Investigation on a No Trial Weight Spray Online Dynamic Balancer

In order to suppress the spindle vibration with high efficiency and high precision, a no without trial weight spray online balance method is proposed in this paper. By analyzing the relationship between the unbalanced excitation and the unbalanced response of the spindle, the relationship between the dynamic influence coefficient and the system model is studied. A high-speed spindle finite element analysis model was established, and the dynamic influence coefficient matrix was identified. A no trial weight spray online dynamic balancing system was developed, which has the advantages of without trial weight and high-precision loading. A new type of integrated balancing terminal that was formed using 3D printing technology was first proposed by our research group, and its advantages in various aspects are significantly higher than traditional assembly balanced terminals. *e experimental verification of the without trial weight spray online dynamic balancing system was performed on a high-speed spindle test stand. Experiments show that the no trial weight spray online balancing method proposed in this paper can achieve high-efficiency and high-precision vibration suppression, greatly reducing balance time and cost of the spindle. At the same time, the online balance test also verified the reliability of the integrated balanced terminal.


Introduction
High-speed spindle is the core functional component of highend CNC machine tools, and its dynamic characteristics will directly affect the machining level of machine tools.However, due to uneven material and manufacturing and assembly errors, the center of mass of the spindle and the center of rotation may be inconsistent.e unbalance of the spindle is inevitable.Unbalance can cause a disastrous accident in many engineering applications [1,2].erefore, strict offline balancing will be performed when the spindle leaves the factory [3,4].However, the offline balance cannot compensate the unbalance mass caused by high-speed expansion, tool or grinding wheel wear, and the offline balance whose balance efficiency is low and requires the disassembly of the main shaft or grinding wheel.erefore, it is very meaningful to study the high-speed and precision online dynamic balance [5,6].
Spindle online balancing includes two main processes: (1) unbalanced vector identification and (2) vibration online suppression.A large number of researchers have studied the unbalanced vector identification method [7].e unbalanced vector identification methods mainly include influence coefficient method (ICM) [8], modal balance method (MBM) [9], holospectral balance method (HBM) [10], and no trial weight balance method (NTWBM) [11].Because NTWBM has the characteristics of without trial weight and high balancing precision, it has been widely applied in the intelligent spindle dynamic balancing [12,13].
e research team headed by Mr. Qu Liang Sheng studied the no trial weight balancing method based on holospectral technique [14][15][16].In 2009, Niebsch and Ramlau [17,18] used wind turbines as the research object to study illconditioned phenomena in the inverse problem of the no trial weight balance method.Khulief et al. [19] and Li et al. [20] simulated the functional expressions of the principal modes of the various stages of the rotor based on the finite element theory, and a good balancing effect is achieved.Research on spindle online vibration suppression technology began in the middle of the last century.Canadian scholars have proposed an active balancing device that is balanced by controlling the weight position.e mass block is driven by the motor [21].Since then, many scholars have studied the way of changing the rotor mass distribution by balancing terminals.In 2006, Hredzak and Guo [22] developed an electromagnetic automatic balancing device.A movable electromagnetic adsorption device on the outside of the disc is used to adjust the position of the steel ball.e device has achieved good results in the experiment.Moon et al. [23] proposed an electromagnetic balance device based on the similar principle.He achieved the spindle balance experiment by means of the influence coefficient method at 14400 rpm.In 2008, Japan Nakamoto et al. [24] designed a new type of balance device that uses magnetic fluid as a balancing mass.By changing the magnetic field distribution in the periphery of the balance terminal, the purpose of changing the magnetic fluid distribution in the terminal was verified.
e device was validated at 6,000 rpm.A new electromagnetic ring balancer for active imbalance compensation of rotating machinery was proposed by Fan et al. [25] in 2014.He verified the feasibility and effectiveness of balancing devices.Although they have studied different implementations, most of them use the ICM to verify the device.e ICM is a kind of test weight method.Test weight is needed to calculate the unbalance vector of the rotor system and then readjust the counterweight to achieve the purpose of correcting the vector.However, this method does not apply to weighted online balancing due to the test weight, such as spray online balancing.e no trial weight balance method can effectively solve the above problems.In addition, the spray dynamic balancing balance has high precision and can perform multiple online dynamic balancing, which is very suitable for high-precision loading.
e no trial weight spray online dynamic balancing technology is studied in this paper.is method can quickly identify unbalanced vectors and achieve high-precision loading.
erefore, it can achieve high-efficiency and high-precision online vibration suppression.An integrated balanced terminal (IBT) using 3D rapid prototyping technology was first proposed by our research group compared with the traditional spray online balancing technology.e IBT perfectly solved the problems of sealing and high-speed spin-off of traditional assembled balanced terminals.is technology is important for improving the accuracy of precision rotating machinery.

No Trial Weight Balance Method of Spindle
e influence coefficient method used in the industrial field is a kind of method that causes the response of the target rotor system by adding trial weights.
e aggravated influence coefficient matrix of the rotor system can be obtained.However, the influence coefficient matrix is not a virtual matrix.It is a characteristic parameter that can reflect the response of the rotor system when excited by energy. is shows that this matrix is an inherent characteristic of the system.In other words, the unbalance vector [α] can be calculated by identifying the characteristic matrix of the rotor system.We combine the high-speed electric spindle designed by our laboratory to study the WTWBM.
e structure is shown in Figure 1. is high-speed spindle system was developed specifically for studying online balancing.e maximum rotational speed of the spindle is 40,000 rpm.A pair of bearings SNFA VEX35 are mounted on the spindle and arranged in a backto-back layout and preloaded by two sleeves and a locknut.
e bearings are lubricated with oil mist, and the spindle is cooled by circulating water cooling.It should be noted that 1 : 7 taper is designed at the front and rear ends of the spindle to mount the counterweight disc.
It is necessary to establish the dynamic model of the spindle in order to identify the unbalance vector.e rotor system of spindle can be discretized into discs, elastic shaft sections, and bearing units.
e rotor is divided into 50 nodes to form 49 units.e coordinate system is established with the rotor axis as the axis.e finite element model of the rotor system is shown in Figure 2.
e beam element has a simple structure and is easy to program, so it is still the first choice for establishing a finite element model.e Timoshenko beam element consisting of two nodes is shown in Figure 3. e motion of each node consists of five degrees of freedom, namely, three translational degrees of freedom δ x , δ y , δ z and two rotational degrees of freedom c y , c z , where the X-axis torsional freedom is not considered [26,27].
Regardless of the internal damping of the beam, the equation of motion of the beam can be expressed as follows [28]: where ] P is additional stiffness matrix due to axial loading, {q} is system displacement vector, and q   � {δ x , δ y , δ z , c y , c z } T , {F b } is external force vector.e superscript b stands for the beam unit and Ω is the speed.
Similarly, the equation for the motion of the disk unit is given by the following eqaution: where [M d ] is mass matrix, [G d ] is gyro matrix, the superscript b stands for the beam unit.Assuming that the unbalanced mass is m D and the eccentricity is e, the unbalance force is as follows: 2 Shock and Vibration In addition, the generalized forces acting on the bearings can be written as follows [29] where [C c ] is the bearing damping matrix, [K c ] is the bearing stiffness matrix, and Q c { } is the vector of generalized forces of the bearings.
In combination with the above derivation, the model of the spindle rotor, disc, and bears is integrated.e differential equation of the spindle system is as follows [30]: where system mass matrix is   can be expressed as follows: where r is unbalanced response, u is unbalanced mass, and φ is lag angle.Substituting Eqs. ( 6) and ( 7) into Eq.( 5) results in e above formula can be written as [α] • r � u, where [α] is a aggravated influence coefficient matrix of the rotor system, which can be derived from the above equation: From the above equation, it can be seen that the aggravated influence coefficient of the rotor system is determined by the mass, stiffness, lag angle, and rotation speed.
at is to say, the aggravated influence coefficient of the rotor system is determined by the rotor structure parameters.It is an inherent parameter to reflect the energy transfer characteristics of the rotor system.
where [Q] is balance vector, A is unbalanced vibration response obtained through testing, and the negative sign indicates that the phase difference between the balance vector and the imbalance is 180 °.After calculating the balance vector, we apply it to the online dynamic balance by modifying the matching process, and the online weighted correction scheme can be obtained by rapid orientation iteration matching.e detailed online balance process is described later.

Online Balancing Experiment System
3.1.System Structure.e spray dynamic balance system developed by our research group was used to perform experimental research on the no trial weight online balance method.e structure of the spray online balancing system is shown in Figure 4. e system is mainly composed of balanced terminals, hydraulic systems, and measurement and control systems.e balance terminal is the key component of the dynamic balancing system, and its function is to store the correction mass.e main function of the hydraulic system is the quantitative delivery of liquid.e function of the measurement and control system is unbalanced vector calculation and hydraulic system control.

New Integrated Balance Terminal.
Balancing terminals are the core components of spray dynamic balancing.e main function is to store the correction mass.rough investigation, it has been found that the balance terminal in the world is the assembled balanced terminal shown in Figure 5.
As can be seen from Figure 5, the traditional assembled balance terminal is assembled from the inner ring and from the outer ring through interference fit.However, the deformation of the outer ring of the balanced terminal is larger than that of the inner ring under the action of centrifugal force.When the balanced terminal rotates with the spindle at a high speed, the magnitude of interference between the inner and outer rings of the balanced terminal will decrease.is will lead to a drop in the sealing performance of the balanced terminal or even complete failure.According to the theory of 4 Shock and Vibration elasticity [31], regardless of the tangential displacement component of the inner and outer rings in the rotation process, the radial centrifugal force of the inner and outer rings is applied as a unit volume force, which can be equivalent to the problem of axisymmetric plane stress.At any radius r, the deformation displacement u r is [32] where ω is angular velocity; ρ is material density; E is elastic modulus; ] is Poisson's ratio; and A and B are the diameters of the inner and outer rings, respectively.en, the minimum allowable amount of interference δ is shown as follows: where R F is the critical radius of the inner and outer rings (R F � 33 mm), R O is the outer radius of the outer ring of the dynamic balance terminal (R O � 47 mm), and R I is inner radius of the inner ring of the balancing terminal (R I � 28 mm).Balanced terminal material is titanium alloy ().When the magnitude of interference is 20 μm, the relationship between the radial expansion of the inner and outer rings and the rotation speed is shown in Figure 6.It can be seen that when the rotational speed reaches the loose critical speed, the interference connection will fail.So, the assembled balance terminal not only loses the capacity of the chamber seal but also has potential safety hazards.Of course, the magnitude of interference cannot be too large.If the magnitude of interference is too large, assembly will be difficult.Pressing in force will even cause fatigue damage to the inner and outer rings.Because of the above problems in the traditional assembly balanced terminals, an integrated balanced terminal that combines the needs of spray online dynamic balancing and the technical features of rapid prototyping was proposed by our research group as shown in Figure 7. e integrated balancing terminal eliminates the traditional design of the inner and outer rings.It is mainly composed of the terminal body, flow path, offline weight holes, tapered holes, and chambers.e integrated balanced terminal is formed in one time using 3D printing technology.e tapered holes are used to connect with the spindle.e function of offline weight holes is to eliminate the initial eccentricity of the balanced terminal in the manufacturing process.is will not cause new additional imbalance to the spindle.e function of the flow path is to direct the fluid to the corresponding chamber.
e capacity of the integrated balanced terminal determines its balance capacity.In order to calculate the balance capacity of the balanced terminal, the volume of chamber must be calculated.In order to simplify the calculation, the corner is ignored and the cross section shown in Figure 8(a) is obtained; the integral microelement is shown in Figure 8(c).Shock and Vibration 5 en, the volume of a single liquid chamber is given by where h 1 � 14.5 mm, h 2 � 26 mm, l 1 � l 2 � 5.75 mm, R 1 � 33 mm so, V ≈ 13230 mm 3  Assume that the density of the weighted liquid is ρ, the maximum balance capacity of a single liquid chamber: where ρ � 0.8 g/mm 3 , so, Q ≈ 413 g•mm.It can be known from the balanced terminal cavity distribution structure that when the adjacent two chambers are filled with the balance liquid and the other chambers are all empty, the balancing capacity of the balanced terminal will reach the maximum value at this time.e maximum is � 2 √ Q � 584 g•mm.e balancing terminal rotates with the spindle at a high speed.So, balancing the liquid in the terminal chamber will stress the chamber due to centrifugal force.Assuming the rotational speed is 30000 rpm, the cavity of the balance terminal is filled with the weight fluid with density 0.8 g/mm 3 , and the centrifugal force produced by the weight fluid in a single cavity can be calculated by the following equation: where m is the liquid mass in the cavity, r is the radial distance between the center of gravity of the counterweight fluid and the axis of rotation, ω is the angular velocity, and When the balanced terminal rotates at a high speed, most of the water pressure generated by the centrifugal force will act on the outer surface of the cavity, and the outer surface of the cavity is not a regular plane.To simplify the calculation, the pressure area on the outer surface of the chamber is equivalent to the projection surface as shown in Figure 9.When the balanced terminal is filled with the counterweight fluid, the pressure on the outer surface of the chamber is calculated by the following equation: where centrifugal force F � 4076 N, α � 84 °, radius R 2 � 44.5 mm, and thickness h 2 � 26 mm; thus, the pressure P � 2.4 MPa.
In order to select the thickness of the cavity wall of the balanced terminal, the analysis was carried out under the extreme conditions (full-chamber liquid and 30,000 rpm).When the pressure is 2.4 MPa, different thicknesses of the cavity arc surface force analysis model are built in ANSYS as shown in Figure 10.
e relationship between the material thickness and the stress and deformation of the titanium alloy balance terminal surface is shown in Figure 11.It can be seen that when the material thickness is selected as 2 mm under the extreme conditions, the stress value and deformation tend to be stable.erefore, in order to meet the requirement of the balanced terminal with the spindle acceleration and deceleration, the wall thickness of the force surface should be greater than 2 mm.We have designed the thickness of the balanced terminal force surface to be 2.5 mm.
e assembled balance terminal of different materials was trial-produced by us as shown in Figure 12.By analyzing the requirements of 3D printing technology and strength, titanium alloy is selected as the manufacturing material for the integrated balance terminal.
So, the integrated balanced terminal manufactured according to the design parameters is shown in Figure 13(a).For the nonblocking of the liquid flow and balancing the  Shock and Vibration signal test on the terminal surface, the flow channel and the surface were modified as shown in Figure 13(b).e balancing terminal is balanced to eliminate the initial mass eccentricity on the balancing machine. is will not cause new additional imbalance to the spindle.e 1/4 model of assembled balanced terminals and integrated balanced terminals is shown in Figure 14. e parameters of the two balanced terminals are shown in Table 1.It can be seen that the integrated balanced terminal designed in this paper has obvious advantages in comparison with the assembled balanced terminal.
e maximum balance capacity of balanced terminal increased from 489 g•mm to 584 g•mm.On the basis of a 19.4% increase in balance capacity, thickness dropped from 38 mm to 28 mm, a drop of 26.3%.e volume of balanced terminal dropped from 160549 mm 3 to 83780 mm 3 with 47.8% drop ratio.e mass of balanced terminal has dropped from 562 g to 387 g with a drop ratio of 31.1%,so that the additional mass added by the balanced terminal to the spindle will be effectively reduced.

Balanced Procedure and Software.
e no trial weight balance method is applied to the spray online balance system, which can achieve efficient online balance, thus maximizing the protection automation processing link.e flow diagram of the no trial weight spray online dynamic is shown in Figure 15, in which three main steps are included: (1) acquisition process of rotor structural parameters; (2) acquisition process of counterweight scheme; and (3) online balance process.e balancing procedure is as follows: (1) Establish the physical model and finite element model of the spindle rotor system by separating the rotor model l 2 = 2.5 mm  (2) Obtain the structural parameters of the spindle rotor system, such as mass matrix M, stiffness matrix K, and damping matrix C by the finite element simulation and simulate the inherent modal characteristics of the spindle (3) Substitute the correction matrices of structural parameters in Equation ( 9) to identify the aggravated influence coefficient matrix (4) Extract the original vibration signal of the spindle and the phase information corresponding to the vibration signal; substitute aggravated influence coefficient matrix into Equation (10) to calculate the unbalance vector (5) e weighting scheme is searched according to the cavity angle and the unbalance vector (6) e spray online dynamic balancing system is injected into the corresponding cavity to correct the unbalance vector.
According to the balance flow chart, a measurement and control system based on virtual instruments was developed by our research group.e operation interface is shown in Figure 16.e no trial weight online balance measurement and control software consists of vibration signal monitoring, trend of vibration displacement change, vibration status display and monitoring, liquid state display and monitoring, and online balance information records.
Vector synthesis principle for weight matching

Methods and System Validation
4.1.Experimental Setup.In order to verify the application of no trial weight spray online dynamic on high-speed spindle, the high-speed spindle comprehensive performance test bed was designed by our research group as shown in Figure 17.
e test bed comprises a spindle 150 SD40Q7 (Figure 1) whose maximum rotational speed is 40,000 rpm, and two pairs of SNFA VEX35 bearings lubricated by oil mist are mounted on the spindle with a clearance tolerance to avoid the preload variation.e water cooling system controls the temperature variation.e balancing discs are installed on the ends of the spindle to suppress vibration.A rotating encoder mounted on the shaft feedbacks the rotational speed.
e spindle rotor is designed with a ratio of 1 : 7 tapered to install the integrated balancing terminal, and the balancing terminal is fixed with a locknut.In the front of the rotor, a HSK C32 tool holder interface is designed.e measuring equipment is composed of a synchronous acquisition system developed by our research group.A National Instrument PXI-4472 was used to extract vibration and phase information.German Micro-Epsilon U-05 eddycurrent sensor (range 0.5 mm; resolution 0.025 μm) is used to measure vibration displacement.e NI-PXI6123 is used to control the hydraulic system.A photoelectric sensor was applied to measure the phase.e layout of the experimental setup is shown in Figure 18.

Measurement Principles.
e system carries out realtime synchronous acquisition of vibration using eddycurrent displacement sensors mounted along the X-axis

Shock and Vibration
(S 1 ) and the Y-axis (S 2 ); the layout is shown in Figure 18(a).e photoelectric sensor (S 3 ) mounted on the X-axis measured the phase (Figure 18(b)).
e X-direction displacement sensor S 1 and the phase sensor S 3 are aligned with the zero-hole position to simplify the complex angular relationship and eliminate the influence of the initial phase angle.en, the lag angle (φ) can be obtained from Equation (17).In Figure 18(b), A 1 is vibration peak point, A 2 is phase point, and A 3 is equivalent point of the rotor unbalance force.

Experimental Results
Analysis.e pressure of the hydraulic system in the experiment is 0.12 MPa and spraying unit time is 100 ms.e X-direction vibration displacement is used as the balance reference.In order to accurately extract the imbalance characteristics, harmonic wavelet technology was used to extract the frequency characteristics [33].e online balance experiments at multiple speeds were verified in the experimental system as shown in Figure 17.
Comparison of rotation frequency amplitude before and after balancing at 12,000 rpm, 15,000 rpm, and 18,000 rpm is shown in Figures 19-21.It can be seen from Figure 19 that the vibration amplitude in the X direction decreases from 2.86 μm to 0.21 μm, the drop ratio reaches 92.66%, and the vibration amplitude in the Y direction decreases from 2.87 μm to 0.51 μm with 82.23% drop ratio at 12,000 rpm.It can be seen from the online balance process that the spray weight is only needed once, and the total online balancing time is 5 seconds.
It can be seen from Figure 20 that the vibration amplitude in the X direction decreases from 3.37 μm to 0.82 μm, the drop ratio reaches 75.67% and the vibration amplitude in the Y direction decreases from 2.79 μm to 1.23 μm with 55.91% drop ratio at 15,000 rpm.It can be seen from the online balance process that the spray weight is only needed once and the total online balancing time is 3 seconds.
It can be seen from Figure 21 that the vibration amplitude in the X direction decreases from 4.66 μm to 0.52 μm, the drop ratio reaches 88.84%, and the vibration amplitude in the Y direction decreases from 4.98 μm to 1.31 μm with 73.69% drop ratio at 18,000 rpm.It can be seen from the online balance process that the spray weight is two times and the total online balancing time is 7 seconds.

Conclusions and Further Work
A no trial weight spray online balance method is proposed.A high-speed spindle finite element analysis model was established, and the dynamic influence coefficient matrix was identified.A no trial weight spray online dynamic balancing system was developed, which has the advantages of no trial weight and high-precision loading.A new type of integrated balancing terminal that was formed using 3D printing technology was first proposed by our research group, and its advantages in various aspects are significantly higher than traditional assembly balanced terminals.Online balance experiments demonstrate the effectiveness of balancing methods and integrated balancing terminals.
It is important to note that the shaft is rigid.And, the focus is on one end of the bearing which is treated as a singleplane balancing problem.But the proposed method in this paper can be extended to both planes.We will further develop in-depth research on the method of no trial weight online dynamic balance considering dynamic stiffness, and we will further improve the loading accuracy of online balance system.

Figure 4 :
Figure 4: Spray online balance system structure.

Figure 11 :Figure 12 :
Figure 11: e relationship between thickness and stress and deformation.

Figure 20 :
Figure 20: Comparison of balance information in the X-Y direction at 15,000 rpm.(a) X-time domain.(b) Y-time domain.(c) X-frequency domain.(d) Y-frequency domain.(e) Online balance process.
is research was supported by the National Science and Technology Major Project of China (Grant Numbers 2015ZX04005001 and 2017ZX04013001).

Figure 21 :
Figure 21: Comparison of balance information in the X-Y direction at 18,000 rpm.(a) X-time domain.(b) Y-time domain.(c) X-frequency domain.(d) Y-frequency domain.(e) Online balance process.

Table 1 :
Comparison of two balanced terminal parameters.