Title page A mathematical calculation method for hysteresis characteristics analysis of arc-shaped finger seal

In order to accurately predict the hysteresis characteristics of finger seal, the minimum hysteresis which can directly reflect the hysteresis of finger seal is proposed to characterize the hysteresis of finger seal. The mathematical model for calculating the minimum hysteresis of finger seal is established, the correction coefficient in the mathematical model is determined, and the mathematical model is verified by experiments. The influence of the structure and working condition parameters of finger laminates on the hysteresis characteristics is studied based on the modified calculation model, and the rule of influence is obtained in the end. Research results show that the maximum error between the leakage characteristics numerical calculation of finger seal base on modified calculation model and the experiment results is 7.64%, and the mathematical model of the minimum hysteresis is reasonable and reliable. The descending order of influence degree of structural parameters on the hysteresis characteristics


Introduction
With the improvement of fuel rate, sealing, reliability and operation cost of the aero-engine, a new type of contact sealing technology which named finger seal appear. Experimental studies show that the leakage rate can be reduced by 20%~70% through using finger seal instead of the traditional labyrinth seal [1]. The manufacturing cost of finger seal is approximate 1/2 of the brush seal in the case of similar sealing effect [2], and it can effectively avoid broken wires of brush seal caused by brush wire friction. However, under the influence of the pressure difference between the upper and lower reaches during the finger seal operation, there is a great friction force between the finger laminates and the aft cover plates, which hinders the recovery of the radial deformation of the finger laminates following the rotor. Then, the contact ·3· gap increases between the finger laminates and the rotor, and hysteresis phenomenon occurs. Leakage of finger seal increases cause by hysteresis phenomenon, and sealing performance of finger seal reduced. Therefore, it has practical significance to study the hysteresis characteristics of finger seal for improving finger seal technology.
According to the literature, the research of finger seal mainly includes the following aspects, such as leakage characteristics, heat transfer characteristics, hysteresis characteristics and wear characteristics. As the most important performance evaluation index of finger seal, the leakage characteristic has been studied extensively by scholars at home and abroad.In the field of numerical computation, the spring-mass-damping equivalent model was firstly proposed by M. J. Braun [3] .Wang Xu [4] proposed the fluid-solid coupling model for calculating leakage of finger seal; Bai LH [5] deduced porous media model suitable for calculating leakage characteristics of finger seal base on the model of Chew and Hogg [6]. At present, the porous media model is the most commonly numerical calculation method used by scholars at home and abroad. For example, the porous media model was used to conduct numerical studies on the leakage characteristics of finger seals by Hu TX [7], Wang Q [8], Bai HL [9], Cao J [ [10][11] et al. Zhang YC [12] established a quasi-dynamic performance model for quantified expression of finger sealing performance with finite element analysis, which solved the problem of difficulty in calculation of leakage characteristics and large computational workload. Wang L [13] proposed an optimization model aiming at low leakage rate and low wear rate. In the field of experimental research, Du CH [14][15] set up a test bench to analyze and study the leakage characteristics of finger seal. The researches on hysteresis characteristics of finger seal are less, compared with the researches on leakage characteristics. Chen GD [16] analyzed the hysteresis characteristics of finger seals with different finger beam lines under different pressure differences, and was the first to propose the concept of hysteresis rate, and use hysteresis rate to represent the advantages and disadvantages of the hysteresis characteristics of finger seal. On the basis of research on hysteresis rate, Zhu SP [17] optimized the structural parameters of finger seal with a low hysteresis rate as the target. Su H [18] studied the mechanism of hysteresis of finger seal from relationship of power and energy. The results show that the hysteresis is caused by the difference of work done by frictional in the finger sealing process. Wang LN [19][20] used the spring-mass-damping equivalent model to study the leakage characteristics of finger seal. Bai Hl [21] studied the hysteresis characteristics of the circular-arc line finger seal by using finite element technique. The results show that the hysteresis performance of the circular arc structure is better than that of the logarithmic spiral structure. And the shorter the arc length is, the better the hysteresis performance is.
A comprehensive analysis of the current research status of brush seal, some scholars have conducted numerical studies on the hysteresis characteristics of finger seal, but mathematical model has not been established to predict the hysteresis of finger seal. According to the structural characteristics and theoretical analysis of the finger seal, the author propose-d the concept of the minimum hysteresis of the arc-shaped finger seal and established the mathematical model of minimum hysteresis of finger seal according to force analysis of single finger of finger laminates. On the basis, the author study on influence law of the structure parameters and operating parameters on hysteresis of the arc-shaped finger seal. The study results can provide the theoretical basis for improving the arc-shaped finger seal performance.

Finger seal structure
The finger seal structure is composed of forward cover plates, spacer, finger laminates,and aft cover plates fixed by rivets. The forward cover plates is located on the high pressure side of the airflow, and the aft cover plates is located on the high pressure side of the airflow.The finger laminates is composed of finger arranged along the circumferential direction of the rotor. Its working principle is as follows: the fingers of one finger laminate cover the gap between the fingers on the adjacent finger laminate to form a sealed structure. The finger foot (the free end of finger beam) and the rotor interference fit to form contact, which blocks fluid axial leakage. The structure is as shown in FIG. 1. Do is the outer diameter of the Finger Seal, Di is the inner diameter of the Finger Seal, Df is the diameter of the finger foot upper circle,Db is the diameter of the finger base circle, Rc is the arc radius of the finger beam arcs' centers, Rs is the arc radius of finger beam, θ is the finger repeat angle, θ' is the finger foot repeat angle, CL is the width of the interstice between fingers, Z is number of fingers for each finger laminate, b is the thickness of each finger laminate, Lsp is the length of finger, h is the height of the finger foot. The basic structural parameter and values are given in Table 1.

Minimum hysteresis
Finger seal is a contacting types seal, which is mainly used in high-pressure and high-rotating speed working conditions. Rotor will occur radial run out, centrifugal deformation and thermal expansion deformation with the action of friction heat at high-speed rotating. Thus, the surface of the rotor occur radial displacement, and the finger laminate which contact with surface of the rotor occur radial deformation along with the rotor radial displacement.
Due to the stiffness of the finger beam, certain restoring force can be generated when deformation of finger beam occurs. When the rotor speed decreases, the radial displacement of the rotor surface decreases, so that the finger boots will return to the original position following the radial displacement change of the rotor surface. However, under the influence of the pressure difference between the up and downstream of finger seal, there is greater friction force between between the finger laminates and the aft cover plates, which blocks the recovery of the deformation of the finger beam and makes the finger boot unable to return to the original position. When the friction is balanced with the restoring force generated by the finger beam, the deformation of the finger beam cannot be recovered following the reduction of the rotor surface displacement, so that a certain amount of residual radial deformation is left. In this case, the amount of deformation is called the minimum hysteresis of the finger seal.
The " hysteresis gap " of the finger seal is the difference between minimum hysteresis of finger seal and the rotor radial displacement, and its size is the main factor that determines the influence of hysteresis on the leakage characteristics. The size of the "hysteresis gap" depends on minimum hysteresis of finger seal, which is the remaining deformation remaining after finger seal restore. When there is a same radial displacement in the stage of radial displacement of the rotor surface decreases, the greater minimum hysteresis of the finger seal, the greater hysteresis gap is, and the greater influence of hysteresis on the leakage characteristics is. The minimum hysteresis of the finger seal not only reflects the hindrance effect of the friction on deformation of finger laminates between the finger laminates and the aft cover plate, but also can indirectly reflect the degree of the influence of the hysteresis on the leakage. Therefore, this paper proposes to use the minimum hysteresis to measure the hysteresis performance index of the finger seal, and to characterize the hysteresis characteristic of the finger seal. The greater the minimum hysteresis, the more serious hysteresis of the finger seal is, and the worse finger seal performance is.

Mathematical calculation model of minimum hysteresis
(1) Basic assumption The main reason for hysteresis of the finger seal is the result of the friction between finger laminates and the aft cover plate cause by the pressure difference between the upstream and downstream. Therefore, in order to calculate the minimum hysteresis of the finger seal, the first step is to analysis the force of finger laminate under the pressure difference between the upstream and downstream. FIG. 2 shows the axial force of the finger laminates under the pressure difference between the upstream and downstream. Pu is the uniformly distributed pressure of the upstream act on seal laminates. Pd is the uniformly distributed pressure of the downstream act on the part of seal laminates below the aft cover plate protection height. The analysis shows that the uniformly distributed pressure of the upstream act on the seal laminates is equal to the uniformly distributed pressure of the downstream act on the part of seal laminates below the aft cover plate protection height and the pressure of the aft cover plate act on the seal laminates. Force condition of a single finger beam as shown in figure 3. qu is the uniform load of the upstream pressure act on the finger beam, qd is the uniform load of the downstream pressure act on the finger beam, qf is the uniform load of the aft cover plate act on the finger beam, Ff is the friction between aft cover plate and finger beam, Fd is the restore force after the beam deformed.  The protective height of the forward cover plate and aft cover plate of finger seal is usually very small. Thus, the axial force of the downstream act on the part of seal laminates below the aft cover plate protection height can be ignored in force analysis. The aft cover plate is contact with the whole finger beam and part of finger foot. However, the contact area between the aft cover plate and finger foot is very small. The contact force between aft cover plate and part of finger foot is ignored and only consider the contact force between aft cover plate and whole finger beam in force analysis. In the analysis and calculation, the finger beam is regarded as a cantilever beam with a certain shape. According to the formation principle of finger beam and structural characteristics of the circular-arc finger seal known the width of the finger beam continuously changes from the outer diameter of the finger beam to the diameter of the finger foot upper circle. But the maximum variation of the width of the finger beam is very small, only 5% of the width of the finger beam, which can be ignored. When the rotor is assembled eccentrically, the amount of deformation of finger beam is ·6· different at different positions of finger laminates. And the finger laminates are staggered and overlapped together, so the radial deformation of two adjacent finger beams is different in the axial direction. However, the variation of deformation is not very small,which is only 5.17% of the amount of deformation, and the variation of deformation can be ignored.
To sum up, the following assumptions can be made when derive the mathematical model of the minimum hysteresis of finger seal.
Assume that the force act on the part of seal laminates below the aft cover plate protection height is zero.
1) Assume that the finger beam of finger seal is a cantilever beam with a certain shape.
2) Assume that the width of finger beam is not change from top circle of finger beam to foot circle of finger beam.
3) The radial deformation of finger laminates of each finger laminate is synchronized, and there is no relative displacement between finger laminates.
(2) Friction moment between finger seal laminate and after cover plate According to basic assumptions and axial force analysis of finger laminates, the contact force act on finger laminate by aft cover plate is the product of upstream pressure and contact area. Firstly, author takes the micro-length finger beam for analysis. The calculation formula of friction force between micro-length finger beam and the aft cover plate is as follows.
In formula (1), dFf is the friction of micro-length finger beam, Cf is the correction coefficient of friction between finger beam and aft cover plate, which need to be determined by the experiment, dN is the force act on micro-length finger beam by the aft cover plate, pu is the pressure of the upstream, hl is the width of finger beam, dl is the length of micro-length finger beam, f is the friction coefficient between finger laminates and aft cover plate.
According to the basic assumption, the finger beam is regarded as a cantilever beam with a certain shape. The friction torque of micro-length finger beam is analyzed though the analysis model shown in figure 4. The friction torque formula of the micro-length finger beam shown as formula (2 (2) In formula (1), θx is the chord tangent angle of the arc at micro-length finger beam, β is the angle between the arc normal and the radial direction of finger seal at micro-length finger beam.
The friction torque of the whole finger beam is obtained though by integrating formula (2)  The finger beam has restoring force to restore the finger beam to its original state during the radial deformation occurs. The finger beam is regarded as a cantilever beam during analyze the restoring force of finger beam. Figure 5 is the analysis model of restoring force of finger beam. The radial deformation of the finger beam is very small. Thus, the change in the chord tangent angle of the finger beam arc is also small, and the change can be ignored. The expression of the restoring torque of finger beam is shown as formula (4

·7·
In formula (4), Ceff is the correction coefficient of equivalent stiffness considering the friction between finger laminates, which is related to the shape of finger beam and the friction force between the finger laminates, and is a comprehensive correction factor. keff is the equivalent stiffness of finger beam, △r is the radial deformation of finger seal, α is the angle between the restoring force and the radial direction of brush seal. The formula for calculating α is shown as formula (5). According to definition of the minimum hysteresis and stress analysis of finger seal known, the friction torque between finger laminates and aft cover plate is equal to restoring torque of finger beam while finger beam restore to the minimum hysteresis position. Therefore, the minimum hysteresis of finger seal can be obtained though making friction torque between finger laminates and aft cover plate equal to restoring torque of finger beam. Calculation model of the minimum hysteresis is shown as formula (6). In formula (6), △rh-min is the minimum hysteresis of finger seal, Ch-min is comprehensive correction coefficient of the minimum hysteresis,which is Ch-min = Cf/Ceff..
Ch-min is a comprehensive correction coefficient which takes into account friction force between finger laminates, the shape of the finger beam, friction force between finger laminates and aft cover plate. In order to eliminate the error caused by no considering the friction between finger laminates in the basic assumption, and to make the numerical calculation results more accurate, this paper use the experimental data in literature 14 to modify the comprehensive correction coefficient.

Correction coefficient of Mathematical calculation model
The hysteresis of finger seal mainly occurs in the rotor speed drop phase. Lower the rotor speed is, more severe the hysteresis is，and bigger the leakage of finger seal is. When only considering the centrifugal expansion deformation of rotor, the radial displacement of rotor surface reduce to zero as rotor speed decreases from maximum to zero. At this moment, the radial deformation of finger laminate is equal to the maximum radial displacement of rotor surface in rotor speed rise and fall cycle process or the minimum hysteresis of finger seal. Therefore, it is possible to use the leakage of finger seal at zero rotor speed in the rotor speed drop phase to correct the minimum hysteresis, and determine the correction coefficient in mathematical model of minimum hysteresis. In this paper, the correction coefficient in the mathematical model of minimum hysteresis is determined by using the experimental data of finger seal with the interference of 0.068mm in literature 14. Figure 6 shows numerical calculation and test results of leakage along with different pressures difference at zero rotor speed in rotor speed drop phase. Author uses interpolation method to calculate finger seal recovery position, and determines the size of the minimum hysteresis according to analyzes of interference fit state and deformation. After that, Author determines the comprehensive correction coefficient at different pressure differences, according to the minimum hysteresis determined by experiment and analyze of calculation model. At last, Author obtains the relationship between Ch-min and pressure difference by nonlinear fitting, shown as formula (7). In formula (7), C0 = 0.19354，△pc=0.1281, w = 0.60086, A = 0.17782.  Figure 7 shows the comprehensive correction coefficient determined by experiment and fitting curve of comprehensive correction coefficient in the model for calculating minimum hysteresis of finger seal. Results show that the fitting curve is perfectly consistent with Ch-min determined by experiment, and the maximum error is 4.75%.

Experimental Validation of Mathematical calcul ation model
The finger laminate cannot restore following expansion deformation of rotor surface during the rotor speed rise and fall. Therefore, finger seal appears hysteresis gap, and increases leakage of the finger seal. The hysteresis gap is an important parameter that reflects the hysteresis characteristics of finger seal. Due to the limitation of the test conditions, the size of hysteresis gap cannot be directly detected in leakage experiment. However, the hysteresis gap of finger seal can be reflected though change of leakage along with rotor speed rise and fall. Therefore, the model for calculating minimum hysteresis of is verified by comparing the leakage of numerical calculation and experimental results in this paper. Figure 8 is leakage varies of finger seal along with rotor speed rise and fall. Results shown in figure, the error between result of numerical calculation without considering hysteresis and experiment result is very big, and the maximum error is 31.95%. The error between result of numerical calculation with considering hysteresis and experiment result is very small, and the maximum error is only 7.64%. The result in figure 8 proves that the numerical calculation method of finger seal which with considering the hysteresis is reasonable and correct. And also indirectly proves the calculation method of the minimum hysteresis is reasonable and correct.

Result analysis
The hysteresis characteristics of the finger seal are analyzed and studied by using the calculation model of the minimum hysteresis of finger seal established in this paper. The main research content is influence of the diameter of finger foot upper circle Df, the diameter of the finger base circle Db, the arc radius of the finger beam arcs' centers Rc, the arc radius of finger beam Rs, the finger repeat angle θ, the width of the interstice between fingers CL, the thickness of each finger laminate b etc Structural parameters and pressure difference, friction coefficient etc working condition parameters on the hysteresis characteristics of fingertip seal, and obtained corresponding influence rules. This paper also analyzes the magnitude of influence of various structural parameters on the hysteresis characteristics of finger seals.
In the analysis process, the value of different structural parameter of finger seal varies greatly. Thus, it is inconvenient to compare and analyze hysteresis characteristics of finger seals at the same level. In order to expediently study the influence law and degree of working condition parameters and structure parameters on hysteresis characteristic of finger seal, parameters are made being dimensionless in this paper. The dimensionless parameter C is introduced, and its ·9· calculation formula is shown as formula (8). In formula (8), Vmax is the maximum value of parameters, Vmin is the minimum value of parameters, V is practical value of parameters, are the maximum value, minimum value, and value of each parameter during calculation min max min Figure 9 is the variation of the minimum hysteresis of finger seal along with variation of structural dimensionless parameters. The results in figure 9 reflects the influence of structural parameters on hysteresis characteristics of finger seal. The influence of structural parameter on hysteresis of finger seal is analyzed in detail as follows. The minimum hysteresis of finger seal increases along with increase of interstice between fingers, but the amplitude of increase is very small. The primary cause is finger beam become narrow along with increase of interstice between fingers while the other parameters remain unchanged. Thereby, stiffness of finger beam become smaller, and restoring force of finger beam reduce. Thus, the minimum hysteresis of finger seal increases along with increase of interstice between fingers. The restoring force of finger seal can be improve by reducing width of interstice between fingers, so as to reduce the minimum hysteresis of finger seal. Due to the reduction of interstice between fingers reduces the allowable radial deformation of finger seal, so interstice between fingers should not be too small.

Effect of structural parameters on hysteresis characteristics
(2) Thickness of each finger laminate The minimum hysteresis of finger seal decreases along with increase of thickness of finger laminates, and the amplitude of increase is relatively large. The main reason is that stiffness of finger beam become bigger when thickness of each finger laminate increases when other parameters remain unchanged. Thus, restoring force of finger beam along with increase of thickness of finger laminates, and the minimum hysteresis reduce. Therefore, increase thickness of each finger laminate can be used to reduce hysteresis of finger seal, and it is faster and more effective to reduce hysteresis of the finger seal than changing other parameters. However, thickness of each finger laminate is not the bigger the better. The reasonable selection of thickness of each finger laminate requires comprehensive considering leakage and wear characteristics of finger seal.
(3) Diameter of the finger foot upper circle The minimum hysteresis of finger seal decreases along with increase of diameter of finger foot upper circle, and its change trend is relatively large. The effect of diameter of finger foot upper circle on the minimum hysteresis is smaller than effect of each finger laminate, but greater than effect of other parameters on the minimum hysteresis. Length and span of finger beam become shorter while increase of diameter of finger foot upper circle when the other parameters remain unchanged. If stiffness of finger beam remain the same, the reduction of span of finger beam increases the restoring force of the finger seal. Thereby, that can reduce hysteresis of finger Seal. Increasing diameter of the top circle of finger beam is also one of effective methods to reduce hysteresis of finger seal.
(4) Diameter of the finger base circle The minimum hysteresis of finger seal increases along with increase of diameter of finger base circle, and increase rate is slightly less than increase rate of minimum hysteresis along with increase of finger foot upper circle' s diameter. Length and span of finger beam become longer while increase of diameter of finger base circle when the other parameters remain unchanged. If stiffness of finger beam remain the same, the increase of span of finger beam reduces the restoring force of the finger seal. Thereby, that can increase hysteresis of finger Seal. As diameter of finger base circle increases, the increase in length of finger beam is slightly smaller than increase in length of finger beam along with increase of finger foot upper circle.

·10·
Therefore, the increase rate of the minimum hysteresis caused by increase in diameter of finger base circle is slightly less than the decrease rate of the minimum hysteresis caused by increase of diameter of finger foot upper circle. Reducing diameter of finger base circle can effectively reduce hysteresis of finger seal.

(5) Arc radius of finger beam
The minimum hysteresis of finger seal slightly decreases along with increase of arc radius of finger beam. The reason is that the increase of arc radius of finger beam not only increases the length of finger beam, but also changes the angle between finger beam profile line and radial direction of finger seal. These factors affect restoring force of finger beam, and the influence rule on the minimum hysteresis is complex. Therefore, it is necessary to calculate minimum hysteresis of finger seals with different structures before determining influence rule of arc radius of finger beam on minimum hysteresis. The minimum hysteresis decreases along with the increases arc radius of finger beam in this paper. When designing a finger seal, it is necessary to determine the change trend of the minimum hysteresis of finger seal along with arc radius of finger beam according to structural parameters of finger seal, and then adjust arc radius of finger beam to reach a reasonable value.

(6) Arc radius of the finger beam arcs´ centers
The minimum hysteresis of finger seal decreases with increase of arc radius of the finger beam arcs´ centers, and variation rate of the minimum hysteresis caused by increase of arc radius of the finger beam arcs´ centers is smaller than caused by variation of thickness of finger laminates, but larger than caused by arc radius of finger beam and width of the interstice between fingers. To analyze the reason, when other parameters remain unchanged, the length and span of finger beam decreases cause by increase of arc radius of the finger beam arcs´ centers. As a result, restoring force of finger beam increases after occurring deformation. Thus, hysteresis of finger seal reduce along with increase of arc radius of the finger beam arcs´ centers. In addition, length of finger beam depends on arc radius of the finger beam arcs´ centers and arc radius of finger beam. In the structural design, the influence of the two factors needs to be considered comprehensively.

(7) Finger repeat angle
The minimum hysteresis of finger seal decreases with increase of finger repeat angle. When other structural parameters remain unchanged, number of finger decreases while finger repeat angle increase, and finger beam become wider. Thus, stiffness of finger beam also become bigger, and restoring force of finger beam after occurring deformation increase. Therefore, hysteresis of finger seal decreases with increase of finger repeat angle. The increase of finger repeat angle can effectively reduce the hysteresis of finger seal, but the greater finger repeat angle is not the better. As a result, determining finger repeat angle need to comprehensively considered leakage characteristics and wear performance.

Comparison of leakage characteristics between carbon fiber and metal brush wire brush type seal
According to the above analysis, hysteresis of finger seal reduces along with increase of thickness of each finger laminate, diameter of the finger foot upper circle, arc radius of the finger beam arcs ' centers , finger repeat angle, arc radius of finger beam etc structure parameters of finger laminates when other parameters remain unchanged. And hysteresis increases along with increase of width of the interstice between fingers and the diameter of the finger base circle. According to analysis results in Figure 9, and the higher slope of the curve, the greater influence of the structural parameters on hysteresis of finger seal, the degree of influence on hysteresis from large to small is finger repeat angle, thickness of each finger laminate, diameter of the finger base circle, arc radius of finger beam arcs´ centers, diameter of finger foot upper circle, arc radius of finger beam, width of the interstice between fingers. This result is based on variation of dimensionless parameters. However, this result can not accurately reflect the degree of influence on hysteresis of the structural parameters, because the value variation of each parameter is different. Therefore, this paper uses variation of the minimum hysteresis of finger seal under change of unit structural parameters to accurately analyze influence of the structural parameters on the hysteresis characteristics. Figure 10 is change trend of the minimum hysteresis of finger seal along with the practical value of structure parameters. And table 2 is the variation of the minimum hysteresis of finger seal under change of the unit parameter. In figure 10 and table 2 show, the degree of influence on hysteresis from large to small is thickness of each finger laminate, finger repeat angle, arc radius of the finger beam arcs centers, diameter of finger base circle, width of interstice between fingers, arc radius of finger beam.

Effect of working conditions and structural parameters on leakage characteristics
This paper not only analyzes influence of structure parameters on hysteresis characteristic of finger seal, but also studies influence of pressure difference and friction coefficient on the hysteresis characteristic of finger seal, and obtains the corresponding influence rule. Figure 11 is variation trend of the minimum hysteresis of finger seals along with pressure difference and friction coefficient. The calculation results in the figure show that the minimum hysteresis finger seal increases along with increase of pressure difference and friction coefficient, and it is a linear increase trend. The slope of straight line of variation trend along with friction coefficient changes is smaller than along with pressure difference. The reason is that increase of pressure difference and friction coefficient can increase the friction between finger laminates and aft cover plate, then make hysteresis of finger seal increase. According to the analysis, reducing the pressure difference and friction coefficient can effectively reduce hysteresis of finger seal. The friction coefficient between finger laminates and aft cover plate can be reduced by choosing the suitable material, and then reduce hysteresis of finger seal.

Figure 11
Variation trend of the minimum hysteresis of finger seals along with pressure difference and friction coef ficient

Conclusion
(1) The author proposes that the hysteresis characteristics of finger seal can be characterized by using the minimum hysteresis. The minimum hysteresis c-an directly describe hysteresis characteristics of finger seal. This conclusion provides theoretical basis for further research on the method, which used to calculate influence of hysteresis on leakage characteristics of finger seal.
(2) The author establishes mathematical mode-l for calculating the minimum hysteresis, and get out the determining method of correction coefficient in the model in this paper. The author also determines the correction coefficient based on the experiment data in literature, and validates the model by using experiment data. The verification results show that the largest error between numerical calculation of finger seal leakage base on the corrected model and the experiment results is 7.64%, which proves the rationality of the calculation method of the minimum hysteresis.
(3) The author uses the mathematical model for calculating the minimum hysteresis of finger seal to study hysteresis characteristics of finger seal. The author study influence of structure parameters of finger laminates and working condition parameters on hysteresis characteristics of finger seal, and obtains the influence rule. The degree of influence on hysteresis from large to small is thickness of each finger laminate, finger repeat angle, arc radius of the finger beam arcs' centers, diameter of the finger foot upper circle, diameter of the finger base circle, width of the interstice between fingers, arc radius of finger beam .The research results provide theoretical basis for the structure design of finger seal.