转子平衡：静态和动态不平衡、共振及实用步骤
本指南介绍 转子平衡 为 刚性转子: what “unbalance” means, how static and dynamic unbalance differ, why resonance and non-linearity can prevent a quality result, and how balancing is typically performed in one or two correction planes.
内容
- 什么是转子？平衡可以纠正什么问题？
- 转子类型和不平衡类型
- 机械振动：平衡可以消除什么，不能消除什么？
- 共振：阻碍平衡的因素
- 线性模型与非线性模型：计算何时失效
- 平衡装置和平衡机
- 平衡刚性转子（实用技巧）
- 如何进行动态平衡（三次运行法）
- 评估平衡质量的标准
- 标准和参考
- 常见问题
什么是转子？平衡可以纠正什么问题？
转子是一个围绕某个轴线旋转的主体，由其轴承表面固定在支架上。转子的轴承表面通过滚动轴承或滑动轴承将负载传递到支架上。轴承表面是耳轴的表面或替代耳轴的表面。
在完全平衡的转子中，其质量围绕旋转轴对称分布，也就是说，转子的任何元件都可以与位于旋转轴对称位置的另一个元件相匹配。在平衡转子中，作用于任何转子元件的离心力与作用于对称元件的离心力相平衡。例如，大小相等、方向相反的离心力 F1 和 F2 作用于元件 1 和 2（图 1 中绿色标记）。所有对称的转子元件都满足此条件，因此作用于转子的总离心力为 0，转子处于平衡状态。
但如果转子的对称性被破坏（图 1 中红色标出的不对称元件），则不平衡离心力 F3 会作用在转子上。转子旋转时，这个力会随转子转动而改变方向。由此产生的动态载荷会传递到轴承上，导致磨损加速。
In addition, under the influence of this variable-direction force there is a cyclic deformation of supports and foundation, on which the rotor is fixed, i.e. there is vibration. In order to eliminate rotor unbalance and the accompanying vibration, balancing masses must be installed to restore symmetry to the rotor.
Rotor balancing is an operation to correct unbalance by adding balancing masses. In other words, the goal of balancing is to bring the principal central axis of inertia of the rotor as close as possible to its axis of rotation, so that the residual unbalance falls within specified limits.
平衡的任务是找到一个或多个平衡质量的大小和位置（角度）。
转子类型和不平衡类型
考虑到转子材料的强度和作用在其上的离心力的大小，转子可以分为两种类型——刚性转子和柔性转子。
在工作模式下，刚性转子在离心力作用下的变形很小，因此可以忽略这种变形对计算的影响。
Deformation of flexible rotors can no longer be neglected. Deformation of flexible rotors complicates the solution of balancing problem and requires application of other mathematical models in comparison with the problem of balancing of rigid rotors. It should be noted that the same rotor at low speeds can behave as rigid, and at high speeds - as flexible. The practical criterion is the service speed relative to the rotor’s first critical (bending) speed: a rotor is treated as rigid — ISO 21940-11 speaks of a rotor with “rigid behaviour” — when it runs well below that speed, in practice below roughly 50–70% of the first critical speed. Above that, the rotor bends into a mode shape that changes with speed: it is flexible and must be balanced by modal or multi-plane methods (ISO 21940-12). In the following, we will consider only the balancing of rigid rotors.
Depending on how the unbalanced masses are distributed along the rotor, ISO 21940-2 distinguishes several states of unbalance:
- 静态不平衡 — the principal inertia axis is displaced parallel to the shaft axis; it can be detected without rotation, because the rotor turns under gravity until its heavy spot is at the bottom. A single correction mass in one plane removes it;
- couple (moment) unbalance — the principal inertia axis intersects the shaft axis at the center of mass; the two equal unbalances lie in different planes and 180° apart. It appears only during rotation and requires two correction masses in two planes;
- 动态不平衡 — the general, real-world case: a combination of static and couple unbalance. The principal inertia axis neither is parallel to, nor intersects, the shaft axis. Two correction planes are necessary and sufficient for a rigid rotor.
The older term “moment unbalance” is a synonym of couple unbalance; it should not be confused with dynamic unbalance, which is the sum of the static and couple components. An example of a rotor with static unbalance is shown in Fig. 2.
Couple unbalance appears only when the rotor is rotating.
An example of a rotor with couple unbalance is shown in Fig. 3.
In this case, the unbalanced equal masses M1 and M2 are in different planes - in different places along the length of the rotor. In static position, i.e. when the rotor does not rotate, only gravity acts on the rotor and the masses balance each other. In dynamics, when the rotor rotates, centrifugal forces Fc1 and Fc2 start acting on the masses M1 and M2. These forces are equal in magnitude and opposite in direction. However, since they are applied at different places along the length of the shaft and are not on the same line, these forces do not compensate each other. The forces Fc1 and Fc2 create a moment applied to the rotor — this is why couple unbalance is also called moment unbalance. Accordingly, uncompensated centrifugal forces act on the bearing positions, which can greatly exceed the calculated values and reduce the service life of the bearings.
Since this type of unbalance appears only during the rotation of the rotor, it cannot be corrected in static conditions by balancing "on knives" or similar methods. In order to eliminate couple unbalance, two compensating weights must be installed, which produce a moment equal in magnitude and opposite in direction to the moment arising from the masses M1 and M2. The compensating masses do not have to be set opposite and equal in magnitude to the masses M1 and M2. The main thing is that they produce a moment that fully compensates for the unbalance moment.
In general, the masses M1 and M2 may not be equal to each other, so there will be a combination of static and couple unbalance — this general case is exactly what ISO 21940-2 calls dynamic unbalance. It is theoretically proven that for a rigid rotor, two weights spaced apart along the length of the rotor are necessary and sufficient to eliminate its unbalance. These weights will compensate both the moment resulting from the couple unbalance and the centrifugal force resulting from the asymmetry of the mass relative to the rotor axis (static unbalance). Typically, couple unbalance is characteristic of long rotors, such as shafts, and static unbalance is characteristic of narrow rotors. However, if the narrow rotor is skewed relative to the axis, or deformed ("figure eight"), then couple unbalance will be difficult to eliminate (see Fig. 4), because in this case it is difficult to install correcting weights that create the necessary compensating moment.
力 F1 和 F2 不在同一条直线上，不能相互补偿。
Due to the fact that the arm available to create the compensating moment is small due to the narrow rotor, large correction weights may be required. However, this also results in an "induced unbalance" due to the deformation of the narrow rotor by centrifugal forces from the correction weights. (see, for example, Methodological instructions for balancing rigid rotors to GOST 22061-76 — the modern international counterpart is ISO 21940-11, formerly ISO 1940-1 — Section 10, "Rotor–supports system").
This is noticeable on narrow fan impellers, where, in addition to mass unbalance, an aerodynamic unbalance is also present: unequal blade geometry produces an unequal blade loading and hence a net radial force. Like the centrifugal force of a correction weight, this force scales with the square of speed, but it also depends on the operating point — air density, damper position, duct resistance — so a correction weight balanced at one duty point will not stay optimal at another. The aerodynamic component must therefore be corrected by restoring the blade geometry, not by adding mass.
电磁力 in an electric machine (unbalanced magnetic pull from an eccentric air gap, broken bars, shorted laminations) behave differently again: they are governed by the air-gap flux, not by rotational speed, and they mostly excite the machine at twice the line frequency and at pole-pass sidebands rather than at 1×. Because they act at frequencies other than the rotation frequency, balancing cannot compensate them at all. In short, balancing removes the 1× mass-related excitation only — it cannot eliminate every source of vibration in a machine.
机构振动
振动是机构设计对周期性激振力影响的反应。这种力可以是不同性质的。
不平衡转子产生的离心力是作用在"重心"上的未补偿力。通过平衡转子可以消除这种力及其引起的振动。
配合部件的制造和装配误差会产生"几何"性质的相互作用力。例如，轴颈不圆、齿轮齿廓误差、轴承滚道波纹、配合轴不对中等都可能导致此类力的产生。如果轴颈不圆，则轴线会根据轴的旋转角度发生偏移。虽然这种振动在转子转速下也会出现，但几乎无法通过平衡来消除。
风扇叶轮和其他叶片装置旋转产生的空气动力。液压泵、涡轮机等叶轮旋转产生的水动力。
电机运行时产生的电磁力，如不对称转子绕组、短路绕组等。
The magnitude of the vibration (e.g. its amplitude Av) depends not only on the excitatory force Fv acting on the mechanism with circular frequency ω, but also on the rigidity k of the mechanism, its mass m, as well as the damping coefficient C, as formula (1) below shows.
各种类型的传感器可用于测量振动和平衡机制，包括
- 绝对振动传感器，用于测量振动加速度（加速度计）和振动速度传感器；
- 相对振动传感器——涡流式或电容式，用于测量振动位移；
- 在某些情况下（当机构设计允许时），力传感器也可用于评估其振动载荷；特别是，它们被广泛用于测量硬轴承平衡机支架的振动载荷。
因此，振动是机器对外力作用的反应。振动的大小不仅取决于作用在机械上的力的大小，还取决于机械设计的刚性。一个相同的力会导致不同的振动。在硬轴承机械中，即使振动很小，轴承也可能承受很大的动态载荷。这就是为什么在平衡硬轴承机床时要使用力传感器而不是振动传感器（振动加速度计）的原因。
振动传感器用于具有相对柔性支撑的机构，此时不平衡离心力的作用会导致支撑的明显变形和振动。力传感器用于刚性支架，此时，即使由于不平衡而产生很大的力，也不会导致明显的振动。
共振是妨碍平衡的一个因素
前面我们提到转子分为刚性和柔性两种。转子的刚度或柔度不应与安装转子的支架（地基）的刚度或移动性相混淆。当转子在离心力作用下的变形（弯曲）可以忽略不计时，转子被认为是刚性的。柔性转子的变形相对较大，无法忽略。
在本文中，我们只考虑刚性转子的平衡问题。刚性（不变形）转子可以安装在刚性或活动（柔性）支架上。显然，支架的刚度/可悬挂性也是相对的，取决于转子的速度和由此产生的离心力的大小。转子支架的自然振动频率是一个有条件的界限。
For mechanical systems, the shape and frequency of natural vibrations are determined by the mass and the elasticity of the elements of mechanical system. That is, the frequency of natural vibrations is an internal characteristic of the mechanical system and does not depend on external forces. Being deflected from the state of equilibrium, supports due to elasticity tend to return to the position of equilibrium. But due to the inertia of the massive rotor, this process is in the nature of damped oscillations. These vibrations are the natural vibrations of the rotor-support system. Their frequency depends on the ratio of the mass of the rotor to the elasticity of the supports, as formula (2) below shows.
当转子开始旋转，其旋转频率接近自然振动频率时，振动振幅会急剧增加，从而导致结构破坏。
The phenomenon of mechanical resonance occurs. Near resonance the response is amplified by the quality factor Q = 1/(2ζ), typically 3–17 for machine structures, and the peak can be narrow: a speed change of the order of a few percent may change the vibration level several-fold. Across the resonance the phase lag swings by 180°, passing through 90° at the peak.
If the design of the mechanism is unsuccessful and the operating frequency of the rotor is close to the frequency of natural vibrations, then the operation of the mechanism becomes impossible because of the inadmissibly high vibration. Balancing by the usual methods is then impossible, because even a small change in speed drastically changes the vibration parameters. For balancing in the area of resonance, special methods not considered in this article are used.
可以通过滑行（关闭转子旋转）或冲击法确定机构的自然振动频率，然后对系统的冲击响应进行频谱分析。
For mechanisms, which working frequency of rotation is above the resonance frequency, i.e. working in the supercritical (post-resonant) regime, the supports are considered to be moving and vibration sensors are used for measurement, mainly vibration accelerometers, measuring acceleration of structural elements. For mechanisms operating in pre-resonance mode, the supports are considered rigid. In this case, force sensors are used.
机械系统的线性和非线性模型。非线性是阻碍平衡的一个因素。
在平衡刚性转子时，平衡计算使用的数学模型称为线性模型。线性模型是指在这种模型中，一个量与另一个量成正比（线性）。例如，如果转子上未补偿的质量增加一倍，那么振动值也将增加一倍。对于刚性转子，可以使用线性模型，因为它们不会变形。
对于柔性转子而言，线性模型已不再适用。对于挠性转子，如果重点的质量在旋转过程中增加，就会产生额外的变形，除了质量外，重点位置的半径也会增加。因此，对于柔性转子而言，振动将增加两倍以上，通常的计算方法将不起作用。
Another source of non-linearity is a change in support stiffness at large deflections: at small deflections one set of structural elements carries the load, at large ones others come into play. This is why you cannot balance mechanisms that are not fixed on a foundation, but, for example, simply placed on the floor. With significant vibrations, the force of the unbalance can pull the mechanism off the floor, thereby significantly changing the stiffness characteristics of the system. Motor feet must be securely fastened, bolt mounts must be tightened, washer thickness must provide sufficient mounting rigidity, etc. If the bearings are broken, significant shaft misalignment and shocks are possible, which will also result in poor linearity and an inability to perform a quality balance.
平衡装置和平衡机
Recall that balancing is the process of aligning the main central axis of inertia with the rotor's axis of rotation.
这一过程可以通过两种方法进行。
The first method involves machining the rotor trunnions in such a way that the axis passing through the centers of the trunnions coincides with the main central axis of inertia of the rotor. Such a technique is rarely used in practice and will not be discussed in detail in this article.
第二种（最常见的）方法是移动、安装或拆除转子上的校正砝码，使转子的惯性轴尽可能靠近其旋转轴。
在平衡过程中移动、添加或移除校正砝码可以通过各种技术操作来完成，包括：钻孔、铣削、堆焊、焊接、拧紧或拧松、激光或电子束灼烧、电解、电磁堆焊等。
平衡过程可以通过两种方式实现：
- Field balancing (in situ) — the assembled rotor is balanced in its own bearings, on its own foundation, at its own operating speed, using a portable balancing kit;
- Shop balancing — the rotor is dismounted and balanced on a dedicated balancing machine.
For balancing of rotors in their own bearings, specialized balancing devices (kits) are usually used, which allow measuring the vibration of the balanced rotor at its frequency of rotation in vector form, i.e. to measure both the amplitude and the phase of vibration. At present, the above devices are manufactured on the basis of microprocessor technology and (apart from vibration measurement and analysis) provide automatic calculation of parameters of correcting weights, which should be installed on the rotor to compensate its unbalance.
这些设备包括：
- 基于计算机或工业控制器的测量和计算单元；
- 两个（或更多）振动传感器；
- a phase angle sensor;
- 用于在现场安装传感器的配件；
- 专用软件，设计用于在一个、两个或多个校正平面上执行全周期转子振动参数测量。
目前最常见的平衡机有两种：
- Soft-bearing machines (with pliable supports);
- Hard-bearing machines (with rigid supports).
Soft-bearing (above-resonance) machines have relatively pliable supports, for example, based on flat springs. The frequency of natural vibrations of these supports is usually 2-3 times lower than the rotation frequency of the balanced rotor, which is mounted on them, so the machine runs above resonance. Vibration sensors (accelerometers, vibration velocity sensors, etc.) are usually used to measure the motion of the supports of these above-resonance machines.
Hard-bearing (pre-resonance) machines use relatively rigid supports, whose natural frequencies of vibration should be 2-3 times higher than the rotation frequency of the rotor being balanced, so the machine runs below resonance. Force transducers are usually used to measure the dynamic load on the supports of the pre-resonance machine.
The advantage of pre-resonance (hard-bearing) balancing machines is that balancing on them can be performed at relatively low rotor speeds (up to 400 - 500 rpm), which greatly simplifies the design of the machine and its foundation, and increases the productivity and safety of balancing.
平衡刚性转子
重要！
- 平衡只能消除转子质量相对于其旋转轴分布不对称所造成的振动。其他类型的振动无法通过平衡消除！
- 技术机构的设计应确保在旋转工作频率下没有共振，可靠地固定在地基上，安装在可维修的轴承上，并进行平衡。
- 有缺陷的机器必须在平衡前修复。否则，无法实现高质量的平衡。
平衡不能代替维修！
平衡的主要任务是找出补偿配重的质量和位置，以抵消离心力。
As mentioned above, for rigid rotors, it is generally necessary and sufficient to install two compensating weights. This will eliminate both the static and the couple components of the rotor unbalance. The general scheme for measuring vibration during balancing is as follows.
Vibration sensors are installed on the bearing supports at points 1 and 2. A revolution mark is attached to the rotor, usually with reflective tape. The revolution mark is used by the laser tachometer to determine the rotor speed and phase of the vibration signal.
如何进行动态平衡（三次运行法）
In most cases dynamic balancing is carried out by the method of three starts. The method is based on the fact that trial weights of known mass are placed on the rotor in series in plane 1 and 2 and the weights and the location of the balancing weights are calculated based on the results of changes in the vibration parameters.
The plane in which a correction weight is installed is called a 校正平面. Correction planes are located on the rotor itself — typically at the two ends of the rotor body, on the fan or impeller disks, or on dedicated balancing rings. They should be chosen as far apart along the shaft as the design allows, so that a moderate weight produces a sufficient correcting moment. This is not the same as the measuring points, which are on the bearing housings (see Fig. 6).
At the first start-up the initial vibration is measured (in the Balanset software this is Run 0). Then a trial weight of known mass is placed on the rotor closer to one of the bearings. A second start-up is carried out (Run 1) and the vibration parameters are measured, which should change due to the test weight installation. Then the test weight in the first plane is removed and installed in the second plane. A third test run is performed (Run 2) and the vibration parameters are measured. The test weight is removed and the software automatically calculates the masses and installation angles of the balance weights.
The calculated correction weights are then installed in their planes and a check run is made — in the Balanset software this is Run T (Trim). The residual vibration is compared with the tolerance. If the result is still above the target, the software reuses the influence coefficients already determined, so no new trial-weight runs are needed — only a small additional trim correction is computed and installed.
The point of installing the test weights is to determine how the system reacts to changes in unbalance. The weights and locations of the test weights are known, so the software can calculate so called influence coefficients, showing how introducing a known unbalance affects the vibration parameters. The influence coefficients are characteristics of the mechanical system itself and depend on the rigidity of the supports and the mass (inertia) of the rotor-support system.
对于相同设计的同类型机构，影响系数会比较接近。可以将其保存在计算机内存中，用于同类型机构的平衡，而无需进行试运行，从而大大提高了平衡的效率。需要注意的是，在选择试验砝码的质量时，应确保在安装试验砝码时振动参数会发生明显变化。否则，影响系数的计算误差会增大，平衡质量也会下降。
如图 1 所示，离心力沿径向作用，也就是垂直于转子轴线。因此，振动传感器的灵敏轴也必须指向径向。通常，基础在水平方向的刚度较低，因此水平方向的振动更高。为了提高灵敏度，传感器应安装成其灵敏轴也指向水平方向。不过这并无根本性差别。除径向振动外，还必须监测沿转子旋转轴线方向的轴向振动。这种振动通常不是由不平衡引起，而是由其他原因造成，主要与通过联轴器连接的轴的不对中有关。
This vibration cannot be eliminated by balancing, in which case alignment is required. In practice, such machines usually have both rotor unbalance and shaft misalignment, which makes the task of eliminating vibration much more difficult. In such cases, it is necessary to center the machine first and then balance it. (Although with strong torque unbalance, vibration also occurs in the axial direction due to "twisting" of the foundation structure.)
相关文章（平衡支架示例）
评估平衡机制质量的标准
The balancing quality of rotors (mechanisms) can be evaluated in two ways. The first method involves comparing the amount of residual unbalance determined during the balancing process with the tolerance for residual unbalance. These tolerances for the different rotor classes are specified in ISO 21940-11 (formerly ISO 1940-1).
How the tolerance is computed (ISO 21940-11). The standard specifies a balance quality grade G, which is the product of the permissible specific unbalance e每 and the service angular velocity ω, expressed in mm/s:
- ω = 2π·n / 60 [rad/s], where n is the service speed in rpm;
- e每 = G · 1000 / ω [g·mm/kg] (numerically equal to µm of center-of-mass offset) — equivalently e每 = 9549 · G / n;
- U每 = e每 · m [g·mm], where m is the rotor mass in kg.
Worked example. Rotor m = 50 kg, service speed n = 3000 rpm, grade G 6.3 (fans, pumps, standard electric motors): ω = 2π·3000/60 = 314.2 rad/s; e每 = 6.3 · 1000 / 314.2 = 20.1 g·mm/kg (cross-check: 9549 · 6.3 / 3000 ≈ 20.1); U每 = 20.1 · 50 ≈ 1000 g·mm for the whole rotor.
Splitting the tolerance between two planes. For a rotor whose center of mass lies between the correction planes, the total tolerance is divided in inverse proportion to the distance from the center of mass to each plane; for a symmetrical rotor this is simply half in each plane — about 500 g·mm per plane in the example above. Neither plane should be allocated more than 70% or less than 30% of U每.
Typical grades: G 0.4 — gyroscopes, spindles of precision grinders · G 1 — grinding-machine spindles, precision armatures · G 2.5 — turbines, turbo-generators, machine-tool drives · G 6.3 — general engineering: fans, pump impellers, flywheels, standard electric motors · G 16 — cardan shafts with special requirements, agricultural machinery, crushers · G 40 — car wheels, drive shafts (cardan shafts) · G 100 — crankshaft drives of high-speed diesel engines.
然而，遵守规定的公差并不能完全保证机构的运行可靠性，这与实现最低振动水平有关。这是因为机械装置的振动幅度不仅取决于转子残余不平衡力的大小，还取决于其他几个参数，包括：机械装置结构元件的刚度 k、质量 m、阻尼系数以及旋转频率。因此，在许多情况下，为了估算机械装置的动态质量（包括其平衡质量），建议估算机械装置的残余振动水平。
The most widely used standard for permissible vibration levels of industrial machines is ISO 20816-3 (formerly ISO 10816-3). It covers machines above 15 kW running at 120–15,000 rpm, and it classifies them in two dimensions: by power group (Group 1 — above 300 kW; Group 2 — 15 to 300 kW) and by support type (rigid or flexible). Each combination has its own A/B, B/C and C/D zone boundaries in mm/s RMS. Machines outside this scope have dedicated parts of the series (turbine sets — ISO 20816-2, hydraulic machines, reciprocating machines, pumps) or product standards such as ISO 14694 for industrial fans.
For general machines evaluated on non-rotating parts, the classic ISO 10816-1 zones (now part of ISO 20816-1) give the following boundaries of vibration velocity, mm/s RMS:
| 班级 | A/B | B/C | C/D |
|---|---|---|---|
| Class I (small machines, up to 15 kW) | 0.71 | 1.80 | 4.50 |
| Class II (medium machines, 15–75 kW) | 1.12 | 2.80 | 7.10 |
| Class III (large machines, rigid foundation) | 1.80 | 4.50 | 11.20 |
| Class IV (large machines, flexible foundation) | 2.80 | 7.10 | 18.00 |
Zone A corresponds to the vibration of new machines; zone B is acceptable for unrestricted long-term operation; zone C allows only restricted operation; zone D indicates vibration severe enough to cause damage.
标准和参考
- ISO 21940-11:2016 — Mechanical vibration — Rotor balancing — Part 11: Procedures and tolerances for rotors with rigid behaviour. (Replaces ISO 1940-1, which is withdrawn.) G-grades and tolerance calculator →
- ISO 21940-2 — Mechanical vibration — Rotor balancing — Part 2: Vocabulary. (Definitions of static, couple, quasi-static and dynamic unbalance.)
- ISO 20816-1:2016 — Mechanical vibration — Measurement and evaluation of machine vibration — Part 1: General guidelines. (Replaces ISO 10816-1 and ISO 7919-1.) Evaluation zones →
- ISO 20816-3:2022 — Mechanical vibration — Measurement and evaluation of machine vibration — Part 3: Industrial machines with nominal power above 15 kW and nominal speeds between 120 r/min and 15 000 r/min. (Replaces ISO 10816-3:2009.)
- ISO 14694:2003 — Industrial fans — Specifications for balance quality and vibration levels.
常见问题
平衡能消除所有振动吗？
不。动平衡可以消除因转子质量相对于其旋转轴的不对称分布而引起的振动。由不对中、轴承缺陷、空气动力/流体动力、电磁力和其他原因引起的振动需要单独的诊断和纠正措施。
为什么在共振附近平衡会失效？
在共振附近，即使是微小的速度变化也会导致振幅的大幅变化和180°的相位偏移。在这种情况下，测量结果会变得不稳定，如果没有特殊方法，常规的平衡程序可能无法收敛。
何时需要单平面动平衡，何时需要双平面动平衡？
One plane is enough for disk-shaped rotors, where the axial length of the rotor is small compared with the diameter — as a rule of thumb L/D < 0.5 — and the service speed is well below the first critical speed. Typical examples: a grinding wheel, a single-disk fan impeller, a pulley, a car wheel. Such a rotor carries almost purely static unbalance.
两架飞机 are required for elongated rotors (L/D ≥ 0.5), for any rotor with two or more impellers or disks spaced along the shaft, and whenever the vibration phase at the two bearings differs markedly — a sign of a couple component. A rigid rotor never needs more than two planes.
When in doubt, measure both bearings: if a one-plane correction reduces the vibration at one bearing and increases it at the other, the rotor has a couple component and needs two-plane balancing.
平衡之前应该做什么？
确保机器处于良好工作状态：牢固地安装在基础上，轴承状况良好，无严重松动，且无明显的非线性因素。平衡并不能代替维修。
要点总结
- 动平衡可以纠正质量相关的（离心）激励；但它无法解决不对中、轴承损坏或电磁/空气动力源的问题。
- 共振和非线性会导致传统的平衡方式失效或不安全。
- For rigid rotors, two-plane balancing is the general solution for dynamic unbalance (the combination of static + couple).
