A method for dynamic balancing of a two-mass system

Abstract

A kind of method for dynamic balancing of dual-rotor system, systematically establishes the modal dynamic balancing method of N1+N2 plane of aero-engine dual-rotor system, carries out orthogonal correction mass group calculation in the mode of high pressure as main excitation and in the mode of low pressure rotor as main excitation, and the balance of constant speed ratio condition is controlled by double parameter input method when the aero-engine dual-rotor system is actually running.In dynamic balancing, the mode of high pressure excitation and the mode of low pressure excitation are balanced separately;in the modal dynamic balancing of aero-engine dual-rotor system, the influence coefficient method is combined with the modal dynamic balancing method to calculate the test weight group and counterweight group of each balancing surface of dual-rotor system, which comprehensively has the advantages of not affecting the modal vibration of other orders in modal dynamic balancing and simple algorithm in influence coefficient method, so that the dynamic balancing effect is more obvious.The present application has control and inhibition effect on vibration exceeding standard of dual-rotor aero-engine, and provides reliable and effective guarantee for safe operation of aero-engine.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of aero-engines, and is a method for N1+N2 plane modal dynamic balancing of a double-rotor system by replacing rotor modes at an actual working speed line with rotor modes at a constant speed ratio. BACKGROUND

[0002] Modern aero-engines are developing towards higher and higher rotational speeds and more and more flexible structures in the pursuit of high performance. At present, the rotor structures of aero-engines at home and abroad are mainly double-rotor systems, and most aero-engines have intermediate bearings, which are complex in structure, strongly coupled in dynamics, and work in a harsh environment. Vibration problems have always been a "bottleneck" in the development of engines. Statistics show that about 60% of the vibration problems of aero-engines are caused by rotor imbalance. Before assembly, the high- and low-pressure rotors of the engine are respectively dynamically balanced at a high precision. After being assembled into a complete machine, the engine works at high speed, high load and high temperature, and the balance accuracy of the double-rotor system is easily destroyed. Moreover, in the working speed range of the engine, the double-rotor system has several orders of modes, and the rotor vibration is very sensitive to rotor imbalance and its changes. In addition, for future high-performance engines, "tolerable modes" are an important performance indicator, and the engine needs to "tolerate resonance" at several orders of modes, and the vibration of the rotor system will be more sensitive to imbalance. During the use of the engine, the rotor imbalance state will change frequently, and the rotor dynamic balancing needs to be performed frequently. However, for a double-rotor system, it has two groups of vibration modes of low-pressure rotor excitation and high-pressure rotor excitation, and the orthogonality of the two groups of vibration modes is affected by the rotational speed ratio. In the conventional dynamic balancing method, the modal orthogonality of the double-rotor system is ignored, which leads to mutual influence between the balanced modes. Therefore, the present application proposes a method for N1+N2 plane modal dynamic balancing of a double-rotor system by replacing rotor modes at an actual working speed line with rotor modes at a constant speed ratio.

[0003] Seve F in "Balancing of Machinery with a Flexible Variable-speed Rotor[J]", Journal of Sound and Vibration, 2003, 204:287-302, on the basis of rotor dynamics theory, combined with the finite element method and the influence coefficient balancing method, and compiled a numerical calculation method for flexible rotors. However, for a double-rotor system, due to the influence of the intermediate bearing and the gyroscopic effect, the influence coefficient of the double-rotor system under the unknown mode cannot remain constant when dynamic balancing is performed by the influence coefficient method. Therefore, a dynamic balancing method is needed, which can decouple the coupling effect of the intermediate bearing, keep the influence coefficient constant, and not affect other modes when balancing.

[0004] In the article "Modal Orthogonality and Unbalance Response of Aeroengine Dual-rotor System" by Huang Jiangbo, published in "Vibration and Shock", 2022, 41(21): 176-189, the orthogonality of high-pressure excitation mode and low-pressure excitation mode of dual-rotor system with intermediate bearing is analyzed. The conditions for modal orthogonality of dual-rotor system are derived using complex modal method. It is found that when the speed ratio of dual-rotor system is constant, the modes of dual-rotor system under their respective excitation are orthogonal with respect to the stiffness matrix and the inertia matrix, but there is no orthogonality between the modes under different excitation. The modal decomposition of unbalance response of dual-rotor system using modal orthogonality shows that the unbalance on the low-pressure rotor only excites the mode vibration of the low-pressure rotor under the main excitation; the unbalance on the high-pressure rotor only excites the mode vibration of the high-pressure rotor under the main excitation. Based on the above research conclusion, the conditions for satisfying the modal orthogonality of dual-rotor system are determined, i.e. the speed ratio of high-pressure rotor and low-pressure rotor of engine is kept constant during balancing, and during balancing, the mode under high-pressure excitation is balanced on the high-pressure rotor, and the mode under low-pressure excitation is balanced on the low-pressure rotor, which ensures that the added counterweight during balancing of high-pressure rotor does not affect the low-pressure rotor and does not affect other modes under high-pressure excitation. The low-pressure balancing still follows this rule.

[0005] In the article "Aeroengine Flexible Rotor Dynamic Balancing Method" by Wang Siqi, published in "Noise and Vibration Control", 2011, 06: 91-94+115, (DOI: 10.3969 / j.issn.1006-1355-2011.06.020), according to the characteristics of aeroengine, a forward and backward flexible rotor mixed balancing method is established by combining influence coefficient method and modal dynamic balancing method. However, this method can only be used for dynamic balancing of single-rotor system. Although this method can effectively balance the rotor system, reduce the start-stop times of engine and the number of added test weights, it cannot be implemented for complex dual-rotor system structure due to the problem of multiple source excitation and coupled vibration.

[0006] The existing aeroengine dual-rotor system modal judgment and expression method has the above-mentioned deficiencies, which restricts the research on modal characteristics of dual-rotor system. SUMMARY

[0007] In order to overcome the problems in the prior art that balancing a certain order mode within the balancing speed range affects other order modes, the modal shape is affected by the speed ratio and cannot meet the orthogonality condition, and the coupling effect of high-pressure rotor and low-pressure rotor of dual-rotor system, the present application provides a method for dynamic balancing of dual-rotor system.

[0008] The specific process of the present application is as follows:

[0009] Step 1, determine the modal order and the number of balancing surfaces of the dual-rotor system that needs to be dynamically balanced, and obtain the modal shape data.

[0010] When the low-pressure rotor excitation is the main excitation among the modal orders, the low-pressure rotor excitation modal order is determined, wherein the number of low-pressure rotor excitation modal orders that need to be dynamically balanced is NL2; when the high-pressure rotor excitation is the main excitation, the high-pressure rotor excitation modal order is determined, wherein the number of high-pressure rotor excitation modal orders that need to be dynamically balanced is NH1.

[0011] The modal order is a fourth-order modal, which is a high-pressure excitation first-order modal, a second-order modal, and a low-pressure excitation first-order modal, and a second-order modal.

[0012] Among the determined number of balancing surfaces: the number of balancing surfaces on the high-pressure rotor is N1; the number of high-pressure rotor balancing surfaces N1 is the same as the number of high-pressure rotor excitation modal orders NH1; the number of balancing surfaces on the low-pressure rotor is N2, and the number of low-pressure rotor balancing surfaces N2 is the same as the number of low-pressure rotor excitation modal orders NL2.

[0013] The determined number of balancing surfaces is four, which are high-pressure primary compressor disc x1, high-pressure turbine disc x2, low-pressure fan disc x3, and low-pressure turbine disc x4.

[0014] Among the obtained modal shape data, the kth-order high-pressure rotor excitation modal shape data is r hk , and the i th-order low-pressure rotor excitation modal shape data is r Li . For the convenience of subsequent step calculation, each order modal shape data is integrated into a matrix form. The modal shape data r hk of the high-pressure rotor as the main excitation is integrated into a modal data matrix r h , and the modal shape data r Li of the low-pressure rotor as the main excitation is integrated into a modal data matrix r L . The number of rows of the modal data matrix r h of the high-pressure rotor as the main excitation and the number of rows of the modal data matrix r L of the low-pressure rotor as the main excitation are consistent with the modal orders that need to be dynamically balanced; the number of columns of the modal data matrix r h of the high-pressure rotor as the main excitation is consistent with the selected number of high-pressure rotor balancing surfaces N1; and the number of columns of the modal data matrix r L of the low-pressure rotor as the main excitation is consistent with the selected number of low-pressure rotor balancing surfaces N2. The kth-order high-pressure rotor excitation modal data r hk is shown in formula (3), wherein each parameter represents the modal shape data of the r hk th-order modal at each balancing surface:

[0015] r hk =[rhkh1 r hkh2 … r hkN1 ] T , hk = 1,2,...,NH1 (3)

[0016] The i-th order low pressure rotor excited modal data r Li As shown in equation (4), where each parameter represents the r Li th order modal modal shape data at each balance surface:

[0017] r Li = [r LiL1 r LiL2 … r LiN2 ] T , Li = 1,2,...,NL2 (4)

[0018] The i-th order modal shape data at the four balance surfaces obtained.

[0019] The amplitude and the corresponding phase radian value of each order modal at the balance surface are shown in Table 1, where the phase is in radian and the amplitude is a dimensionless value

[0020] Table 1 Modal shape data used in dynamic balancing of a double rotor experimental machine

[0021]

[0022]

[0023] Step 2, determine the initial unbalance:

[0024] I. Determine the Li-th order modal unbalance of the double rotor system under low pressure rotor excitation

[0025] When the low pressure rotor speed of the double rotor system approaches the Li-th order critical speed of the low pressure rotor excitation ω Li , the unbalance response r(x,t) is:

[0026]

[0027] In equation (5), r Li (x) is the Li-th order modal shape data of the low pressure excitation, is the frequency response function about the speed Ω Li obtained from the prior art, is the Li-th order modal unbalance.

[0028] According to the influence coefficient of the unbalance response and the unbalance, the Li-th order modal unbalance of the rotor system is calculated The specific process is as follows:

[0029] When the low-pressure rotor rotates at an angular velocity of Ω Li and the high-pressure rotor rotates at an angular velocity of Ω hk , the vibration of the double-rotor system is measured: when no test mass is added, the initial unbalance is denoted as u0, and the unbalance response of the double-rotor system is denoted as r0; when a test mass of u1 is added, the unbalance response of the rotor system is measured as r1. The following relationship is established:

[0030] r1-r0=r Li (x)F Li (Ω Li )u1 (7)

[0031] Since the unbalance response of the double-rotor system without the test mass is r0=r Li (x)F Li (Ω Li )u0, the initial unbalance is obtained by substituting equation (7) as follows:

[0032]

[0033] In the equation, u0 is the Li-th order modal unbalance

[0034] II. Determine the hk-th order modal unbalance of the double-rotor system under the excitation of the high-pressure rotor

[0035] When the high-pressure rotor of the double-rotor system approaches the hk-th order critical speed of the high-pressure rotor ω hk , the high-pressure rotor rotates at an angular velocity of Ω hk =ω hk , the constant speed ratio is a, and the low-pressure rotor rotates at an angular velocity of Ω

[0036] The Li-th order modal unbalance is determined according to the above method The hk-th order modal unbalance of the double-rotor system under the excitation of the high-pressure rotor is determined according to the same method

[0037] Step 3. Determine the orthogonal correction mass set:

[0038] The distribution of the correction mass set t is similar to the balanced modal shape and is orthogonal to the remaining order modal shapes.

[0039] For the balanced Li-th order modal under the low-pressure excitation, the correction mass set is denoted as t Li , and for all NL2-th order modal shapes under the low-pressure rotor excitation, there are n2 orthogonal correction mass sets, where n2=NL2. The n2 orthogonal correction mass sets can be obtained from the following equation set:

[0040]

[0041] For the balanced high pressure excited mode of order hk, let the correction mass set be t hk The row vector, for all NH1 order mode shapes under high pressure rotor excitation, there are n1 orthogonal correction mass sets, where n1 = NH1. The n1 orthogonal correction mass sets are obtained by the following equation set:

[0042]

[0043] Step 4, establish the N1+N2 plane modal dynamic balance balancing method:

[0044] In the N1+N2 plane modal dynamic balance balancing method, for the unbalance of the balanced high pressure excited mode of order hk, the modal order of the balanced high pressure excitation is NH1, the number of balancing planes on the high pressure rotor system is N1 selected by step 1, and the orthogonal correction mass obtained by step 3 is t hk ; The counterweight is The counterweight added on the N1 balancing correction planes. The modal unbalance determined in step 2 The counterweight satisfies the following relationship:

[0045]

[0046] In the formula, r hk (x hk )t hk =-1, so:

[0047]

[0048] The double rotor hk order mode is the mode under high pressure rotor excitation, and the counterweight is added on the high pressure rotor correction plane to balance the rotor hk order mode unbalance. Since the selected t hk only satisfies r hk (x hk )t hk =-1, so:

[0049]

[0050] Therefore, the added high pressure rotor balancing counterweight will affect the modal unbalance of the remaining high pressure rotor excitation, that is, the hk+1 order mode unbalance state of the double rotor high pressure excitation will be affected. The counterweight The excitation frequency of the first order mode is the low pressure rotor speed, so the counterweight will not affect the modal unbalance under the low pressure rotor excitation. The process of balancing the first order mode under the low pressure rotor excitation is the method of balancing the first order mode under the low pressure rotor excitation.

[0051] After balancing the first order mode under the low pressure rotor excitation, the second order mode under the low pressure rotor excitation is balanced. When balancing the second order mode under the low pressure rotor excitation, N2 balancing surfaces are selected on the low pressure rotor to add counterweights In order to make the added counterweight not affect the balancing result of the first order mode, t hk+1 The following conditions need to be met:

[0052]

[0053] At this time, the counterweight is At this time, the counterweight is Not only contains the influence of the initial unbalance, but also contains the influence of the first order mode unbalance correction under the low pressure rotor excitation. According to the proportional relationship determined by formula (14), the counterweight Each balancing surface is added in the form of each balancing surface component.

[0054] The process of balancing the first order mode under the low pressure rotor excitation is the method of balancing the first order mode under the low pressure rotor excitation.

[0055] The method of balancing the unbalance of each order mode under the low pressure rotor excitation is similar to the method of balancing the first order and second order modes under the high pressure rotor excitation.

[0056] The dual rotor system of the aero-engine includes two balancing surfaces of the high pressure rotor and two balancing surfaces of the low pressure rotor. Within the working speed, two order modes under the high pressure excitation and two order modes under the low pressure excitation occur, and the modes to be balanced are the first order mode and the second order mode under the high pressure excitation, and the first order mode and the second order mode under the low pressure excitation.

[0057] I. Balance the first order mode under the high pressure rotor excitation.

[0058] In the dual rotor system, the orthogonal test recombination when balancing the first order mode under the high pressure excitation is the 0 degree test weight of the high pressure compressor and the 180 degree test weight of the high pressure turbine disc. The orthogonal counterweight recombination obtained according to the above process is the 150 degree high pressure compressor disc and the 10 degree high pressure turbine disc.

[0059] II. Balance the second order mode under the high pressure rotor excitation and the first order mode under the low pressure rotor excitation

[0060] If the rotor continues to speed up before the second order mode imbalance under high pressure excitation is corrected, the first order critical speed under low pressure excitation can be reached. At the same time, the second order mode under high pressure rotor excitation and the first order mode under low pressure rotor excitation are balanced. According to the method of balancing the hkth order and the hk+1th order high pressure excitation modes of the high pressure rotor, the orthogonal counterweight combination of the second order mode under high pressure excitation and the orthogonal counterweight combination of the first order mode under low pressure excitation are obtained. The orthogonal counterweight combination of the second order mode under high pressure excitation and the orthogonal counterweight combination of the first order mode under low pressure excitation are added in the form of components on the balancing surface of the high pressure rotor and the balancing surface of the low pressure rotor.

[0061] If the rotor cannot continue to speed up to the first order critical speed under low pressure excitation due to excessive vibration before the second order mode imbalance under high pressure excitation is corrected, the second order mode under high pressure excitation is balanced first, and then the first order mode under low pressure excitation is balanced. The second order mode under high pressure rotor excitation is balanced according to the method of balancing the hk+1th order mode of the high pressure rotor. The first order mode under low pressure rotor excitation is balanced according to the process of balancing the first order mode of the high pressure rotor

[0062] In the dual rotor system, when balancing the first order mode under low pressure excitation, the orthogonal test weight combination is 180 degrees for the low pressure fan disc and 0 degrees for the low pressure turbine disc. The obtained orthogonal counterweight combination is 110 degrees for the low pressure turbine disc, and no test weight is added to the fan disc.

[0063] III. Balancing the second order mode under low pressure rotor excitation

[0064] The second order mode under low pressure rotor excitation is balanced according to the similar method of balancing the second order mode under high pressure rotor excitation. Under the condition of the second order critical speed of the low pressure rotor excitation, the initial operating vibration response of the rotor and the vibration response after adding the test weight are measured. According to the method of balancing the hk+1th order mode of the high pressure rotor under high pressure excitation, the second order mode imbalance correction mass under low pressure rotor excitation is obtained, and each balancing surface is added in the form of components of each balancing surface.

[0065] At this point, the N1+N2 plane mode dynamic balancing of the dual rotor system is completed.

[0066] Compared with the prior art, the beneficial effects obtained by the present application are:

[0067] The dual rotor system of the aviation engine proposed in the present application solves the problems in the prior art that balancing a certain order mode of the dual rotor aviation engine affects other order modes, the number of engine start-stop times is large, and the coupling effect of the high and low pressure rotors during balancing is difficult to eliminate:

[0068] 1. This invention systematically establishes a modal dynamic balancing method for the N1+N2 plane of a dual-rotor system of an aero-engine. Through mathematical formulas, the orthogonal correction mass group of the dual-rotor system of the aero-engine under the modal conditions of high pressure as the main excitation and low pressure rotor as the main excitation is clearly calculated. The constant speed ratio condition of the balance can be controlled by the dual-parameter input method when the dual-rotor system of the engine is actually running.

[0069] 2. When performing dynamic balancing, the high-pressure excitation mode and the low-pressure excitation mode are balanced separately. When adding test weights and counterweights, the test weight for the high-pressure rotor excitation mode only needs to be selected on the balance surface of the high-pressure rotor, and the same applies to the low-pressure rotor. This can effectively avoid mutual interference between the balancing effects of the high-pressure rotors and avoid the coupling effect between the high-pressure and low-pressure rotors. This is because the existing technology has proposed that the imbalance distribution on the high-pressure rotor will only affect the modal vibration of the high-pressure rotor excitation mode, and the imbalance distribution on the low-pressure rotor will only affect the modal vibration of the low-pressure rotor excitation mode.

[0070] 3. In the modal dynamic balancing of the dual-rotor system of aero-engine, the combination of the influence coefficient method and the modal dynamic balancing method is used to calculate the trial and reconfiguration of each equilibrium surface of the dual-rotor system. This method combines the advantages of modal dynamic balancing not affecting the modal vibration of other orders and the simple algorithm of the influence coefficient method, making the dynamic balancing effect more obvious.

[0071] After solving the above key problems, the method of this invention is applied to the dynamic balancing of a dual-rotor system of an aero-engine. This method controls and suppresses excessive vibration in dual-rotor aero-engines, providing a reliable and effective guarantee for the safe operation of aero-engines. Figure 2 The unbalanced response at the high-pressure turbine disk was selected for comparison, and the vibration reduction ratio reached 72.4%. Figure 3 The unbalanced response at the low-pressure turbine disk was selected for comparison, and the vibration reduction ratio reached 41.4%, which shows that the method has a significant control and suppression effect on the excessive vibration of the rotor aero-engine. Attached Figure Description

[0072] Figure 1 This is a structural diagram of a dual-rotor test system for a certain type of aero-engine, which is the object of this invention. This dual-rotor test system for aero-engine is derived from the rotor structure of a certain type of aero-engine.

[0073] Figure 2 This is a diagram showing the balancing effect of this embodiment during the first-order modal dynamic balancing under high-pressure excitation. Figure 2 The vibration amplitude of the high-pressure rotor turbine disk, after undergoing first-order modal dynamic balancing under high-pressure excitation, showed a significant decrease, with the peak-to-peak value decreasing from 66.7 μm to 36.4 μm.

[0074] Figure 3is the balance effect diagram of the embodiment when the first order modal dynamic balance of the low pressure excited rotor is completed, Figure 3 The vibration amplitude of the low pressure rotor turbine disk after the first order modal dynamic balance of the low pressure excited rotor is significantly reduced, in which the peak-to-peak value is reduced from 81.5 μm to 47.5 μm.

[0075] Figure 4 is the flow chart of the present application.

[0076] In the figure: 1. low pressure rotor fan disk; 2. low pressure rotor; 3. high pressure rotor compressor disk; 4. high pressure rotor; 5. high pressure rotor compressor disk; 6. intermediate bearing; 7. low pressure rotor turbine disk; 8. unbalance response of high pressure rotor turbine disk before balancing; 9. unbalance response of high pressure rotor turbine disk after balancing; 10. unbalance response of low pressure rotor turbine disk before balancing; 11. unbalance response of low pressure rotor turbine disk after balancing. DETAILED DESCRIPTION

[0077] The embodiment is an N1+N2 plane modal dynamic balancing method established for a certain type of aero-engine double rotor system.

[0078] In the aero-engine double rotor system, the main excitation rotor is a low pressure rotor, and the speed of each main excitation rotor is the low pressure rotor speed Ω L , with the unit of r / min; the speed of each non-main excitation rotor is the high pressure rotor speed Ω h , with the unit of r / min. The line segment formed by the change relationship between the high pressure rotor speed Ω h and the low pressure rotor speed Ω L during engine operation is called the working speed line. The working speed line of the aero-engine double rotor system is determined by consulting the "Aero-engine Design Manual".

[0079] When the engine is working, the ratio of the high pressure rotor speed Ω h to the low pressure rotor speed Ω L is the speed ratio, and when the speed ratio is a constant value, the speed ratio is called a constant speed ratio; the constant speed ratio is a. In the embodiment, the value range of the low pressure rotor speed Ω L is [0, 4000]. The high pressure rotor speed Ω h is the dependent variable with the low pressure rotor speed Ω L as the independent variable. In the piecewise function expression (1) of the working speed line, 2200 is the low pressure slow speed, 4000 is the low pressure working speed, and 5450 is the high pressure slow speed, with the unit of r / min; when the low pressure rotor speed Ω L ≤ 2200 r / min, the high pressure rotor speed Ω h is equal to the low pressure rotor speed Ω LThe speed ratio of the low-pressure rotor is 2.478; when the speed of the low-pressure rotor is 2000r / min L ≤4000r / min, the speed ratio is 0.444.

[0080] The speed ratio of the low-pressure rotor is 2.478; when the speed of the low-pressure rotor is 2000r / min h ≤4000r / min, the speed ratio is 0.444. L The speed ratio of the low-pressure rotor is 2.478; when the speed of the low-pressure rotor is 2000r / min

[0081]

[0082] The speed ratio of the low-pressure rotor is 2.478; when the speed of the low-pressure rotor is 2000r / min L ≤4000r / min, the speed ratio is 0.444. h The speed ratio of the low-pressure rotor is 2.478; when the speed of the low-pressure rotor is 2000r / min

[0083] Ω h =1.3Ω L (2)

[0084] The specific process of establishing N1+N2 plane modal dynamic balance in the aero-engine double-rotor system is as follows:

[0085] Step 1, determine the modal order and the number of balance planes of the double-rotor system to be dynamically balanced, and obtain the modal vibration mode data.

[0086] I. Determine the modal order: the modal order of the double-rotor system can be obtained by the existing finite element technology. Because there are two excitation conditions of low-pressure rotor excitation and high-pressure rotor excitation in the double-rotor system, when the low-pressure rotor excitation is the main excitation, it is the low-pressure rotor excitation modal order, and the number of low-pressure rotor excitation modal orders to be dynamically balanced is NL2; when the high-pressure rotor excitation is the main excitation, it is the high-pressure rotor excitation modal order, and the number of high-pressure rotor excitation modal orders to be dynamically balanced is NH1.

[0087] The determined modal order in this embodiment is four modal orders, which are high-pressure excitation first-order modal, second-order modal, low-pressure excitation first-order modal and second-order modal.

[0088] II. Determine the number of balance planes: the number of balance planes is determined by the modal order and is consistent with the modal order to be dynamically balanced. Among them: the number of balance planes on the high-pressure rotor is N1; the number of high-pressure rotor balance planes N1 is the same as the number of high-pressure rotor excitation modal orders NH1; the number of balance planes on the low-pressure rotor is N2, and the number of low-pressure rotor balance planes N2 is the same as the number of low-pressure rotor excitation modal orders NL2.

[0089] The number of balancing surfaces determined by the embodiment is four, which are high-pressure first-stage compressor disc x1, high-pressure turbine disc x2, low-pressure fan disc x3 and low-pressure turbine disc x4.

[0090] III. Obtain modal shape data: The modal shape data is measured by actual test. The modal shape data of the aero-engine double-rotor system includes the modal shape data at the balancing surfaces. The modal shape data at the balancing surfaces includes the amplitude and the corresponding phase radian value of each order modal at the balancing surfaces. The kth order high-pressure rotor excitation modal shape data is rk hk ; the ith order low-pressure rotor excitation modal shape data is ri Li . For the convenience of subsequent calculation, the modal shape data of each order is integrated into a matrix form. The modal shape data rk hk of the high-pressure rotor as the main excitation is integrated into a modal data matrix rH h ; the modal shape data ri Li of the low-pressure rotor as the main excitation is integrated into a modal data matrix rL L . The number of rows of the modal data matrix rH h of the high-pressure rotor as the main excitation and the number of rows of the modal data matrix rL L of the low-pressure rotor as the main excitation are consistent with the modal order number to be dynamically balanced; the number of columns of the modal data matrix rH h of the high-pressure rotor as the main excitation is consistent with the number of selected balancing surfaces N1 of the high-pressure rotor; and the number of columns of the modal data matrix rL L of the low-pressure rotor as the main excitation is consistent with the number of selected balancing surfaces N2 of the low-pressure rotor. In the modal shape data, the kth order high-pressure rotor excitation modal shape data rk hk is as shown in formula (3), wherein each parameter is the modal shape data of the r hk th order high-pressure rotor excitation modal at each balancing surface:

[0091] r hk = [r hkh1 r hkh2 … r hkN1 ] T , hk=1, 2, …, NH1 (3)

[0092] The ith order low-pressure rotor excitation modal shape data ri Li is as shown in formula (4), wherein each parameter is the modal shape data of the r Li th order modal at each balancing surface:

[0093] r Li = [r LiL1 r LiL2 … r LiN2 ] T , Li=1, 2, …, NL2 (4)

[0094] The mode shape data at the four balancing planes are shown in Table 1, where the phase is in radian and the amplitude is a dimensionless value.

[0095] Table 1 Mode shape data for dynamic balancing of a two-rotor experimental machine

[0096]

[0097] Step 2, determine the initial unbalance:

[0098] From the prior art, the unbalance response of a rotor has a clear relationship with the unbalance of the rotor. At a fixed measuring point, when the rotational speed is constant, the unbalance response depends only on the size and phase of the unbalance of the rotor.

[0099] I. determine the Li-th order modal unbalance of the two-rotor system under the excitation of the low-pressure rotor

[0100] When the rotational speed of the low-pressure rotor of the two-rotor system approaches the Li-th order critical speed ω Li of the low-pressure rotor excitation, the rotational speed of the low-pressure rotor Ω Li = ω Li , the constant speed ratio value is a, a = 1.3, the rotational speed of the high-pressure rotor Ω h = aω Li , at this time, the first order modal response of the low-pressure rotor excitation is absolutely dominant in the vibration response of the two-rotor system, and the unbalance response r(x, t) is:

[0101]

[0102] In formula (5), r Li (x) is the Li-th order modal shape data of the low-pressure excitation, is the frequency response function obtained from the prior art about the rotational speed Ω Li , is the Li-th order modal unbalance.

[0103] An arbitrary phase reference is marked on the two-rotor system, and the amplitude and phase of the vibration of the two-rotor system at the Li-th order critical speed of the low-pressure excitation when carrying the initial unbalance can be measured at the same time, and the phase is expressed as:

[0104]

[0105] In formula (6): is the frequency response function at the rotational speed Ω Li = ω Li , Ω h = aω Li . First-order mode shape r Li (x M The product of the calibrated coefficients of the measurement system and the rotor system. The dual-rotor system under steady-state conditions... The unbalance is kept constant, meaning the measured unbalance response is proportional to the unbalance of the dual-rotor system. The Li-th modal unbalance of the rotor system is calculated based on the influence coefficients between the unbalance response and the unbalance. The process is as follows:

[0106] At a low-pressure rotor speed of Ω Li And the high-voltage rotor speed is Ω hk When measuring the vibration of a dual-rotor system: During measurement, without trial rearrangement, the initial unbalance is denoted as u0, and the unbalance response of the dual-rotor system is denoted as r0; after adding trial rearrangement and reassembling u1, the measured unbalance response of the rotor system is r1. Then the following relationship holds:

[0107] r1-r0=r Li (x)F Li (Ω Li )u1 (7)

[0108] Since the unbalanced response of the dual-rotor system without additional recombination is r0 = r Li (x)F Li (Ω Li Substituting u0 into equation (7), the initial unbalance is:

[0109]

[0110] In the formula, u0 is the unbalance quantity of the Li-th order mode.

[0111] II. Determine the hk-th order modal imbalance of the dual-rotor system under high-voltage rotor excitation.

[0112] When the high-voltage rotor speed of the dual-rotor system approaches the hk-th order critical speed ω of the high-voltage rotor excitation. hk At that time, the high-voltage rotor speed Ω hk =ω hk The constant speed ratio is a, and the low-pressure rotor speed is...

[0113] The Li-th order modal imbalance is determined as described above. The same method was used to determine the hk-th order modal imbalance of the dual-rotor system under high-voltage rotor excitation.

[0114] Step 3, determine the orthogonal calibration quality group:

[0115] According to the orthogonality of the modal shapes, the unbalance distribution similar to the modal shape only affects the corresponding rotor modal vibration, so the added correction mass distribution t is similar to the balanced modal shape and orthogonal to the remaining modal shapes.

[0116] For the balanced low pressure excited Li order modal, let the correction mass distribution be t Li The row vector, for all NL2 order modal shapes under low pressure rotor excitation, there are n2 orthogonal correction mass distributions, where n2 = NL2. The n2 orthogonal correction mass distributions can be obtained from the following equation set:

[0117]

[0118] For the balanced high pressure excited hk order modal, let the correction mass distribution be t hk The row vector, for all NH1 order modal shapes under high pressure rotor excitation, there are n1 orthogonal correction mass distributions, where n1 = NH1. The n1 orthogonal correction mass distributions are obtained from the following equation set:

[0119]

[0120] The embodiment is represented by the orthogonal correction mass distributions of the low pressure excitation and the high pressure excitation as shown in Table 2.

[0121] Table 2 orthogonal correction mass distribution data of low pressure excitation and high pressure excitation

[0122]

[0123] Step 4, establish the modal dynamic balance balancing method of N1+N2 planes:

[0124] For the balanced high pressure excited hk order modal unbalance, the balanced high pressure excited modal order is NH1 order, the number of balancing planes on the high pressure rotor system is N1 selected by step 1, and the orthogonal correction mass obtained by step 3 is t hk ; the counterweight is The counterweight added on the N1 balancing correction planes. The modal unbalance determined in step 2 makes the counterweight satisfy the following relationship:

[0125]

[0126] In the formula, r hk (x hk )t hk =-1, so:

[0127]

[0128] The double rotor hkth order mode is the mode under the excitation of the high pressure rotor, and the counterweight is applied on the correction surface of the high pressure rotor The rotor hkth order mode unbalance can be balanced. Since t hk Only r hk (x hk )t hk =-1, so:

[0129] r hk+1 (x hk )t hk ≠0 (13)

[0130] Therefore, the added high pressure rotor balance weight will affect the remaining high pressure rotor excited mode unbalance, i.e. the hk+1th order mode unbalance state under the high pressure excitation of the double rotor will be affected. The excitation frequency of the counterweight is the high pressure rotor speed, so the counterweight will not affect the mode unbalance under the low pressure rotor excitation. After the hkth order high pressure excitation mode is balanced, the process is the method for balancing the hk+1th order high pressure excitation mode under the high pressure rotor excitation.

[0131] After the hkth order high pressure excitation mode is balanced, the hk+1th order mode under the high pressure rotor excitation is continued to be balanced. When the hk+1th order mode under the high pressure rotor excitation is balanced, N1 balance surfaces are selected on the high pressure rotor to apply counterweights In order to make the added counterweight not affect the hkth order mode balance result, t hk+1 needs to satisfy the following conditions:

[0132]

[0133] At this time, the counterweight is And at this time, the not only contains the influence of the initial unbalance, but also contains the influence of the hkth order mode unbalance correction amount of the high pressure rotor excitation. According to the proportional relationship determined by formula (14), the counterweight is added to each balance surface in the form of each balance surface component.

[0134] After the hk+1th order high pressure excitation mode is balanced, the process is the method for balancing the hk+1th order high pressure excitation mode under the high pressure rotor excitation.

[0135] The method for balancing the unbalance amount of each order mode under the low pressure rotor excitation is similar to the method for balancing the hkth order and hk+1th order high pressure excitation modes under the high pressure rotor excitation.

[0136] As Figure 1As shown, the aero-engine double rotor system includes two balance surfaces of the high-pressure rotor and two balance surfaces of the low-pressure rotor. Within the working speed, two high-pressure excited modes and two low-pressure excited modes occur. The modes to be balanced are the first and second high-pressure excited modes and the first and second low-pressure excited modes.

[0137] I. Balancing the first mode excited by the high-pressure rotor.

[0138] In the double rotor system, the orthogonal test recombination for balancing the first mode excited by the high-pressure rotor is 0 degree 1.5g test weight for the high-pressure compressor and 180 degree 5.45g test weight for the high-pressure turbine disk. According to the above process, the orthogonal test recombination is 150 degree 1.5g for the high-pressure compressor disk and 10 degree 5.41g for the high-pressure turbine disk. The dynamic balance effect of the first mode excited by the high-pressure rotor is shown in Figure 2 As shown. The balancing speed is 2200r / min for the high-pressure rotor. Therefore, the unbalance response at the high-pressure turbine disk is selected for comparison, and the damping ratio can reach 72.4%.

[0139] II. Balancing the second mode excited by the high-pressure rotor and the first mode excited by the low-pressure rotor.

[0140] If the unbalance of the second mode excited by the high-pressure rotor is not corrected before the rotor continues to increase the speed, the first critical speed excited by the low-pressure rotor can be reached. Because the test weight for the high-pressure rotor does not affect the mode unbalance excited by the low-pressure rotor, the second mode excited by the high-pressure rotor and the first mode excited by the low-pressure rotor are balanced at the same time. Because the test weight for the high-pressure rotor does not affect the mode unbalance excited by the low-pressure rotor, the orthogonal test recombination of the second mode excited by the high-pressure rotor and the orthogonal test recombination of the first mode excited by the low-pressure rotor are added at the balance surfaces of the high-pressure rotor and the low-pressure rotor. According to the method for balancing the first and second modes excited by the high-pressure rotor, the orthogonal test recombination of the second mode excited by the high-pressure rotor and the orthogonal test recombination of the first mode excited by the low-pressure rotor are obtained. The orthogonal test recombination of the second mode excited by the high-pressure rotor and the orthogonal test recombination of the first mode excited by the low-pressure rotor are added at the balance surfaces of the high-pressure rotor and the low-pressure rotor in the form of components.

[0141] If the unbalance of the second mode excited by the high-pressure rotor is not corrected before the rotor continues to increase the speed, the first critical speed excited by the low-pressure rotor can be reached. Because the test weight for the high-pressure rotor does not affect the mode unbalance excited by the low-pressure rotor, the second mode excited by the high-pressure rotor and the first mode excited by the low-pressure rotor are balanced at the same time. Because the test weight for the high-pressure rotor does not affect the mode unbalance excited by the low-pressure rotor, the orthogonal test recombination of the second mode excited by the high-pressure rotor and the orthogonal test recombination of the first mode excited by the low-pressure rotor are added at the balance surfaces of the high-pressure rotor and the low-pressure rotor. According to the method for balancing the first and second modes excited by the high-pressure rotor, the orthogonal test recombination of the second mode excited by the high-pressure rotor and the orthogonal test recombination of the first mode excited by the low-pressure rotor are obtained. The orthogonal test recombination of the second mode excited by the high-pressure rotor and the orthogonal test recombination of the first mode excited by the low-pressure rotor are added at the balance surfaces of the high-pressure rotor and the low-pressure rotor in the form of components.

[0142] In the dual-rotor system, the orthogonal test recombination for balancing the first order mode under low pressure excitation is 180 degrees 0g for the low pressure fan disk and 0 degrees 2.23g for the low pressure turbine disk. The obtained orthogonal counterweight recombination is 110 degrees 2.98g for the low pressure turbine disk and no addition for the fan disk. The dynamic balance effect of the first order mode under low pressure excitation is shown in FIG. 6. The balancing speed is 2127r / min for the low pressure rotor. Therefore, the unbalance response at the low pressure turbine disk is selected for comparison, and the damping ratio is 41.4%. Figure 3

[0143] III. Balancing the second order mode under low pressure rotor excitation

[0144] The second order mode under low pressure rotor excitation is balanced in a similar manner to the second order mode under high pressure rotor excitation. In the state of the second order critical speed under low pressure rotor excitation, the initial operation vibration response of the rotor and the vibration response after adding the test weight are measured, the unbalance correction mass of the second order mode under low pressure rotor excitation is obtained according to the method of balancing the hk+1 order mode under high pressure rotor excitation, and each balancing surface is added according to the component form of each balancing surface.

[0145] By now, the N1+N2 plane mode dynamic balancing method for the dual-rotor system is completed.

[0146] The embodiment uses 4 balancing surfaces, 2 balancing surfaces on the high pressure rotor and 2 balancing surfaces on the low pressure rotor, and balances the first four order modes of the dual-rotor.

[0147] The embodiment is applied to the vibration control and damping of the dual-rotor system of an aero-engine, provides a guarantee for the safe operation of the dual-rotor aero-engine, is applicable to the case where the main excitation mode is the low pressure rotor excitation mode or the high pressure rotor excitation mode, and provides a new idea for the dynamic balancing method research of the dual-rotor system with intermediate bearings.​

Claims

1. A method for dynamic balancing a dual-rotor system, characterized in that, The specific process is as follows: Step 1: Determine the modal order and number of balancing surfaces of the dual-rotor system requiring dynamic balancing and obtain the modal shape data: Step 2, determine the initial imbalance: Ⅰ. Determine the Lith-order modal imbalance of the dual-rotor system under low-voltage rotor excitation. ; When the low-pressure rotor speed of the dual-rotor system approaches the Lith-order critical speed ω of the low-pressure rotor excitation. Li At that time, the unbalanced response r(x,t) is: (5) In equation (5), r Li (x) represents the Li-th mode shape data under low-voltage excitation. Regarding rotational speed The frequency response function, This is the unbalance quantity of the Li-th order mode; The Li-th order modal unbalance of the rotor system is calculated based on the influence coefficients of both the unbalance response and the unbalance quantity. ; II. Determine the hk-th order modal imbalance of the dual-rotor system under high-voltage rotor excitation. ; When the high-voltage rotor speed of the dual-rotor system approaches the hk-th order critical speed ω of the high-voltage rotor excitation. hk At that time, the high-voltage rotor speed The constant speed ratio is a, and the low-pressure rotor speed is... ; The unbalance of the Li-th order mode is determined as described above. The same method was used to determine the hk-th order modal imbalance of the dual-rotor system under high-voltage rotor excitation. ; Step 3, determine the orthogonal calibration quality group: The corrected mass group distribution t is similar to the equilibrium mode shape and remains orthogonal to the other mode shapes; For the Li-th order mode of balanced low-voltage excitation, let the corrected mass group be t. Li For all NL2 mode shapes under low-pressure rotor excitation, the row vector has n2 orthogonal corrective mass sets, where n2 = NL2; these n2 orthogonal corrective mass sets can be obtained from the following system of equations: (9) For the hk-th order mode of balanced high-voltage excitation, let the corrected mass group be t. hk For all NH1 order mode shapes under high-pressure rotor excitation, there are n1 orthogonal corrective mass sets, where n1 = NH1; these n1 orthogonal corrective mass sets are obtained through the following system of equations: (10) Step 4, establish the modal dynamic balancing method for the N1+N2 plane: The dual-rotor system includes two balance surfaces of the high-pressure rotor and two balance surfaces of the low-pressure rotor. Within the operating speed range, there are two modes of high-pressure excitation and two modes of low-pressure excitation. The modes that need to be balanced are the first and second modes of high-pressure excitation and the first and second modes of low-pressure excitation. Ⅰ First-order mode under balanced high-voltage rotor excitation In a dual-rotor system, the orthogonal test combination under the first-order mode of balancing high-pressure excitation is a 0-degree test weight for the high-pressure compressor and a 180-degree test weight for the high-pressure turbine disk; the orthogonal combination obtained according to the above process is a 150-degree test weight for the high-pressure compressor disk and a 10-degree test weight for the high-pressure turbine disk. II. Balancing the second-order mode under high-voltage rotor excitation and the first-order mode under low-voltage rotor excitation If the rotor continues to accelerate before the imbalance of the second-order mode under high-pressure excitation is corrected, it can reach the first-order critical speed under low-pressure excitation; at the same time, the second-order mode under high-pressure rotor excitation and the first-order mode under low-pressure rotor excitation are balanced. Following the method of balancing the hk-th and hk+1-th high-voltage excitation modes under high-voltage rotor excitation, the orthogonal combination of the second-order mode of high-voltage excitation and the orthogonal combination of the first-order mode of low-voltage excitation are obtained; the orthogonal combination of the second-order mode of high-voltage excitation and the orthogonal combination of the first-order mode of low-voltage excitation are added in component form on the high-voltage rotor balance surface and the low-voltage rotor balance surface. If, before the imbalance of the second-order mode under high-pressure excitation is corrected, the rotor cannot continue to accelerate to the first-order critical speed under low-pressure excitation due to excessive vibration, then the second-order mode under high-pressure excitation should be balanced first, followed by the first-order mode under low-pressure excitation; the second-order mode under high-pressure excitation should be balanced according to the method described for balancing the (hk+1)th-order mode under high-pressure rotor excitation; and the first-order mode under low-pressure rotor excitation should be balanced according to the process described for balancing the first-order mode under high-pressure rotor excitation. III. Second-order mode under balanced low-voltage rotor excitation Under the second critical speed state of low-pressure rotor excitation, the initial running vibration response and the vibration response after adding test weight of the rotor are measured. According to the method of balancing the hk+1th high-pressure excitation mode under high-pressure rotor excitation, the second mode unbalance correction mass under low-pressure rotor excitation is obtained, and it is added to each balance surface in the form of each balance surface component. At this point, the dynamic balancing of the N1+N2 planar modes of the dual-rotor system was completed.

2. The method for dynamic balancing of a dual-rotor system as described in claim 1, characterized in that: The modal order mentioned in step 1 is four modes, namely the first and second modes of high voltage excitation and the first and second modes of low voltage excitation.

3. The method for dynamic balancing of a dual-rotor system as described in claim 1, characterized in that: In step 1, when determining the modal order, if the low-voltage rotor excitation is the main excitation, it is the low-voltage rotor excitation modal order, and the number of low-voltage rotor excitation modal orders that need to be dynamically balanced is NL2; if the high-voltage rotor excitation is the main excitation, it is the high-voltage rotor excitation modal order, and the number of high-voltage rotor excitation modal orders that need to be dynamically balanced is NH1. The number of balance surfaces determined in step 1 is four, namely high-pressure first-stage compressor disk x1, high-pressure turbine disk x2, low-pressure fan disk x3, and low-pressure turbine disk x4; wherein: the number of balance surfaces on the high-pressure rotor is N1; the number of balance surfaces on the high-pressure rotor N1 is the same as the number of excitation mode orders of the high-pressure rotor NH1; the number of balance surfaces on the low-pressure rotor is N2, the number of balance surfaces on the low-pressure rotor N2 is the same as the number of excitation mode orders of the low-pressure rotor NL2; The modal vibration data obtained in step 1 includes the vibration data at the equilibrium surface of the dual rotor system; the vibration data at the equilibrium surface includes the amplitude and corresponding phase radian value of each mode at the equilibrium surface.

4. The method for dynamic balancing of a dual-rotor system as described in claim 3, characterized in that: In the modal vibration data, the k-th order high-voltage rotor excitation mode data r hk As shown in formula (3), where each parameter is r hk Mode shape data of the high-voltage rotor excitation mode at each equilibrium plane: (3) The i-th order low-pressure rotor excitation mode data r Li As shown in formula (4), where each parameter is r Li Modal shape data of the first mode at each equilibrium plane: (4)。 5. The method for dynamic balancing of a dual-rotor system as described in claim 1, characterized in that, In the modal data obtained in step 1, the k-th high-voltage rotor excitation mode data is r. hk The i-th order low-voltage rotor excitation mode shape data is r Li To facilitate subsequent calculations, the modal data of each order are integrated into a matrix form; among which, the modal data r of the high-pressure rotor as the main excitation are... hk Integrate into modal data matrix r h Mode shape data r of low-voltage rotor as the main excitation Li Integrate into modal data matrix r L The modal data matrix r of the high-voltage rotor main excitation h The number of rows and the modal data matrix r of the low-voltage rotor main excitation L The number of rows in the matrix is ​​consistent with the modal order requiring dynamic balancing; the modal data matrix r of the high-voltage rotor main excitation h The number of columns is consistent with the number of high-pressure rotor balance surfaces N1; the low-pressure rotor main excitation mode data matrix r L The number of columns is consistent with the selected number of low-pressure rotor balance surfaces N2.

6. The method for dynamic balancing of a dual-rotor system as described in claim 4, characterized in that, Table 1 shows the amplitude and corresponding phase radian values ​​of each mode at the equilibrium plane; in the table, the phase is in radians and the amplitude is a dimensionless value. Table 1. Mode shape data used for modal dynamic balancing of the dual-rotor experimental setup 7. The method for dynamic balancing of a dual-rotor system as described in claim 1, characterized in that, In step 2, when the low-pressure rotor speed of the dual-rotor system approaches the Lith-order critical speed ω of the low-pressure rotor excitation... Li At that time, the low-pressure rotor speed The constant speed ratio value a = 1.3, the high-pressure rotor speed .

8. The method for dynamic balancing of a dual-rotor system as described in claim 1, characterized in that, Step 2 describes the calculation of the Li-th modal imbalance of the rotor system. The process is as follows: At a low-pressure rotor speed of Ω Li And the high-voltage rotor speed is Ω hk When measuring the vibration of the dual-rotor system: during measurement, if no reassembly is performed, the initial unbalance is recorded as u0, and the unbalance response of the dual-rotor system is recorded as r0; when... After adding a trial configuration of u1, the measured unbalanced response of the rotor system is r1; then the following relationship holds: (7) Because the unbalanced response of the dual-rotor system without additional recombination is: Substituting into equation (7), the initial imbalance is: (8) In the formula, u0 is the unbalance quantity of the Li-th order mode. .

9. The method for dynamic balancing of a dual-rotor system as described in claim 1, characterized in that, In step 4, the modal dynamic balancing method for establishing the N1+N2 plane is described. For the hk-th order modal unbalance of the high-voltage rotor excitation, the modal order of the balanced high-voltage excitation is NH1. The number of balancing surfaces on the high-voltage rotor system is selected as N1 in step 1, and the orthogonal correction mass obtained in step 3 is... The counterweight is ; The counterweights added to the N1 balance correction surfaces; the modal imbalance determined in step 2. , to make the counterweight The following relationship must be satisfied: (11) In the formula, Therefore: (12) The hk-th order mode of the dual rotor is the mode under high-voltage rotor excitation, with a counterweight applied to the high-voltage rotor correction surface. This can balance the hk-th order modal imbalance of the rotor; due to the selection Only meet Therefore: (13) Therefore, the added high-pressure rotor counterweight This will affect the modal imbalance of the remaining high-voltage rotor excitation, that is, the hk+1th order modal imbalance state of the dual-rotor high-voltage excitation will be affected; counterweight The excitation frequency is the high-voltage rotor speed, so the counterweight will not affect the modal imbalance under low-voltage rotor excitation; the process of balancing the hk-th high-voltage excitation mode is the method of balancing the hk-th high-voltage excitation mode under high-voltage rotor excitation. After balancing the hk-th high-voltage excitation mode, continue balancing the hk+1-th mode under high-voltage rotor excitation. When balancing the hk+1-th mode under high-voltage rotor excitation, N1 balancing surfaces need to be selected on the high-voltage rotor to apply counterweights. To ensure that the added weights do not affect the hk-th order modal equilibrium results, The following conditions must be met: (14) At this time, the counterweight is , And at this time It not only includes the influence of the initial imbalance, but also the influence of the hk-order modal imbalance correction of the high-voltage rotor excitation; according to the proportional relationship determined by equation (14), the counterweight Add them to each equilibrium surface in the form of each equilibrium surface component; The process of balancing the (hk+1)th high-voltage excitation mode is the method for balancing the (hk+1)th high-voltage excitation mode under high-voltage rotor excitation. The method for balancing the unbalance of each mode under low-pressure rotor excitation is similar to the method for balancing the hk-th and hk+1-th high-pressure excitation modes under high-pressure rotor excitation.

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