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I. Static Balance: Static balance is achieved by performing balancing on one correction surface of the rotor; the remaining unbalance after this process ensures that the rotor is within the permitted limits for unbalance in a static state. This is known as static balance, or single-sided balance. II. Dynamic Balancing: Dynamic balancing involves performing balance adjustments on two or more correction surfaces of the rotor simultaneously. The remaining imbalance after these adjustments is kept within the allowable limits specified for dynamic operation; this process is known as dynamic balancing, or two-sided/multi-sided balancing. III. Selection and determination of rotor balancing. How to choose the method of rotor balancing is a key issue. There is a principle guiding this choice: as long as the requirements of the application after rotor balancing are met, static balancing should be used instead of dynamic balancing; and if dynamic balancing is possible, then static and dynamic balancing together should not be employed. The reason is simple: static balance is easier to achieve than dynamic balance, saving effort, labor, and costs. So how is the rotor balancing pattern determined? It needs to be determined based on the following factors and criteria: 1. The geometric shape and structural dimensions of the rotor, in particular the ratio of the rotor’s diameter D to the distance b between its two opposite surfaces, as well as the spacing between the supports of the rotor. 2. Operating speed of the rotor: Technical standards related to the requirements for rotor balancing, such as GB3215, API610, GB9239, and ISO1940, etc. 3. Conditions for static balance of the rotor: In the GB9239 balancing standards, the conditions for static balance of a rigid rotor are defined as follows: If the spacing between the supports of a disc-shaped rotor is large enough and the axial runout of the disc-like portion during rotation is very small, so that couple imbalance (dynamic balance) can be ignored, then the imbalance can be corrected using a single correction surface, achieving unilateral (static) balance. It is necessary to verify whether these conditions are met for a specific rotor. After balancing a large number of rotors of a certain type on a plane, the maximum residual couple imbalance can be determined and divided by the support distance. If, under the most unfavorable conditions, this value is not greater than half of the allowable remaining unbalance, then single-sided (static) balancing is sufficient. It is easy to see from this definition that the conditions under which the rotor can achieve unidirectional (static) balance are mainly threefold: (1) one of them is that the rotor has a disc-shaped geometry ； (2) One is that the support spacing when the rotor is balanced on a balancing machine should be large ； (3) Another requirement is that the end-face runout of the correction surface must be very small when the rotor is rotating. The above three conditions are explained as follows: (1) What is meant by a disc-shaped rotor is mainly determined by the ratio of the rotor’s diameter D to the distance b between its two opposite surfaces. According to the API610 standard, when D/b < 6, it is sufficient to perform single-sided balancing of the rotor ； When D/b≥6, it can be used as a criterion to determine whether the rotor is disk-shaped, but this cannot be taken as an absolute rule, as the type of balancing required for the rotor also depends on its operating speed. (2) There is no specific parameter specified for the support spacing; however, a ratio of this spacing to the rotor alignment distance b of ≥5 is considered to indicate that the support spacing is sufficiently large. (3) The axial runout of the rotor refers mainly to the end-face runout of the reference surface during rotation; since any rotor undergoing balancing tests is finely machined, the manufacturing process ensures that the tolerance between the rotor’s holes and the reference surface is very small, resulting in minimal end-face runout. Based on the conditions for single-sided (static) balancing of the rotor mentioned above, and in conjunction with the technical standards related to pumps (such as GB3215 and API610), the requirements for rotors that are to be balanced only statically are as follows: (1) For the rotors of single-stage and two-stage pumps, when the operating speed is < 1800 revolutions per minute, static balancing alone is sufficient, regardless of whether D/b < 6 or D/b ≥ 6. However, if dynamic balancing is required, it is necessary to ensure that D/b < 6; otherwise, only static balancing can be performed. (2) For the rotors of single-stage and two-stage pumps, when the operating speed is ≥ 1800 revolutions per minute, static balancing is sufficient if D/b ≥ 6. However, the remaining unbalance after balancing must be equal to or less than 1/2 of the allowable unbalance. If dynamic balancing is required, it is necessary to check whether the balance of the two correction surfaces can be achieved separately on a balancing machine; if not, only static balancing can be performed. (3) For some rotors such as open impellers, if end support cannot be achieved, only static balancing is sufficient. Since the ends cannot provide support, a cantilever structure is necessary; as a result, performing dynamic balancing on a balancing machine is very dangerous, and it is only possible to carry out single-sided (static) balancing on a balancing stand. 4. The conditions for achieving dynamic balance of a rotor are specified in standard GB9239: For any rigid rotor that does not meet the conditions required for static balance of a disc-shaped rotor, it is necessary to achieve balance in two planes, that is, through dynamic balance. The conditions for rotors that require only static balancing are as follows (a balance accuracy of G0.4 represents the highest precision; generally, for pump impellers, a dynamic balancing accuracy of G6.3 or G2.5 is chosen): (1) For single-stage and two-stage pumps, when the operating speed is ≥ 1800 revolutions per minute, dynamic balancing should be performed as long as D/b < 6. (2) For multi-stage pumps and combined rotors (3 stages or more), dynamic balancing of the combined rotor shall be performed regardless of the operating speed. IV. Why is dynamic balancing necessary? During the manufacturing and assembly processes of the rotor, the finally assembled rotor can never achieve perfect mechanical axisymmetry (known as shaft eccentricity); as a result, there is always a certain amount of imbalance. This amount of imbalance is usually referred to as the initial imbalance. There are various reasons that can cause a rotor to have an initial degree of imbalance, but if such imbalance is present, it will generate a centrifugal force as the rotor rotates. This centrifugal force increases gradually as the rotation speed rises. Centrifugal force is transmitted to the machine through the bearings, causing vibration throughout the machine, generating noise, accelerating bearing wear, reducing the machine’s lifespan, and even leading to control failures and serious accidents. Dynamic unbalance is the most common type of unbalance. It is a combination of static imbalance and couple imbalance. After dynamic balancing, an unbalanced rotor not only has its couple imbalance eliminated but also its static imbalance removed; at this point, the rotor’s principal axis of inertia coincides exactly with its axis of rotation, thereby achieving balance for the rotor. But ideals are plump while reality is gaunt; it is impossible to balance an unbalanced rotor so that its imbalance amount becomes zero. Because it is affected by the precision of the dynamic balancing equipment and the limitations of the rotor. Thus, the concept of balance accuracy arises – that is, the most reasonable order of magnitude that we can achieve under the existing conditions, thereby meeting both the requirements of production and daily life as well as those related to economic efficiency. The dynamic balancing test is the process of performing dynamic balance checks and corrections on a rotor to meet its operational requirements. Parts that undergo rotational motion, such as various drive shafts, spindles, fan blades, pump impellers, cutting tools, and rotors of electric motors and turbines, are collectively referred to as rotors. Under ideal conditions, the pressure exerted on the bearings is the same whether the rotating body is rotating or not; such a rotating body is a balanced rotating body. However, various rotating bodies in machinery suffer from factors such as uneven material composition or defects in the blanks, errors that occur during processing and assembly, and even asymmetric geometric shapes from the design stage. As a result, when these rotating bodies spin, the centrifugal inertial forces generated by each tiny particle on them cannot cancel each other out. These centrifugal inertial forces act through the bearings on the machinery and its foundation, causing vibrations and noise, accelerating bearing wear, and shortening the lifespan of the machinery. In severe cases, this can lead to destructive accidents. To this end, the rotor must be balanced to achieve the permissible level of balance accuracy, or to keep the resulting amplitude of mechanical vibrations within allowable limits. V. Accuracy requirements for dynamic balancers: The minimum achievable residual unbalance is the smallest amount of residual unbalance that a balancer can achieve when balancing a rotor; it serves as a performance indicator reflecting the balancer’s maximum balancing capability. A rigidly supported balancing machine can be expressed directly by the minimum residual unbalance amount on the balancing surface, with the unit being grams per millimeter; some also use grams per centimeter. The minimum achievable residual unbalance is influenced by factors such as the type of balancing machine, the measurement method, the transmission mechanism, the type of bearings, the planar separation ratio of the correction surfaces, and the sensitivity of the balancing machine. To achieve a high level of rotor balance accuracy (i.e., a very small residual unbalance), it is necessary to eliminate as many factors as possible that affect this accuracy. Among these influencing factors, the drive method of the balancing machine and the imbalance of its drive components have the greatest impact. The journal accuracy of the rotor must also be strictly controlled, as well as in the case of rotors equipped with impellers, rotors whose center of gravity is not located at the center of the rotor, and rotors that have bearing housings and packing for sealing purposes. Accuracy class G, in g.mm/kg. Examples of rotor types: G4000 – 4000; crankshaft drives for low-speed marine diesel engines with a single number of cylinders and rigid installation. G1600 – 1600; crankshaft drives for large two-stroke engines with rigid installation. G630 – 630; crankshaft drives for marine diesel engines with rigid installation ； Rigidly mounted crankshaft drive units for large four-stroke engines: G250 250. Rigidly mounted crankshaft drive units for high-speed four-cylinder diesel engines: G100 100. Crankshaft drive units for six-cylinder and multi-cylinder diesel engines. Complete (gasoline, diesel) engines for cars, trucks, and locomotives. G40 40 Automotive wheels, wheel rims, complete wheels ； Driving components for engines used in automobiles, trucks, and locomotives. G16 16 crusher, parts for agricultural machinery ； Individual parts for (gasoline, diesel) engines used in cars, trucks, and locomotives. G6.3 6.3 Gas and steam turbine engines, including the rigid turbine engine rotors of marine (commercial) vessel main turbines ； Turboscharger ； Machine tool drive components ； Rotors for medium and large motors with special requirements ； Small motor rotor ； Turbopump. G2.5 2.5 Gear for the main turbine of seagoing vessels (merchant ships) ； Centrifuges, pump impellers ； fan ； Rotor components of aero-gas turbines ； flywheel ； General parts of machine tools ； Ordinary motor rotor ； Individual parts of engines with special requirements. G1 1 Tape recorder and record player drive components ； Grinding machine drive component ； Small armatures with special requirements. G0.4 0.4 Spindle, grinding wheel, armature, and gyroscope of precision grinders. Considering technological advancement and economic feasibility, the International Organization for Standardization (ISO) established the globally recognized ISO 1940 balance grade in 1940. This standard divides rotor balance grades into 11 levels, with increments of 2.5 times between each level; balance machines range from the highest requirement of G0.4 to the lowest of G4000. The unit is grams per millimeter per kilogram (gmm/kg), and it represents the eccentric distance of the imbalance from the rotor’s axis. VI. Methods of rotor dynamic balancing The main causes of unbalanced forces in rotors are: (1) the initial imbalance present during rotor manufacturing ； (2) Bending caused by frictional heating due to long-term operation of the rotor ； (3) Gradual imbalance generated during rotor operation ； (4) The rotor is affected by external changes, resulting in rotor bending. To address the imbalance issue of the rotor, offline dynamic balancing correction methods and online dynamic balancing correction methods are commonly used. The former usually involves removing the rotor of the equipment and then sending it to a balancing machine for correction ； The biggest difference between the latter and the former is that the rotor does not need to be removed; instead, the imbalance is corrected directly on site. 6.1 Offline dynamic balancing correction method: The offline dynamic balancing of the rotor is also known as process dynamic balancing. It involves installing the rotor to be tested on a balancing machine, aligning the sensors with the reflective tags on the rotor, and feeding the resulting signals back to the balancing machine. Technicians then make the necessary adjustments and repairs based on the level of imbalance and the angle indicated on the balancing machine. The main process is as follows: First, shut down the rotating equipment and remove the faulty rotor ； Next, the rotor is taken to a dynamic balancing machine to determine the rotor bearing positions ； At this point, fix the rotor, set the parameters, and start the balancer ； The periodic vibration signal of the rotor to be tested is then converted into an inductive signal, thereby determining the imbalance amount and phase corresponding to the rotor’s alignment surface ； Finally, the imbalance is corrected and eliminated through deduplication or weighting, thereby meeting the requirements for safe production. In general, the offline dynamic balancing correction method is a technique for achieving overall balance of the rotor itself. 6.2 Online Dynamic Balancing Correction Method Online dynamic balancing, also known as on-site dynamic balancing or full-machine dynamic balancing, means conducting inspections and performing dynamic balancing operations on the equipment at its operating speed right at the site, as the name implies. Online dynamic balancing refers to a method of reducing vibrations by performing dynamic balancing on the rotor while the unit is operating normally, without removing it. Online dynamic balancing technology refers to the process of calculating and analyzing the imbalance and phase of the rotor’s vibration signals captured by sensors, as well as using changes in the inputs of relevant control systems to eliminate that imbalance. Specifically, this is achieved by using sensors to detect the vibration signals of the rotor. The equipment frame serves as the base for the balancing machine; the vibrations of the rotor and bearings constitute a linear system, with the resulting vibrations being linearly additive. The collected vibration data is processed to determine the amount of imbalance and the corresponding angles for each balancing surface of the rotor. Finally, methods such as adding weight or removing weight are used to eliminate the imbalance, thereby achieving the purpose of correction. Therefore, the online dynamic balancing correction method has advantages such as avoiding errors caused by disassembly and installation, reducing unnecessary labor, being easy to operate, facilitating automation, and ensuring the safe and orderly operation of the unit. Therefore, online dynamic balancing is an important measure to eliminate equipment vibration, and it is also the future development trend for industries such as China’s petrochemical industry and mechanical equipment sector. In general, both online and offline rotor dynamic balancing can correct the imbalance in rotors, with high precision in correction in both cases; the rotors thus become suitable for safe operation after correction. Therefore, when performing dynamic balancing on a rotor, we need to conduct a comprehensive analysis of its efficiency, cost, and other factors based on the actual situation, determine whether online or offline dynamic balancing is more suitable, and then choose the most appropriate method for the task. On the one hand, it can **improve efficiency ； On the other hand, it also helps improve the accuracy of the rotor dynamic balancing process. However, due to the diversity of equipment support structures and rotors, online dynamic balancing is much more complex than offline dynamic balancing. Moreover, the requirement to achieve the desired level of dynamic balance in the shortest possible time adds to the difficulty. Therefore, the difference between online and offline dynamic balancing is: (1) the testing instruments used in the two are different ； (2) There are fewer correction surfaces available for online dynamic balancing compared to offline methods ； (3) A large number of people are present during online dynamic balancing operations ； (4) The criteria for balancing the two are different ； (5) The two have different economic and time requirements. VII. Balancing processes and methods: For an unbalanced rotor, its degree of imbalance is measured and corrections are applied to eliminate that imbalance; this is the process of rotor balancing, also known as a balancing test. It is an important process in rotor machining. 1. Selection of the correction surface: The process of eliminating the imbalance of a rotor and bringing it into a balanced state is called balance correction. Balance correction is carried out in a plane perpendicular to the rotor axis, which is known as the correction plane. The method of correcting balance within a single correction plane is called single-sided balance or static balance, while the method that requires correction in two or more correction planes is called double-sided balance or multi-sided balance, also known as dynamic balance. For rotors with a large initial imbalance and excessive vibration during rotation, one-sided balancing must be performed first before dynamic balancing to eliminate the static imbalance. Sometimes, due to an improper selection of the alignment plane (that is, the center of gravity is not within the selected alignment plane), correcting the static balance can actually increase the couple imbalance. Therefore, it is best to perform the alignment within the plane where the center of gravity lies, in order to reduce couple imbalance. If overlapping is not allowed in the plane where the center of gravity lies, it should generally be done on two planes located on either side of that plane. For rigid rotors, there are generally static imbalance and couple imbalance. To achieve balance, the imbalance can be corrected in any two arbitrarily chosen correction planes perpendicular to the axis, which is known as double-sided balancing. Correction methods generally involve adding weight (such as using counterweights) or removing weight (such as drilling holes). The position of the correction plane is generally determined by the structure of the rotor. To reduce the time and effort required for balancing operations, it is necessary to minimize the amount of correction. To this end, the distance between the two correction surfaces as well as the correction radius should be increased as much as possible under feasible conditions, in order to achieve good balancing results. 2. Verification method: The imbalance of a rotor arises because its principal axis of inertia does not coincide with the axis of rotation. Balance correction is achieved by altering the mass distribution of the rotor so that its principal axis of inertia aligns with the axis of rotation, thereby achieving balance. Common correction methods include adjusting the correction weight, adding weight or removing weight, etc. To increase stiffness, screw connections, riveting, or welding can be used; these methods enable the rotor to achieve better balance and higher precision, while also being convenient and safe. De-duplication often employs methods such as drilling, grinding, chiseling, and milling. For the rotor components and impellers of mechanical pumps, weight reduction is achieved through grinding, or weights are installed on the couplings. The choice of correction method depends on the rotor structure and process requirements, as well as the geometric shape of the correction surface. Generally, the location for adding weight or removing weight is considered during the design of the rotor. 3. Correction error: During the balancing process, in addition to the measurement errors of the balancer, there are also errors resulting from inaccurate balance correction (including the magnitude and position of the correction amount); such errors are referred to as correction errors. It can be divided into correcting angle error, correcting amplitude error, correcting radius error, and correcting plane position error, etc. In actual calibration, the above four types of errors usually occur together, and they should be taken into account comprehensively during analysis. Additionally, the ratio between the initial imbalance and the remaining imbalance, as well as the impact of the imbalance reduction rate of the balancing machine, should also be considered.