CN115014637B - Modal dynamic balance method based on low-rotation-speed measurement - Google Patents

Modal dynamic balance method based on low-rotation-speed measurement Download PDF

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CN115014637B
CN115014637B CN202210411949.5A CN202210411949A CN115014637B CN 115014637 B CN115014637 B CN 115014637B CN 202210411949 A CN202210411949 A CN 202210411949A CN 115014637 B CN115014637 B CN 115014637B
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邓振鸿
罗华耿
张保强
吴太欢
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Xiamen University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M1/00—Testing static or dynamic balance of machines or structures
    • G01M1/14—Determining imbalance
    • G01M1/16—Determining imbalance by oscillating or rotating the body to be tested
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Abstract

A modal dynamic balance method based on low-rotation-speed measurement relates to the field of rotary machinery and vibration test. In order to eliminate unbalanced response of the former N-order modes, test weights are added on N planes, unbalanced calibration quantities to be added are solved, N balance planes are taken for balancing the former N-order modes, synchronous vibration response of measuring points is obtained when a rotor runs at any rotating speed, the former N+M-order modes are obtained at a lower rotating speed, different rotating speeds are measured in a range lower than a first-order critical rotating speed of the rotor, known test weights are added on the first balance planes, influence coefficients are obtained after the measurement of different rotating speeds are also carried out, and unbalanced calibration quantities are solved repeatedly for the rest balance planes. The rotor can be measured at a lower rotating speed, and the balance parameters are fitted by combining with the rotor modal parameter information, so that modal dynamic balance at a low speed is realized, resonance caused by the supercritical rotating speed in the testing process is avoided, and the rotor is ensured to have a more reliable balance state in a wider rotating speed range.

Description

Modal dynamic balance method based on low-rotation-speed measurement
Technical Field
The invention relates to the field of rotary machinery and vibration testing, in particular to a modal dynamic balance method based on low-rotation-speed measurement, which can be applied to dynamic balance of a flexible rotor.
Background
Industrial rotating machinery such as aeroengines and large gas turbines are high in running speed, rotors of the industrial rotating machinery often need to work in a supercritical speed state, and in the process of passing through the supercritical speed, the rotors of the industrial rotating machinery often cause remarkable vibration and deformation due to structural resonance due to unbalanced rotor mass, so that the running performance and safety of equipment are seriously affected.
In order to ensure that the rotor system can pass through the critical rotation speed more stably, the conventional modal dynamic balance method generally requires dynamic balance test to be carried out near the critical rotation speed, but has the problem that vibration is too large when approaching or passing through the critical rotation speed in the measurement process, so that the dynamic balance test cost is higher, and a larger potential safety hazard exists. Therefore, a new dynamic balance method which can be safer and more reliable is urgently needed.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provide a modal dynamic balance method based on low-rotation-speed measurement, so that a rotor can be measured at a lower rotation speed, resonance is avoided in the test process as much as possible, modal balance is realized, and a rotor is ensured to have a more reliable balance state in a wider rotation speed range.
The specific steps of the invention are as follows:
1) For a flexible rotor system, it is in the axial direction x s The synchronous vibration response at the point due to mass imbalance is as follows:
Figure BDA0003604084170000011
wherein ,mr ，ω r ，ξ r ，ψ r (x) Respectively representing the r-th order modal mass, modal frequency, modal damping and modal shape, wherein omega is the shaft rotation speed, U (x) is the unbalanced distribution of the rotor, and l is the length of the rotor; typically, as the rotational speed approaches the r-th order modal frequency, the order modal response dominates, written as:
Figure BDA0003604084170000021
to eliminate the unbalanced response of the first N-order modes, the test weight lambda needs to be added on N planes k K=1, 2,..n, such that
Figure BDA0003604084170000022
The method comprises the following steps:
Figure BDA0003604084170000023
in the formula ,
Figure BDA0003604084170000024
representing the integral of the modal unbalance component of the r-th order over the entire axis, which may be referred to as the modal unbalance factor of the r-th order, then equation (3) is written in the form of a matrix as: />
Figure BDA0003604084170000025
The required added unbalance calibration amount can be solved according to the above formula:
Λ＝-[Ψ] -1 u (5)
wherein Λ= [ λ ] 1 ,λ 2 ,...λ N ] T Represents the set of unbalance calibration quantities that need to be added, [ ψ ]] r ＝[ψ r (x 1 ),ψ r (x 2 ),...,ψ r (x N )]Represents the r-order vibration mode vector, u= [ u ] 1 ,u 2 ,...u N ] T Representing a vector of the first N-order modal unbalance factors.
2) Taking N balance surfaces to balance the front N-order modes, and setting a measurement point to be positioned at x s Where each order mode shape is ψ i (x s ) Each row [ ψ ] of the vibration mode matrix] i Multiplying by
Figure BDA0003604084170000026
Then write as:
Figure BDA0003604084170000027
when the rotor runs at any rotation speed, the rotor is at a measuring point x s The synchronous vibration response of (2) is:
Figure BDA0003604084170000028
wherein ,
Figure BDA0003604084170000029
defined as a complex parameter term related to rotational speed, when considering the balanced pre-N-order mode, consider that the mode after the n+m-th order contributes negligible to the pre-N-order, at lower rotational speeds Ω (Ω < ω) 1 ＜ω N ＜＜ω N+M+1 ) Then, the mode is truncated to the former N+M order mode to obtain:
Figure BDA0003604084170000031
when the different rotational speeds Ω= { Ω are performed in the first-order critical rotational speed range below the rotor 1 ,Ω 2 ,...Ω p Measurement of } (p.gtoreq.N+M), then the above formula is written as:
Figure BDA0003604084170000032
for simplicity and convenience, record
Figure BDA0003604084170000033
Z＝[Z 1 (Ω) Z 2 (Ω)...Z N+M (Ω)]When the critical rotation speed omega of each order of the rotor is known i (i=1, 2,) n+m and critical damping ratio ζ i (i=1, 2,..n+m), then solved by the formula:
Figure BDA0003604084170000034
in the formula ,(·)+ Representing the pseudo-inverse of the matrix, y= [ Y (x s ,Ω 1 ),Y(x s ,Ω 2 ),...,Y(x s ,Ω p )] T 。
Adding a known test weight Q to the first equilibrium surface 1 The same applies to different rotational speeds Ω= { Ω 1 ,Ω 2 ,...Ω p The influence coefficient is obtained after the measurement of the (p is more than or equal to N+M):
Figure BDA0003604084170000035
writing the above in the form of a matrix:
H 1 ＝ZX 1 (12)
wherein ,
Figure BDA0003604084170000036
representative and vibration modeTerms related to modal mass, so:
Figure BDA0003604084170000041
repeating the steps for the rest balance surfaces to obtain:
Figure BDA0003604084170000042
wherein ,
Figure BDA0003604084170000043
the unbalance calibration quantity Λ is solved.
Compared with the prior art, the invention has the beneficial effects that:
the method has the advantages that the mode shape is known on the premise that the mode balance is required to be carried out by the rotor when the rotor is operated to be close to each order of critical rotation speed, but the method is limited by practical conditions, such as complex rotor structure, the mode shape cannot be obtained accurately, and the operation of the rotor to the critical rotation speed is dangerous and difficult to realize.
Drawings
FIG. 1 is a schematic diagram of a rotor imbalance distribution.
Fig. 2 is a front 5-order mode shape curve of the rotor.
Fig. 3 is a vibration response curve of the rotor due to unbalance.
Fig. 4 is a graph of vibration response curve measurements at initial imbalance.
Fig. 5 is a comparison of vibration response amplitudes before and after balancing.
Detailed Description
The invention is further illustrated with reference to the following examples.
The embodiment adopted by the invention is a numerical simulation case, and is specifically described as follows: a rotor of uniform cross section is shown in fig. 1, the rotor length being 1m, there being unbalanced masses at different cross section positions, the specific positions and sizes being shown in table 1. The maximum working speed of the rotor is set between the third order critical speed and the fourth order critical speed, and the contribution of the modes of the sixth order and above to the vibration response of the rotor in the whole working speed is considered to be negligible, so that the superposition result of the mode components of the first 5 orders is taken to represent the real vibration response of the rotor, the first 5 orders critical speed, the damping ratio and the mode quality of the rotor are set as shown in the table 2, and the first 5 orders mode shape curve of the rotor is shown in the figure 2. From the given parameters, an imbalance response curve of the rotor over the first 5 th order critical speed range can be calculated as shown in fig. 3.
Table 1 unbalance amount size and distribution
Figure BDA0003604084170000051
TABLE 2 critical speed to damping ratio of the first 5 th order of the rotor
Figure BDA0003604084170000052
In this embodiment, the target is to balance the modal unbalance response of the first 3 rd order, and by estimating the first 5 th order critical rotation speed and damping ratio information of the rotor through simulation or experiment, the remaining information (unbalance distribution, modal mass and modal shape are all unknown amounts) of the rotor is implemented as follows:
1) Selected U 1 、U 2 and U3 The plane is used as a balance surface, and the sensor positions are arranged at U 2 The plane is located;
2) Imbalance response Y to rotor in low speed range (Ω=200-600 RPM) 0 (Ω,
Figure BDA0003604084170000056
) The amplitude and phase of (a) are measured to obtain an unbalanced vibration response curve as shown in figure 4;
3) Let us estimate the first 5 th order critical rotation speed and damping ratio information of the rotor through simulation or experiment (see table 2), using
Figure BDA0003604084170000053
Calculating Z in the measured rotation speed range omega=200-600 RPM r (Ω), r=1, 2,3,4,5, further solved:
Figure BDA0003604084170000054
4) In U 1 The plane is added with a known test block Q 1 (1 g.mm in magnitude, 0 ° in phase), measured at Ω=200-600 RPM, gives an imbalance response
Figure BDA0003604084170000055
Calculating H 1 (Ω), and solve: />
Figure BDA0003604084170000061
5) Take off Q 1 In U 2 The plane is added with a known test block Q 2 (magnitude 1 g.mm, phase 0 °) and likewise measured at Ω=200-600 RPM, gives an imbalance response
Figure BDA0003604084170000062
Calculating H 2 (Ω), solving:
Figure BDA0003604084170000063
6) Take off Q 2 In U 2 The plane is added with a known test block Q 3 (magnitude 1 g.mm, phase 0 °) and likewise measured at Ω=200-600 RPM, gives an imbalance response
Figure BDA0003604084170000064
Calculating H 3 (Ω), solving:
Figure BDA0003604084170000065
7) Γ, X obtained in the steps 3) to 6) 1 、X 2 and X3 Substituting the values into the table 3, the balance calibration amounts to be added to the three balance surfaces were obtained:
TABLE 3 Table 3
Calibration quantity λ 1 λ 2 λ 3
Size (g, mm) 0.44 1.69 0.36
Phase (°) 14.961 -89.86 -27.28
8) Balancing according to the balancing calibration quantity obtained in the step 7), and finishing balancing of the rotor. As shown in fig. 5, it can be seen that the imbalance response curve (broken line in the figure) after balancing by the method according to the present invention is significantly lower in the entire operating rotation speed range than before balancing (solid line in the figure), and the vibration response due to modal imbalance in the rotation speed range can be largely eliminated.

Claims (1)

1. The modal dynamic balance method based on low-rotation-speed measurement is characterized by comprising the following specific steps of:
1) For a flexible rotor system, it is in the axial direction x s The synchronous vibration response at the point due to mass imbalance is as follows:
Figure FDA0004168416530000011
wherein ,mr ，ω r ，ξ r ，ψ r (x) Respectively representing the r-th order modal mass, modal frequency, modal damping and modal shape, wherein omega is the shaft rotation speed, U (x) is the unbalanced distribution of the rotor, and l is the length of the rotor; typically, as the rotational speed approaches the r-th order modal frequency, the order modal response dominates, written as:
Figure FDA0004168416530000012
to eliminate the unbalanced response of the first N-order modes, the test weight lambda needs to be added on N planes k K=1, 2,..n, such that
Figure FDA0004168416530000017
The method comprises the following steps:
Figure FDA0004168416530000013
in the formula ,
Figure FDA0004168416530000014
the form written in matrix is:
Figure FDA0004168416530000015
solving the unbalance calibration quantity required to be added according to the above formula:
Λ＝-[Ψ] -1 u
wherein Λ= [ λ ] 1 ,λ 2 ,...λ N ] T ，[Ψ] r ＝[ψ r (x 1 ),ψ r (x 2 ),...,ψ r (x N )]，u＝[u 1 ,u 2 ,...u N ] T ；
2) Taking N balance surfaces to balance the front N-order modes, and setting a measurement point to be positioned at x s Where each order mode shape is ψ i (x s ) Each row [ ψ ] of the vibration mode matrix] i Multiplying by
Figure FDA0004168416530000016
Then write as:
Figure FDA0004168416530000021
when the rotor runs at any rotation speed, the rotor is at a measuring point x s The synchronous vibration response of (2) is:
Figure FDA0004168416530000022
wherein ,
Figure FDA0004168416530000023
when the N-th order mode is considered before balancing, the mode after the n+m-th order is considered to contribute negligible to the N-th order before, at a lower rotation speed Ω (Ω < ω) 1 ＜ω N <<ω N+M+1 ) Then, the mode is truncated to the former N+M order mode to obtain:
Figure FDA0004168416530000024
when the different rotational speeds Ω= { Ω are performed in the first-order critical rotational speed range below the rotor 1 ,Ω 2 ,...Ω p Measurement of } (p.gtoreq.N+M), then the above formula is written as:
Figure FDA0004168416530000025
recording device
Figure FDA0004168416530000026
Z＝[Z 1 (Ω) Z 2 (Ω)...Z N+M (Ω)]When the critical rotation speed omega of each order of the rotor is known i (i=1, 2,) n+m and critical damping ratio ζ i (i=1, 2,..n+m), then solved by the formula:
Figure FDA0004168416530000027
in the formula ,(·)+ Representing the pseudo-inverse of the matrix, y= [ Y (x s ,Ω 1 ),Y(x s ,Ω 2 ),...,Y(x s ,Ω p )] T ；
Adding a known test weight Q to the first equilibrium surface 1 The same applies to different rotational speeds Ω= { Ω 1 ,Ω 2 ,...Ω p The influence coefficient is obtained after the measurement of the (p is more than or equal to N+M):
Figure FDA0004168416530000031
writing the above in the form of a matrix:
H 1 ＝ZX 1
wherein ,
Figure FDA0004168416530000032
therefore:
Figure FDA0004168416530000033
repeating the steps for the rest balance surfaces to obtain:
Figure FDA0004168416530000034
wherein ,
Figure FDA0004168416530000035
the unbalance calibration quantity Λ is solved. />
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