KR19990039105A - Method for measuring tuning response of rotating body considering initial deformation and balancing method of rotating body - Google Patents

Abstract

Disclosed are a method of measuring a tuning response of a rotating body in which initial deformation is considered and a method of balancing the rotating body. The disclosed measurement methods are: dividing the desired rotor into any number of nodes, and corresponding to all nodes, M (mass, inertia), G (gyro), K (rigid), C (attenuation) Obtaining a matrix; Obtaining an initial deformation vector δ according to the initial deformation of the rotating body; Obtaining a forward turning response vector (P _f ) and a rear turning response vector (P _b ) in which an initial deformation vector (δ) and an imbalance vector (W1) are considered while rotating the rotating body at an arbitrary speed (Ω1); Calculating an imbalance vector W1 corresponding to an arbitrary speed? 1 from the swing response vector P _f and the rear swing response vector P _b ; Unbalance distribution vector (1) considering the unbalance vector (W1) satisfying the forward swing response vector (P _f ') and the rear swing response vector (P _b ') of the given specification at the speed (Ω2) to balance the rotating body ( Calculating W2); Determining the position, magnitude, and direction of the compensation mass with respect to the rotating body from the imbalance vector W2; Balancing the rotating body according to the determined position, size, and direction of the compensation mass; By including it, the balance with respect to a rotating body can be performed quickly, correctly, and efficiently.

Description

Method of measuring tuning response of rotating body considering initial deformation and balancing method of rotating body

The present invention relates to a method for measuring the tuning response of a rotating body in consideration of the initial deformation of the rotating body and a balancing method of the rotating body. The size, position, and direction of the corrected mass according to the initial deformation and imbalance of the rotating body can be effectively calculated. It relates to a method and a balancing method of the rotating body.

Rotor balancing is basically necessary for smooth operation of the rotating body, in particular the rotating body rotating at high speed. In the case of a rotor in an ideal state, the axis of inertia of the rotor coincides with the axis of rotation of the rotor in all motions. This is not the case in the real world, which creates centrifugal forces and moments, which transmit large forces to the bearings or support structures that support the rotor. If the rotor is largely unbalanced, the movement becomes large in correspondence with the degree of imbalance, thereby destroying the bearings and the supporting structure supporting the rotor. Unbalance and shaft alignment are known to be two of the main factors leading to machine malfunctions and catastrophic failures. Unbalance of the rotor in the machine can be caused by tolerances and material unevenness in the initial fabrication process, or it can be caused by corrosion in use, thermal effects (deformation), and accumulation of material on its surface. Multistage rotors such as compressors, pumps, turbines, etc. have a structure in which many rotors are complex, so that imbalances may appear in the rotors. In the initial fabrication of most rotors, from simple motors to complex pumps, compressors and aircraft engines, manufacturers typically apply complex procedures to ensure the initial balancing of machines.

Balancing on the rotor includes the initial balancing of all components and the final balancing of the assembly. This balancing procedure includes a commercial balancing device based on the concept of flexible bearings or rigid bearing support. Thus, the rotors are balanced with extremely high precision in that respect. For highly flexible rotors, the manufacturer can use a high-speed rotary pit facility for final trim balancing. Any situation during which the rotor is operating can cause an imbalance in the system. As such, the balancing of the rotor can vary after initial installation, where the balancing procedure can be quite different from that initially applied by the manufacturer. In order to correct this imbalance, it is costly and time consuming to remove the rotor and return the equipment to the manufacturer. Therefore, in most cases, balancing is performed on-site with the facility installed. The purpose of in-situ balancing is to add correction mass to one or several axial planes of the rotor in order to maintain an acceptable level of vibration. Field balancing has been studied for quite some time.

The International Standards Organization (ISO) published a document on the quality of rotating bodies and the classification of rotors in 1973. Rotors can be classified as either rigid or ductile, depending on their dynamic behavior at operating speed. The classification of the rotor is determined by performing critical velocity analysis in the system. If the deformation or potential energy of the bearing exceeds 80% of the total strain energy of the system, the rotor is generally classified as rigid. The rigid rotor can be balanced in two arbitrary planes. The rotor can be balanced through the operating speed section. On the other hand, when the strain energy of the shaft exceeds 20% of the strain energy of the system and the rotor is operated at one or more critical speeds, it may be considered a soft or semi-ductile rotor. In this environment, two-plane rigid body balancing is not appropriate, and at what speed it may be necessary to place additional trim mass along the axis to minimize vibration amplitude. The ISO describes the type and quality of balance required for each specific case.

Five basic classifications were introduced by the ISO. First, a rigid rotor of class 1 can be balanced in two arbitrary axial planes and balanced in the operating speed range. Secondly, class 2 quasi-soft rotors are not completely rigid but can be properly balanced in low speed balancing machines and maintain smooth operation in the speed range. Third, a class 3 flexible rotor cannot be balanced in a low speed balance machine and requires one or more high speed trim plane corrections. Fourth, the flexible addition rotors of class 4 can be classified as classes 1,2 or 3, but they have components or flexible couplings. Fifth, a class 5 single-speed flexible rotor can be classified as a class 3 flexible rotor, but balanced to operate at only one speed. And single-sided or two-sided constant balancing is usually suitable for Class 1 and 2. Classes 3 and 4 often use least square error coefficients or combined modal (MODAL) techniques for flexible rotors.

In the balancing type, balancing of rotary machines can be generally classified into two general areas. The first general area is factory balancing performed on rigid or soft bearing balancing machines. This method is carried out by the producer and the components are individually balanced as well as the final assembly is balanced as a rigid rotor. If components are balanced before final assembly, as in final assembly, even very long flexible rotors can be operated very successfully without additional adjustments in the field. In particular, a rotating body, such as a generator, may need to be further balanced in the rotating pit installation. This is usually done for class 3 rotors that must be operated over several critical speeds. This balancing procedure is very expensive and time consuming. There are not many facilities available for this balancing. The second wider balancing area is field balancing where the rotor is balanced at the site of use. In field balancing, the turbo rotor is balanced using its own bearings and suitable devices are installed on the shaft and bearing housing or foundation to monitor motion.

The following briefly summarizes the conventional field balancing types and their procedures.

-Single face balancing by influence coefficient method

In balancing, the single-sided single velocity coefficient of influence is the simplest of all procedures. The phase angle depends on the measured value of the displacement probe and can be obtained directly from the strobe light. Some balancing devices also have different conventions for initial phase measurements. In this method the trial mass (calibration mass) is installed on the shaft. The new vibration response of the rotating body with the corrected mass is recorded after mounting the trial mass. The change in vibration is divided by the trial mass and becomes the coefficient of influence. Here, the coefficient of influence represents the rotational response generated by mounting the unit trial mass at the specified velocity. It should be recognized that this coefficient of influence is a vector quantity representing amplitude and phase angle. The phase angle represents the relative phase between the excitation function and the vibration measurement. Field balancing by coefficient of influence requires a full understanding of the phase measurement. Phase measurements can be made with contactless or displacement probes, speed pickups, or accelerometers. The phase can measure the phase angle of the axis with an electronic key pager or strobe light. Familiarize yourself with the phase measurement procedures you will apply before attempting single-plane or multi-plane balancing. In single level balancing by the coefficient of influence method, the system is considered to be linear and based on the calculated coefficient of impact, a correction mass is placed on the axis. This procedure is called trim balancing and is a commonly used method in 90% of the field facing balancing. By applying a single plane balance correction, the vibration of the rotor can be made zero at that specific speed. The main problem with the single-sided single velocity coefficient of influence balancing method is that even if the amplitude of vibration decreases at that particular position and speed, it can show large vibration elsewhere. At other speeds, the rotor may appear to be unbalanced. The advantage of this method is that it can be implemented quickly in the field and the coefficient of influence can be calculated as a general procedure. Once the coefficient of influence has been calculated for a particular rotor, the machine can be balanced without adding additional trial mass. This balancing method is called a one-shot balancing method and is based on a predetermined coefficient of influence value.

Two-sided Balancing by Influence Coefficient Method

Many unstable rotors in one unbalanced phase. Balance correction on one side can cause excessive amounts of vibration elsewhere. At the same time under these circumstances, the coefficient of impact balancing procedure should be applied. This method is a schematic method for determining the axial influence coefficient and the two-sided balance correction. This method has been very widely used and is still used in industry. This method can be simply programmed with a hand calculator. In a two-sided balancing procedure, the trial mass is mounted on the first or near plane and the response is measured on both sides. The mass is then removed and placed in the second or distant plane and again measured on two sides. Two more coefficients of influence are obtained. This constitutes a 2x2 complex coefficient of influence matrix that must be inversed to balance. The 2x2 complex coefficient matrix is the same as the 4x4 real matrix. Therefore, balancing is equivalent to taking the inverse of the 4x4 matrix to determine the four quantities representing the corresponding rotor balancing magnitude and relative phase angle position. This method is boring with counting and is very simple for calculators and computers.

-Single face balancing with static and dynamic components

This method is a variation of the single-faced influence factor procedure except that vibrations are measured at both ends of the axis. The two vibration records are vectored to subtract and determine the static and dynamic components. If the average static component appears to be high, then one or two masses are installed in phase. This method is a simplified mode procedure using the influence coefficient method.

Single-face Balancing by Influence Coefficient Method Using Linear Regression

In single-sided balancing, the rotor can be over-balanced at one speed, resulting in extremely high vibrations at other speeds. Linear regression balancing is a least-squares error method for rotor balancing. Instead of measuring the rotor response at one speed, vibration characteristics are measured in one speed section. Balancing by linear regression does not cause large excitation at other speeds. The magnitude of the balancing calculated by this method is always smaller than the magnitude calculated from the single-faced influence coefficient method. This method ensures that too large a balance mass is applied and not overcorrected. This method is useful for balancing rotors with axial deflection and large initial runout.

-Generalized impact coefficients using PSEUDO-INVERSION

This method is generalized by using two or more planes and multi-speed measurements and is called 'generalized influence coefficient'. This method, also called the quasi-orthogonal or least-squares error method, is called EXACT SPEED POINT METHOD OF BALANCING. Is the best fit and is the preferred method for large turbine generators. Generalized least-squares error balancing requires no dynamic prior knowledge of the rotating body of the system. It has been successfully used in multi-plane balancing of turbine-generators over five critical speeds. Several variations of this procedure have been developed, such as the weighted least-squares error method, which emphasizes the weighted score of a particular vibration to minimize or ignore it. If the balance plane is inappropriately selected, the least square error method in the adjacent plane that is 180 degrees or more (OUT OF PHASE) creates an extremely large balance mass that is difficult to apply to the rotor. The least-squares error method cannot limit the magnitude of the predicted balance mass for a particular plane.

Multi-plane balancing using linear programming

This method proceeds in a manner similar to the least-squares error balancing method to obtain the coefficient of influence. However, linear programming can impose a limit or upward bound on the magnitude of the balancing mass calculated at a particular station. However, the result of linear programming and generalized least squares error balancing theory are very similar in the best working system.

Mode balancing

Flexible rotors that operate over several critical speeds, such as generators, are best balanced by this method. In this method, the critical velocity mode form of the rotor should be known through theoretical or experimental measurements. The mass installed on the rotor is proportional to the rotor mode. Each mode distribution is used to balance one mode at a time. The goal of this method is to balance the rotor with a large magnification factor without compromising the balance in other modes. The amount of mode balance correction applied to a rotor is best calculated using an influence coefficient method that predicts the mode effect balancing coefficients and measures the rotor response. M. Darlow introduced many documents in 1989 about combined modes and coefficients of influence, called 'unified methods of rotor balancing'. The accuracy of this method relies on knowledge of the rotor mode form, which can be very complex above the second critical speed. This method has been successfully applied to generators and long flexible centrifuges.

-3 trial mass balancing

In general, it is often not possible to obtain an accurate phase characteristic, so by applying the trial mass to three different locations, it is possible to generate a trajectory of the balance point to determine the magnitude of the imbalance and the location of the balance. This method has been applied very successfully to fans that have a constant rotation speed and beats. The calculation method is schematic and requires no computer.

-Phase free of the flat balancing

The three attempted mass balancing method has been extended to balance the flexible rotor over several critical speeds. In this expansion, the mass is installed on the rotor with a modal distribution. The amplitude of the rotor is measured at the critical velocity using an FFT analyzer or using a peak hold data acquisition system that records the maximum amplitude at a specific duration. Each mode can then be balanced individually without having the previous critical speed mode. This method is advantageous for balancing flexible rotors that have very large magnification factors and are difficult or impossible to balance with least-squares errors or linear programming methods. This is advantageous for precisely balancing an axis with significant axial bending. The disadvantage of this method is that the rotor mode type must be known and three runs must be made for the angular velocity to be balanced. After this method is done, it may be possible to start the operation twice using the rotor mode sensitivity derived from the initial practice.

Rotor balancing without trial mass

This method is called a one-time balancing method. Once the coefficient of influence is determined for a particular rotor, the imbalance can be predicted by the current coefficient of influence and the measured vibration. In this method, the Nyquist plot of the rotor amplitude is used to determine the rotor expansion coefficient in a particular mode. By calculating the rotor expansion coefficient, the mode imbalance eccentricity can be determined at a particular critical speed. From the pole plot or the Nyquist plot, the phase position of the mode mass distribution can be determined to balance a specific critical velocity. The disadvantage of this method is that you need to know the rotor mode mass and the critical velocity mode. The advantage of this method is that the balance predictor can be used during initial trial operation to further improve by the coefficient of influence balancing method.

-Coupling Trim Balancing

In some cases, the two machines are not completely balanced, but they can have a large vibration when connected by coupling. It is not advisable to attempt to rebalance the unit since the two rotors are balanced. In the coupling balancing method developed by Winkler in 1983, the coupling is rotated 180 degrees. From the new vibration measurement, the optimum position of the coupling can be calculated for trim balancing.

The conventional methods described above are for determining the size, position, and direction of the imbalance, and by determining the influence coefficient due to the attachment of the trial mass, the size, position, and direction of the imbalance can be known and can be balanced by correcting the balancing mass. Do. The trial mass is intended to estimate this because we do not know the current state of the balancing object. You can add more trial masses on more than one face to make it more accurate, or measure multiple places to balance the root sum square (RSS) of the least squares of the responses from the least squares. However, this process is more difficult and time consuming because the actual environment has a deflection of the shaft in addition to the imbalance.

Theoretically, when there is a perfect model, one attempted mass gives the magnitude, position, and direction of the imbalance, as well as the balancing of systems operating at different speeds, such as mode balancing. Since the speed of the rotor changes the effects of the imbalance and the inherent response tendency (called the mode) of the system, balancing must know the mode of the system, whether experimental or simulation, and then balance for each mode.

In general, axial deflection in the rotating body is necessarily generated to some extent and difficult to correct. Even if we rebuild the axis, we don't know if it will be necessary. In other words, in the actual rotating body in many cases due to the initial deformation of the rotating body, the vibration characteristics due to the imbalance and other synchronous vibration phenomenon occurs, causing confusion in the actual measurement. For example, in the balancing of the rotating body, it is assumed that all of the vibration characteristics synchronized with the rotational speed are due to the imbalance, thereby causing an error in the balancing efficiency.

The present invention clearly identifies the vibration characteristics due to imbalance and other tuning vibrations by the initial deformation of the rotating body, and provides a method of measuring the tuning response of the rotating body in consideration of the initial deformation of the rotating body and a balancing method of the rotating body. There is this.

In addition, another object of the present invention is to provide a method for measuring the amount of deformation of a rotating body in consideration of the initial deformation of the rotating body to enable effective balancing of the rotating body and a balancing method of the rotating body.

1 is a structural diagram of a general motor bearing system with an initial deformation.

Fig. 2A is a schematic configuration diagram of a rotating device for experimentally carrying out the method of the present invention.

FIG. 2B is an orbital plot of the rotating body of the apparatus shown in FIG. 2A.

FIG. 3A is a diagram showing a mode shape by a mode test of the rotating body illustrated in FIG. 2.

FIG. 3B is a diagram showing an initial deformation amount for each node (position, node) of the rotating body shown in FIG. 2.

4 is a diagram showing a tuning vibration response by experiment of the rotating body shown in FIG. 3.

FIG. 5 is a diagram illustrating a theoretical tuning response for each node in which only the initial deformation is considered in the rotating body illustrated in FIG. 3.

FIG. 6 is a diagram showing a theoretical tuning response for each node considering only mass imbalance in the rotor shown in FIG. 3.

FIG. 7 is a diagram illustrating a theoretical tuning response for each node in which the initial deformation and the mass imbalance are considered together in the rotor shown in FIG. 3.

8 illustrates a rotor for numerically analyzing vibration characteristics of a general rotor having an initial deformation.

FIG. 9 is a diagram illustrating an initial strain amount of each node of the rotating body illustrated in FIG. 8.

10A and 10B show initial deformation (a), imbalance (b), and initial deformation and imbalance (c) at node # 3 of the rotor shown in FIGS. 8 and 9 with an isotropic bearing. This is a diagram showing the real and imaginary parts of the considered tuning vibration.

FIG. 11 illustrates an orbital plot for each node of a rotor according to a predetermined rotation speed in the rotor illustrated in FIG. 10.

12A and 12B are rotation speed-amplitude (imaginary part and real part) sizes respectively showing the unbalance response functions H _ff , F _bf , F _fb , and F _{bb of} the rotating body shown in FIG. 8. Is leading.

Figures 13 (A) and 13 (B) show the tuning vibration (a), unbalance response (b), and initial stage according to the initial deformation when the bearings are replaced by anisotropic bearings in the rotating bodies shown in Figs. Rotational speed-amplitude (imaginary part, real part) magnitude diagram showing the tuning response (c) due to deformation and imbalance, respectively.

In order to achieve the above object, according to the present invention, the desired rotating body is divided into any number of nodes, and M (mass, inertia), G (gyro), and K (corresponding to all nodes) Stiffness), C (attenuation) matrix; Obtaining an initial deformation vector δ according to the initial deformation of the rotating body; Obtaining a forward turning response vector (P _f ) and a rear turning response vector (P _b ) in consideration of an initial deformation vector (δ) and an unbalance vector W1 while rotating the rotating body at an arbitrary speed Ω 1; Obtaining an imbalance vector (W1) corresponding to an arbitrary speed (Ω1) from the turning response vector (P _f ) and the rear turning response vector (P _b ), at a speed (Ω2) to balance the rotor, The imbalance vector W1 that satisfies the forward turning response vector P _f ′ and the rear turning response vector P _b ′ from the forward turning response vector P _f ′ and the rear turning response vector P _b ′ is taken into account. Determining an unbalanced distribution vector (W2), comprising: determining the position, magnitude, and direction of the compensation mass with respect to the rotor from the imbalance vector (W2); balancing the rotor according to the position, magnitude, and direction of the determined compensation mass; Provided is a method for measuring the amount of deformation of a rotating body considering the initial deformation comprising a step; The.

In order to achieve the above object, according to another type of the present invention, the desired rotating body is divided into any number of nodes, and M (mass, inertia) and G (gyro) corresponding to all nodes. Obtaining a K (rigid), C (decay) matrix; Obtaining an initial deformation vector δ according to the initial deformation of the rotating body; Obtaining a forward turning response vector (P _f ) and a rear turning response vector (P _b ) in consideration of an initial deformation vector (δ) and an unbalance vector W1 while rotating the rotating body at an arbitrary speed Ω 1; Obtaining an imbalance vector (W1) corresponding to an arbitrary speed (Ω1) from the turning response vector (P _f ) and the rear turning response vector (P _b ), at a speed (Ω2) to balance the rotor, The imbalance vector W1 that satisfies the forward turning response vector P _f ′ and the rear turning response vector P _b ′ from the forward turning response vector P _f ′ and the rear turning response vector P _b ′ is taken into account. Determining an unbalanced distribution vector (W2), comprising: determining the position, magnitude, and direction of the compensation mass with respect to the rotor from the imbalance vector (W2); balancing the rotor according to the position, magnitude, and direction of the determined compensation mass; There is provided a method of balancing a rotating body comprising a step.

The present invention obtains a theoretical model to model the effects of imbalance and axial deflection on a system modeled with finite elements. According to the present invention, by measuring the axial deflection and measuring the response at any one speed, the proposed equation can be used to estimate the response due to both unbalance and warp at any speed. Similarly, the influence of the corrected mass can be seen. This saves time and effort. This method is for flexible rotors (all objects are flexible but classified for rigidity for convenience and economic reasons and are not classified as stiffness even if the approach of the rigid system is not very wrong). Can be evaluated in multiple ways.

Method for measuring the deformation amount of the rotating body according to the present invention includes the following sequence.

1. The desired rotor is divided into any number of nodes, and M (mass, inertia), G (gyro), K (stiffness), and C (attenuation) matrix corresponding to all nodes are obtained. .

2. Find the initial strain vector δ according to the initial deformation of the rotor.

3. Influence coefficient method for the forward turning response vector (P _f ) and the rear turning response vector (P _b ) considering the initial deformation vector (δ) and the imbalance vector W1 while rotating the rotor at an arbitrary speed (Ω1). Get by

4. The imbalance vector W1 corresponding to the arbitrary speed? 1 is obtained from the swing response vector P _f and the rear swing response vector P _b .

5. The time at speed (Ω2) to balance the whole, the front turning response of a given specification vector (P _f ') and the rear turning response vector (P _b' forward turning response from) vector (P _f ') and the rear An imbalance distribution vector W2 considering the imbalance vector W1 satisfying the turning response vector P _b ′ is obtained.

6. Determine the position, magnitude, and direction of the compensation mass relative to the rotor from the unbalance vector (W2).

7. Finally, balance the rotor according to the position, size, and direction of the determined compensation mass.

Hereinafter, the present invention will be described in more detail.

All rotation shafts inevitably generate more than a certain amount of deformation in the manufacturing process, and deformation may occur during operation due to deformation factors such as uneven temperature distribution depending on the operating conditions of the rotating body. As shown in FIG. 1, the initial deformation state may be expressed such that the geometric center of the rotating body deviates from the drive shaft line. If the displacement of the rotational axis from the linear axis of the drive shaft or the rotational axis is taken as the coordinate, the finite element equation of motion of the elastic rotor bearing system with initial deformation is defined by the following equation (1) or (2).

In Equations 1 and 2, M is an inertia (mass) matrix, G is a gyroscopic matrix, K is a stiffness matrix, C is an attenuation matrix, and superscripts r and c are rotating bodies and connecting supports (bearings, etc.), respectively. Q is the finite element coordinate vector, f is the external force vector, and Ω is the velocity. Herein, the finite element coordinate vector q and the external force vector f are expressed by Equation 3 below.

In Equation 3, y and z are the y- and z-direction coordinate vectors at the nodes, respectively, and f _y and f _z are corresponding force vectors. If the external force vector f _w is caused by an imbalance, it is expressed as Equation 4 below.

Where W is a complex imbalance vector and n is the number of nodes in the finite element model. In addition, Δ is an initial deformation vector and is expressed by Equation 5 below. The unbalance vector (W) can be obtained by a general influence coefficient method, and inversely obtains the unbalance distribution according to the initial strain and the unbalance by the nodes of the rotor from the forward swing response (P _f ) and the rear swing response (P _b ) described later. Thus, the position and size of the correction mass relative to the rotating body can thus be determined.

Where δ is the initial deformation vector due to the axial deflection expressed in complex numbers, and δ _i , δ _{θ i} (i = 1, 2, n) represent the linear and angular displacements of the axis at the corresponding nodes (sampling positions), respectively. Indicates. The above δ is obtained from each node of the rotating body rotating at a low speed such that there is no response due to a stationary state or an imbalance during the process of the present invention.

The external force term due to the imbalance of Equation 4 gives only the excitation force for the linear displacement coordinates and is proportional to the square of the rotational speed, while the initial deformation effect expressed in Equation 5 also provides the excitation force for each displacement coordinate. However, there is a characteristic that is independent of the rotational speed. However, since the excitation force due to the initial deformation gives the actual excitation force by the product of the stiffness matrix, the excitation force may be affected by the rotation speed by the rotation speed dependency of the bearing. The stiffness and attenuation matrices K ^c (Ω) and C ^c (Ω) of the connection are both asymmetric in general and are dependent on the rotational speed (Ω) by the bearing. The inertia matrix M ^r and the rigid matrix K ^r are symmetrical, and the gyro matrix G ^r is skew symmetric. The system-wide matrices are all 2N × 2N, where N is the dimension of the y or z coordinate vector (N = 2 × finite element model nodes). If the matrix of equations is expressed as partial matrix, it is expressed as Equation 6 below.

In Equation 6, the partial matrices M, G, and K are symmetrical. According to Equation 2, the influence of the initial strain on the rotating body is shown to be an external force applied to the ideal linear axis, the distribution load for the given initial strain occurs. Therefore, the initial deformation in the general rotating body shows the characteristics of the external force distributed along the rotation axis, and eventually the effect depends on the main mode affecting the rotation speed range.

In order to identify the initial deformation, the components tuned to the rotational speed along the axis of rotation are measured while rotating the rotor at low speed. When measuring the initial strain, eddy currents or capacitive displacement sensors, which are affected by the material properties of the rotating shaft, do not cause electrical runout or precisely determine the initial strain in the exact sense, such as laser scan micrometers. It is desirable to use equipment that can track the geometric center position. The initial deformation is expected to have a relatively simple shape along the axis of rotation, so it can be used by interpolation only with data measured at several preselected positions. Initial deformation in the case of deformation due to its own weight can be excluded from the model by defining the driving center axis according to deformation due to its own weight.

Conventionally, the modal component analysis of the excitation force was possible by the mode analysis because the analysis was performed in the state of ignoring the gyro effect or ignoring the bearing anisotropy and rotational speed dependence on the simple rotor. However, in the general rotor bearing system, there is a problem that it is difficult to apply the expansion theorem with eigenvalues and eigenvectors calculated at a specific rotational speed. Therefore, the tuning vibration response equation is described based on the direct calculation method. Since the use of a complex coordinate system has many advantages in the vibration analysis of the axisymmetric rotating body, the motion equation of the complex coordinate system is used here. First, the complex coordinate system may be defined as in Equation 7 below, and Equation 2 may be expressed as in Equation 8.

p = y + jz, f = fy + jfz

Subscripts f and b in Equation 8 are forward and backward, respectively, and "-" means a conjugate complex number. The diagonal complex bearing damping and stiffness matrix is

In Equation 8 above, the external force vector f may be expressed as Equation 10 below.

The tuning vibration response for the rotation speed tuning excitation given in Equation 10 is expressed as Equation 11 below with the front and rear components.

In Equation 11, P _f and P _b mean the forward and backward whirl response vectors of the unbalanced response, respectively. In this case, the tuning vibration response is represented by Equation 12 below by substituting Equation 10 and 11 into Equation 8.

From the forward swing response P _f and the rear swing response P _b obtained through Equation 12 above, the above-described unbalance distribution of the unbalance vector W can be known. Therefore, the magnitude of the correction mass for the rotating body, Position and direction can be determined.

And the dynamic stiffness matrix is expressed as Equation 13.

In Equation 12, H _ff , H _bf , H _fb , and H _bb are unbalance response functions, and when the unbalance response function is obtained, an unbalanced response or influence coefficient can be easily obtained. As can be seen in Equation 12, the initial deformation will have both the forward and rear vibration forces at the same time, in particular the rear excitation force is dependent on the initial deformation amount in the bearing position. In general, the unbalanced excitation force is not affected by H _fb and H _bb , but the initial deformation provides the rear excitation force, so the characteristics due to this can be observed. However, even in the case of many rotors, even if the initial deformation at the bearing position is small, the excitation force applied by the initial deformation can be increased by the bearing stiffness coefficient because the stiffness coefficient of the position is multiplied.

In the case of the isotropic rotating body, the tuning vibration response can be obtained from Equation 12 as shown in Equation 14 below.

Therefore, in the case of isotropic rotors, no backward swing response is observed, and the initial deformation also affects only the forward swing response.

On the other hand, assuming that there is no initial strain δ at the bearing position, the following relation can be obtained from Equation 12.

Therefore, only two unbalanced response functions, H _ff and H _bf , which have a direct influence on the unbalanced response appear.

Experimental Example

In order to measure the tuning vibration response due to the initial deformation and the practical verification of the theory presented through the above equation, there is an initial deformation in which a leaf spring type vibration isolator is connected on the bed mounted in the air mount as shown in FIG. The test rotating body was constructed. The rotating shaft used had a diameter of 0.025m, a total length of 1.2m, and a total of three disks were combined and supported by two bearings. One supported the ball bearings and the other used vibration isolation. The ball bearings were located at node # 3 (0.1 m) and the vibration isolation device was located at node # 19 (0.9 m). Two displacement sensors were placed in the horizontal and vertical directions of the rotating shaft, respectively, and the synchronized vibration signals of the rotating shaft were measured. By changing the sampling rate, the number of samples per revolution based on the trigger signal was kept constant at 256, regardless of the rotational speed, and averaged five times. Initial strain was obtained by measuring the tuning response during rotation at 100 RPM as shown in FIG. 2B. The obtained data was analyzed by FFT (Fast Fourier Transform), and the frequency response function was obtained by analyzing only 1X component that exhibits vibration characteristics. The mode test was carried out to understand the characteristics of the test rotor. For theoretical analysis, the finite element method is used and the nodes are divided into 13 models. The natural frequencies by the finite element model and the natural frequencies by experiment are shown in Table 1 below.

First mode 2nd mode 3rd mode location Perpendicular level Perpendicular level Perpendicular level Theoretical (Hz) 24.3 24.3 53.1 53.1 151.7 151.7 Experimental value (Hz) 24.3 24.3 53.1 53.1 160.8 158.8

As shown in Table 1 above, it can be seen that the natural frequency by the finite element model and the natural frequency by the experiment coincide well and can be considered as isotropic.

3 (A) shows a comparison between the mode shape obtained by the frequency response function by the mode test and the theoretical mode shape obtained by the finite element model. 3B shows an initial deformation amount, which shows that the initial deformation near the node to which the vibration isolation device is connected is exceptionally large and the phase of the tuning signal of the right node of the vibration isolation device has not changed.

4 shows the tuning vibration response measured along the axis. Although some posterior components were observed, only the front components are shown in FIG. 4. 5, 6 and 7 respectively show the tuning response at each selected node when there is only initial strain, only mass imbalance, and both initial strain and mass imbalance. Since the distribution of the imbalance is unknown, the magnitude and phase are adjusted to correspond to the response, assuming that the imbalance is only in the disc. Comparing FIG. 4 with FIG. 7, it can be seen that the proposed model predicts the experimental results well. On the other hand, the characteristic that a response appears small at one rotational speed from the response only of initial deformation is observed. This property is clearly different from the effect due to the imbalance only shown in FIG. 6 and is closely related to the shape of the initial deformation.

Example of analysis by numerical value

8 and 9 show the vibration characteristics of the general rotor by numerical analysis. The initial deformation was expressed as a simple cubic function to determine the initial deformation vector. In addition, detailed specifications of the considered system are shown in Table 2 below.

Axis of rotation Length 1.2m diameter 8.0cm Young's modulus 2.0 × 10 ¹¹ N / m ² density 8000 Kg / m ³ disk mass 20kg Polar moment of inertia 0.63 Kg-m ² Radial moment of inertia 0.085 Kg-m ² location Node # 5, 6, 13 bearing Load 29.42, 78.84 Kg location Node # 1, 10 Width / diameter, gap / radius 0.5, 2/1000 viscosity 9.37 mPa.s

The finite element model for the axis of rotation of the system is twelve beam elements of the same size, three rigid disks and two oil film type bearings. Here, the bearings are assumed to be two types of film bearings, and they are assumed to have a rotational speed dependency. The bearings are approximated by a second function in the rotational speed range.

Analysis example 1

In this analytical example, in order to examine the difference between the initial deformation and the imbalance effect in the rotating body having 4 tilting pad bearings, only the unbalanced and the initial deformation of the same rotating body were compared. Both bearings have an LBP type preload angle of 80 and a preload coefficient of zero. Figure 10 shows an example of the tuning vibration by the initial deformation. FIG. 11 depicts this tuning vibration as an Orbital Plot at several rotational speeds.

FIG. 10 shows the tuning vibration response measured at the same position when there is only an initial strain (a), only an imbalance (b) and both an imbalance and an initial strain (c). As expected, no backward response is observed. At low speeds, the responses are very different. At low speed, the initial deformation shape is observed as a response, but the higher the speed, the greater the influence of the imbalance. It can be seen that the effect of the initial deformation can be neglected at high speed unless the effect of the initial deformation is more pronounced than the imbalance effect in the low speed range.

Analysis Example 2

In this analytical example, the effect of initial deformation on the tuning vibration was examined by considering bearings with anisotropic characteristics. Here, we analyzed the case where the LBP type preload angle was 80 and the preload coefficient was 0 with 5 tilting pad bearings only anisotropic for the same rotating body as the previous analysis example. FIG. 12 shows an unbalance response function, and FIG. 13 shows experimental and theoretical tuning in the case of initial strain (a) alone, mass imbalance (b), initial strain and mass imbalance (c), etc. Show each response. 12 and 13, when the initial deformation is applied, not only the front excitation but also the rear excitation is shown. In other words, if there is an initial variation, it reflects all unbalanced response functions. However, when there is a mass imbalance, as shown in FIG. 13 (b), only two unbalance response functions H _ff and H _bf are reflected.

According to the present invention, the effect of the initial deformation on the tuning vibration response of a general rotating body is analyzed. In the analysis of a rotor with an initial deformation, deformation in the lateral and angular directions should be taken into account because the initial deformation is related to the stiffness matrix of the rotor bearing system. As can be seen from the foregoing examples and analytical examples, the tuning vibration at any speed can be zero but depends on the position of measurement or the shape of the initial deformation. The response due to imbalance dominates at high speed, while the response due to early deformation dominates at low speed. It is practically impossible to distinguish the initial and unbalanced responses from tuning responses. According to the present invention, it is possible to characterize the tuning response caused by the initial deformation. Therefore, since the magnitude of the correction mass with respect to the rotating body and the position of the correction mass with respect to the rotating body can be obtained accurately, the effect of balancing with respect to the rotating body becomes high.

Claims (2)

Dividing the desired rotating body into any number of nodes, and obtaining M (mass, inertia), G (gyro), K (rigid), and C (attenuation) matrices corresponding to all nodes; Obtaining an initial deformation vector δ according to the initial deformation of the rotating body; Obtaining a forward turning response vector (P _f ) and a rear turning response vector (P _b ) in which an initial deformation vector (δ) and an imbalance vector W1 are considered while rotating the rotating body at an arbitrary speed Ω 1; Obtaining an imbalance vector W1 corresponding to an arbitrary speed? 1 from the swing response vector P _f and the rear swing response vector P _b ; The time at speed (Ω2) to balance the whole, the front turning response of a given specification vector (P _f ') and the rear turning response vector (P _b') from the forward pivot response vector (P _f ') and the rear turning response vector Obtaining an imbalance distribution vector W2 including the imbalance vector W1 satisfying (P _b '); Determining the position, magnitude, and direction of the compensation mass with respect to the rotating body from the imbalance vector W2; Method for measuring the amount of deformation of the rotating body is considered to include the initial deformation. Dividing the desired rotating body into any number of nodes, and obtaining M (mass, inertia), G (gyro), K (rigid), and C (attenuation) matrices corresponding to all nodes; Obtaining an initial deformation vector δ according to the initial deformation of the rotating body; Obtaining a forward turning response vector (P _f ) and a rear turning response vector (P _b ) in which an initial deformation vector (δ) and an imbalance vector W1 are considered while rotating the rotating body at an arbitrary speed Ω 1; Obtaining an imbalance vector W1 corresponding to an arbitrary speed? 1 from the swing response vector P _f and the rear swing response vector P _b ; The time at speed (Ω2) to balance the whole, the front turning response of a given specification vector (P _f ') and the rear turning response vector (P _b') from the forward pivot response vector (P _f ') and the rear turning response vector Obtaining an imbalance distribution vector W2 including the imbalance vector W1 satisfying (P _b '); Determining the position, magnitude, and direction of the compensation mass with respect to the rotating body from the imbalance vector W2; Balancing the rotating body according to the determined position, size, and direction of the compensation mass; Balancing method of the rotating body comprising a