A finite element-based novel approach to undamped vibrational analysis of complex curved beams with arbitrary curvature using explicit interpolation functions

doi:10.1007/s10483-025-3315-6

Abstract

Abstract:

Curved beams with complex geometries are vital in numerous engineering applications, where precise vibration analysis is crucial for ensuring safe and effective designs. Traditional finite element methods (FEMs) often struggle to accurately represent the dynamic characteristics of these structures due to the limitations in their shape function approximations. To overcome this challenge, the current study introduces an innovative finite element (FE)-based technique for the undamped vibrational analysis of curved beams with arbitrary curvature, employing explicitly derived interpolation functions. Initially, the exact interpolation functions are developed for circular arc elements with the force method. These functions facilitate the creation of a highly accurate stiffness matrix, which is validated against the benchmark examples. To accommodate arbitrary curvature, a systematic transformation technique is established to approximate the intricate curves with a series of circular arcs. The numerical findings indicate that increasing the number of arc segments enhances accuracy, approaching the exact solutions. The analysis of free vibrations is conducted for both circular and non-circular beams. Mass matrices are derived using two methods: lumped mass and consistent mass, where the latter is based on the interpolation functions. The effectiveness of the proposed method is confirmed through the comparisons with the existing literature, demonstrating strong agreement. Finally, several practical cases involving beams with diverse curvature profiles are analyzed. Both natural frequencies and mode shapes are determined, providing significant insights into the dynamic behavior of these structures. This research offers a dependable and efficient analytical framework for the vibrational analysis of complex curved beams, with promising implications for structural and mechanical engineering.

Key words: arbitrary curvature, curved beam, interpolation function, free vibration, mode shape, finite element method (FEM)

2010 MSC Number:

O327

A. KESHMIRI, T. H. MOTTAGHI, A. R. MASOODI. A finite element-based novel approach to undamped vibrational analysis of complex curved beams with arbitrary curvature using explicit interpolation functions. Applied Mathematics and Mechanics (English Edition), 2025, 46(11): 2177-2198.

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Figures/Tables 26

Fig.?1

Fig.?2

Fig.?3

Fig.?4

Fig.?5

Fig.?6

Fig.?7

Fig.?8

Fig.?9

Table?1

Comparison of the results obtained from the proposed method with the exact solutions derived from Castigliano’s method"

$n$	Present study		Explicit answer		Error $v /%$	Error $R x /%$
$n$	$v /mm$	$R x /t$	$v /mm$	$R x /t$	Error $v /%$	Error $R x /%$
2	$− 9.620 3$	13.706	-10.497	17.387	8.35	21.20
4	$− 10.046 0$	16.475	-10.497	17.387	4.30	5.25
6	$− 10.348 0$	16.976	-10.497	17.387	1.42	2.36
8	$− 10.446 0$	17.153	-10.497	17.387	0.49	1.35
10	$− 10.483 0$	17.235	-10.497	17.387	0.13	0.87
12	$− 10.498 0$	17.281	-10.497	17.387	0.01	0.61

Table?1

Fig.?10

Table?2

Fig.?11

Table?3

Fig.?12

Fig.?13

Table?4

First four dimensionless natural frequencies of the circular beam under S-S and C-C support conditions"

$φ /(°)$	S-S				C-C
$φ /(°)$	$λ 1$	$λ 2$	$λ 3$	$λ 4$	$λ 1$	$λ 2$	$λ 3$	$λ 4$
120	6.857	16.95	32.52	42.42	11.57	21.81	40.24	42.42
150	3.833	10.44	20.40	31.78	6.851	13.98	25.56	35.53
180	2.256	6.845	13.73	22.11	4.337	9.416	17.38	25.94
210	1.359	4.680	9.698	15.98	2.874	6.612	12.40	19.04
240	0.816	3.290	7.086	11.94	1.971	4.790	9.157	14.37
270	0.473	2.355	5.307	9.152	1.389	3.551	6.944	11.10
285	0.350	2.001	4.625	8.075	1.177	3.079	6.094	9.838
300	0.250	1.704	4.046	7.156	1.003	2.680	5.372	8.755
315	0.168	1.452	3.551	6.365	0.860	2.341	4.754	7.822
330	0.101	1.238	3.125	5.681	0.742	2.051	4.222	7.013
340	0.063	1.113	2.874	5.276	0.676	1.881	3.907	6.533
350	0.030	1.000	2.645	4.905	0.617	1.727	3.620	6.094

Table?4

Fig.?14

Fig.?15

Table?5

First four dimensionless natural frequencies of the radical and third-degree parabolic beams under S-S and C-C support conditions"

Geometry of arch	Mode	S-S	C-C
Radical,	1	23.967	39.833
$y = {x, 0 ⩽ x ⩽ 4, 8 − x, 4 < x ⩽ 8$	2	43.855	48.640
	3	69.894	79.335
	4	85.791	103.20
Parabolic,	1	44.825	75.947
$y = 12 × {(x 3 + x), − 1 ⩽ x ⩽ 1, ((2 − x) 3 + (2 − x)), 1 < x ⩽ 3$	2	49.628	76.502
	3	184.85	217.03
	4	185.12	217.09

Table?5

Fig.?16

Fig.?17

Fig.?18

Fig.?19

Table?6

Fig.?20

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Support condition	Mode	Present study	Ref. [19]	Ref. [41]	Ref. [57]	Error/%
S-S	1	29.192	29.285	29.306	29.264	0.317 6
	2	33.255	33.321	33.243	33.280	0.079 1
	3	66.721	67.202	67.123	66.935	0.545 1
	4	79.796	80.049	79.950	79.817	0.178 5
C-C	1	36.694	36.716	36.657	36.674	0.031 8
	2	42.250	42.278	42.289	42.228	0.035 5
	3	82.160	82.361	82.228	81.930	0.015 8
	4	84.452	84.565	84.471	84.328	0.003 2

Geometry of arch	Mode	Present study	Ref. [41]	Error/%
Elliptic	1	34.860	34.892	0.091 7
	2	56.727	56.766	0.068 7
	3	81.265	81.420	0.190 4
	4	123.91	124.29	0.305 7
Sinusoid	1	56.064	56.083	0.033 9
	2	66.035	66.047	0.018 2
	3	113.32	113.41	0.079 4
	4	179.03	179.26	0.128 3

Mode	S-S	C-C
1	2.458 0	5.518 1
2	6.515 8	9.861 4
3	11.849	14.945
4	14.846	15.729