Numerical and experimental investigations on a friction ring damper for a flywheel

A damping strategy using a friction ring damper for an industrial flywheel was numerically and experimentally investigated. The friction ring damper, located on the arms of the flywheel, was experimentally found to effectively reduce the vibration amplitude of the flywheel. The vibration energy was dissipated when relative motions occur at the friction contact interfaces. Nonlinear dynamic analysis based on a lumped-parameter model of a flywheel equipped with a friction ring damper was conducted. The normal load, N, was used to evaluate the damping performance of the friction ring damper. For several values of N, steady-state responses under harmonic excitation and nonlinear modes were obtained using the harmonic balance method combined with the alternating frequency–time domain method. The forced response analysis proved the existence of an optimal value of N, which could minimize the vibration amplitude of the flywheel. The nonlinear modal analysis showed that all the damping ratio–frequency curves were completely coincident even for different values of N, and the frequency corresponding to the maximum damping ratio was equal to the frequency at the intersection of the forced response curves under the fully slip state and the fully stick state of the friction contact interface. By analyzing the behaviors of the friction contact interface, it was shown that the friction contact interface provides damping in the combined stick–slip state. The forced response under random excitation was calculated using the Runge–Kutta method, and the friction interface behaviors were analyzed. Finally, spectral testing was conducted to verify the numerical results.


Introduction
This study focused on the vibration suppression of flywheels. Flywheels are widely used to control the attitude and maintain the stability of satellites. The cycle of working life of a satellite has gone through two phases: the launch phase and the in-orbit operation phase. When running in orbit, the flywheel is in the working state and is rotating. The vibration of the flywheel in orbit operation phase is mainly the micro-vibration, which is caused by the unbalanced mass of the wheel body and the irregularity of the bearing. In the launch phase, the flywheel does not rotate. The vibration of the flywheel in the launch phase is mainly excited by the random excitation of the rocket. This study is aimed at vibration suppression of the flywheel in the launch phase where the flywheel does not rotate. In the launch phase, the flywheel is excited to a violent resonance under severe random vibrations. According to estimation based on the resonance intensity of the flywheel, the transmitted force acting on the bearing caused by the vibration of the wheel body greatly exceeds the static load capacity of the bearing, putting the bearing at risk of damage. Meanwhile, the inherent damping in the flywheel is very low, and there is nearly no source of energy dissipation. Therefore, the flywheel's ability to withstand these vibrations mainly depends on the external inputs of damping; therefore, it is necessary to develop new methods and devices to damp the vibration of the flywheel.
In previous studies, the use of a flexible dynamic vibration absorber (DVA) was proposed to suppress the high-frequency micro-vibrations of the flywheel [1][2][3]; in this method, rubber pads were used to increase the stiffness and damping of the DVA. In this study, friction pads made of epoxy phenolic glass cloth are used to replace the rubber pads in the damping ring, and the elastic modulus of the friction pad is 1 GPa. Thus, the original flexible DVA becomes a dry friction damper mounted on the arms of the flywheel, in which contact is permanently maintained by the preload of the bolts. Friction thus occurs at the friction contact interfaces when relative motions occur between the friction pads and arms of the flywheel to dissipate the vibration energy. The friction at the contact interface is the most widely used source of external damping in bladed disk assemblies [4] and blisks (integrally bladed disks) [5], which include shroud contact in blades, platform dampers, and friction rings. Our work drew upon numerous methods of design and analysis of these devices reported in the literature. The HBM [6] is the most widely used method for the nonlinear steadystate response prediction of bladed disk assemblies and blisks with friction dampers [7][8][9][10][11], which is a computationally efficient and accurate method to perform dynamic analyses quantitatively. The nonlinear modal analysis of the bladed disks with friction contact interfaces [12,13] is based on the concept of complex nonlinear modes [14] and the frequency domain formulation of the dynamical system's equations of motion [15]. The AFT method [16] is used for calculating the implicit nonlinear friction forces. Recently, the HBM combined with the AFT to calculate the friction force in the time domain and the arc-length continuation technique to improve the convergence of the method [17] are used when performing nonlinear forced response analysis [18,19] and nonlinear modal analysis [20].
Although the bladed disk assemblies and blisks with friction dampers have been extensively studied, little numerical or experimental research has been performed on the friction ring damper for flywheels. Motivated by this gap in research, in this study, an equivalent lumped-parameter model of the flywheel with friction contact interfaces was established to study the dynamic behavior of the system. The vibration suppression performance of the friction ring damper was discussed based on nonlinear forced response analysis and nonlinear modal analysis. In addition, spectral testing was conducted to verify the numerical results.
The remainder of this paper is organized into five sections: the lumped-parameter model of the flywheelfriction ring damper system is described, and the governing equation of the system is presented in Sect. 2. The numerical analysis (nonlinear forced response analysis and nonlinear modal analysis based on HBM combined with AFT) is described in Sect. 3. The numerical results and related discussions are presented in Sect. 4. The spectral testing on the flywheel conducted without and with the friction ring damper to verify the numerical results is discussed in Sect. 5. The conclusions are presented in Sect. 6.

Dynamic modeling
The structures of the flywheel and the friction ring damper are shown in Fig. 1a, b, respectively. The structures of the flywheel equipped with the friction ring damper are shown in Fig. 1c, d, respectively. Figure 1a, d shows that the flywheel is composed of the wheel body, the support bearing assembling, a disk spring, and the flexible housing. Figure 1b, d shows that the friction ring damper consists of two damping rings, ten friction pads, five sleeves, and ten rubber rings, which is placed on the arms of the flywheel. The friction pads are stuck and positioned on the arms of the flywheel through grooves, the upper and the lower damping rings are connected by bolts and sleeves, and the friction pads are pressed on the arms of the flywheel by the two damping rings to form the friction contact interfaces, where the bolts provide normal loads.
As shown in Fig. 2a, the characteristic mode of the flywheel obtained from the finite element model is an umbrella-shaped flapping mode, which is dominated by the axial movement of the wheel body and the bearing resting on the disk spring [1]. The umbrellashaped flapping mode is the main contributor to the transmitted force acting on the bearing owing to a unit force applied at the rim of the flywheel. As shown in Fig. 2b, the aim of this study is to reduce the vibration amplitude of the flywheel and mitigate the axial force acting on the bearing by using the friction ring damper. The working principle of the friction ring damper is that, when the vibration of the flywheel at the umbrella-shaped flapping mode is excited, relative motions will occur at the friction interfaces (in Fig. 1d) between the arms and the friction pads, thus friction damping is generated, which dissipates the vibration energy and reduces the vibration of the flywheel.
According to the dynamic model shown in Fig. 2b, an equivalent 11-degree of freedom (DOF) lumpedparameter model of the flywheel-bearing system equipped with the friction ring damper is developed as shown in Fig. 2c. Figure 2c is the top view of the dynamic model, which is used to characterize the vertical direction (Z-direction shown in Fig. 2b) vibration, in which all the spring elements only provide forces in the vertical direction. In Fig. 2c, m f,i = m f (i = 1,2,…,5) and k f denote the masses and support stiffness of the flywheel, respectively; k f0 denotes the connection stiffness; m b and k b denote the mass and support stiffness of the bearing, respectively; m r,i = m r (i = 1,2,…,5) and k r0 denote the masses and the connection stiffness of the damping ring, respectively; c r denotes the viscous damping; and k g denotes  Table 1.
As shown in Fig. 2b, the 'Transmitted force' is the disturbance force transmitted to the bearing, which can be represented by the mass of the wheel body multiplied by the acceleration. Thus, a decrease in the acceleration of the rim can represent the decrease in the transmitted force. In Fig. 2c, the transmitted force can also be denoted as F b = k b x b . Thus, the decrease in F b can also represent the decrease in the transmitted force on the bearing, that is, to make x b as small as possible.
The corresponding relationships between the parameters of the two dynamic models shown in The distributions of friction contact nodes on the arms are shown in Fig. 3a, the angle between the projection of the first arm on the XY-plane and the X-axis is c 1 ¼ /; thus, the angle between the radial direction of the i-th (i = 1, 2, …, 5) arm and the X-axis is c i ¼ / þ ði À 1Þh, where h ¼ 2p=5. Five nodes at the center of the friction contact interfaces on the upper surface of the arms are taken out and used as the friction contact points in the lumped-parameter model. In addition, there are no relative motions between the arms and the friction pads in the circumferential direction shown in Fig. 3a, i.e., the friction forces in the circumferential direction are not considered.
As shown in Fig. 3b, the angle between the friction contact interface and the XY-plane is a. When the flywheel is subjected to the excitation in the axial direction, due to the umbrella-shaped flapping mode of the flywheel, the friction between the arms and the friction pads will occur, and the friction force acts in the tangential direction and dissipates energy.
A finite element model of the flywheel-bearing system is established to calculate the vibration displacements of the friction contact nodes. The finite element model of the flywheel-bearing system under the base acceleration excitation is shown in Fig. 4. In Fig. 4, the nodes on the bottom of the flywheel are coupled to a node with a mass of M = 1e4 kg, and an excitation force Ma(t)g is applied at this node, where a(t) is the acceleration of the base (as shown in Fig. 18), g = 9.80665 m/s 2 is the gravitational acceleration.
The dynamic explicit analysis is performed in ABAQUS, and the vibration displacements x f,i,x , x f,i,y , and x f,i of the i-th friction contact node along the X, Y, and Z directions are obtained, which are shown in Fig. 5a. Figure 5a only shows the vibration displacements of the first friction contact node, and the vibration displacements of the other four friction contact nodes are similar to Fig. 5a.
As shown in Fig. 3, in the finite element model, all friction contact nodes only move in the tangential direction, the friction forces of all friction contact nodes only act in the tangential direction, and the Table 1 Parameter values of the lumped-parameter model 0.9105 kg 2.824 9 10 6 N/m 1 9 10 7 N/m 1.0710 kg 1.6150 9 10 7 N/m m r k g k r0 c r -0.0990 kg 1 9 10 10 N/m 1 9 10 6 N/m 0 -friction force for the i-th friction contact node is f nl;i;tangential . As shown in Fig. 5a, during vibration, the displacement of the flywheel is very small; thus, it is considered that the angle a between the friction contact interface and the XY-plane in Fig. 3b does not change. By decomposing the friction force f nl;i;tangential in the vertical direction and in the radial direction, the friction force in the vertical direction is f nl;i;vertical ¼ f nl;i;tangential Á sin a, and the friction force in the radial direction is f nl;i;radial ¼ f nl;i;tangential Á cos a.
The function of f nl;i;vertical is to suppress the vibration of the flywheel in the vertical direction, and the function of f nl;i;radial is to suppress the expansion and contraction deformation of the rim in the radial direction. Therefore, in suppressing the vertical vibration of the flywheel and reducing the force transmitted to the bearing, it is actually the friction force f nl;i;vertical in the vertical direction that works. For the finite element model shown in Fig. 3, the vibration displacement of the i-th friction contact node on the flywheel along the tangential direction is It can be seen from Fig. 5b that when the flywheel is excited by the base acceleration excitation in the vertical direction, the vibration displacement x f;i;tan;xy is much smaller than x f;i;tan;z . Therefore, Eq. (1a) is simplified as Thus, in the finite element model, the relative displacement between the flywheel and the damping ring along the tangential direction is where x r;i;tangential and x r;i are the vibration displacements of the i-th friction contact node on the damping ring along the tangential direction and along the vertical direction, respectively.
In the finite element model, the friction contact interfaces between the flywheel and the damping ring are modeled based on the Coulomb friction model [18,21]. The Coulomb friction force along the tangential direction is where k t is the tangential stiffness of the friction contact interface; f c = l f N is the critical friction force, where l f = 0.26 is the coefficient of friction, and N is the normal load on the friction contact interface, i.e., the preload of each bolt. In this study, the normal load N is assumed to be constant during vibration, which will not contribute to the excitation force of the flywheel-friction ring damper system; x d;i;tangential is the displacement of the i-th friction contact point along the tangential direction where x d;i is the displacement of the i-th friction contact point along the vertical direction. From Eqs. (3)(4)(5), the friction force along the tangential direction (in Fig. 3b) is expressed by the vertical vibration displacement (in Fig. 2c) as follow Thus, the friction force acting in the vertical direction is where z i ¼x f;i À x r;i is the relative displacement between the flywheel and the damping ring along the vertical direction. The governing equation of the equivalent 11-DOF lumped-parameter model of the system is where M, C, and K are the mass matrix, damping matrix, and stiffness matrix, respectively; x, _ x, and € x are the displacement vector, velocity vector, and acceleration vector, respectively; and F nl and F ext are the nonlinear friction force vector and external force vector, respectively.
The matrices and vectors in Eq. (8) are given by Àk r0 0 0 Àk r0 2k r0 ; Àk f0 0 0 Àk f0 k f þ 2k f0 ; ð12Þ 3 Numerical analysis In this section, the nonlinear forced response and nonlinear mode of the flywheel equipped with the friction ring damper are derived using the HBM [11] combined with AFT [16]. Without loss of generality, the HBM-AFT method is presented for a general multi-DOFs dynamic system with a general nonlinear force.

Forced response analysis
Forced response analysis is performed based on the HBM by assuming that where N h is the retained number of harmonics, k = 1, 2, 3, …, N h . The vectors of the Fourier coefficients in Eqs. (15)(16)(17)(18)(19) are where Then, Eq. (8) can be expressed as where is the Kronecker product, and The AFT procedure is employed to obtain the frequency-domain components of the nonlinear friction force This gives the following set of the nonlinear algebraic equations Then, the Newton-Raphson method with the arclength continuation-based prediction-correction process [17][18][19] is employed to solve Eq. (28) and obtain the nonlinear forced responses, i.e., the amplitudes of all harmonics of the displacement, velocity, and acceleration of the i-th degree of freedom where A dis;i , A vel;i , A acc;i are amplitudes of the displacement, velocity, and acceleration, respectively. For the forced response analysis, the continuation parameter is the excitation frequency x.

Nonlinear modal analysis
Nonlinear modal analysis is performed based on the HBM for F ext ¼ 0 in Eq. (8), which gives the following set of the nonlinear algebraic equations Then, mass normalization of the harmonics of the eigenvector Q is used in this study where a is the modal mass of all harmonics, which is used to represent the vibration level.
Using the Newton-Raphson method with the arclength continuation-based prediction-correction process to solve Eqs. (32) and (33), the normalized harmonics of the eigenvector w can be obtained.
For the nonlinear modal analysis, the continuation parameter is the logarithm of the modal mass of all harmonics log 10 a, which is defined as the modal amplitude.

Nonlinear forced responses
When the friction contact interface is in the fully stick state (N = ?), the only connector between the flywheel and damping ring is the tangential stiffness spring k t . The nonlinear forced responses of the flywheel under harmonic excitation for several values of k t were obtained and plotted, as shown in Fig. 6. It can be seen that when the friction contact interface is in the fully stick state, the frequency corresponding to the peak of the response increases monotonically as k t increases. A modal test (in Sect. 5) was performed on the system to determine the value of k t , and the experimental results showed that the frequency of the flapping mode was 199.4 Hz. Thus, k t is taken as 8e5 N/m in the lumped-parameter model, and the frequency obtained from simulation is 200.7 Hz when the friction contact interface is in the fully stick state (N = ?) and 199.4 Hz when it is in the fully slip state (k t = 0).
Usually, the coefficient of friction is constant. In order to obtain the optimal vibration reduction performance of the friction ring damper, the normal load N on the friction contact interface needs to be adjusted to adapt to the change of the excitation force. Thus, by assigning different values to the normal load N, the influence of the normal load on the vibration reduction effect of the friction ring damper can be reflected.
The nonlinear forced responses of the flywheel were obtained with several values of N, as shown in Fig. 7 (Fig. 7b is an enlarged view of Fig. 7a). It can be seen that as N gradually increases from 0 to ?, the vibration amplitude of the flywheel first decreases and then increases, and the friction contact interface undergoes a transition from the fully slip state (N = 0) to the fully stick state (N = ?). The qualitative results are similar to those for bladed disks with under-platform dampers or friction rings [6]. When N is in the range of 4 -48, the friction ring damper greatly and effectively reduces the vibration amplitude of the flywheel. When N is in the range of 6 -12, an optimal value of N exists that can minimize the vibration amplitude of the flywheel. It can be seen from Fig. 7b-d that the friction ring damper can suppress the vibration displacement, transmitted force and acceleration of the flywheel at the same time. The optimal value of N was found to be 7.8123 N by employing a genetic algorithm [3].
As shown in Fig. 8a, b, the 2nd-30th order harmonics (k = 2, N h = 30) and only the first-order harmonic (k = 1, N h = 1) are used to obtain the response of the flywheel at the optimal value of N under harmonic excitation, respectively. It can be seen that the contribution of the 2nd-30th order harmonics to the response of the flywheel is much smaller than that of the first-order harmonic to the response of the flywheel. Forced response of the flywheel is also obtained by using the fourth-order Runge-Kutta method solving Eq. (8), which is used to verify the result obtained from the first-order HBM. The result is shown in Fig. 9. It can be seen that the result obtained from the first-order HBM is in good agreement with the result obtained from the fourthorder Runge-Kutta method. Therefore, the response of the flywheel can be accurately approximated by using only the first-order harmonic, i.e., the retained number of harmonics in Eqs. (15)- (19) could be N h = 1. Figure 10a, b shows the nonlinear modes of the system, i.e., the frequency-modal amplitude and damping ratio-modal amplitude diagrams, respectively, for several values of normal load N. As N increases, the nonlinear modes translate in the direction in which the modal amplitude becomes larger. The damping ratio-frequency diagrams at the modal amplitude shown in Fig. 10a, b are presented in Fig. 6 Effect of k t on the response of the flywheel Fig. 10c. It can be seen from Fig. 10c that all the damping ratio-frequency curves are fully coincident even for different values of N, and the frequency at the maximum damping ratio is equal to the intersection frequency of the forced response curves for N = 0 and N = ? in Fig. 10d. The nonlinear forced responses of the flywheel at N = 0, N = ?, and N = opt (the optimal value of N) are shown in Fig. 11a, where f slip and f stick are the frequencies when the friction contact interface is in the fully slip state and the fully stick state, respectively. The nonlinear modes at N = opt are shown in Fig. 11b, where some feature points on the nonlinear modes are marked out using small squares (named as points a-f). The time history of the friction force, hysteresis loop of the friction force, and time history of the displacement of the friction contact point at points a-e are plotted in Fig. 12.

Nonlinear modes and behaviors of the friction interface
It can be seen from Fig. 11a that the vibration amplitude of the flywheel is the smallest when N = opt, and the frequency at the peak response (maximum vibration amplitude) is equal to the frequency at the intersection of the response curves for N = 0 and N = ?. At the intersection frequency, the response reaches its maximum, from the previous analysis in Fig. 9c, the damping ratio also reaches its maximum. As shown in Figs. 11b and 12, when the vibration amplitude of the flywheel is less than the modal amplitude at point a, the friction contact interface is in the fully stick state, which results in zero damping. At point a, the time history of the friction force is harmonic, the hysteresis loop of the friction force is a line, and the displacement of the friction contact point is 0. Point a is a state transition point, where the state of the friction contact interface begins to change from the fully stick state to the combined stick-slip state as the modal amplitude increases. At points b and c, the time history of the friction force is combined of the horizontal section and the inclined section, which are corresponding to the slip state and the stick state, respectively; the hysteresis loop of the friction force is a closed loop; the time history of the displacement of the friction contact point is combined of the horizontal section and the inclined section, which are corresponding to the stick state and the slip state, respectively. The friction contact interface is in the combined stick-slip state, i.e., the stick state and the slip state coexist, which causes a softening and a damping effect. At points d and e, the inclined section of the time history of the friction force is extremely steep; the relative displacement between the flywheel and the damping ring is very large, resulting in a wide hysteresis loop; the time history of the displacement of the friction contact point is harmonic. The combined stick-slip state gradually approaches the fully slip state. When the vibration amplitude of the flywheel exceeds the modal amplitude at point f, the friction contact interface is in the fully slip state, the modal frequency approaches the eigenfrequency of the linear system without friction.
In the state transition of the friction contact interface from the fully stick state to the fully slip state, the frequency decreases monotonously as the modal amplitude increases from point a to point f. Correspondingly, the damping ratio increases rapidly from 0 (the fully stick state at point a) to the maximum (a) (b) Fig. 8 Contribution of the 2nd to 30th order harmonics (a) and only the first-order harmonic (b) to the response of the flywheel at the optimal value of N under harmonic excitation Fig. 9 Comparison of forced response of the flywheel obtained from the fourth-order Runge-Kutta method and from the firstorder HBM at the optimal value of N under harmonic excitation value (the combined stick-slip state at point c) and then slowly decreases to 0 (the fully slip state at point f). The nonlinear modes give directly rise to a measure of the friction damping effectiveness and reveal the stick-slip state transition behavior at the friction contact interface and the damping-enhancing mechanism of friction. The forced response of nonlinear system depends on excitation. In order to study the influence of the excitation on the nonlinear forced response of the flywheel-friction ring damper system, the nonlinear forced responses of the flywheel are depicted in Fig. 13a for different excitation levels (i.e., different amplitudes of the external force F = F 0 cos(xt)) at the optimal value of N. The motivation to compute and exploit the nonlinear modes is that the forced responses in the nonlinear systems occur in their neighborhoods. The nonlinear modes can provide valuable insight into the flywheel-friction ring damper system of the resonances, a feature of considerable engineering importance. For illustration, in Fig. 13a, the nonlinear mode at the optimal value of N in Fig. 11b is superposed to the nonlinear forced responses. It can be observed that the backbone (i.e., NM, nonlinear mode) traces the locus of the nonlinear forced response peaks, which reveals the frequency- amplitude characteristic of the nonlinear mode. When the amplitude of the excitation force increases, due to the existence of friction, the frequency of the flywheelfriction ring damper system gradually decreases, which reveals the softening characteristic of this nonlinear system, and the decrease in the damping ratio results in a narrower resonance peak for larger amplitude of the excitation force.
The nonlinear forced response and the nonlinear mode at N = opt (Fig. 11a, b) are plotted on the same graph in Fig. 13b. The vibration amplitudes of the flywheel at point 1 and point 5 on the nonlinear forced response curve are the same as the modal amplitude at point a on the nonlinear mode diagram. As discussed earlier, the friction contact interface is in the fully stick state at point 1 and point 5. Point 3 is the peak on the nonlinear forced response curve, at which point the vibration amplitude is the same as the modal amplitude at point c on the nonlinear mode curve, the damping ratio just reaches its maximum value at the peak (point 3) of the nonlinear forced response curve. The frequency range between f s at point 1 and f e at point 5 is the non-coinciding section for the forced response curves at N = opt and N = ? (stick state). The vibration amplitudes in the non-coinciding section lie between the modal amplitudes at point a and point c. Therefore, in the non-coinciding section in Fig. 13b, the friction contact interface is in the combined stick-slip state and is able to provide friction damping.
The time history of the friction force and the corresponding hysteresis loop at points 1-5 in Fig. 13b are showed in Fig. 14a, b, respectively. The time history of the displacement of the friction contact point and the slip ratio at different frequencies are displayed in Fig. 14c, d, respectively. As shown in Fig. 14a, b, the horizontal section and the inclined section of the friction force curves correspond to the slip state and the stick state, respectively. In contrast, for the displacement curves of the friction contact point, the horizontal section and the inclined section correspond to the stick state and the slip state, respectively (Fig. 14c). The slip ratio is defined to reveal the damping effect of the friction ring damper. In Fig. 14c, the slip ratio is defined as the ratio of the time that the friction contact point is in the fully slip state to the total period of vibration, the shape of the slip ratio curve in Fig. 14d is similar to the shape of the damping ratio-modal amplitude curve in Fig. 11b, and the slip ratio can be observed to be in the range of 0-45%. At point 3, the slip ratio close to the maximum value. At point 1 and point 5, the friction contact point is in the fully stick state, i.e., no slip occurs. Therefore, the slip ratios, areas of the hysteresis loops, and displacements of the friction contact point are all zero, and the time histories of the friction forces are harmonic. As shown in Fig. 14, as the frequency increases from point 1 to point 5, the area of the hysteresis loop, displacement of the friction contact point, and slip ratio of the friction contact point firstly increase rapidly from zero to the maximum and then decrease slowly to zero. This observation is consistent with the law of damping ratio change with modal amplitude (Fig. 11b).

Modal testing
Modal test setup of the flywheel equipped with the friction ring damper is shown in Fig. 15. The test equipment used are the LMS Test. Lab vibration and noise testing and analysis system, an impact hammer and a computer. The test method is the impact hammer testing. An acceleration sensor is firmly attached to the rim of the flywheel, and the mode of the flywheel is obtained by the method of multi-point hammering.
The distribution of hammering points is shown in Fig. 16a, and the umbrella-shaped flapping mode of the flywheel equipped with the friction ring damper obtained from modal test is shown in Fig. 16b. The frequency range obtained is [199 Hz,199.8 Hz]; thus, the average value of 199.4 Hz is taken as the inherent frequency of the flywheel with friction ring damper.

Spectral testing
Spectral testing of the flywheel was conducted without and with the friction ring damper to confirm its vibration suppression effect. Photographs of the spectral testing setup are shown in Fig. 17. Random excitation was produced through the vibrating table (Fig. 17a), and acceleration data were measured using the accelerometers (Fig. 17b). As shown in Fig. 17b, we arrange an accelerometer on the bearing to measure the acceleration. The total inertial force of the flywheel transmitted to the bearing can be represented by the mass of the flywheel multiplied by the acceleration at the bearing; thus, a decrease in acceleration at the bearing can represent the decrease in the transmitted force. The experimental results are presented in Figs. 18 and 19.  Figure 19a, b compares the acceleration power spectral density (PSD) of the flywheel and the bearing without and with the friction ring damper, respectively. According to Fig. 19a, the maximum acceleration PSD of the flywheel without the friction ring damper is 1.353 g 2 /Hz at the frequency 206.8 Hz, and the maximum acceleration PSD of the flywheel with the friction ring damper is 0.8296 g 2 /Hz at the frequency 200.7 Hz. The maximum acceleration PSD of the flywheel is reduced by 38.68% with the friction ring damper as compared to that without the friction ring damper. According to Fig. 19b, the maximum acceleration PSD of the bearing without the friction ring damper is 0.2791 g 2 /Hz at the frequency 206.8 Hz, and the maximum acceleration PSD of the bearing with the friction ring damper is 0.1746 g 2 /Hz at the frequency 200.7 Hz. The As shown in Fig. 20, we removed the harmonic excitation force F in the lumped-parameter model in Fig. 2c and replaced the fixed foundation under the bearing stiffness with a mass of M = 1e4 kg. An excitation force Ma(t)g is applied at this mass point, where a(t) is the acceleration of the base (as shown in Fig. 18), g = 9.80665 m/s 2 is the gravitational acceleration.
The governing equation of the equivalent 12-DOF lumped-parameter model of the system is The matrices and vectors in Eq. (34) are given by In Eqs. (35-37), the expressions of M r , M f , C r , K rr , K ff are shown in Eqs. (9)(10)(11)(12).
The relative displacement during vibration between the flywheel and the damping ring in Eq. (7) becomes where x M is the displacement of mass M. By taking the acceleration of the vibrating table in the time domain as input, and directly solving Eq. (34) using the fourth-order Runge-Kutta method, the displacement, velocity and acceleration of the system in the time domain, the friction force and displacement of the friction contact point can be obtained at the same time.  Fig. 22. It can be seen that the numerical results are consistent with the experimental results.
The friction forces under base excitation of the vibrating table are shown in Fig. 23. As shown in Fig. 23a, b (Fig. 23b is an enlarged view of Fig. 23a), the friction force is no longer periodic, when the stick force F kt of the tangential stiffness spring k t exceeds the critical friction force, slip occurs at the friction contact interface. As shown in Fig. 23c, the friction force curves are a family of hysteresis loops. The Fast Fourier Transform is performed on the friction force to obtain the friction force in the frequency domain, as shown in Fig. 23d, where it can be seen that the friction force peaks at the first resonance frequency (200.5 Hz) of the flywheel and at its triple resonance frequencies (596.8 Hz).  The time history of the displacement of the friction contact point under base excitation of the vibrating table is obtained and shown in Fig. 24a, b, where Fig. 24b is an enlarged view of Fig. 24a. The displacement curve of the friction contact point shows a staircase shape, where the horizontal section and the inclined section correspond to the stick state and the slip state of the friction contact interface, respectively.
In each step of the stair, we define the duration of the inclined section divided by the sum of the duration of the horizontal and the inclined sections as the slip ratio in a stair step. The slip ratios of all steps are obtained, as shown in Fig. 24c, d, where Fig. 24d is an enlarged view of Fig. 24c. It can be seen that the slip ratios of all stair steps are in the range of 0-40%, which is similar to the range of slip ratio shown in Fig. 14d.

Conclusions
A damping strategy for an industrial flywheel using a friction ring damper is investigated numerically and experimentally. The vibration reduction performance of the damper is predicted based on the established lumped-parameter model of the flywheel with friction contact interfaces. The model is numerically analyzed using the HBM combined with AFT to obtain the nonlinear forced responses under harmonic excitation, nonlinear modes of the flywheel equipped with the friction ring damper, and the behaviors of the friction contact interface. The damping performance of the friction ring damper is discussed based on numerical results. The results show that when the friction contact interface is in the fully stick state, the frequency increases monotonically as the tangential stiffness increases. The normal load N is used to evaluate the damping performance, and it is shown that an optimal value of N exists, which can minimize the vibration (a) (b) (d) (c) Fig. 22 Comparison of acceleration PSD of the flywheel (a, b) and bearing (c, d) obtained by Runge-Kutta method and spectral testing without/with the friction ring damper amplitude of the flywheel. In addition, all the damping ratio-frequency curves are completely coincident even with different values of N, and the frequency at the maximum damping ratio is equal to that at the intersection of the forced response curves under the fully slip state and the fully stick state. When the friction contact interface is in the combined stick-slip state, the friction contact interface is able to provide friction damping. The friction interface behaviors under random excitation are also analyzed. The slip ratio is defined as the ratio of the time when the friction contact point is in the fully slip state to the total period of vibration, which is shown to be in the range of 0-45% at the optimal value of N. Finally, spectral testing is conducted to confirm the vibration suppression effect of the friction ring damper and experimentally verify the numerical results.