A dynamic balancing method and device for a flexible support steam turbine generator set

Abstract

The present application belongs to the field of power machinery engineering technology, and specifically relates to a dynamic balancing method and device for a flexible support steam turbine generator set; the method comprises: establishing a mathematical model of a rigid support rotor and a flexible support rotor, simulating and calculating the dynamic balancing response coefficients of the rigid support rotor and the flexible support rotor under the influence of different vibration modes in the mathematical model, and obtaining a difference coefficient curve of the dynamic balancing response coefficients between the flexible support steam turbine generator set and the rigid support; according to the actual dynamic balancing influence coefficient of the rigid support rotor obtained in the past, the dynamic balancing influence coefficient of the flexible support rotor is obtained by correcting the difference coefficient curve; using the obtained dynamic balancing influence coefficient, while considering the vibration mode influence in the vibration data, a dynamic balancing plan for the flexible support rotor is formulated. This method can effectively correct the weighting data of the flexible support rotor on site to avoid the time and economic losses caused by multiple weighting attempts.

Description

Technical Field

[0001] The present application belongs to the field of power machinery engineering technology, and specifically relates to a dynamic balancing method and device for a flexible support steam turbine generator set. Background Art

[0002] In recent years, flexible-support steam turbine generator sets have become widely used in nuclear power plants. However, due to the short time the units have been in operation and the limited number of on-site dynamic balancing operations, the accumulated experience coefficients for on-site dynamic balancing of flexible-support steam turbine generator sets are relatively limited. On the other hand, similar rigid-support units have been in use in China for many years, and a large amount of on-site dynamic balancing experience and technology has been accumulated. The current challenge is to successfully apply this experience data to the dynamic balancing issues that may arise in the future for flexible-support units.

[0003] Currently, the most common approach to balancing rotors in flexible-support steam turbine generator sets is to perform trial weighting based on the empirical coefficients of rigid units, determining the actual impact coefficients through trial weighting. This method requires a long construction period and fails to fully utilize the empirical coefficients of rigid units. Summary of the Invention

[0004] The purpose of the present application is to provide a method and device for dynamic balancing of a flexible supported steam turbine generator set, so as to solve the problem that it takes a long time to perform trial weighting according to the empirical coefficient of the rigid set and to determine the actual influence coefficient by trial weighting.

[0005] Technical solution to achieve the purpose of this application:

[0006] An embodiment of the present application provides a dynamic balancing method for a flexible support steam turbine generator set, the method comprising:

[0007] Step 1: Establish mathematical models of rigid support rotors and flexible support rotors, simulate and calculate the dynamic balance response coefficients of the rigid support rotors and flexible support rotors under the influence of different vibration modes in the mathematical models, and obtain the difference coefficient curve of the dynamic balance response coefficient between the flexible support steam turbine generator set and the rigid support;

[0008] Step 2: Based on the actual dynamic balance influence coefficient of the rigid support rotor obtained in advance, the dynamic balance influence coefficient of the flexible support rotor is obtained by correcting the difference coefficient curve;

[0009] Step 3: Use the obtained dynamic balancing influence coefficients and consider the influence of mode shapes in the vibration data to develop a dynamic balancing plan for the flexibly supported rotor.

[0010] Optionally, step 1 specifically includes:

[0011] Statistically calculate the stiffness and damping coefficient matrix of each support bearing of the rotor, and form a sequence X according to the bearing number(0) ;

[0012] X (0) ={X (0) (1),X (0) (2),X (0) (3),…,X (0) (11)}

[0013] By establishing the mathematical model of each bearing oil film, the stiffness and damping matrix X of each bearing oil film is calculated. (1) ;

[0014] X (1) ={X (1) (1),X (1) (2),X (1) (3),…,X (1) (11)

[0015] The rotor support dynamic matrix X is obtained when the rigid support is used. (1) and the dynamic coefficient matrix X when the flexible support (2) , X (2) =X (0) +X (1) ;

[0016] The rotor model is established by finite element method, and the support stiffness and damping coefficient are loaded in the rotor model. (1) and X (2) Input it to calculate the dynamic balance response coefficient at different speeds and different positions and curve;

[0017] When the weight is increased at the same speed and position and Divide by to get the difference coefficient The difference coefficient curve of

[0018] Optionally, step 2 specifically includes:

[0019] The actual dynamic balance influence coefficient of the rigid support rotor in the past is the empirical coefficient accumulated during the dynamic balancing process of the rigid support unit in the past;

[0020] By comparing the difference coefficient curve obtained in step 1, the corresponding Calculate the response coefficient of the flexible support unit corresponding to the speed and weighted position

[0021] Optionally, step 3 specifically includes:

[0022] Vibration data measured by two sensors at the front and rear support bearings of each rotor in the shaft system The first-order vibration mode component of the rotor vibration vector can be decomposed using the formula: Second-order modal components The third-order vibration mode component

[0023] in,

[0024] When the calculated speed condition is between the first and second critical speeds of the rotor:

[0025]

[0026] Where: They are first-order and second-order balancing weights respectively; is the influence coefficient of the first-order counterweight under the flexible support on the first-order vibration mode; is the influence coefficient of the second-order counterweight on the second-order vibration mode under flexible support; is the interference influence coefficient of the first-order counterweight on the second-order vibration mode of the flexible support; is the interference influence coefficient of the first-order vibration mode of the second-order counterweight under flexible support;

[0027] When the calculated speed condition is between the second and third critical speeds of the rotor:

[0028]

[0029] Where: They are second and third order balance weights respectively; is the influence coefficient of the second-order counterweight on the second-order vibration mode; is the influence coefficient of the third-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order vibration mode of the third-order counterweight.

[0030] The present application also provides a dynamic balancing device for a flexible support steam turbine generator set, the device comprising:

[0031] A curve acquisition module is used to establish a mathematical model of a rigid support rotor and a flexible support rotor, simulate and calculate the dynamic balance response coefficients of the rigid support rotor and the flexible support rotor under the influence of different vibration modes in the mathematical model, and obtain a difference coefficient curve of the dynamic balance response coefficient between the flexible support steam turbine generator set and the rigid support;

[0032] A coefficient acquisition module is used to obtain the dynamic balance influence coefficient of the flexible support rotor by correcting the difference coefficient curve based on the actual dynamic balance influence coefficient of the rigid support rotor obtained in advance;

[0033] A solution formulation module is used to use the obtained dynamic balancing influence coefficients and consider the influence of mode shapes in the vibration data to formulate a dynamic balancing solution for the flexible supported rotor.

[0034] Optionally, the curve acquisition module is specifically used to:

[0035] Statistically calculate the stiffness and damping coefficient matrix of each support bearing of the rotor, and form a sequence X according to the bearing number (0) ;

[0036] X (0) ={X (0) (1),X (0) (2),X (0) (3),…,X (0) (11)

[0037] By establishing the mathematical model of each bearing oil film, the stiffness and damping matrix X of each bearing oil film is calculated. (1) ;

[0038] X (1) ={X (1) (1),X (1) (2),X (1) (3),…,X (1) (11)

[0039] The rotor support dynamic matrix X is obtained when the rigid support is used. (1) and the dynamic coefficient matrix X when the flexible support (2) , X (2) =X (0) +X (1) ;

[0040] The rotor model is established by finite element method, and the support stiffness and damping coefficient are loaded in the rotor model. (1) and X (2) Input it to calculate the dynamic balance response coefficient at different speeds and different positions and curve;

[0041] When the weight is increased at the same speed and position and Divide by to get the difference coefficient The difference coefficient curve of

[0042] Optionally, the coefficient acquisition module is specifically used to:

[0043] The actual dynamic balance influence coefficient of the rigid support rotor in the past is the empirical coefficient accumulated during the dynamic balancing process of the rigid support unit in the past;

[0044] By comparing the difference coefficient curve obtained in step 1, the corresponding Calculate the response coefficient of the flexible support unit corresponding to the speed and weighted position

[0045] Optionally, the plan formulation module is specifically used to:

[0046] Vibration data measured by two sensors at the front and rear support bearings of each rotor in the shaft system The first-order vibration mode component of the rotor vibration vector can be decomposed using the formula: Second-order modal components The third-order vibration mode component

[0047] in,

[0048] When the calculated speed condition is between the first and second critical speeds of the rotor:

[0049]

[0050] Where: They are first-order and second-order balancing weights respectively; is the influence coefficient of the first-order counterweight under the flexible support on the first-order vibration mode; is the influence coefficient of the second-order counterweight on the second-order vibration mode under flexible support; is the interference influence coefficient of the first-order counterweight on the second-order vibration mode of the flexible support; is the interference influence coefficient of the first-order vibration mode of the second-order counterweight under flexible support;

[0051] When the calculated speed condition is between the second and third critical speeds of the rotor:

[0052]

[0053] Where: They are second and third order balance weights respectively; is the influence coefficient of the second-order counterweight on the second-order vibration mode; is the influence coefficient of the third-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order vibration mode of the third-order counterweight.

[0054] The beneficial technical effects of this application are:

[0055] (1) The embodiments of the present application provide a dynamic balancing method and apparatus for a flexible support steam turbine generator set, which simulates and calculates the dynamic balancing response coefficients of rigid and flexible support rotors under the influence of different vibration modes in a mathematical model, and solves to obtain a difference coefficient curve of the dynamic balancing response coefficients between the flexible support and rigid support rotors;

[0056] 2) The embodiments of the present application provide a method and apparatus for dynamic balancing of a flexibly supported steam turbine generator set. The method summarizes the actual dynamic balancing influence coefficients of rigidly supported rotors in the past and obtains the dynamic balancing influence coefficients of the flexibly supported rotor by correcting the difference coefficients of different vibration modes.

[0057] 3) The embodiments of the present application provide a dynamic balancing method and apparatus for a flexible-support steam turbine generator set, which uses the corrected dynamic balancing influence coefficient in combination with the rotor vibration mode parameters to formulate a dynamic balancing plan for the flexible-support rotor on-site. DETAILED DESCRIPTION

[0058] In order to enable those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below. Obviously, the embodiments described below are only a part of the embodiments of the present application, rather than all of them. Based on the embodiments recorded in this application, all other embodiments obtained by those skilled in the art without paying creative work are within the scope of protection of this application.

[0059] The embodiments of the present application provide a dynamic balancing method and device for a flexible-supported steam turbine generator set, which can effectively correct the weighting data of the flexible-supported rotor on site, thereby avoiding the time and economic losses caused by multiple weighting attempts. Compared with the existing technology, the dynamic balancing method for a flexible-supported steam turbine generator set provided in the embodiments of the present application uses a mathematical model to calculate a curve of the difference coefficients of rigid and flexible supports. This method not only takes into account the influence of the flexible support, but also fully utilizes the support of historical empirical data of rigid supports. The probability of a successful weighting in one go is greatly improved, avoiding the time and economic losses caused by multiple weighting attempts.

[0060] Based on the above content, in order to clearly and in detail illustrate the above advantages of the present application, the specific implementation methods of the present application will be described below.

[0061] An embodiment of the present application provides a dynamic balancing method for a flexible support steam turbine generator set, the method comprising:

[0062] Step 1: Establish mathematical models of rigid support rotors and flexible support rotors, simulate and calculate the dynamic balance response coefficients of the rigid support rotors and flexible support rotors under the influence of different vibration modes in the mathematical models, and obtain the difference coefficient curve of the dynamic balance response coefficient between the flexible support steam turbine generator set and the rigid support;

[0063] Step 2: Based on the actual dynamic balance influence coefficient of the rigid support rotor obtained in advance, the dynamic balance influence coefficient of the flexible support rotor is obtained by correcting the difference coefficient curve;

[0064] Step 3: Use the obtained dynamic balancing influence coefficients and consider the influence of mode shapes in the vibration data to develop a dynamic balancing plan for the flexibly supported rotor.

[0065] In one example, step 1 specifically includes:

[0066] Statistically calculate the stiffness and damping coefficient matrix of each support bearing of the rotor, and form a sequence X according to the bearing number (0) ;

[0067] X (0) ={X (0) (1),X (0) (2),X (0) (3),…,X (0) (11)

[0068] By establishing the mathematical model of each bearing oil film, the stiffness and damping matrix X of each bearing oil film is calculated. (1) ;

[0069] X (1) ={X (1) (1),X (1) (2),X (1) (3),…,X (1) (11)

[0070] The rotor support dynamic matrix X is obtained when the rigid support is used. (1) and the dynamic coefficient matrix X when the flexible support (2) , X (2) =X (0) +X (1) ;

[0071] The rotor model is established by finite element method, and the support stiffness and damping coefficient are loaded in the rotor model. (1) and X (2) Input it to calculate the dynamic balance response coefficient at different speeds and different positions and curve;

[0072] When the weight is increased at the same speed and position and Divide by to get the difference coefficient The difference coefficient curve of

[0073] As an example, step 2 specifically includes:

[0074] The actual dynamic balance influence coefficient of the rigid support rotor in the past is the empirical coefficient accumulated during the dynamic balancing process of the rigid support unit in the past;

[0075] By comparing the difference coefficient curve obtained in step 1, the corresponding Calculate the response coefficient of the flexible support unit corresponding to the speed and weighted position

[0076] In one example, step 3 specifically includes:

[0077] Vibration data measured by two sensors at the front and rear support bearings of each rotor in the shaft system The first-order vibration mode component of the rotor vibration vector can be decomposed using the formula: Second-order modal components The third-order vibration mode component

[0078] in,

[0079] When the calculated speed condition is between the first and second critical speeds of the rotor:

[0080]

[0081] Where: They are first-order and second-order balancing weights respectively; is the influence coefficient of the first-order counterweight under the flexible support on the first-order vibration mode; is the influence coefficient of the second-order counterweight on the second-order vibration mode under flexible support; is the interference influence coefficient of the first-order counterweight on the second-order vibration mode of the flexible support; is the interference influence coefficient of the first-order vibration mode of the second-order counterweight under flexible support;

[0082] When the calculated speed condition is between the second and third critical speeds of the rotor:

[0083]

[0084] Where: They are second and third order balance weights respectively; is the influence coefficient of the second-order counterweight on the second-order vibration mode; is the influence coefficient of the third-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order vibration mode of the third-order counterweight.

[0085] A precise dynamic balancing method for a flexible support steam turbine generator set is divided into three steps: calculation of the dynamic balancing coefficient difference coefficient curve, determination of the dynamic balancing coefficient of the flexible support rotor, and formulation of a dynamic balancing plan. These three steps together achieve accurate and efficient on-site dynamic balancing.

[0086] Based on the dynamic balancing method of a flexible support steam turbine generator set provided in the above embodiment, an embodiment of the present application further provides a dynamic balancing device of a flexible support steam turbine generator set.

[0087] The embodiment of the application provides a dynamic balancing device for a flexible support steam turbine generator set, comprising:

[0088] A curve acquisition module is used to establish a mathematical model of a rigid support rotor and a flexible support rotor, simulate and calculate the dynamic balance response coefficients of the rigid support rotor and the flexible support rotor under the influence of different vibration modes in the mathematical model, and obtain a difference coefficient curve of the dynamic balance response coefficient between the flexible support steam turbine generator set and the rigid support;

[0089] A coefficient acquisition module is used to obtain the dynamic balance influence coefficient of the flexible support rotor by correcting the difference coefficient curve based on the actual dynamic balance influence coefficient of the rigid support rotor obtained in advance;

[0090] A solution formulation module is used to use the obtained dynamic balancing influence coefficients and consider the influence of mode shapes in the vibration data to formulate a dynamic balancing solution for the flexible supported rotor.

[0091] In one example, the curve acquisition module is specifically configured to:

[0092] Statistically calculate the stiffness and damping coefficient matrix of each support bearing of the rotor, and form a sequence X according to the bearing number (0) ;

[0093] X (0) ={X (0) (1),X (0) (2),X (0) (3),…,X (0) (11)

[0094] By establishing the mathematical model of each bearing oil film, the stiffness and damping matrix X of each bearing oil film is calculated. (1) ;

[0095] X (1) ={X (1) (1),X(1) (2),X (1) (3),…,X (1) (11)

[0096] The rotor support dynamic matrix X is obtained when the rigid support is used. (1) and the dynamic coefficient matrix X when the flexible support (2) , X (2) =X (0) +X (1) ;

[0097] The rotor model is established by finite element method, and the support stiffness and damping coefficient are loaded in the rotor model. (1) and X (2) Input it to calculate the dynamic balance response coefficient at different speeds and different positions and curve;

[0098] When the weight is increased at the same speed and position and Divide by to get the difference coefficient The difference coefficient curve of

[0099] As an example, the coefficient acquisition module is specifically used to:

[0100] The actual dynamic balance influence coefficient of the rigid support rotor in the past is the empirical coefficient accumulated during the dynamic balancing process of the rigid support unit in the past;

[0101] By comparing the difference coefficient curve obtained in step 1, the corresponding Calculate the response coefficient of the flexible support unit corresponding to the speed and weighted position

[0102] In one example, the plan formulation module is specifically used to:

[0103] Vibration data measured by two sensors at the front and rear support bearings of each rotor in the shaft system The first-order vibration mode component of the rotor vibration vector can be decomposed using the formula: Second-order modal components The third-order vibration mode component

[0104] in,

[0105] When the calculated speed condition is between the first and second critical speeds of the rotor:

[0106]

[0107] Where: They are first-order and second-order balancing weights respectively; is the influence coefficient of the first-order counterweight under the flexible support on the first-order vibration mode; is the influence coefficient of the second-order counterweight on the second-order vibration mode under flexible support; is the interference influence coefficient of the first-order counterweight on the second-order vibration mode of the flexible support; is the interference influence coefficient of the first-order vibration mode of the second-order counterweight under flexible support;

[0108] When the calculated speed condition is between the second and third critical speeds of the rotor:

[0109]

[0110] Where: They are second and third order balance weights respectively; is the influence coefficient of the second-order counterweight on the second-order vibration mode; is the influence coefficient of the third-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order vibration mode of the third-order counterweight.

[0111] The present application has been described in detail above with reference to the embodiments. However, the present application is not limited to the above embodiments. Various modifications can be made within the scope of knowledge possessed by a person skilled in the art without departing from the purpose of the present application. Any content not described in detail in the present application may be based on existing technologies.

Claims

1. A dynamic balancing method for a flexible support steam turbine generator set, characterized in that: The method comprises: Step 1: Establish mathematical models of rigid support rotors and flexible support rotors, simulate and calculate the dynamic balance response coefficients of the rigid support rotors and flexible support rotors under the influence of different vibration modes in the mathematical models, and obtain the difference coefficient curve of the dynamic balance response coefficient between the flexible support steam turbine generator set and the rigid support; Step 2: Based on the actual dynamic balance influence coefficient of the rigid support rotor obtained in advance, the dynamic balance influence coefficient of the flexible support rotor is obtained by correcting the difference coefficient curve; Step 3: Use the obtained dynamic balancing influence coefficients and consider the influence of mode shapes in the vibration data to develop a dynamic balancing plan for the flexibly supported rotor.

2. The dynamic balancing method of a flexible support steam turbine generator set according to claim 1, characterized in that: The step 1 specifically includes: Statistically calculate the stiffness and damping coefficient matrix of each support bearing of the rotor, and form a sequence X according to the bearing number (0) ; X (0) ={X (0) (1),X (0) (2),X (0) (3),…,X (0) (11)} By establishing the mathematical model of each bearing oil film, the stiffness and damping matrix X of each bearing oil film is calculated. (1) ; X (1) ={X (1) (1),X (1) (2),X (1) (3),…,X (1) (11)} The rotor support dynamic matrix X is obtained when the rigid support is used. (1) and the dynamic coefficient matrix X when the flexible support (2) , X (2) =X (0) +X (1) ; The rotor model is established by finite element method, and the support stiffness and damping coefficient are loaded in the rotor model. (1) and X (2) Input it to calculate the dynamic balance response coefficient at different speeds and different positions and curve; When the same speed and position are increased and Divide by to get the difference coefficient The difference coefficient curve of 3. The dynamic balancing method of a flexible support steam turbine generator set according to claim 2, characterized in that: The step 2 specifically includes: The actual dynamic balance influence coefficient of the rigid support rotor in the past is the empirical coefficient accumulated during the dynamic balancing process of the rigid support unit in the past; By comparing the difference coefficient curve obtained in step 1, the corresponding Calculate the response coefficient of the flexible support unit corresponding to the speed and weighted position 4. The dynamic balancing method of a flexible support steam turbine generator set according to claim 3, characterized in that: The step 3 specifically includes: Vibration data measured by two sensors at the front and rear support bearings of each rotor in the shaft system The first-order vibration mode component of the rotor vibration vector can be decomposed using the formula: Second-order modal components The third-order vibration mode component in, When the calculated speed condition is between the first and second critical speeds of the rotor: Where: They are first-order and second-order balancing weights respectively; is the influence coefficient of the first-order counterweight under the flexible support on the first-order vibration mode; is the influence coefficient of the second-order counterweight on the second-order vibration mode under flexible support; is the interference influence coefficient of the first-order counterweight on the second-order vibration mode of the flexible support; is the interference influence coefficient of the first-order vibration mode of the second-order counterweight under flexible support; When the calculated speed condition is between the second and third critical speeds of the rotor: Where: They are second and third order balance weights respectively; is the influence coefficient of the second-order counterweight on the second-order vibration mode; is the influence coefficient of the third-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order vibration mode of the third-order counterweight.

5. A dynamic balancing device for a flexible support steam turbine generator set, characterized in that: The device comprises: A curve acquisition module is used to establish a mathematical model of a rigid support rotor and a flexible support rotor, simulate and calculate the dynamic balance response coefficients of the rigid support rotor and the flexible support rotor under the influence of different vibration modes in the mathematical model, and obtain a difference coefficient curve of the dynamic balance response coefficient between the flexible support steam turbine generator set and the rigid support; A coefficient acquisition module is used to obtain the dynamic balance influence coefficient of the flexible support rotor by correcting the difference coefficient curve based on the actual dynamic balance influence coefficient of the rigid support rotor obtained in advance; A solution formulation module is used to use the obtained dynamic balancing influence coefficients and consider the influence of mode shapes in the vibration data to formulate a dynamic balancing solution for the flexible supported rotor.

6. The dynamic balancing device of the flexible support steam turbine generator set according to claim 5, characterized in that: The curve acquisition module is specifically used to: Statistically calculate the stiffness and damping coefficient matrix of each support bearing of the rotor, and form a sequence X according to the bearing number (0) ; X (0) ={X (0) (1),X (0) (2),X (0) (3),…,X (0) (11)} By establishing the mathematical model of each bearing oil film, the stiffness and damping matrix X of each bearing oil film is calculated. (1) ; X (1) ={X (1) (1),X (1) (2),X (1) (3),…,X (1) (11)} The rotor support dynamic matrix X is obtained when the rigid support is used. (1) and the dynamic coefficient matrix X when the flexible support (2) , X (2) =X (0) +X (1) ; The rotor model is established by finite element method, and the support stiffness and damping coefficient are loaded in the rotor model. (1) and X (2) Input it to calculate the dynamic balance response coefficient at different speeds and different positions and curve; When the same speed and position are increased and Divide by to get the difference coefficient The difference coefficient curve of 7. The dynamic balancing device of a flexible support steam turbine generator set according to claim 6, characterized in that: The coefficient acquisition module is specifically used to: The actual dynamic balance influence coefficient of the rigid support rotor in the past is the empirical coefficient accumulated during the dynamic balancing process of the rigid support unit in the past; By comparing the difference coefficient curve obtained in step 1, the corresponding Calculate the response coefficient of the flexible support unit corresponding to the speed and weighted position 8. The dynamic balancing device of a flexible support steam turbine generator set according to claim 7, characterized in that: The program formulation module is specifically used to: Vibration data measured by two sensors at the front and rear support bearings of each rotor in the shaft system The first-order vibration mode component of the rotor vibration vector can be decomposed using the formula: Second-order modal components The third-order vibration mode component in, When the calculated speed condition is between the first and second critical speeds of the rotor: Where: They are first-order and second-order balancing weights respectively; is the influence coefficient of the first-order counterweight under the flexible support on the first-order vibration mode; is the influence coefficient of the second-order counterweight on the second-order vibration mode under flexible support; is the interference influence coefficient of the first-order counterweight on the second-order vibration mode of the flexible support; is the interference influence coefficient of the first-order vibration mode of the second-order counterweight under flexible support; When the calculated speed condition is between the second and third critical speeds of the rotor: Where: They are second and third order balance weights respectively; is the influence coefficient of the second-order counterweight on the second-order vibration mode; is the influence coefficient of the third-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order counterweight on the third-order vibration mode; is the interference influence coefficient of the second-order vibration mode of the third-order counterweight.

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