Field balancing of vehicle transmission shaft based on the influence coefficient method

Abstract

Imbalance of transmission shaft may result in serious bearing failures and poor vehicle noise, vibration, and harshness performance. Usually, shafts in a multi-shaft system are balanced separately before been assembled in vehicle. This causes potential out of balance issue in the full vehicle level in the field due to tolerance stacking up and dynamic imbalance coupling. In this article, a system level balancing method, the influence coefficient method, is introduced and applied to solve the problem of excessive floor panel vibration. The balancing result shows that the first-order vibration from vehicle transmission shaft imbalance obviously decreased. Method applied in this article is simple and practical and could be used for other applications to prevent bearing failure or noise, vibration, and harshness issue due to vehicle level imbalance.

Keywords

Vehicle transmission shaft imbalance field balancing the influence coefficient method

Introduction

Imbalance in a rotary machine is a common source of noise, vibration, and harshness (NVH) issue or bearing failure.^1,2 In a four-wheel-drive sport utility vehicle (SUV), the transmission shaft is the main rotational part. The shaft is typically long and sometimes has multiple sections. Balancing of the shaft has significant effect on vehicle NVH performance.^3,4 Thus, a lot of effect has been made to control the dynamic balance of the transmission shaft during the development stage. Normally, for a multi-section shaft system, each section of the shaft is balanced to the specification as per the G40 dynamic balance standard in ISO-1940-1:2003.⁵ Unfortunately, due to the assembly errors and dynamic balance coupling, the transmission shaft assembled may not meet the dynamic balance requirement under the whole vehicle condition. Liu and Orzechowski⁶ made it clear to us that the imbalance levels of the full assemblies can be higher than those of the subassemblies by as much as three times. Faced with this problem, Steyer et al.⁷ achieved an optimized driveline NVH performance through finite element analysis studies. Besides, very little literature is found in view of vehicle level balance in the field.

A similar issue occurred for a SUV, which experiences excessive floor panel vibration in the field due to imbalance of the multi-sections transmission shaft. In this article, a full vehicle level balancing method, that is, the influence coefficient method, is applied. Theoretical background of the influence coefficient method based on trial masses⁸ is introduced. It is then being applied to determine the optimal mass and location angle to be added on the shaft. Order analysis for the vibration signal is conducted for evaluating the phase and amplitude of the first-order component.⁹ Balancing result shows that the application of this field-balancing method improves the vehicle NVH performance effectively.

The influence coefficient method

Basic principle of the influence coefficient method can be expressed as the relationship between the vectors shown in Figure 1.

Figure 1.

Basic principle of the influence coefficient method.

Assuming that a rotor has an initial imbalance and the vibration response on the measurement plane under normal working condition is $v_{o}$ . Its first-order amplitude and phase can be calculated from the vibration and rotation signals. After a known trial mass $q_{t}$ is being applied at a set position on the correction plane, the vibration response changes to a different vector $v_{r}$ . The response variation due to the trial mass is

v_{t} = v_{r} - v_{o}

(1)

The influence coefficient, $a$ , is defined as the variation due to unit mass. Thus

a = \frac{v_{r} - v_{o}}{q_{t}}

(2)

To get ideal balancing result, the correction mass $q_{c}$ should meet the equation as

a q_{c} + v_{o} = 0

(3)

Substitute equation (2) into equation (3), $q_{c}$ can be obtained as

q_{c} = - \frac{v_{o}}{v_{r} - v_{o}} q_{t}

(4)

Under normal circumstances, we assume that a rotor to be balanced has K measuring planes and M correction planes, the equation of the correction mass vector can be written as

A Q + V = 0

(5)

where $A$ is the influence coefficient matrix, and each column of this matrix can be obtained by adding a different trial mass, $V$ is the initial vibration response vector and can be obtained without any trial masses added, and $Q$ is the final correction mass vector to be calculated. And for practical calculation, equation (5) can be expressed as

\begin{array}{l} [\begin{matrix} \begin{matrix} a_{11 x} + i a_{11 y} & a_{12 x} + i a_{12 y} \\ a_{21 x} + i a_{21 y} & a_{22 x} + i a_{22 y} \end{matrix} & \begin{matrix} \dots & a_{1 M x} + i a_{1 M y} \\ \dots & a_{2 M x} + i a_{2 M y} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \\ a_{K 1 x} + i a_{K 1 y} & a_{K 2 x} + i a_{K 2 y} \end{matrix} & \begin{matrix} ⋱ & ⋮ \\ \dots & a_{K M x} + i a_{K M y} \end{matrix} \end{matrix}] \\ [\begin{matrix} \begin{matrix} q_{1 x} + i q_{1 y} \\ q_{2 x} + i q_{2 y} \end{matrix} \\ \begin{matrix} ⋮ \\ q_{M x} + i q_{M y} \end{matrix} \end{matrix}] + [\begin{matrix} \begin{matrix} v_{1 x} + i v_{1 y} \\ v_{2 x} + i v_{2 y} \end{matrix} \\ \begin{matrix} ⋮ \\ v_{K x} + i v_{K y} \end{matrix} \end{matrix}] = 0 \end{array}

(6)

where $k = 1, 2, \dots, K$ , and $m = 1, 2, \dots, M$ . $a_{k m x} + i a_{k m y} = a_{k m}$ , which contains the amplitude and phase information of influence coefficient. $q_{m x} + i q_{m y} = q_{m}$ , which contains the quantity and phase information of correction mass. $v_{k x} + i v_{k y} = v_{k}$ , which contains the amplitude and phase information of vibration response on measuring plane k under working condition.

To verify the effect of balancing, the calculated correction masses $q_{m}$ should be added to the corresponding correction planes and repeat the evaluation test. It is worth noting that to get a unique solution of equation (6), the number of measuring planes should be more than that of the correction planes, namely $K \geq M$ . The process of this method in practical engineering is shown in Figure 2.

Figure 2.

Specific process of the influence coefficient method for field balancing.

Field balancing of a transmission shaft

To decrease the excessive floor panel vibration caused by the imbalance of a vehicle transmission shaft, the influence coefficient method is applied. The transmission shaft assembly to be balanced is a three-section structure. It has front, middle, and rear sections and been assembled to the vehicle at four attachments. Accelerometers were installed at the support brackets which measure the responses due to imbalance of the shaft. An optical tachometer probe was used to capture the rotation speed of the shaft. Figure 3 shows the assembled shaft in vehicle and instrument locations.

Figure 3.

The transmission shaft assembly and instrumentation: (a) the front shaft and (b) the rear shaft.

The four attachments are set as measurement planes, which are named from measurement plane 1 to measurement plane 4 from front to rear. In view of the short middle shaft, the correction planes are set near the four ends of the front and rear shafts, and named from correction plane 1 to correction plane 4 from front to rear. The tachometer probe is installed at the rear shaft and the reflector is pasted on the shaft to serve as the rotation speed pulse trigger. The rotation pulse signal is used to conduct the order analysis of the vibration signal, thus getting the amplitude and phase of the first-order component.

Before balancing, it is necessary to set the angle of reflector and correction planes. Then, a test run is needed to determine the rotation speed of the transmission shaft. In this project, the vehicle balancing speed is about 40 km/h, for it is the highest speed under four-wheel-drive condition. The corresponding rotation speed of the transmission shaft is about 590 r/min. During the balancing, the fluctuation of rotation speed should be no more than 5%.

The balancing is normally started step by step according to Figure 2. An initial condition vibration response was measured first without any condition balance weight added. Tests were repeated four times with trial mass 1, 2, 3, and 4 been installed each time. The information of trial masses is shown in Table 1, and the vibration responses of measurement planes with different trial masses are shown in Table 2. Trial masses are added to the outer surface of the shaft, whose radius is 33 mm.

Table 1.

Information of trial masses.

Run	Correction plane 1		Correction plane 2		Correction plane 3		Correction plane 4
Run	Mass (g)	Angle ( $°$ )	Mass (g)	Angle ( $°$ )	Mass (g)	Angle ( $°$ )	Mass (g)	Angle ( $°$ )
Trial 1	20	270	–	–	–	–	–	–
Trial 2	–	–	15	0	–	–	–	–
Trial 3	–	–	–	–	15	315	–	–
Trial 4	–	–	–	–	–	–	20	315

Table 2.

Vibration responses of measuring planes in different conditions.

Run	Measurement plane 1		Measurement plane 2		Measurement plane 3		Measurement plane 4
Run	Amp (mm/s²)	Phase ( $°$ )	Amp (mm/s²)	Phase ( $°$ )	Amp (mm/s²)	Phase ( $°$ )	Amp (mm/s²)	Phase ( $°$ )
Initial	92.5	78	32.2	225	35.4	127	35.8	169
Trial 1	51.2	321	26.4	82	27	64	30.9	183
Trial 2	117	41	49.1	262	44.4	255	46.5	209
Trial 3	93.6	45	46.7	67	42.7	44	37.6	282
Trial 4	117	57	37.6	293	27.3	281	13	33

According to the influence coefficient method, the influence coefficient matrix $A$ can be calculated from the trial masses in Table 1 and vibration responses in Table 2

A = [\begin{matrix} \begin{matrix} 6.221, ∠ 9.5 ° & 4.695, ∠ 348.77 ° \\ 2.78, ∠ 151.6 ° & 2.025, ∠ 301.65 ° \end{matrix} & \begin{matrix} 3.525, ∠ 17.64 ° & 2.257, ∠ 54.75 ° \\ 5.167, ∠ 103.05 ° & 1.965, ∠ 27.46 ° \end{matrix} \\ \begin{matrix} 1.669, ∠ 83.11 ° & 4.789, ∠ 277.851 ° \\ 0.474, ∠ 26.89 ° & 1.993, ∠ 259.34 ° \end{matrix} & \begin{matrix} 3.47, ∠ 46.53 ° & 3.056, ∠ 340.71 ° \\ 4.081, ∠ 359.57 ° & 2.302, ∠ 45.31 ° \end{matrix} \end{matrix}]

Initial vibration response vector on measuring planes is

V = [\begin{matrix} \begin{matrix} 92.5, ∠ 78 ° \\ 32.2, ∠ 225 ° \end{matrix} \\ \begin{matrix} 35.4, ∠ 127 ° \\ 35.8, ∠ 169 ° \end{matrix} \end{matrix}]

And according to equation (5), the information of correction masses $Q$ can be calculated as

Q = [\begin{matrix} \begin{matrix} 21.02, ∠ 235 ° \\ 4.67, ∠ 51.7 ° \end{matrix} \\ \begin{matrix} 8.8, ∠ 55.4 ° \\ 9.46, ∠ 252.1 ° \end{matrix} \end{matrix}]

This means that correction masses are $21.02 g$ at angle of $235 °$ on correction plane 1, $4.67 g$ at angle of $51.7 °$ on correction plane 2, $8.8 g$ at angle of $55.4 °$ on correction plane 3, and $9.46 g$ at angle of $252.1 °$ on correction plane 4 for ideal. However, in practice, standard mass blocks are used as correction masses. There is a slight difference between theory and practice. In this project, the added correction masses are shown in Table 3.

Table 3.

Correction masses in theory and practice.

	Correction plane 1		Correction plane 2		Correction plane 3		Correction plane 4
	Mass (g)	Angle ( $°$ )	Mass (g)	Angle ( $°$ )	Mass (g)	Angle ( $°$ )	Mass (g)	Angle ( $°$ )
Theory	21.02	235	4.67	51.7	8.8	55.4	9.46	252.1
Practice	20	240	5	45	10	60	10	255

After adding the correction masses to the corresponding correction planes, the vehicle is restarted to the set working condition. Vibration responses on measuring planes before and after balancing are shown in Table 4. Vibration polar plot of measuring planes during balancing are shown in Figure 4.

Figure 4.

Vibration polar plot of measurement planes: (a) measurement plane 1, (b) measurement plane 2, (c) measurement plane 3, and (d) measurement plane 4. Purple circles indicate the first-order vibration responses at initial state; blue circles named Trial: T1, Trial: T2, Trial: T3, and Trial: T4 indicate the first order of vibration responses with trial 1, 2, 3, and 4 added, respectively; and green circles indicate the first-order vibration responses of final state with correction masses added.

Table 4.

Vibration responses before and after balancing.

Run	Measurement plane 1 (Amp (mm/s²))	Measurement plane 2 (Amp (mm/s²))	Measurement plane 3 (Amp (mm/s²))	Measurement plane 4 (Amp (mm/s²))
Before	92.5	32.2	35.4	35.8
After	54	30.2	25.3	16.6
Reduced (%)	41.6	6.2	28.5	53.6

After the balancing, the first-order vibration amplitude at the four support attachments decreases from 92.5, 32.2, 35.4, and 35.8 mm/s² to 54, 30.2, 25.3, and 16.6 mm/s², which are about 41.6%, 6.2%, 28.5%, and 53.6% reduction, respectively. The field balancing method of vehicle transmission shaft reduces the vibration amplitude caused by the imbalance obviously and improves the NVH performance of the vehicle effectively.

Conclusion

A severe floor panel vibration issue due to excessive imbalance of the transmission shaft was discussed in this article. A system level dynamic balancing was conducted based on the applied influence coefficient method. This avoided the limitation of the traditional component-level balancing. Balancing result shows that the first-order vibration amplitude at four shaft attachment locations decreased 41.6%, 6.2%, 28.5%, and 53.6%, respectively. Vehicle floor panel vibration is decreased, and its NVH performance is improved. This proves the efficiency of the proposed method and provides a reference for other balancing issues in the field.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was supported by grants from the National Natural Science Foundation of China (grant no.: 11704040), the Fundamental Research Funds for the Central Universities (grant nos.: 106112017CDJQJ338810 and CDJXZ 2016003), and the Opening Fund of State Key Laboratory of Vehicle NVH and Safety Technology (grant no.: NVHSKL-201504).

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