Keywords

1 Introduction

Gas foil bearings (GFBs) are increasingly used in oil-free and high-speed as well as high-temperature fields and show great potential in the nuclear field [1, 2]. The bump-type GFB, as shown in Fig. 1, is the most studied and widely used type of GFBs [3]. Its top foil is supported by one or more compliant corrugated foil strips, and the end of the top foil and bump foil are welded to the bearing sleeve. The hydrodynamic pressure generated between the rotor and top foil acts on the foil structure while suspending the rotor. The hydrodynamic pressure deforms the foil structure, and the relative motion between the bump foil and top foil as well as the bearing sleeve is generated. Then, the friction force arises to prevent the relative motion. The friction provides energy dissipation for the system and stiffens the foil structure, thus significantly affecting the mechanical characteristics of the GFB system [4, 5].

Fig. 1.
figure 1

The configuration of the bump-type GFB system.

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The early numerical investigation on the friction inside the foil structure was carried out by Ku and Heshmat [6]. They found that the bumps near the welded end have a much higher static stiffness, and a higher friction coefficient could increase the static stiffness. Then experimental results of hysteresis curves for bump strips pressed between two parallel surfaces were presented in [7]. It was found that the local stiffness is higher than the global stiffness, and the local stiffness and damping are dependent on the excitation amplitude and static load. Following their previous work, Ku and Heshmat [8, 9] developed a theoretical model to calculate the equivalent viscous damping coefficient and structural stiffness for nonrotating journal acting on bump foil strips. Then, the experiments were carried out to investigate the dynamic characteristics of the bump foil, and the results were compared to the theoretical model with good agreement [10, 11]. The results showed that along with the increase of excitation amplitude or decrease of the static load, the local structural stiffness and equivalent viscous damping coefficient decrease, and an increase of the excitation frequency was found to increase the local structural stiffness while decrease the equivalent viscous damping coefficient. Subsequently, similar experiments were performed by many authors to investigate the structural stiffness and equivalent viscous damping coefficient [12,13,14,15,16].

The understanding of hysteresis behaviors of the bump foil structure was improved by Lez et al. [17, 18]. The simulation results evidenced the presence of stick and slip states in the foil structure, and then, the variations of the equivalent stiffness and damping coefficients could be explained by stick-slip behaviors of contact nodes. During the loading and unloading processes, the high-slope and low-slope curves correspond to stick states and slip states of most contact nodes, respectively. Besides, only the few bumps close to the free end of the bump foil structure participate in the change of stick and slip states, while the other bumps keep stuck all the time.

Larsen et al. [19] investigated the mechanical behaviors of bump foil strips pressed between two parallel surfaces, and experimental results showed complex dynamic phenomena which cannot be fully captured by classical friction model. Feng et al. [20] first introduced the original LuGre friction model into the modeling of bump foil structure. This model is capable of capturing the presliding behavior, Stirbeck effect and associated stick-slip motion [21]. The original LuGre friction model was then adopted by Zhou et al. [5] to develop a fully coupled dynamic model of the GFB system. However, this friction model was developed for cases of fixed normal contact force and may give nonphysical results under variable normal contact force conditions, which is caused by the implicit expression of normal contact force in the calculation of the friction force.

Therefore, in this paper, to further investigate the hysteresis behaviors of the bump-type foil bearing structure, the amended LuGre friction model was adopted to couple with the bump foil structure. The modeling and solution were validated by experimental data. Then, the presilding behavior, the stick-slip states during loading-unloading process and the Stribeck effect were investigated in sequence.

2 Modeling and Solution

The purpose of this work is to investigate the hysteresis behaviors of the bump foil structure. So, the system is modeled as a flexible bump foil strip pressed between two parallel rigid surfaces, as shown in Fig. 2(a). The reduction of the bump foil structure and the frictional contact between the bump foil and two surfaces are implemented, and finally, the system is solved in a fully coupled manner.

Fig. 2.
figure 2

Modeling of the system: (a) the schematic of the system and the reduction scheme of the bump foil and (b) the diagram of the frictional contact.

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2.1 Modeling of the Bump Foil Structure

The standard two-node Euler beam elements are used to model the bump foil structure. The governing equation for an element \(e\) is expressed as.

$${\mathbf{M}}^{e} {\ddot{\mathbf{q}}}^{e} + {\mathbf{C}}^{e} {\dot{\mathbf{q}}}^{e} + {\mathbf{K}}^{e} {\mathbf{q}}^{e} = {\mathbf{F}}^{e} ,$$
(1)

where \({\mathbf{q}}^{e}\) is the nodal deflection vector of the element, \({\mathbf{M}}^{e}\), \({\mathbf{C}}^{e}\) and \({\mathbf{K}}^{e}\) are the element mass matrix, damping matrix and stiffness matrix, respectively, and \({\mathbf{F}}^{e}\) is the element external force vector. The proportional damping \({\mathbf{C}}^{e} = \beta {\mathbf{K}}^{e}\) is adopted in which \(\beta\) is the proportional damping coefficient. Besides, a correction of the Young’s modulus is implemented due to the plane stress assumption.

In view of that the natural frequency of the bump foil structure is much higher than the excitation frequency in this work, the dynamic characteristics of the bump foil can be ignored. Besides, most of the nodes of the bump foil elements are unloaded. Therefore, the Guyan reduction method [22] can be adopted to improve the computational efficiency while still guarantee the accuracy. Based on the geometric configuration of the bump foil, only the highest nodes and corner nodes of the bumps are assumed to be potentially loaded. These potentially loaded nodes are defined as boundary nodes of substructures and the left over nodes are interior nodes, as shown in Fig. 2(a).

By assembling the corresponding elements, the governing equation of a substructure can be obtained as

$${\mathbf{M}}_{s} {\ddot{\mathbf{q}}}_{s} + {\mathbf{C}}_{s} {\dot{\mathbf{q}}}_{s} + {\mathbf{K}}_{s} {\mathbf{q}}_{s} = {\mathbf{F}}_{s} ,$$
(2)

where \({\mathbf{q}}_{s}\) is the nodal deflection vector of the substructure, \({\mathbf{M}}_{s}\), \({\mathbf{C}}_{s}\) and \({\mathbf{K}}_{s}\) are the mass matrix, damping matrix and stiffness matrix of the substructure, respectively, and \({\mathbf{F}}_{s}\) is the external force vector of the substructure. Based on the mechanical characteristics of the bump foil, the governing equation of a substructure can be approximated as a static equilibrium equation

$${\mathbf{K}}_{s} {\mathbf{q}}_{s} = {\mathbf{F}}_{s} .$$
(3)

By partitioning the degrees of freedom (DOFs) of the substructure into the boundary DOFs \({\mathbf{q}}_{s,b}\) and interior DOFs \({\mathbf{q}}_{s,i}\), the static equilibrium equation can be rewritten as

$$\left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{bb} } & {{\mathbf{K}}_{bi} } \\ {{\mathbf{K}}_{ib} } & {{\mathbf{K}}_{ii} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{q}}_{s,b} } \\ {{\mathbf{q}}_{s,i} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{F}}_{s,b} } \\ {\mathbf{0}} \\ \end{array} } \right],$$
(4)

where the subscripts \(b\) and \(i\) denote the boundary and interior, respectively. The second row of Eq. (4) can be reformulated as

$${\mathbf{q}}_{s,i} = - {\mathbf{K}}_{ii}^{ - 1} {\mathbf{K}}_{ib} {\mathbf{q}}_{s,b} .$$
(5)

Then, the substructure DOFs \({\mathbf{q}}_{s}\) can be represented by the boundary DOFs \({\mathbf{q}}_{s,b}\) as

$${\mathbf{q}}_{s} = \left[ {\begin{array}{*{20}c} {\mathbf{I}} \\ { - {\mathbf{K}}_{ii}^{ - 1} {\mathbf{K}}_{ib} } \\ \end{array} } \right]{\mathbf{q}}_{s,b} = \Phi {\mathbf{q}}_{s,b} ,$$
(6)

where \(\Phi\) is the transformation matrix. Substituting Eq. (6) into Eq. (3) and premultiplying both sides by \(\Phi^{\text{T}}\), the static equilibrium equation is reduced to

$${\tilde{\mathbf{K}}}_{s} {\mathbf{q}}_{s,b} = {\tilde{\mathbf{F}}}_{s} ,$$
(7)

where \({\tilde{\mathbf{K}}}_{s}\) and \({\tilde{\mathbf{F}}}_{s}\) are reduced substructure stiffness matrix and external force vector, respectively. Note that, considering the dynamic friction model, although the dynamic characteristics of the bump foil is negligible, the mass matrix and damping matrix of the bump foil are needed to be incorporated to avoid a fully implicit solution strategy which greatly reduces the computational efficiency. The mass matrix and damping matrix of the substructure are reduced to the subspace using the same base as the stiffness matrix; that is

$${\tilde{\mathbf{M}}}_{s} = \Phi^{\text{T}} {\mathbf{M}}_{s} \Phi ,$$
(8)
$${\tilde{\mathbf{C}}}_{s} = \Phi^{\text{T}} {\mathbf{C}}_{s} \Phi .$$
(9)

Then, the reduced substructure can be viewed as a super element and is governed by

$${\tilde{\mathbf{M}}}_{s} {\ddot{\mathbf{q}}}_{s,b} + {\tilde{\mathbf{C}}}_{s} {\dot{\mathbf{q}}}_{s,b} + {\tilde{\mathbf{K}}}_{s} {\mathbf{q}}_{s,b} = {\tilde{\mathbf{F}}}_{s} .$$
(10)

By assembling all the substructure equations, the governing equation of the reduced bump foil structure can be organized as

$${\mathbf{M}}_{f} {\ddot{\mathbf{q}}} + {\mathbf{C}}_{f} {\dot{\mathbf{q}}} + {\mathbf{K}}_{f} {\mathbf{q}} = {\mathbf{F}}_{c} \left( {{\mathbf{q}},{\dot{\mathbf{q}}},{\mathbf{z}}} \right),$$
(11)

where \({\mathbf{q}}\) is the generalized nodal deflection vector of the reduced bump foil structure, \({\mathbf{M}}_{f}\), \({\mathbf{C}}_{f}\) and \({\mathbf{K}}_{f}\) are the generalized mass matrix, damping matrix and stiffness matrix of the reduced bump foil structure, respectively. \({\mathbf{F}}_{c} \left( {{\mathbf{q}},{\dot{\mathbf{q}}},{\mathbf{z}}} \right)\) is the generalized force due to the frictional contact, in which \({\mathbf{z}}\) is the generalized coordinate vector of the internal friction states of contacts.

2.2 Modeling of the Frictional Contact

In view of the reduction scheme of the bump foil, a general node-to-element contact framework is adopted in this work to model the frictional contact behavior between the bump foil and two parallel surfaces. The boundary nodes of the reduced bump foil are chosen to be the slave nodes of the contacts, and the two parallel surfaces are the master segments. For a contact pair, the contact force is composed of the normal contact force \(f_{n}\) and tangential friction force \(f_{t}\).

The normal contact force \(f_{n}\) is governed by the Hertz contact model as

$$f_{n} = k\delta + c\dot{\delta },$$
(12)

where \(\delta\) is the penetration depth, as shown in Fig. 2(b), \(k\) and \(c\) are the normal contact stiffness and damping, respectively.

The tangential friction force \(f_{t}\) is governed by the amended LuGre dynamic friction model [23]. This model is capable of capturing the Stribeck effect and dealing with conditions of significant variations in the normal contact force [24]. A typical form of the amended LuGre friction model is given as

$$f_{t} = \left( {\sigma_{0} z_{t} + \overline{\sigma }_{1} \dot{z}_{t} + \sigma_{2} v_{t} } \right)f_{n} ,$$
(13)
$$\dot{z}_{t} = v_{t} - \sigma_{0} \frac{{\left| {v_{t} } \right|}}{{g\left( {v_{t} } \right)}}z_{t} ,$$
(14)

where \(z_{t}\) is the internal friction state of the contact point, which can be seen as the average microscopic deflection of the contact surface, while \(v_{t}\) is the macroscopical tangential velocity of the contact point, as illustrated in Fig. 2(b). The function

$$\overline{\sigma }_{1} = \sigma_{1} e^{{ - \left( {{{v_{t} } \mathord{\left/ {\vphantom {{v_{t} } {v_{c} }}} \right. \kern-0pt} {v_{c} }}} \right)^{2} }} ,$$
(15)

with a sufficiently small transition velocity \(v_{c}\) is applied to make the friction model passive and well micro-damped [25]. \(\sigma_{0}\), \(\sigma_{1}\) and \(\sigma_{2}\) are the friction stiffness, micro damping and viscous damping coefficients per unit of normal force, respectively. \(g\left( {v_{t} } \right)\) is a function that captures the Stribeck effect and is expressed as

$$g\left( {v_{t} } \right) = \mu_{k} + \left( {\mu_{s} - \mu_{k} } \right)e^{{ - \left( {{{v_{t} } \mathord{\left/ {\vphantom {{v_{t} } {v_{s} }}} \right. \kern-0pt} {v_{s} }}} \right)^{2} }} ,$$
(16)

where \(\mu_{k}\) and \(\mu_{s}\) are the kinetic friction coefficient and maximum static friction coefficient, respectively, and \(v_{s}\) is called the Stribeck velocity. For a constant velocity, the steady-state friction force \(f_{t,s}\) is written as

$$f_{t,s} = \left( {g\left( {v_{t} } \right){\text{sgn}} \left( {v_{t} } \right) + \sigma_{2} v_{t} } \right)f_{n} .$$
(17)

Denoting the generalized coordinate vector of all the internal friction states as \({\mathbf{z}}\), the governing equation of the dynamic friction is established as

$${\dot{\mathbf{z}}} = {\mathbf{R}}\left( {{\mathbf{q}},{\dot{\mathbf{q}}},{\mathbf{z}}} \right).$$
(18)

2.3 System Coupling and Solution

Eventually, the governing equations of the system comprise Eq. (11) and Eq. (18). Denoting the state vector of the system as

$${\mathbf{y}} = \left[ {\begin{array}{*{20}c} {\mathbf{q}} \\ {{\dot{\mathbf{q}}}} \\ {\mathbf{z}} \\ \end{array} } \right],$$
(19)

the governing equations of the system can be arranged in a fully coupled form as

$${\mathbf{M}}\left( {\mathbf{y}} \right){\dot{\mathbf{y}}} = {\mathbf{F}}\left( {\mathbf{y}} \right).$$
(20)

All the state variables are solved simultaneously by the variable-step and variable-order implicit integrator ode15s in MATLAB [26], and the analytical Jacobian matrix \({{\partial {\mathbf{F}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{F}}} {\partial {\mathbf{y}}}}} \right. \kern-0pt} {\partial {\mathbf{y}}}}\) is implemented to improve the computation efficiency.

3 Results and Discussions

In this section, the hysteresis behaviors of the bump foil are investigated. The parameters of the bump foil are listed in Table 1 and the parameters of the friction and simulation are listed in Table 2.

Table 1. Parameters of the bump foil.
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Table 2. Parameters of the friction and simulation.
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3.1 Modeling Verification

The modeling of the bump foil structure and the solution strategy are verified in this subsection by comparing with the test results under dynamic excitation in Ref. [19]. The excitation is implemented through the sinusoidal amplitude perturbation of the upper surface. The kinetic and maximum static friction coefficients are both set as 0.2 and the excitation frequency is 1 Hz. The simulation is carried out with different friction stiffness per normal force \(\sigma_{0}\), and the results are given in Fig. 3.

Fig. 3.
figure 3

Comparison of hysteresis curves under different friction stiffness per normal force with test result.

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As can be seen, with a high friction stiffness per normal force, the hysteresis curve obviously possesses four stages during one loading-unloading process. Along with the decrease of the friction stiffness per normal force, the boundaries between the two stages in loading and unloading processes are increasingly blurred, and the dissipation of energy (the area of the hysteresis curve) is accordingly decreased. Besides, the simulation results are in good agreement with the test data, especially for the case of \(\sigma_{0} = 80\,{\text{mm}}^{ - 1}\).

3.2 Presliding Behavior

To explain the variation trend of hysteresis curves in Fig. 3, the details of the friction are illustrated in Fig. 4 in the form of normalized friction and velocity of node 2 with the first half period being the loading process and the second half period being the unloading process. The friction force is normalized by the maximum static friction force (equal to the kinetic friction force in this case), and the tangential velocity is normalized by the product of excitation amplitude and excitation angular frequency.

Fig. 4.
figure 4

Normalized friction force and tangential velocity of node 2 during loading-unloading process under different friction stiffness per normal force.

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As can be seen, for a high friction stiffness per normal force, the stick and slip states are quite distinguishable; that is, when the friction force is less than the maximum static friction force, the tangential velocity stays low to approximate the stick state. Along with the decrease of the friction stiffness per normal force, it is more and more difficult to distinguish the stick and slip states, as there is a considerable tangential velocity even though the friction force is less than the maximum static friction force. This phenomenon is physical and is due to the presliding behavior of the contact point. Therefore, the friction stiffness per normal force \(\sigma_{0}\) should be chosen based on the experimental data to capture reasonable presliding behavior.

3.3 Stick-Slip States During Loading-Unloading Process

As indicated in Fig. 4, a low friction stiffness per normal force would make it difficult to distinguish the stick and slip states. So, to investigate the changes in friction states of contact nodes, a high friction stiffness per normal force \(\sigma_{0} = 1000\,{\text{mm}}^{ - 1}\) is adopted in this subsection. The stick state is recognized when the normalized friction force is higher than 0.99, and the results are plotted in Fig. 5 with four key points.

Fig. 5.
figure 5

Changes in stick-slip states of all the contact nodes during loading-unloading process.

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As can be seen, the high-slope loading curve from key point 1 to 2 is caused by the stick states of all contact nodes, while the low-slope loading curve from key point 2 to 3 corresponds to slip states of almost all contact nodes. The unloading process is similar to the loading one except for a longer stick stage. Around key point 2 and 4, the contact nodes change from the stick state to slip state in sequence, beginning at the free end of the bump foil. Note that, there would be one or more nodes close to the welded end of the bump foil that are always stuck.

3.4 Stribeck Effect

In this subsection, the influence of the Stribeck effect on the hysteresis curves is investigated with \(\sigma_{0} = 80\,{\text{mm}}^{ - 1}\), \(\upmu_{k} = 0.1\) and \(\upmu_{s} = 0.2\). The simulation is carried out with different excitation amplitudes under an excitation frequency of 10 Hz. The results are given in Fig. 6.

Fig. 6.
figure 6

Influence of the Stribeck effect on the hysteresis curves with different excitation amplitudes.

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As can be seen, when taking into account the Stribeck effect, the hysteresis curves are dominated by different friction coefficients in different stages. The low-slope loading and unloading curves are obviously dominated by the kinetic friction coefficient as most of the nodes are sliding, while the high-slop loading and unloading curves are governed by the maximum static friction coefficient in the form of extended length. Near the end of the loading and unloading processes, the tangential velocities of contact nodes decrease and the friction forces increase, resulting an increasing slope of the curve. This phenomenon is observed in [19], which means that the Stribeck effect is a prevalent phenomenon in bump foil structure.

4 Conclusions

The hysteresis behaviors of a bump-type foil bearing structure were investigated in this paper. The bump foil was modeled by two-node Euler beam elements and reduced by Guyan reduction method. The amended LuGre friction model was adopted to deal with the cases of variable normal contact force. All the state variables were solved simultaneously in the fully coupled scheme, and the modeling and solution were validated by experimental data.

It was found that the friction stiffness per normal force controls the presliding behavior of the contact point. Along with the decrease of the friction stiffness per normal force, the presliding behavior is enhanced and it is more and more difficult to distinguish the stick and slip states, finally resulting a decrease of the energy dissipation of hysteresis curves. Besides, the high-slope loading and unloading curves correspond to the stick states of all contact nodes, while the low-slope ones correspond to the slip states of partial contact nodes close to the free end of the bump foil. Finally, the Stribeck effect was taken into account and the results showed that the high-slope and low-slope parts of hysteresis curves are dominated by maximum static friction coefficient and kinetic friction coefficient, respectively. Moreover, the comparisons of simulation and experimental hysteresis curves indicated that the presliding behavior and Stribeck effect are prevalent in bump foil structure and should be taken into account in the modeling of the GFB system.