This research has experimentally investigated the influence on vibration characteristics of thin cantilever cylindrical shell (TCS) with hard coating under cantilever boundary condition. Firstly, the theoretical model of TCS with hard coating is established to calculate its natural frequencies and modal shapes so as to roughly understand vibration characteristic of TCS when it is coated with hard coating material. Then, by considering its nonlinear stiffness and damping influences, an experiment system is established to accurately measure vibration parameters of the shell, and the corresponding test methods and identification techniques are also proposed. Finally, based on the measured data, the influences on natural frequencies, modal shapes, damping ratios, and vibration responses of TCS with hard coating are analyzed and discussed in detail. It can be found that hard coating can play an important role in vibration reduction of TCS, and for the most modes of TCS, hard coating will result in the decrease of natural frequencies, but the decreased level is not very big, and its damping effects on the higher frequency range of the shell are weak and ineffective. Therefore, in order to make better use of this coating material, we must carefully choose the concerned antivibration frequency range of the shell; otherwise it may lead to some negative effects.
Thin cylindrical shell (TCS) has long been an important structural component due to its high stiffness to weight and strength to weight ratios, which is widely used in the engineering fields, such as aircraft casings, pipes and ducts, rotary drums in granulator, and aircraft engine [
The hard coating is a kind of coating materials prepared by the metal, ceramic, or their mixtures, which is used as surface treatment for antifriction, antierosion, vibration reduction, and other engineering application fields [
At present, great efforts have been made to study vibration character of thin-walled structures with hard coating, such as beams, plates, and shells including the corresponding modeling and analysis techniques, antivibration design, and evaluation methods. For example, Ivancic et al. [
Besides, many scholars also studied nonlinear vibration analysis methods for composite cylindrical shells, and these analytical techniques can provide important reference for studying vibration characteristics of the shell with coating materials. For example, Li et al. [
However, most of researches done by the above scholars and researchers are mainly based on theory or simulation, experimental studies on the influence on vibration characteristics of TCS with hard coating are still scarce, and as a lack of the related test conclusion, theoretical analysis result cannot be effectively verified, let alone validating some advanced composite shell theories. Besides, due to the complicated vibrational properties of TCS, such as very closed modes, the small vibration levels and abundant local vibration, especially when the hard coating material is applied on its surface, such composite shell structure is turned into a nonlinear system with variable stiffness and damping under different excitation levels, which would inevitably increase the experimental difficulties. Up until now, either from the perspective of test accuracy, or from the test efficiency, these test problems have not been well solved. In order to meet the needs of engineering applications of such damping material, it is necessary to adopt some experimental techniques to study vibration characteristics of TCS with hard coating.
In this research, the influence on vibration characteristics of thin cylindrical shell with hard coating has been investigated experimentally under cantilever boundary condition. Firstly in Section
In this section, in order to deeply understand vibration characteristic of TCS with hard coating, the natural frequencies and modal shapes of coated TCS are calculate based on the established theoretical model. Although this model can not calculate damping results and also may inevitably contain some calculation errors, it is helpful for us to determine measured frequency range, build experimental model, understand geographic distributions of some nodes or nodal lines, and so forth.
The TCS studied in this research is made of structural steel with the length of 100 mm and an average thickness of 2 mm, as shown in Figure
Dimension parameters of thin cylindrical shell.
Dimension name | Dimension values |
---|---|
Length (mm) | 100 |
Thickness (mm) | 2 |
Internal radius (mm) | 142 |
External radius (mm) | 144 |
Extension edge radius (mm) | 150 |
Thickness of extension edge (mm) | 3 |
Material parameters of TCS and hard coating.
Name | Elastic modulus (Pa) | Poisson’s ratio | Density (kg/m3) |
---|---|---|---|
Structural steel |
|
0.3 | 7850 |
NiCrAlCoY + YSZ |
|
0.3 | 4176 |
TCS coated with and without hard coating and its clamping-ring used in cantilever boundary condition.
The theoretical model of TCS with hard coating is shown in Figure
The theoretical model of TCS with hard coating.
The cross section of TCS with hard coating is shown in Figure
Schematic of cross section of TCS with hard coating.
The strain-displacement relations of TCS with hard coating can be expressed as
Assuming that the TCS is under the plane stress state, then on the basis of generalized Hooke law its strain-stress relations can be described as
The hard coating and substrate are all isotropic materials, so the elastic matrix of them can be defined as
The stress resultants and bending moment resultants of TCS with hard coating can be expressed as
Substituting (
Based on the theoretical model above, the strain energy
Assume that, in the
This is because the displacement admissible function can be obtained from the Gram-Schmidt process. When the first polynomial
In order to finish the simulation work of cantilever boundary condition of TCS, as seen in Figure
After that, the orthogonal polynomial functions can be settled. And we chose
According to Rayleigh-Ritz method, the Lagrange function
Then, take the partial of shape coefficients
Furthermore, the equation which contains the shape coefficients of
Besides, in (
It should be noted that each submatrix in the above matrixes is
Finally, repeat the above steps, and the natural frequencies and mode shapes of cantilever cylindrical shell with and without hard coating can be gradually calculated out based on the self-written Matlab program, which are listed in Table
The calculated natural frequencies and mode shapes of cantilever cylindrical shell with and without hard coating.
Modal order | TCS without coating | TCS with hard coating | Frequency differences |
||
---|---|---|---|---|---|
Natural frequency |
Mode shape |
Natural frequency |
Mode shape |
||
1 | 923.1 |
|
918.4 |
|
0.5 |
|
|||||
2 | 985.3 |
|
972.3 |
|
1.3 |
|
|||||
3 | 1064.9 |
|
1057.8 |
|
0.7 |
|
|||||
4 | 1194.6 |
|
1171.8 |
|
1.9 |
|
|||||
5 | 1326.5 |
|
1306.3 |
|
1.5 |
|
|||||
6 | 1378.2 |
|
1362.4 |
|
1.1 |
|
|||||
7 | 1650.8 |
|
1644 |
|
0.4 |
|
|||||
8 | 2021.5 |
|
2011.9 |
|
0.5 |
In Section
On the one hand, due to light weight, closed modes, low vibration level, and complicated local vibration of TCS, traditional accelerometer will bring added mass and stiffness to the tested shell [
The disadvantages of different vibration excitation devices for vibration test of TCS with or without hard coating.
Excitation device | Vibration parameters of TCS | Disadvantage | |||
---|---|---|---|---|---|
Natural frequency | Modal shape | Damping ratio | Vibration response | ||
|
√ | √ | × | × | Pulse excitation level cannot be precisely controlled and the excitation force varies for each measurement, and double hit often leads to test errors in damping and vibration response measurement. |
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× | × | × | × | The related force sensor will bring added mass and stiffness to TCS, which will severely affect test results of damping, natural frequency, and so forth. |
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√ | √ | × | × | The excitation energy of piezoelectric ceramic exciter is often insufficient for TCS, which will result in poor response signal with low level of signal noise ratio. |
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√ | √ | √ | √ | Excitation frequencies are not that high, often limited to 1 Hz~3000 Hz, and the test procedures are often complicated. |
The schematic of test system of vibration characteristics of TCS with or without hard coating.
In these devices, LMS SCADAS Mobile Front-End and Dell notebook computer are responsible for recording and saving response signal from laser Doppler vibrometer. Dell notebook computer with Intel Core i7 2.93 GHz processor and 4 G RAM is used to operate LMS Test. Lab 12A software and store measured data. For the natural frequency test, sine sweep excitation is conducted with a closed loop control via accelerometer on the countertop of the vibration shaker, and each natural frequency of TCS with or without hard coating can be precisely determined through each resonant peak in frequency domain. For the damping ratio test, the sliding-envelope method in time domain is employed, and through fitting the attenuation envelope under different sliding-windows, we can identify each damping ratio of TCS from the corresponding attenuation signal of each mode. For the modal shape test, 45° fixed mirror is used to change the light path of laser Doppler vibrometer, and DC rotary motor is to drive the 45° rotation mirror to complete a set of cross-sectional scan with 360° circumferential coverage for the tested shell, so that modal shapes data at certain mode can be obtained efficiently. For random vibration response test, the similar test technique is employed to get circumferential scan data of certain sections of the shell, and the corresponding extracting method is proposed to get the response data of the concerned measuring points (we will describe these test methods in detail in Section
In this section, on the basis of considering nonlinear stiffness and damping properties of hard coating, the experimental test is divided into two phases, that is, Phase I: no coating TCS measurement and Phase II: coated TCS measurement, and the corresponding test methods and identification techniques which are suitable for the thin-walled shell with or without hard coating are also proposed, as seen in the following test procedures.
The time domain signal involving the 5th natural frequency of the tested shell is showed in Figure
The 5th natural frequency of TCS obtained by different FFT processing techniques.
The raw sweep signal
The spectrum by direct FFT
The spectrum by small-segment FFT
The 3rd time-attenuation signal of TCS without hard coating obtained by sliding-envelope method.
Attenuation envelope-curves of the 3rd damping under different sliding-windows obtained by sliding-envelope method.
Window 1
Window 2
Window 3
Window 4
Assuming
At this moment, the concerned scanning time
Then, the time width criterion which is mainly used for the extraction operation of laser scanning data can be established and expressed as
Finally, based on the time width criterion, we can extract vibration response data corresponding to the different measuring points. For example, for
In the measurement, we use the following random excitation spectrum, as seen in Figure
The setting parameters used in random vibration response measurement when the total root mean square value is 0.45 g.
Order | Frequency (Hz) | Left slope (dB/Oct) | Acceleration (g2/Hz) | Right slope (dB/Oct) | High-abort (dB) | Hi-alarm (dB) | Low-alarm (dB) | Low-abort (dB) |
---|---|---|---|---|---|---|---|---|
1 | 1 | 3 | 6 | 3 | −3 | −6 | ||
2 | 100 | 3 | 0.0001 | 6 | 3 | −3 | −6 | |
3 | 1500 | 0.0001 | −3 | 6 | 3 | −3 | −6 | |
4 | 2200 | −3 | 6 | 3 | −3 | −6 |
Random excitation spectrum used in random vibration response measurement.
In this section, based on the established experiment system in Section
Firstly in Phase I when the TCS is not coated with hard coating, several tightening torques are chosen to tighten eight M8 bolts in the preexperiment, and after the repeated comparisons and experiments, the torque value of 50 Nm is finally determined. Then, when the tested shell is well constrained, we use point 1, point 2, and point 3 to get response signal, which are 120° with each other and in the same cross section of the shell and the axial distance from this section to free end of TCS is about 3 mm. Next, the different sweep rates, such as 1 Hz/s, 2 Hz/s, and 4 Hz/s, are compared in the measurement, and the rate of 1 Hz/s is finally employed to accurately get natural frequencies by sine sweep excitation technique. At last, by using the sliding-envelope method and laser rotating scanning method, we can obtain frequencies, shapes, damping, and vibration response results of TCS with high accuracy and efficiency.
Then in Phase II when the TCS is coated with hard coating, we use the same torque tightening of 50 Nm to effectively tighten eight M8 bolts of the shell, and with the same test techniques in Phase I, we can get the corresponding vibration parameters at the same positions under different excitation levels, so that we can investigate nonlinear stiffness and damping properties of the shell coated with hard coating.
For Phase I, the following setups and parameters are chosen to get frequency and damping results: (I) the constant excitation level of 1 g; (II) sweep rate of 1 Hz/s; (III) frequency resolution of 0.5 Hz; (IV) Hanning window for sweep response signal with downward sweep direction; (V) frequency range of 0–2048 Hz. Besides, the following parameters are set to get mode shapes and vibration responses: (I) excitation level of 1 g to get shape data; (II) the total root mean square value of 0.45 g in the random excitation; (III) rectangular window for stable response signal with Hanning window for random signal; (IV) sampling frequency of 8192 Hz; (V) rotated scan speed of 2 r/min; (VI) random excitation time not less than 10 s.
For Phase II, coated TCS measurement, the setups are basically similar to the above parameters determined in Phase I except for the excitation levels, which need to be changed to measure the nonlinear stiffness and damping properties of the shell coated with hard coating. Because it is hard for vibration shaker to control large amplitude when the excitation frequencies are low (which would lead to overload phenomenon), the lower excitation levels such as 0.5 g, 1 g, 2 g, 4 g, and 6 g are chosen to get the first four mode parameters, while for the rest of the modes, the larger excitation levels, such as 1 g, 2 g, 4 g, 8 g, 12 g, are used to get the concerned results. It should be noted that in the mode shape measurement, each mode shape is assembled from two sets of cross-sectional scans, one is in the section which includes point 1, point 2, and point 3, and the other is about 25 mm to the clamped end of the shell, which is restricted by the height of DC rotary motor itself, but does not affect the test results when the number of axial half-waves
The first four natural frequency, damping, and shape results of cantilever cylindrical shell coated with and without hard coating under different excitation levels are listed in Tables
The first 4 natural frequencies of cantilever cylindrical shell coated with and without hard coating under different excitation levels.
Modal order | No coating state |
0.5 g level |
Effect degree |
1 g level |
Effect degree |
2 g level |
Effect degree |
4 g level |
Effect degree |
6 g level |
Effect degree |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 898.5 | 893.0 | −0.6 | 892.0 | −0.7 | 892.0 | −0.7 | 891.5 | −0.8 | 891.5 | −0.8 |
2 | 970.5 | 951.5 | −2.0 | 950.5 | −2.1 | 950.0 | −2.1 | 949.0 | −2.2 | 947.5 | −2.4 |
3 | 1045.0 | 1037.0 | −0.8 | 1036.0 | −0.9 | 1034.5 | −1.1 | 1033.0 | −1.2 | 1032.0 | −1.3 |
4 | 1152.5 | 1144.5 | −0.7 | 1144.0 | −0.7 | 1142.5 | −0.9 | 1138.0 | −1.3 | 1135.0 | −1.5 |
The 5th to 8th natural frequencies of cantilever cylindrical shell coated with and without hard coating under different excitation levels.
Modal order | No coating state |
1 g level |
Effect degree |
2 g level |
Effect degree |
4 g level |
Effect degree |
8 g level |
Effect degree |
12 g level |
Effect degree |
---|---|---|---|---|---|---|---|---|---|---|---|
5 | 1292.0 | 1289.5 | −0.2 | 1289.5 | −0.2 | 1289.0 | −0.2 | 1288.5 | −0.3 | 1287.5 | −0.3 |
6 | 1309.5 | 1307.0 | −0.2 | 1306.0 | −0.3 | 1306.0 | −0.3 | 1306.0 | −0.3 | 1305.5 | −0.3 |
7 | 1606.5 | 1604.5 | −0.1 | 1604.5 | −0.1 | 1604.0 | −0.2 | 1603.0 | −0.2 | 1602.0 | −0.3 |
8 | 1990.0 | 1988.0 | −0.1 | 1988.0 | −0.1 | 1987.5 | −0.1 | 1987.0 | −0.2 | 1986.5 | −0.2 |
The first 4 damping ratios of cantilever cylindrical shell coated with and without hard coating under different excitation levels.
Modal order | No coating state |
0.5 g level |
Effect degree |
1 g level |
Effect degree |
2 g level |
Effect degree |
4 g level |
Effect degree |
6 g level |
Effect degree |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.26 | 0.55 | 111.5 | 0.51 | 96.2 | 0.50 | 92.3 | 0.39 | 50.0 | 0.38 | 46.2 |
2 | 0.17 | 0.48 | 182.4 | 0.48 | 182.4 | 0.43 | 152.9 | 0.39 | 129.4 | 0.36 | 111.8 |
3 | 0.25 | 0.47 | 88.0 | 0.26 | 4.0 | 0.24 | −4.0 | 0.31 | 24.0 | 0.34 | 36.0 |
4 | 0.21 | 0.18 | −14.3 | 0.27 | 28.6 | 0.30 | 42.9 | 0.37 | 76.2 | 0.40 | 90.5 |
The 5th to 8th damping ratios of cantilever cylindrical shell coated with and without hard coating under different excitation levels.
Modal order | No coating state |
1 g level |
Effect degree |
2 g level |
Effect degree |
4 g level |
Effect degree |
8 g level |
Effect degree |
12 g level |
Effect degree |
---|---|---|---|---|---|---|---|---|---|---|---|
5 | 0.23 | 0.35 | 52.2 | 0.27 | 17.4 | 0.25 | 8.7 | 0.22 | −4.3 | 0.17 | −26.1 |
6 | 0.34 | 0.36 | 5.9 | 0.38 | 11.8 | 0.39 | 14.7 | 0.41 | 20.6 | 0.45 | 32.4 |
7 | 0.19 | 0.22 | 15.8 | 0.19 | 0 | 0.16 | −15.8 | 0.19 | 0 | 0.15 | −21.1 |
8 | 0.30 | 0.34 | 13.3 | 0.33 | 10.0 | 0.30 | 0 | 0.19 | −36.7 | 0.20 | −33.3 |
The first 4 mode shapes of cantilever cylindrical shell coated with and without hard coating under different excitation levels.
Modal order | No coating state |
0.5 g level |
1 g level |
2 g level |
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1 |
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2 |
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3 |
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4 |
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The 5th to 8th mode shapes of cantilever cylindrical shell coated with and without hard coating under different excitation levels.
Modal order | No coating state |
1 g level |
2 g level |
4 g level |
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5 |
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6 |
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7 |
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8 |
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The scattergrams of natural frequencies of cantilever cylindrical shell coated with and without hard coating under different excitation levels.
The scattergrams of damping ratios of cantilever cylindrical shell coated with and without hard coating under different excitation levels.
The measured time waveforms of cantilever cylindrical shell without hard coating at 3 measuring points.
Point 1
Point 2
Point 3
The measured time waveforms of cantilever cylindrical shell with hard coating at 3 measuring points.
Point 1
Point 2
Point 3
The measured PSD functions of cantilever cylindrical shell without hard coating at 3 measuring points.
Point 1
Point 2
Point 3
The measured PSD functions of cantilever cylindrical shell with hard coating at 3 measuring points.
Point 1
Point 2
Point 3
In this section, on the basis of the measured results in Section
Because we have calculated the resulting natural frequencies of TCS under the two coating states by the self-written Matlab program in Section
The calculated natural frequencies and mode shapes of cantilever cylindrical shell with and without hard coating and their errors.
Modal order | No coating state | Coating state | ||||
---|---|---|---|---|---|---|
Measured |
Calculated |
Errors |
Measured |
Calculated |
Errors |
|
1 | 898.5 | 923.1 | 2.7 | 893.0 | 918.4 | 2.8 |
2 | 970.5 | 985.3 | 1.5 | 951.5 | 972.3 | 2.2 |
3 | 1045.0 | 1064.9 | 1.9 | 1037.0 | 1057.8 | 2.0 |
4 | 1152.5 | 1194.6 | 3.7 | 1144.5 | 1171.8 | 2.4 |
5 | 1292.0 | 1326.5 | 2.7 | 1289.5 | 1306.3 | 1.3 |
6 | 1309.5 | 1378.2 | 5.2 | 1307.0 | 1362.4 | 4.2 |
7 | 1606.5 | 1650.8 | 2.8 | 1604.5 | 1644 | 2.5 |
8 | 1990.0 | 2021.5 | 1.6 | 1988.0 | 2011.9 | 1.2 |
The relation curves of the measured and calculated natural frequencies of cantilever cylindrical shell with and without hard coating.
Although the calculated frequency results are inaccurate, we can still make good use of their changing trends. From the relation curves of the measured and calculated frequency results in Figure
This research has investigated the influence on vibration characteristics of cantilever cylindrical shell with hard coating experimentally, and the theoretical model of TCS with hard coating is also established to roughly master vibration characteristics of shell structure. Based on the calculated and experimental results, the following conclusions can be drawn: Hard coating will turn shell structure into a nonlinear system with variable stiffness and damping under different excitation levels, and for the most modes of cantilever cylindrical shell, hard coating will result in the decrease of natural frequencies, but the decreased level is not very big, which is within the range of 0.1%~2.4%. Beside, as the hard coating material is very thin, the mode shapes of the shell cannot be easily changed by such coating. For some modes of cantilever cylindrical shell, hard coating will increase their damping obviously, while for a small part of modes of the shell, hard coating will decrease the damping results, which is within the range of 4%~36.7%. Besides, with the increase of excitation levels, damping effects of hard coating are becoming weak, especially for high order damping ratios of the shell. Hard coating can play an important role in vibration reduction of cantilever cylindrical shell based on the measured random vibration response data in time and frequency domain, respectively. But it is found out that its damping effects on the higher frequency range of the shell are weak and ineffective. Therefore, in order to make better use of this coating material, we must carefully choose the concerned antivibration frequency range of the shell; otherwise it may lead to some negative effects.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was supported by the National Natural Science Foundation of China Grant no. 51375079, the National Natural Science Foundation of China Grant no. 51505070, and the Fundamental Research Funds for the Central Universities of China Grant nos. N150304011 and N160313002.