Theoretical and experimental investigation on vibration of bolted-flange-joined conical-cylindrical shells

doi:10.1007/s10483-025-3261-8

Abstract

Abstract:

This study investigates the vibration characteristics of bolted-flange-joined conical-cylindrical shells (BFJCCSs) through both theoretical analysis and experimental testing. The proposed model incorporates the pressure distribution within the bolted joint and accounts for the flange effect. The energy expressions for the conical and cylindrical shells are derived from Donnell's shell theory, while those for the flanges are obtained from the Euler-Bernoulli beam theory. The Lagrange equation is used to derive the dynamic equation, and the experimental studies on the BFJCCS are conducted to validate the accuracy of the model. Subsequently, the comprehensive effects of bolt loosening and bolt number on the frequency parameters are analyzed. Additionally, the effects of the flange dimensions and cone angle on the vibration behavior of the BFJCCS are discussed. In particular, the dynamic differences between the welded conical-cylindrical shell (WCCS) and BFJCCS are investigated. It is found that compared with the WCCS, the fundamental frequency of the BFJCCS is reduced by 7.6%, and the corresponding modal damping ratio is reduced by 21.0%. However, the high-order frequencies of the BFJCCS are higher than those of the WCCS, accompanied by a higher modal damping ratio. Compared with the bolt loosening degree, the bolt number has a more significant effect on frequencies. As the bolt number decreases, the impact of the bolt loosening degree diminishes gradually.

Key words: bolted-flange-joined, conical-cylindrical shell, vibration characteristic, bolt loosening, experiment

2010 MSC Number:

O327

Chunhao ZHANG, Qingdong CHAI, Changyuan YU, Wuce XING, Yanqing WANG. Theoretical and experimental investigation on vibration of bolted-flange-joined conical-cylindrical shells. Applied Mathematics and Mechanics (English Edition), 2025, 46(6): 1049-1068.

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Figures/Tables 24

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 1

The parameters of the BFJCCS specimen"

$E$ /GPa	$ρ$ / $(kg ⋅ m − 3)$	$L co$ /mm	$L cy$ /mm	$R co$ /mm	$R cy$ /mm	$h$ /mm	$d f$ /mm	$h f$ /mm	$α$ /rad	$θ 0$ /rad
204	7 850	198.3	172	71.85	171	2	32	3.5	$π / 6$	$π / 36$

Table 1

Fig. 7

Fig. 8

Fig. 9

Table 2

Identified maximum spring stiffness corresponding to different tightening torques"

Tightening torque/(N·m)	$K u / (N ⋅ m − 1)$	$K v / (N ⋅ m − 1)$	$K w / (N ⋅ m − 1)$	$K θ$ /(N·m·rad^-1)
5	$5.62 × 108$	$2.41 × 108$	$1.01 × 108$	$2.51 × 108$
10	$6.31 × 108$	$2.55 × 108$	$1.13 × 108$	$3.52 × 108$
15	$6.65 × 108$	$2.76 × 108$	$1.21 × 108$	$4.51 × 108$
20	$7.01 × 108$	$3.02 × 108$	$1.31 × 108$	$5.03 × 108$

Table 2

Fig. 10

Table 3

Convergence of the first four natural frequencies of the BFJCCS"

Order	$M = 0$	$M = 1$	$M = 2$	$M = 3$	$M = 4$	$M = 5$
$f 1$ /Hz	832.3	351.3	340.0	337.2	335.4	335.2
$f 2$ /Hz	1 078.2	572.8	560.1	558.4	556.2	556.0
$f 3$ /Hz	1 067.8	808.1	797.0	795.6	792.5	792.3
$f 4$ /Hz	1 329.3	1 164.4	1 142.6	1 137.1	1 132.7	1 132.0

Table 3

Fig. 11

Table 4

Natural frequencies of the BFJCCS and WCCS"

Order	Experiment		Theory
Order	WCCS	BFJCCS	Considering flanges	Neglecting flanges
$f 1$ /Hz	362.5	335.1	335.4	273.1
$f 2$ /Hz	425.0	539.6	556.2	407.0
$f 3$ /Hz	732.3	809.0	792.5	646.9
$f 4$ /Hz	1 006.3	1 095.0	1 132.7	955.4

Table 4

Table 5

The first four mode shapes of the BFJCCS and WCCS"

Mode	$n = 2$	$n = 3$	$n = 4$	$n = 5$
Theory
Experiment
WCCS

Table 5

Fig. 12

Table 6

Comparison of experimental modal damping ratios between the BFJCCS and WCCS"

Order	$ξ 1$ /%	$ξ 2$ /%	$ξ 3$ /%	$ξ 4$ /%
WCCS	1.24	0.10	0.05	0.01
BFJCCS	0.98	0.15	0.12	0.11

Table 6

Table 7

Comparisons of natural frequencies between experiment and theory with different bolt loosening degrees"

Tightening torque/(N·m)		Natural frequency
Tightening torque/(N·m)		$f 1$	$f 2$	$f 3$	$f 4$
	Experiment/Hz	329.0	530.4	800.2	1 086.4
5	Theory/Hz	330.4	550.5	792.2	1 129.1
	Error/%	0.0	3.8	1.0	3.9
	Experiment/Hz	332.0	535.3	804.4	1 092.0
10	Theory/Hz	332.5	552.6	792.3	1 131.0
	Error/%	0.2	3.2	1.5	3.6
	Experiment/Hz	333.9	536.5	805.7	1 092.5
15	Theory/Hz	333.9	554.4	792.5	1 131.9
	Error/%	0.0	3.3	1.6	3.6
Note that the error in the table is defined as the absolute value of the difference between the theoretical and experimental results divided by the experimental results, multiplied by 100%

Table 7

Table 8

Natural frequencies of experiment and theory with different bolt numbers and bolt loosening degrees"

$N$	Tightening torque/(N·m)		Natural frequency
$N$	Tightening torque/(N·m)		$f 1$	$f 2$	$f 3$	$f 4$
16		Experiment/Hz	335.1	539.6	809.0	1 095.0
	20	Theory/Hz	335.4	556.2	796.5	1 132.7
		Error/%	0.0	3.1	1.5	3.4
		Experiment/Hz	329.0	530.4	800.2	1 086.4
	5	Theory/Hz	330.4	550.5	792.2	1 125.1
		Error/%	0.0	3.8	1.0	3.6
12		Experiment/Hz	314.3	526.1	800.8	1 070.0
	20	Theory/Hz	322.7	548.9	792.5	1 127.4
		Error/%	2.7	4.3	1.0	5.4
		Experiment/Hz	313.1	521.2	796.5	1 060.8
	5	Theory/Hz	318.9	543.4	788.7	1 121.6
		Error/%	1.9	4.2	1.0	5.7
8		Experiment/Hz	313.7	510.9	790.0	1 054.1
	20	Theory/Hz	318.1	539.0	787.8	1 107.9
		Error/%	1.4	5.5	0.3	5.1
		Experiment/Hz	311.9	505.4	785.5	1 048.6
	5	Theory/Hz	315.4	533.7	782.6	1 102.7
		Error/%	1.1	5.6	0.4	5.2
4		Experiment/Hz	302.7	491.9	778.8	1 033.0
	20	Theory/Hz	309.0	522.6	785.0	1 044.1
		Error/%	2.1	6.2	0.8	2.6
		Experiment/Hz	302.1	490.7	778.2	1 031.8
	5	Theory/Hz	308.6	520.9	783.7	1 041.8
		Error/%	0.2	6.1	0.7	0.1
Note that the error in the table is defined as the absolute value of the difference between the theoretical and experimental results divided by the experimental results, multiplied by 100%

Table 8

Table 9

Natural frequencies of the BFJCCS with different flange widths (hf=3.5 mm)"

Order	$d f = 15$ mm	$d f = 20$ mm	$d f = 25$ mm	$d f = 30$ mm	$d f = 35$ mm	$d f = 40$ mm
$f 1$ /Hz	287.6	299.4	313.5	329.0	345.1	361.0
$f 2$ /Hz	523.8	536.7	546.8	554.0	558.8	561.8
$f 3$ /Hz	764.2	782.9	790.7	792.9	790.6	782.2
$f 4$ /Hz	1 097.8	1 126.1	1 138.8	1 137.1	1 121.6	1 081.9

Table 9

Table 10

Natural frequencies of the BFJCCS with different flange thicknesses (df=32 mm)"

Order	$h f = 1$ mm	$h f = 2$ mm	$h f = 3$ mm	$h f = 4$ mm	$h f = 5$ mm	$h f = 6$ mm
$f 1$ /Hz	313.6	324.5	332.2	338.5	344.0	349.0
$f 2$ /Hz	543.9	550.5	554.6	557.7	560.3	562.4
$f 3$ /Hz	776.0	780.9	788.5	796.0	801.0	804.3
$f 4$ /Hz	1 082.9	1 083.0	1 114.9	1 146.9	1 152.9	1 154.1

Table 10

Table 11

Natural frequencies and mode shapes of the BFJCCS with positive cone angles"

Cone angle/(°)		$n = 2$	$n = 3$	$n = 4$	$n = 5$
20	Natural frequency/Hz	290.1	387.0	563.6	859.9
20	Mode shape
30	Natural frequency/Hz	330.4	550.5	792.2	1 129.1
30	Mode shape
45	Natural frequency/Hz	603.0	1 022.4	1 122.6	1 136.6
45	Mode shape

Table 11

Table 12

Natural frequencies and mode shapes of the BFJCCS with negative cone angles"

Cone angle/(°)		$n = 2$	$n = 3$	$n = 4$	$n = 5$
$− 20$	Natural frequency/Hz	327.5	235.5	224.1	283.5
$− 20$	Mode shape
$− 30$	Natural frequency/Hz	322.3	194.9	173.2	234.8
$− 30$	Mode shape
$− 45$	Natural frequency/Hz	240.7	143.2	143.8	188.1
$− 45$	Mode Shape

Table 12

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