A multidegreeoffreedom coupling dynamic model, which contains a joint cutterhead, an inner ring gear, a support shield body, and pinions, is established, considering the external stochastic excitations, timevarying meshing stiffness, transmission errors, clearance, and so forth. Based on the parameters of an actual project and the strong impact of external excitations, the modal properties and dynamic responses are analyzed, and the cutterhead joint surface loads are obtained and treated by rain flow count. Numerical results indicate that the low natural frequencies are 57 Hz and 61 Hz, and natural vibration modes are pinionsmotors rotational mode and translationaloverturning coupled mode of cutterhead with inner ring gear correspondingly. Besides, the axial and radial amplitude of dynamic responses are 0.55 mm and 0.25 mm, respectively. The frequencies of radial, torsional, and overturning vibrations are predominantly concentrated in 112 Hz and 120 Hz, which indicates that the vibration responses of cutterhead are mainly affected by the external excitations. Finally, as the rainflow counting results have shown, the standard deviation of the cutterhead joint surface loads in each direction increases by 12–15 times, compared with that of the external excitations; therefore inertia effect should be considered in cutterhead design. The proposed research lays a foundation for dynamic performance optimization and fatigue crack growth life assessment of cutterhead structure.
As a key component of the full face rock tunnel boring machine (TBM), the cutterhead plays the functions of crushing rock, stabilizing excavated opening, and so on, which affects the boring performance and efficiency of the whole machine [
For a long time, a great number of studies about the TBM cutterhead system design have been carried out. Samuel and Seow [
As mentioned above, scholars have studied rock fragmentation mechanism, force models of disc cutters, stochastic loads of excavation, and disc cutters’ plane layout about the TBM cutterhead system, by using the methods of similar model experiment, numerical simulation, and field test. In addition, many design methods and research achievements about multidegreeoffreedom coupling dynamic model have been obtained, which are of valuable reference to our study. However, the problem of vibration in split type of cutterhead with heavy random loads has not been previously performed. Moreover, the dynamic model in [
In the TBM cutterhead system, the multiple pinions are driven by variablefrequency motors via planetary gear reducers and couplings. Then, the pinions drive an inner ring gear clockwise by redundant control, and the inner ring gear and flange are fixed with bolts, so as to drive the cutterhead. The TBM cutterhead system with various components is shown in Figure
Components of TBM cutterhead system. 1: Cutterhead piece. 2: Main bearing. 3: Pinion. 4: Coupling. 5: Motor. 6: Reducer.
The structure of TBM split cutterhead.
In this paper, a dynamic mathematic model of TBM cutterhead system is established by using lumpedparameter method, which is shown in Figure
Multidegreeoffreedom coupling dynamic model of TBM cutterhead system.
Bendtorsion coupled dynamic model of cutterhead system
Torsionaxial coupled dynamic model of cutterhead system
Internal meshing dynamic model of cutterhead system
In the dynamic model of cutterhead system abovementioned, the moving coordinate systems are used for convenient modeling. The coordinates are illustrated in Figure
The system in Figure
Because of the low speed of cutterhead, Coriolis accelerations and centripetal accelerations are ignored in published models. The differential equations of each component are as follows, based on the Lagrange’s equation.
(1) For motors,
(2) For pinions,
Thus,
Then, the dynamic meshing loads can be calculated as
Similarly, the forces of meshing damping are expressed as
(3) For inner ring gear
(4) For cutterhead center block:
(5) For support shield body,
(6) For cutterhead pieces,
The
As discussed above, assembling the system equations in matrix form yields
There are two types of excitations in the dynamic model, which are external and internal excitations, influencing the dynamic characteristics of the cutterhead system. The external excitations depend on the time variability of the parameters of disc cutters’ layout, geological conditions, tunneling parameters, and so forth. Meanwhile, the internal excitations are affected by the timevarying meshing stiffness, comprehensive accumulated errors, bearing stiffness, and so on.
A 3D simulation model with multicutters is established, under typical geological conditions, based on the procedure of LSDYNA, due to complicated geological environments. Then, the dynamic loads between disc cutters and surrounding rock are obtained and modified reference to the field data [
Theoretically speaking, the disc cutters suffer normal forces
The total loads of cutterhead equal to the resultant force of each disc cutter, ignoring the losses in transfer process.
Considering the complexity of actual rock breaking loads, the mean normal force is equal to nominal load of the disc cutter, and it is 0.15 times the mean tangential force and 0.1 times the mean side force [
Forces acting on a normal cutter and a gauge cutter [
Normal cutters
Gauge cutters
Deduced the relationship between the loads of cutterhead and cutters, the formulae of loads in each cutterhead piece are presented by the following expressions.
Also,
In the same way,
The input torque
For spur gears, the meshing stiffness shows obvious nature of the step period, which will change suddenly when the coincidence degree is not an integer, leading to the generation of the dynamic excitation force [
The error excitations can be modeled by harmonic functions [
Each pinion bears the radial load, so the support equivalent stiffness is calculated by the following empirical formula [
According to Mechanics of Materials, the torsional stiffness of shaft connections can be expressed as
Since the positive thrust rollers and negative thrust rollers belong to radialthrust bearing, the axial stiffness is defined as [
The meshing damping coefficient is calculated by the following empirical formula [
The torsional damping coefficient of shaft connections is expressed as [
The stiffness of other components, such as cutterhead and support shield body, is calculated by using the finite element method (FEM). And the damping coefficient is calculated by the following formula [
There are not good theoretical methods for solving the dynamic equation (
Solution flowchart of the dynamic model.
Taking the cutterhead system of the hard rock TBM of a water tunnel project as a background, an application instance is presented. The relative parameters are as follows: (1) cutterhead geometry: the cutterhead diameter
Layout of cutters.
Based on the field data [
Loads time history of cutters.
Normal forces of center cutters
Normal forces of normal cutters
Side forces of normal cutters
Normal forces of gauge cutters
Side forces of gauge cutters
As can be seen in Figure
According to formulas (
Frequency spectrum results of external excitations.
Frequency spectrum results of radial force
Frequency spectrum results of axial force
Frequency spectrum results of torque
Figure
The free vibration of the linear, timeinvariant representation is considered, ignoring the damping coefficient and external excitations in formula (
Natural frequencies and vibration of the cutterhead system.
Vibration modes  Natural frequencies/Hz 

Rigid mode 

Rotational vibration of pinions and motors 

Translational and overturning coupled vibration of cutterhead and inner ring gear 

The vibration modes with normalization are illustrated in Figure
Vibration modes of the cutterhead system.
The main conclusions obtained through Table
The first natural vibration mode is rigid mode, with the rigid motion of inelastic deformation, which keeps the constant transmission ratio to each rotational part.
The amplitudes of free vibration are mainly in the middle order modes, and the low and high order modes are relatively smaller.
The lowest fifteen natural vibration modes are mainly rotational vibration of pinions and motors and translational and overturning coupled vibration around arbitrary axis of cutterhead and inner ring gear, which is consistent with the engineering example in [
With the first 2 s of external excitations, we can obtain the dynamic responses of cutterhead, as shown in Figures
Dynamic responses of cutterhead pieces.
Displacement in
Displacement in
Displacement in
Dynamic responses of the cutterhead center block.
Displacement in
Displacement in
Displacement in
Rotation in
Rotation in
Rotation in
From the dynamic results, Figure
The vibration amplitudes of cutterhead pieces in each direction are less than 1 mm, which shows that the stiffness of cutterhead system is relatively high to resist the impact of external excitations.
The vibration in each direction is similar to the variation of external excitations, which is influenced greatly.
The magnitude of amplitude in each cutterhead piece is identical, where the axial vibration is maximal, with the amplitude being up to about 0.55 mm. And the maximum amplitude of radial direction is close to 0.25 mm. It is shown that although the radial force is much less than axial force, the radial stiffness is also relatively lower, which can explain the cause of identical magnitude in each direction. However, the radial vibrations have stronger influence on the cutterhead driving system, which may cause some engineering problems, such as seal failure and abnormal wears of the bearing raceway.
Similar conclusions may be obtained from Figure
The vibration regularity of center block is consistent with cutterhead pieces, and the amplitude in each direction is slightly smaller, with the maximum amplitude of 0.48 mm in axial direction and 0.2 mm in radial directions. These responses of translational vibration provide input conditions for calculating the joint surface loads of cutterhead.
The maximum angular amplitude around
The comparison of Figure
Thus, as the results mentioned above, it is illustrated that the proposed model and method are effective and correct.
The dynamic responses of cutterhead system are not only related to the time domain but also affected by the frequency; thereby the Fast Fourier Transform (FFT) algorithm is used to generate the frequency responses, as shown in Figures
Frequency responses of the cutterhead pieces.
Frequency responses in
Frequency responses in
Frequency responses in
Frequency responses of the cutterhead center block.
Frequency responses in
Frequency responses in
Frequency responses in
Frequency responses in
Frequency responses in
Frequency responses in
As may be seen from the spectral analysis in Figures
Under the influence of timevarying internal and external excitations, the main frequencies of dynamic responses are as follows: 100–120 Hz, 224 Hz, 236 Hz, 390 Hz, and 693 Hz, which are consistent with external excitations. It is indicated that the vibration type of cutterhead belongs to forced vibration; in other words, the dynamic responses are influenced by the external excitations more greatly.
The frequencies of radial, torsional, and overturning vibrations are concentrated in 112 Hz and 120 Hz, which are in good agreement with the natural frequencies of the translational and overturning coupled vibration mode (listed in Table
The comparison of frequency responses with the natural frequencies and external excitations shows that the frequencies are basically identical, which can further demonstrate the effectiveness of the proposed model and method.
Substituting the dynamic responses of cutterhead into (
Timevarying histories and statistics results of the cutterhead joint surface loads.
Time history of the tangential force
Rain flow counting statistics of the tangential force
Time history of the normal force
Rain flow counting statistics of the normal force
Time history of the axial force
Rain flow counting statistics of the axial force
From the abovementioned results, the distribution types and characteristic values can be estimated and tested by
Distribution statistics of the cutterhead joint surface loads.
Joint surface loads  External excitations  

Mean/kN  Standard deviation/kN  Mean/kN  Standard deviation/kN  
Tangential  −139.69  93.25  139.66  7.40 
Normal  −106.27  130.03  106.32  11.04 
Axial  1410.55  915.48  1410.55  60.00 
From the data in Figure
The cutterhead joint surface loads change rapidly with a considerable discrete degree, under the influence of complex factors.
The mean of joint surface loads are highly consistent with the external excitations (minus represents the direction), while the standard deviation in each direction increases by 12–15 times. It is indicated that the obtained simulation results are correct from the results of mean joint surface loads, and the inertial effect should be considered for the structure design of cutterhead, combined with dynamic analysis. These simulation results can provide boundary conditions for dynamic performance optimization and crack propagation of the cutterhead structure.
In this paper, a multidegreeoffreedom coupling dynamic model is presented for the TBM cutterhead system. Based on the parameters of an actual project and the cutters’ forces, the structured modal properties and dynamic responses are analyzed. The main results are summarized as follows.
The lowest fifteen natural vibration modes of the cutterhead system are classified as rigid mode, rotational vibration modes of pinions and motors, and translational and overturning coupled vibration modes of cutterhead and inner ring gear, and the corresponding natural frequency is 57 Hz and 61 Hz, which is greater than rotation frequency of pinions and meshing frequency of internal excitations. However, the resonance of the cutterhead system may be inevitable due to the overlap frequencies between natural frequencies and external excitations.
The vibration responses of cutterhead are similar to the variation of external excitations, with the identical magnitude of amplitude in each translational direction, where the axial amplitude is about 0.55 mm, the radial amplitude is close to 0.25 mm, the angular amplitude around
The frequencies of dynamic responses are predominantly concentrated in 100–120 Hz, 224 Hz, 236 Hz, 390 Hz, and 693 Hz. And it is suggested that the two frequencies of 112 Hz and 120 Hz should be avoided, while carrying out the structural design of cutterhead and matching the boring parameters.
Considering the influence of internal and external excitations, it is shown that the cutterhead joint surface loads change rapidly with large amplitudes, as well as complex nonlinear characteristics. As the rain flow results have shown, the standard deviation in each direction increases by 12–15 times. It is indicated that the amplification effect of dynamic loads should be mainly considered in cutterhead structural design, so as to lay a foundation for dynamic optimization and fatigue life assessment of the cutterhead structure.
There are some further topics that should be studied, although we have obtained many effective results about TBM cutterhead system. In the next stage, we will study the parameter influence laws about dynamic characteristics, estimate the fatigue life of the cutterhead based on the joint surface loads, and carry out the field test and vibration experiment in the near further.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (Grant no. 51005033 and no. 51375001), Development Program of China (973 Program) (Grant no. 2013CB035400), National Key Technologies R&D Program of Liaoning Province (Grant no. 2011220031), and the Fundamental Research Specific Funded Program of Central University (Grant no. DUT13LK14). The authors gratefully acknowledge the reviewers and editors for their helpful comments and suggestions, which have improved the research.