of Figure 12c 24 Hz is reduced from 4.48 mm to 4.37 mm, and the vibration suppression effect gradually begins to appear. When the damping ratio of the action is 0.05, the amplitude of the 12(a) 7 Hz cutting frequency is reduced from 4.6 mm to 4.18 mm, the amplitude of the 12(b) 12 Hz cutting frequency is reduced from 4.78 mm to 4.57 mm, and the amplitude of the 12(c) 24 Hz cutting frequency is reduced from 4.48 mm to 4.35 mm, and the vibration suppression effect is the most obvious.

From the above analysis, it can be concluded that with the increase of damping, the amplitude under the action of each cutting frequency decreases, among which the amplitude decreases most under the action of the 0.05 damping ratio, and the amplitude decreases very clearly at the 7 Hz cutting frequency. The method of increasing structural damping is used to suppress the vibration of the boring machine, mainly for low-frequency cutting excitation, and the amplitude under high-frequency cutting excitation is basically not affected by damping.

The sensitivity Pareto diagram of damping change under different excitations is shown in Figure 13. The figure shows that when the damping ratio changes, the influence on 7 Hz excitation is 12.92%, the influence on 12 Hz excitation is 4.39%, and the influence on 24 Hz excitation is 2.9%. From the overall impact analysis, the damping ratio change has the greatest impact at 7 Hz, and the cumulative impact reaches 63.9%. 7 Hz belongs to low-frequency vibration, which has a great influence on the machining accuracy of a heavy boring machine. Therefore, by optimizing the damping ratio parameters, the 7 Hz cutting excitation amplitude can be suppressed. It does not change the effect of other cutting frequencies.

5 Conclusion

The tool post system of a heavy boring machine examined in this article is established as a mathematical model of a 5-DOF system and beam coupling. Based on the dynamic theory and Bernoulli beam theory, the theoretical frequency response characteristics of the tool post system are obtained. The correctness of the mathematical model is verified based on the finite element method, and the results show that the error is within the acceptable range. On the basis of the parameters calculated by the mathematical model, based on the finite element transient analysis method, the deformation of each structure of the tool post system under different cutting frequencies is studied. The results show that the maximum amplitude structure is the tool holder, and the low-frequency excitation has the greatest influence on the vibration of the tool post system. In order to suppress the vibration of the tool holder under low-frequency excitation, the damping characteristics are analyzed.

In the future, heavy boring machine design must focus on the optimization of the tool holder, which can suppress vibration by means of material damping or structural damping.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fmech.2026.1881979/full#supplementary-material

Supplementary material Supplementary material at the source

Supplementary material Supplementary material at the source

Footnotes

Edited by: Roman Protasov, Slovenska technicka univerzita v Bratislave Strojnicka fakulta, Slovakia

Reviewed by: Zhenkun Guo, Beijing University of Civil Engineering and Architecture, China

Mengrui Shi, Tianjin Chengjian University, China