Testing and Analysis of Torsional Vibration of Ship Transmission Shafting Based on Five-Point Smoothing Algorithm

Abstract

In this paper, a method is proposed to accurately measure the torsional vibration of a ship’s propulsion shafting, and calculate its instantaneous torque based on instantaneous speed data. The instantaneous speed signal of a ship is measured by a data acquisition instrument indirectly under the deceleration condition, and then filtered by the five-point smoothing algorithm. The filtered instantaneous speed signal is further processed to calculate the maximum amplitude of the torsional vibration, along with the maximum torque of the propulsion shafting under specific operating conditions. The proposed method can effectively improve the calculation accuracy of the torsional vibration of the shafting, and provides reliable theoretical basis for setting the safe speed of the ship’s diesel engine.

1. Introduction

The power plant is the power core of the ship, and the ship’s propulsion shafting is the core of the power plant. Therefore, the ship’s propulsion shafting is an indispensable part of the ship [1,2,3,4,5].

The torsional vibration of the ship’s propulsion shafting is caused by periodic torque excitation, such as the diesel engine and propeller. The periodic torque excitation makes the torsion angle of each shaft segment different. The different torsion angles cause different positions to swing back and forth around the center line, resulting in torsional deformation. The excitation sources of the torsional vibration of the ship’s propulsion shafting mainly include the excitation torque generated by the gas pressure changes in the cylinder of each piston in the diesel engine; the inertial moment generated by the reciprocating mass of the piston-linkage mechanism; the error excitation and meshing impact excitation of the gear system; component installation misalignment, material inhomogeneity, inaccuracy in machining, and excitation torque from propeller rotation. Torsional vibration will cause stress changes in the shaft system, and metal fatigue can cause cracks [6,7,8]. When the frequency of the excitation force is close to or equal to the natural frequency of the propulsion shafting, the torsional vibration will intensify. The torsional vibrations cause workpieces to fail or break. Therefore, accurate measurement of the torsional vibration of the propulsion shafting is an important measure to ensure the effectiveness and reliability of a ship.

In recent years, the research on the measurement method of ships’ shafting torsional vibration has been relatively mature. In order to adapt to the higher monitoring requirements, scholars at home and abroad mainly conduct research on on-site measurement error elimination. A variety of error elimination methods and devices have appeared, which effectively improve the accuracy of torsional vibration measurement. Many scholars’ research on torsional vibration measurement technology is mostly limited to the acquisition of electrical signals. The torsional vibration is obtained from different methods, but the errors brought about in the measurement process are ignored, so that the data accuracy of the torsional vibration measurement cannot be guaranteed. The measurement methods of torsional vibration are mainly divided into two types: the measurement of stress and strain [9], and the measurement of torsional angular displacement [10]. The shafting torsion will cause the instantaneous speed fluctuation of the shafting. Torsional vibration analysis can be carried out by measuring the instantaneous speed of shafting [9,11].

At present, the torsional vibration test mainly adopts the non-contact measurement method. The instantaneous rotational speed becomes an important data record during the torsional vibration test. The measurement accuracy of instantaneous speed directly affects the calculation accuracy of torsional vibration. The acquisition of the instantaneous rotational speed mainly adopts equipment such as the coding disc, coding belt, reflective paper, and so on. When there is eccentricity in the installation of the coding disc or the overlapping of the coding belt is uneven, it will have a great impact on the instantaneous speed signal [12]. Geng Chong et al. [13] provided a qualitative formula about the error, and proposed a signal synthesis processing method to correct the error. However, they did not delve into the error. Dong Dawei et al. [14] theoretically derived the expression of the error. The factors affecting the torsional vibration error were also not clearly pointed out. South China University of Technology [15] analyzed the principle of the error caused by the eccentricity of the coding disc to the torsional vibration test, when using the coding disc to measure the torsional vibration. They proposed a method of using four sensors to measure in order to eliminate the eccentric installation error.

Due to the different diameters of the measured shaft system, the overlap of the coding belt will make the black and white stripes different from the other parts, resulting in inaccurate measurement signals [16,17]. Brian R. Resor et al. [12] studied the influence of the coding belt printing error on torsional vibration test and proposed a correction method using compensation technology. Yan Dandan et al. [18] proposed a generation method of the torsional vibration pulse simulation signal, which was analyzed according to the calculated instantaneous angular velocity and torsion angle. In this paper, the five-point smoothing algorithm is used to correct the error caused by the overlap. The accurate instantaneous speed signal is obtained, so as to calculate the accurate torsional vibration parameters.

2. Principle of Torsional Vibration Measurement

If torsional vibration occurs, the torsional angle of the shafting consists of two parts: rigid body rotation and torsional vibration, and its expression [19] is as follows:

$$\phi (t)={\omega }_{0}t+A\sin ({\omega }_t+{\psi }_k)$$

(1)

In Formula (1), ${\omega }_{0}t$ is the change in the rigid body rotation angle of the shafting; ${\omega }_0$ is the angular velocity of the shafting; the sine component is the torsional vibration angle, $A$ is the torsional vibration angle amplitude, and ${\omega }_t$ is the torsional vibration frequency.

When the torsional vibration contains multiple frequency components, Formula (1) can be expressed as:

$$\phi (t)={\omega }_{0}t+{\displaystyle \sum _k{A}_k\sin ({\omega }_{tk}+{\psi }_k)}$$

(2)

In Formula (2), ${\omega }_{tk}$, ${\psi }_k$, $A_k$ indicate the angular velocity, initial phase, and amplitude of the kth vibration component.

It can be concluded from Formula (2) that when calculating the torsion angle, the instantaneous angular velocity of the shafting needs to be processed. The torsional vibration of the shafting can be calculated by removing the average rotational speed ${\omega }_0$ and analyzing the measured signal.

3. Theoretical Calculations

3.1. Principle of Five-Point Smoothing Algorithm

The five-point smoothing algorithm is a kind of smoothing filtering algorithm. In the smoothing filtering algorithm, the principle of the five-point smoothing method to achieve smooth filtering is to use the polynomial least squares method to approximate the sampling point. The algorithm is simple and the filtering effect is good [20,21]. At the same time, the five-point smoothing algorithm preserves the characteristics of the original curve well, and effectively eliminates the existing interference components.

Set the equidistant nodes X_−n, X_−n+1,…, X₋₁, X₀, X₁,…, X_n−1, X_n on the experimental data as Y_−n, Y_−n+1,…, Y₋₁, Y₀, Y₁,…, Y_n−1, Y_n. For X_n, set the equidistance between the two nodes as h, and do the transformation $t=\frac{x-x_0}{h}$, then the original node is transformed into $t_{-n}=-n$, $t_{-n+1}=-n+1$,…, $t_{-1}=-1$, $t_0=0$, $t_1=1$,…, $t_n=n$.

Fit the obtained experimental data with m degree polynomial, and set the fitting polynomial as:

$$Y(t)=a_0+a_{1}t+a_2{t}^2+\cdots +a_m{t}^m$$

(3)

In Formula (3), the undetermined coefficients are determined using the least squares method, let:

$${\displaystyle \sum _{i=-n}^n{R}_i^2}={\displaystyle \sum _{i=-n}^n{[{\displaystyle \sum _{j=0}^m{a}_j{t}_i^j-}{Y}_i]}^2=\varphi (a_0,}{a}_1,\cdots a_m)$$

(4)

In order to make $\varphi (a_0,a_1,\cdots a_m)$ reach the minimum value, take its partial derivative with respect to $a_k(k=0,1,2,\cdots m)$ and make it equal to 0, the following equation can be obtained:

$${\displaystyle \sum _{i=-n}^n{Y}_i{t}_i^k={\displaystyle \sum _{j=0}^n{a}_j{\displaystyle \sum _{i=-n}^n{t}_i^{k+1}}}}$$

(5)

Formula (5) is called a normal system of formulas.

When n = 2 (that is five nodes) and m = 3, a normal system of equations will be obtained, from which $a_0,a_1,\cdots a_i,a_{it}$ is solved and brought into Formula (5). At the same time, set t = 0, 1, −1, 2, −2, the five-point smoothing formula can be obtained, as follows:

$${\stackrel{-}{Y}}_{-2}=(69Y_{-2}+4Y_{-1}-6Y_0+4Y_1-Y_2)/70$$

(6)

$${\stackrel{-}{Y}}_{-1}=(2Y_{-2}+27Y_{-1}+12Y_0-8Y_1+2Y_2)/35$$

(7)

$${\stackrel{-}{Y}}_0=(-3Y_{-2}+12Y_{-1}+17Y_0+12Y_1-3Y_2)/35$$

(8)

$${\stackrel{-}{Y}}_1=(2Y_{-2}-8Y_{-1}+12Y_0+27Y_1+2Y_2)/35$$

(9)

$${\stackrel{-}{Y}}_2=(-Y_{-2}+4Y_{-1}-6Y_0+4Y_1+69Y_2)/70$$

(10)

In Formulas (6)–(10), ${\stackrel{-}{Y}}_i$ is the smoothed value of $Y_i$. For equally spaced sampling points, only the sampled data $Y_i$ are used, regardless of the node $X_i$ and the size of the node spacing [22].

The five-point smoothing algorithm requires that the number of nodes is at least five nodes. When the number of nodes exceeds five, in order to calculate symmetrically, except for Formulas (6)–(10) at both ends, Formula (8) is used for the smoothing calculation. It is equivalent to using a different cubic least squares polynomial on each subinterval for the smoothing calculation [23].

3.2. Establishment of Torsional Vibration Calculation Model

In this paper, the shafting torsional vibration calculation model adopts the equivalent system of the shafting [24,25,26,27]. The equivalent system is based on the actual situation of the shafting system, which simplifies the shafting into discrete rigid lumped masses and elastic shaft segments without inertia for easy calculation. The transformation principle is that the equivalent system should be able to represent the torsional vibration characteristics of the actual shafting; the natural frequency of free vibration should be basically the same as the actual natural frequency; the mode shape of the torsional vibration is basically similar to the actual shafting’s mode shape. When the difference between the measured natural frequency and the calculated value is more than 5%, and the actual measurement is correct, the equivalent parameters should be adjusted; larger mass components, such as flywheels, flanges, and active and driven parts of elastic couplings or pneumatic clutches, etc., take their center of rotation plane as the concentration point of inertia. The inertia of the elastic member such as the shaft segment between them is equally divided and added to the concentration points at both ends.

For a torsional vibration system with n concentrated moments of inertia, under the action of an external excitation moment M with frequency $\omega$. For any lumped mass K, its vibration differential Formula [28] is:

$$\begin{array}{l}{I}_k{\stackrel{..}{\phi }}_k+C_k{\stackrel{.}{\phi }}_k+{\mu }_{k-1,k}(\stackrel{.}{{\phi }_k}-{\phi }_{k-1})+\\ {\mu }_{k,k+1}({\stackrel{.}{\phi }}_k-{\phi }_{k+1})+K_{k-1,k}({\phi }_k-{\phi }_{k-1})+\\ K_{k,k+1}({\phi }_k-{\phi }_{k+1})=M_k{e}^{i(\omega t+\alpha k)}\end{array}$$

(11)

In Formula (11), $I_k$ is the moment of inertia of the concentrated mass K, and the unit is $\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$.

${\phi }_k$, ${\stackrel{.}{\phi }}_k$, ${\stackrel{..}{\phi }}_k$ are the angular displacement, angular velocity, and angular acceleration of the concentrated mass K.

$C_k$ is the absolute damping coefficient of the concentrated mass K, the unit is $\mathrm{N}\cdot \mathrm{m}\cdot \mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$.

${\mu }_{k-1,k}$, ${\mu }_{k,k+1}$ are the relative damping coefficient of the connecting axis (k − 1)~k and k~(k + 1), the unit is $\mathrm{N}\cdot \mathrm{m}\cdot \mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$.

$K_{k-1,k}$, $K_{k,k+1}$ are the torsional stiffness of the connecting shafts (k − 1)~k and k~(k + 1), the unit is $\mathrm{N}\cdot \mathrm{m}\mathrm{/}\mathrm{r}\mathrm{a}\mathrm{d}$.

$M_k$ is the magnitude of the excitation moment acting on the concentrated mass K, the unit is $\mathrm{N}\cdot \mathrm{m}$.

$\omega$ is the circular frequency of the excitation torque.

$t$ is time, its unit is s.

${\alpha }_k$ is the initial phase of the excitation torque.

When the torsional vibration occurs, the total stress ${\tau }_{tot}$ on the shafting is the superposition value of the basic value of the torsional vibration stress ${\tau }_{bas}$ and the torsional amplitude value ${\tau }_{\begin{array}{l}amp\\ \end{array}}$. Its expression is:

$${\tau }_{tot}={\tau }_{bas}+{\tau }_{amp}$$

(12)

Table 1. Technical parameters of diesel engine.

Diesel Engine Specifications	Parameter
Number of cylinders (Line)	8
Cylinder diameter (mm)	320
Crank radius (mm)	220
Connecting rod length (mm)	860
Rated power (KW)	2206
Rated speed (r/min)	525
Minimum rated speed (r/min)	210
Single cylinder reciprocating mass (KG)	151.5
Crankshaft tensile strength (M·Pa)	600
Fire angle (Deg.)	360-180-270-450-90-630-540-0

gives the technical parameters of the diesel engine. The components of the shafting can be seen in Figure 1. According to the simplification principle of the equivalent system and the corresponding calculation formula, the lumped parameter method is used to build the shafting discrete vibration mathematical model for the shafting of a certain type of bulk carrier. The simplified equivalent system is shown in Figure 2:

In Figure 2, the pairs of black dots represent the lumped mass, and the horizontal lines between the adjacent lumped mass connecting lines represent the elastic link axes.

3.3. Steady-State Torsional Vibration Calculation

$${\tau }_{k,k+1}=A_1{\tau }_0$$

(13)

$${\tau }_0=\frac{i{\omega }^2{\displaystyle \sum _{k=1}^k{I}_k{a}_k^2}}{W_{k,k+1}}$$

(14)

In Formula (14), $W_{k,k+1}=\frac{\pi d^3}{16[1-{(\frac{d_0}{d})}^4]}$, $W_{k,k+1}$ is the section modulus of the $k,k+1$ axis segment.

$d_0$ and $d$ are the inner and outer diameters of the shaft, the unit is mm.

According to the equivalent system model of the shafting in Figure 2 and the equivalent parameters of the system in

Table 2. Shafting equivalent parameters.

No.	Speed Ratio	Moment of Inertia ($\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$)	Connecting Shaft No.	Torsional Flexibility ($\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{N}\cdot \mathrm{m}$)	Outer Diameter/Inner Diameter (mm)
1	0.300	1.711	1–2	8.242 × 10−9	110.0/67.0
2	0.300	0.378	2–3	0.0	0.0/0.0
3	1.000	0.936	3–4	2.101 × 10−7	110.0/0.0
4	1.000	0.184	4–5	1.367 × 10−8	160.0/100.0
5	1.000	0.623	5–6	2.041 × 10−5	0.0/0.0
6	1.000	14.900	6–7	5.000 × 10−7	0.0/0.0
7	1.000	4.854	7–8	1.172 × 10−8	250.0/0.0
8	1.000	31.137	8–9	1.728 × 10−8	250.0/0.0
9	1.000	31.137	9–10	1.728 × 10−8	250.0/0.0
10	1.000	31.137	10–11	1.728 × 10−8	250.0/0.0
11	1.000	31.137	11–12	1.728 × 10−8	250.0/0.0
12	1.000	31.137	12–13	1.728 × 10−8	250.0/0.0
13	1.000	31.137	13–14	1.728 × 10−8	250.0/0.0
14	1.000	31.137	14–15	1.728 × 10−8	250.0/0.0
15	1.000	31.137	15–16	1.128 × 10−8	250.0/0.0
16	1.000	7.342	16–17	1.049 × 10−8	250.0/0.0
17	1.000	245.585	17–18	1.082 × 10−6	0.0/0.0
18	1.000	20.929	18–19	1.082 × 10−6	0.0/0.0
19	1.000	56.486	19–20	2.067 × 10−8	215.0/0.0
20	1.000	20.897	20–21	0.0	0.0/0.0
21	1.135	20.155	21–22	0.0	0.0/0.0
22	1.135	44.904	22–23	8.161 × 10−8	160.0/55.0
23	1.135	2.573	23–24	0.0	0.0/0.0
24	1.135	2.619	24–25	8.161 × 10−8	160.0/55.0
25	1.135	2.573	25–26	0.0	0.0/0.0
26	4.484	16.070	26–27	1.365 × 10−7	320.0/0.0
27	4.484	2.781	27–28	4.481 × 10−7	260.0/0.0
28	4.484	3.370	28–29	1.353 × 10−6	320.0/0.0
29	4.484	231.690	/	/	/

, the torsional vibration stress of the shafting is calculated by Formulas (13) and (14) (only for the normal fire condition of the main engine). Figure 3 shows the relationship between the speed of the main engine and the torsional vibration stress of the intermediate shaft.

It can be seen from torsional vibration stress change of intermediate shaft with diesel engine speed in Figure 3. Under the normal ignition condition of the diesel engine, in the speed range of 220–525 r/min, the torsional vibration stress of the intermediate shaft first decreases and then increases with the increase in the diesel engine speed, and the maximum value of the torsional vibration stress is 5.199 $\mathrm{N}/\mathrm{m}{\mathrm{m}}^2$.

4. Test

4.1. Preparation before the Test

According to the torsional vibration test requirements of the ship shafting, select a suitable test position on the ship shafting. The torsional vibration test position is the intermediate shaft section of the propulsion shafting. After grinding and cleaning the shaft section, stick the coding belt on it and check the pulse number of the coding belt. Install the sensor-fixing bracket on the hull structure, fix the sensor directly above the coding belt through the connecting rod, and keep the extension line of the sensor passing through the center line of the shafting. The distance from the sensor to the surface of the code belt is about 1–2 mm. Connect the sensor to the corresponding channel of the data acquisition instrument, and set the channel parameters of the data acquisition instrument. The test equipment installation site is shown in Figure 4.

4.2. Test Results and Data Analysis

The composition of the test system is shown in Figure 5. In this test, a laser sensor (parameters of the laser sensor shown in

Table 3. Parameters of the laser sensor.

Technical Characteristics	Parameter
Diffuse Reflection: D = f(encoding) = f(Albedo)	1 mm D 5 mm (black and white)
Transmitter	LED IR
Output (square wave)/PNP/	TTL
Supply Voltage	5 VCC
Current Consumption	60 mA
Frequency	0 < F < 250 KHz
Power Adjustment	Big Potentiometer ¾ turn
Temperature Range	Limits: From −50 °C to +120 °C, long life: −5 + 80 °C

) is used to pick up the instantaneous speed of the shafting. The sampling frequency is set to 4096 Hz and the frequency resolution is 1 Hz.

When the ship was sailing normally and the diesel engine was in normal firing condition, the above-mentioned test system was used to conduct the deceleration torsional vibration test of the ship shafting. The speed of the diesel engine was reduced from 525 r/min to 220 r/min. At that point, 525 r/min was the full load speed and 220 r/min was the idle speed. Since the intermediate shaft and the diesel engine were connected through a gearbox, the speed ratio of the gearbox was 4.48, so the speed of the intermediate shaft will be reduced from 117 r/min to 49 r/min. Its instantaneous speed waveform is shown in Figure 6:

A partial amplification of the instantaneous speed of the shafting is performed, as shown in Figure 7. There are obvious burrs in the instantaneous speed signal of the shafting. The main reason for the formation of the burr is the irregular overlap of the coding blet, and the irregular signal is output during signal acquisition. The black and white stripes on the coding belt are of a fixed width. Due to the difference in shaft diameter, the lap joint of the coding belt cannot be perfectly connected after the coding belt is wound once. The black or white stripes will widen or narrow at the lap joint, as shown in Figure 8. The time for the laser sensor to pass the stripe is shorter or longer due to the narrowing or widening of the stripes. The width of the corresponding pulse signal is narrowed or widened, so that the instantaneous rotational speed suddenly increases or decreases.

Burrs can affect the accuracy of torsional vibration calculations. In order to obtain accurate torsional vibration calculation results, this paper uses a five-point smoothing algorithm to filter the instantaneous speed signal. The filtered shafting instantaneous speed signal is shown in Figure 9. After the instantaneous speed signal is filtered, it can be seen from the enlarged view of Figure 9 that the burrs are gone.

Record the number of black stripes on the coding belt installed on the shafting, that were the total number of pulses, n. According to the calculation formula of average angular velocity:

$$\omega =\frac{{\theta }_0}{\Delta t_i}$$

(15)

In Formula (15), ${\theta }_0$ is the angle between two pulses on the coding blet, ${\theta }_0=2\pi /n$; where t is the time interval between two adjacent pulses passing through the laser sensor, $\Delta t_i=t_{n+1}-t_n$.

According to Formula (15), calculate the average speed of the intermediate shaft. The variation relationship of the average speed with time is shown in Figure 10. The speed fluctuation of the shafting can be obtained by subtracting the average speed from the filtered instantaneous speed. Filter and integrate the rotational speed fluctuation signal, and slice the calculation results to obtain the displacement amplitudes of different harmonic torsion angles of the measured shafting, as shown in Figure 11:

Under the set working conditions, the diesel engine was tested for torsional vibration at a reduced speed, and its speed was reduced from 525 r/min to 220 r/min. After signal processing and calculation analysis, it can be seen from Figure 10 that when the diesel engine speed is 226 r/min, the torsional vibration of the intermediate shaft is the largest, and the amplitude of the maximum torsional vibration is 0.1202°.

Instantaneous torque formula:

$$T=\theta G{I}_P$$

(16)

In Formula (16), θ is the angle in radians and $G{I}_P$ is the torsional stiffness.

Torsional shear stress formula:

$$\tau =\frac{Tr}{I_P}$$

(17)

The maximum torsional shear stress occurs on the surface of the circular shaft, as shown in Formula (18):

$${\tau }_{\max }=\frac{TR}{I_P}$$

(18)

Let $W_P\equiv \frac{I_P}{R}$: $W_p$ is the torsional section coefficient.

Then:

$${\tau }_{\max }=\frac{T}{W_P}$$

(19)

From Formula (18), the maximum instantaneous torque of the intermediate shaft is calculated to be 4.6817 $\mathrm{K}\mathrm{N}·\mathrm{m}$. Substituting the instantaneous torque into Formula (19), the maximum torsional vibration stress of the intermediate shaft is 2.713 $\mathrm{N}/\mathrm{m}{\mathrm{m}}^2$. Combined with Figure 3, we can know that the maximum torsional vibration stress of the shafting does not exceed the maximum allowable stress. In other words, within the speed range of 220–525 r/min, the ship does not have a restricted speed zone.

5. Conclusions

• Model the propulsion shafting of a certain type of bulk carrier according to the traditional lumped parameter method. A discrete model of torsional vibration of the ship’s propulsion shafting is constructed, and the variation of the torsional vibration stress of the intermediate shaft with the speed of the diesel engine is calculated. Between 220–525 r/min, the torsional vibration stress of the intermediate shaft first decreases and then increases with the increase in the diesel engine speed, and reaches the maximum value when the speed is 525 r/min, and the maximum value of the torsional vibration stress is 5.199 $\mathrm{N}/\mathrm{m}{\mathrm{m}}^2$;
• Using the five-point smoothing algorithm to filter the instantaneous speed signal, the calculation result of the ship’s propulsion shafting torsional vibration test is more accurate;
• By comparing the measured results and theoretical calculation results. It can be seen that there is no restricted speed zone in the ship’s propulsion shafting within the speed range of 220–525 r/min. The calculation results have certain guiding significance for controlling the vibration of ship shafting and improving the safety of ships.