The whirl-motion dynamics of a rotor on an axis was explained by Jeffcott₁ in 1919 and signified a turning point in rotor dynamics. The focal point of Jeffcott’s study was the assessment of the trajectory of the rotor’s elastic center (in the direction of the line between the spatial center (O) and the elastic center (C) (High Spot)) according to the centrifugal force acting in the direction of the heavy spot, the line between the elastic center (C), and the center of mass (G).

Jeffcott also demonstrated that the dynamic behavior of the high spot, or the unbalanced exciting force, under a normal state is a harmonious motion. A point to note is that the phase angle of the heavy spot is defined from the horizontal line (OX) of the fixed space. It causes that the phase angle of the heavy spot from the key phaser as a system parameter is not expressed in the derived equation of motion. The equations of motion are derived in Jeffcott’s paper as follows:

Where r(= x + yj) is whirl motion of the elastic center. The phase angle of the high spot (∠xOC) was obtained as the relative position angle ϕ of the heavy spot as follows:

Although Jeffcott did not define the heavy spot’s phase angle as the reference of the key phaser and did not indicate clearly the phase angle of the heavy spot, he defined the relative position angle ϕ, thereby making his analysis complete and not debatable.

But the derived Eq. (1) is not able to use for the balancing technique because of neglecting the the phase angle of the heavy spot from the key phaser.

The influence coefficient method was first introduced by Thearle in 1934 and has been widely used in academic and industrial applications, and is now considered to be the classic balancing technique. In 1992, Ehrich described the influence coefficient method by first introducing the Jeffcott rotor’s equations of motion.⁴ Although Ehrich’s equations of motion are almost identical to the ones derived by Jeffcott, Ehrich’s equations are more advanced because the heavy spot is expressed by using complete equations. As shown in Fig. 2, Ehrich defined the phase angle of the heavy spot by using the fixed-space reference line (x Axis) and the rotor reference line (Key Phaser). The whirl motion (r) and the exciting force (F_u) due to mass unbalance were expressed in complex numbers. The phase angle ψ signified the phase angle of the heavy spot with reference to the key phaser in Fig. 2. And Ωt was the phase angle of the key phaser varying in time with reference to the horizontal line. Therefore, F_u was completely defined as follows:

Fig. 2 
Mathematical model for rotor balancing technique

Therefore the equation of motion of the unbalance exciting force is written as follows:

Where r is a position vector from O to C and e_u is a position vector from C to G. Assuming synchronous motion, we have

And by transforming Eq. (6) into the frequency domain, the steady state response of whirl motiom R is derived as follows:

Where A(Ω) is the mechanical impedance of Jeffcott rotor. Eq. (8) represents that the complex response of whirl motion is proportional to the complex mass-unbalance during constant speed, which is basic concept of influence coefficient method. Eq. (9) is referred to as the influence coefficient, and it expresses the relationships among the parameters associated with the mass, damping, and stiffness elements of the rotor dynamics system. However, the influence coefficient method does not use the relationship in Eq. (9) but simply processes the mechanical impedance A(Ω) as a complex coefficient while fixing the operation speed.

In Eq. (8), R is a measured value, and the amount of unbalance Me_u and the influence coefficient A(Ω) are unknown. Since the two unknown values cannot be obtained from Eq. (8) alone, a trial run is performed. By adding a trial mass (U_t), R' is measured again at an identical operation speed to obtain another equation:

Therefore, the influence coefficient A(Ω) and the mass unbalance Me_u can now be obtained from Eqs. (8) and (10). Consequently, the highlight of the influence coefficient method is to identify the magnitude and phase angle of complex eccentricity e_u by adding trial-mass process. Here the merit of the influence coefficient method is that the mechanical impedance A(Ω) can be treated as a constant value. As a result, the mathematical modeling processes are not necessary any more. In some practical aspect, it would be true. But in precision engineering aspect, it might be stupid to use no more the works established in the rotor dynamic analysis field.

This implies that the approach is very useful in field applications because balancing can be achieved by non-experts without an understanding of the mathematical modeling of the rotor dynamics system. On the other hand, it is also true that the advanced rotor dynamics modeling techniques that have been introduced since 1919 cannot be applied to the balancing technique. We believe that balancing can be viewed as an inverse problem, where the whirl motion is measured and the mass unbalance that caused the whirl can be explained. If so, there should be an interaction between the balancing technique and the techniques of analyzing rotor dynamics systems, but the two areas remain very much independent. In order to help establish an essential relationship between these two areas, we propose a new balancing technique that incorporates both Eqs. (8) and (9).

This paper proposes a new balancing technique based on equations of motion. As a sample system, a schematic of a Jeffcott rotor with shaft bow is considered in Fig. 3. For the generality of measurement, a gap sensor that measures the high spot is placed at a phase angle λ on the opposite side of the x axis. A photo sensor that indicates the position of the key phaser is placed along the +x axis. For the elastic center, the equation of motion at the center of mass G is derived as follows:

Fig. 3 
Whirl motion of Jeffcott rotor with shaft bow

where e_b is the position vector from O to B expressed in complex numbers. This value may be either known or unknown according to the situation of the rotor system. The third term of Eq. (11) K(r − e_b) is the restoring force reacting at the elastic center C due to the elastic deformation of the rotor shaft. For the simplicity, in this paper, e_b is assumed to be prescribed one. And the nonhomogeneous term of Eq. (11) F_ue^jΩt represents the centrifugal force by the eccentricity applying out from C to G. The magnitude of the centrifugal force is Me_uΩ² and the phase angle of F_u is the cosine angle between line CG¯ and x-axis, Ωt + ψ_m shown in in Fig. 3. Therefore we have,

Similarly, in Fig. 3, we have

Where e_b and ψ_b can be measured while slow speed. Therefore substituting Eqs. (12)-(14) into Eq. (11), we obtain the final form of equation of motion about the Jeffcott rotor with shaft bow as follows:

Dividing by mass M, finally, we have

Where 2ζΩ_cr and Ω_cr² are unknown natural characteristic parameters of Jeffcott rotor. And also, substituting Eq. (7) into Eq. (15), we have,

Based on Eq. (16), we can identify the natural characteristic parameters 2ζΩ_cr and Ω_cr² as well as the target unknown eccentricity e_u. Then Eq. (16) contains four unknowns and two equations. To obtain these unknown values, more than two operational speeds. Let them be Ω₁ and Ω₂. Then applying them into Eq. (16), we have

Solving (16) and (17), we obtain the unknowns, Ω_cr², 2ζΩ_cr, e_u, and ψ_m without trial mass process.

Compensation mass unbalance U_c(= me_c) is obtained as following relationship:

So, we have

A simple test rig was implemented to verify the efficacy of the new balancing technique proposed in this study. As shown in Fig. 4, the rotor cylinder was placed on a steel axis and was supported by ball bearings across a 580 mm span. The rotor was positioned to the left of the center of the bearing support. The axis was connected directly to the driving motor and the two were linked by using a coupling. The ratio of the cylinder length to its diameter was 1.0, and the critical speeds in the cylindrical and conical modes were 3500 and 73000 rpm, respectively. Twelve holes were drilled at the 40-mm position on the radius to attach trial weights to both sides of the rotor, with each hole capable of holding up to 24 g of trial weight (Fig. 5). The rotation axis was sustained with ball bearings (No. 6004) at both ends, and the length of the support unit was 580 mm. The rotation axis was connected to a 3000-W inductor motor via a rubber coupling. There was some backlash through the coupling between the motor and the rotation axis. Also, there was misalignment in the permanent shaft bow, rotation axis, and motor axis. As shown in Fig. 3, the photo sensor and two gap sensor were installed horizontally (λ = 0°). Two gap sensors were placed in identical locations on both sides to measure the whirl motion, as displayed in Fig. 4. One photo sensor was prepared to trace the position of the key phaser. The whirl motion necessary for single-plane balancing was obtained by calculating the average of the whirl motions as measured by the two gap sensors. The new single-plane balancing technique proposed in this paper involves the following procedure:

Fig. 4 
Test rig

Fig. 5 
The relationship between the phase angle and the wave form on oscilloscope

Step 1. Measurement of whirl motion at slow speed 400 rpm without a trial weight to measure the shaft bow: e_b.

Step 2. Measurement of whirl responses R₁ (Ω₁ is chosen at around 1500 rpm) and R₁ (Ω₂ is chosen at around 2100 rpm)

Step 3. Calculation of Ω_cr², 2ζΩ_cr, e_u, and ψ_m from Eqs. (16) and (17).

Step 4. Caculation of compensation mass unbalance U_c from Eq. (18) to perform balancing.

The complex whirl response R is measured by a photo sensor and a gap sensor. The measured time-domain signal r of Eq. (7) is displayed on the on oscilloscope. The sinusoidal curve shown in Fig. 5 depicts the time domain signal picked up by the gap sensor. And the pulse signal is picked up by a photo sensor. We can read a pick-to-pick value and the phase angle θ₀.

From Fig. 5, the amplitude R is the half of pick-to-pick value 2X and the phase angle of complex value R becomes 2π − θ₀ − λ. That is, we have

We must know the true value of the mass unbalance for the experimental model in advance in order to verify the efficacy of the proposed technique. The true value was obtained based on experiment according to Section 8 “Determination of the residual unbalance” in ISO 1940/1-1986(E). As shown in Fig. 6, after the trial weights are placed into the rotor holes at equal intervals, unbalance is obtained by using the influence coefficient method, and the average unbalance is regarded as the true value.

Fig. 6 
Installation angles of the added masses

The main vibration sources of the experimental model are rotor unbalance, shaft bow, and misalignment between the motor and the axis. The run-out caused by shaft bow and misalignment were measured by using the convergent value as the speed was decreased from 1000 rpm. As indicated in Fig. 7, there was no variation in the whirl motion below 400 rpm. The experimental results indicate that run-out occurred at 0.17(mm)∠75°, which was regarded as e_b in Eq. (14): e_b = 0.17 ej75π180 = 0.164 + 0.044j.

Fig. 7 
Measured run-out

The true value of the mass unbalance of the experimental model eu¯ was obtained according to ISO 1940/1-1986(E). The whirl motions without run-out are measured at sampling speeds of 1500 and 2100 rpm, and the results are depicted on a complex plane in Fig. 8. The average of these values was 0.23(mm) ∠294°, which was the true value of the experimental model: e_u = 0.23 ej294π180 = 0.094 − 0.210j.

Fig. 8 
Measuring process of the true value of eccentricity by using the ISO standard

This section explains the mass eccentricity of the experimental model e_u according to the proposed method. Whirl responses R₁ and R₂ were measured at 1500-2600 rpm. Sampling is carried out with two categories; low speed range (1500-2000 rpm) for R₁ and high speed range (2000-2600 rpm) for R₂. Table 1 lists the magnitudes and phase angles of the whirl responses. Two of the whirl responses were selected in Tabls 1(a) and 1(b), respectively and used in Eqs. (1) and (17).

The identified e_u by proposed method was chose as the average value, 0.27(mm)∠314° in Table 2. Compared with eu¯ = 23(mm)∠294^o measured according to ISO specifications, The percent errors resulting in magnitude and phase discrepancies are 20% and 2%, respectively. Although there was some discrepancy in magnitude, the phase angle coincided excellently. In addition to mass eccentricity, the proposed method provides critical speed and damping ratios of 3113 rpm and 0.012, respectively.

The compensation weight was calculated from Eq. (18) based on the mass eccentricities obtained in this section, and balancing was performed and passed the first critical speed. Fig. 9 displays the whirl response before and after balancing. According to observation, the critical speed was 3157 rpm and the damping ratio was approximately 0.052. After balancing, there was a 1.4% error in critical speed and damping improved by 77%. The experimental results indicate that the proposed method yielded a result close to the true value and the critical speed was predicted with a high level of accuracy.

Fig. 9 
Bode plot of the whirl response

The balancing technique is sensitive toward measurement error. In order to assess the effect of the proposed method on measurement error, the experimental parameters used in the measurement were applied to the influence coefficient method to compare with the results obtained by using the proposed method. The influence coefficient method requires as-is and trial-run measurements. Two measurements at 1538.5 and 2222.2 rpm used in single-plane balancing were set as as-is values, and three separate experiments were conducted at 1500 and 2100 rpm, and their average was used as the trial-run value to obtain mass eccentricity. Fig. 10 displays the mass eccentricity distributions displayed by the proposed method and the influence coefficient method. As the figure indicates, the proposed method converges more accurately to the phase of mass eccentricity than does the influence coefficient method.

Fig. 10 
Deviations in unbalance