Free vibration and transient response of double curved beams connected by intermediate straight beams

doi:10.1007/s10483-025-3197-8

Abstract

Abstract:

This paper investigates the free vibration and transient response of interconnected structures including double curved beams and intermediate straight beams, which are all constructed from functionally graded porous (FGP) materials. The strain potential and kinetic energies of each beam along with the work done by the external force are calculated. Additionally, a higher-order beam element is introduced to derive stiffness and mass matrices, along with the force vector. The curved and straight beams are discretized, and their assembled stiffness, mass matrices, and force vectors, are obtained. Continuity conditions at the joints are used to derive the total matrices of the entire structure. Subsequently, the natural frequencies and transient response of the system are determined. The accuracy of the mathematical model and the self-developed computer program is validated through the comparison of the obtained results with those of the existing literature and commercial software ANSYS, demonstrating excellent agreement. Furthermore, a comprehensive study is conducted to investigate the effects of various parameters on the free vibration and transient response of the considered structure.

Key words: double curved beam, vibration analysis, intermediate straight beam, functionally graded porous (FGP) material, finite element method

2010 MSC Number:

O313

R. A. JAFARI-TALOOKOLAEI, H. GHANDVAR, E. JUMAEV, S. KHATIR, T. CUONG-LE. Free vibration and transient response of double curved beams connected by intermediate straight beams. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 37-62.

TrendMD

Figures/Tables 28

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 1

Physical properties of the FGP beams"

Material	E₁/GPa	$ρ 1$ /(kg·m^-3)	$ν$
Steel	200	7 850	1/3
Aluminium	70	2 702	1/3

Table 1

Table 2

Convergence analysis of the first five dimensionless natural frequencies for the FGP structure with the CCCC boundary condition and porosity distribution type 2 (L1h=10, R2R1=1.5, N=3, θ0-60° and e0=0.5)"

Total number of elements	$ω ¯ 1$	$ω ¯ 2$	$ω ¯ 3$	$ω ¯ 4$	$ω ¯ 5$
51	6.791 0	9.514 9	11.837 0	13.414 6	14.433 2
102	6.752 2	9.411 5	11.707 2	13.150 5	14.122 9
153	6.749 9	9.404 9	11.699 3	13.134 5	14.104 2
204	6.749 5	9.403 7	11.698 0	13.131 8	14.101 0
255	6.749 4	9.403 4	11.697 6	13.131 0	14.100 1
306	6.749 3	9.403 3	11.697 4	13.130 7	14.099 7
357	6.749 3	9.403 2	11.697 4	13.130 6	14.099 6
395	6.749 3	9.403 2	11.697 4	13.130 5	14.099 5
446	6.749 3	9.403 2	11.697 3	13.130 5	14.099 5
497	6.749 3	9.403 2	11.697 3	13.130 5	14.099 5

Table 2

Fig. 7

Table 3

Comparison of the dimensionless fundamental natural frequency Ω¯ of FGP straight beams under different boundary conditions for various slenderness ratios (e0=0.5)"

Boundary condition	$L / h$	Porosity distribution type 1			Porosity distribution type 2
Boundary condition	$L / h$	Present	Ref. [52]	ANSYS^[52]	Present	Ref. [52]	ANSYS^[52]
HH	10	0.279 5	0.279 8	0.277 8	0.258 7	0.259 9	0.254 9
	20	0.142 1	0.142 2	0.141 9	0.131 3	0.131 8	0.129 6
	50	0.057 1	0.057 1	0.057 1	0.052 7	0.052 9	0.052 1
CC	10	0.591 4	0.594 4	0.610 1	0.544 8	0.547 5	0.560 0
	20	0.316 1	0.316 6	0.317 6	0.288 3	0.288 8	0.294 1
	50	0.129 1	0.129 1	0.128 9	0.117 3	0.117 4	0.118 3
CH	10	0.423 0	0.424 2	0.422 7	0.388 4	0.389 8	0.390 5
	20	0.220 1	0.220 3	0.220 1	0.200 9	0.201 3	0.201 5
	50	0.089 1	0.089 1	0.089 1	0.081 2	0.081 3	0.081 3
CF	10	0.100 7	0.100 8	0.100 7	0.091 6	0.091 7	0.092 0
	20	0.050 8	0.050 8	0.050 8	0.046 1	0.046 2	0.046 3
	50	0.020 4	0.020 4	0.020 4	0.018 5	0.018 5	0.018 6

Table 3

Table 4

Comparison of the first three dimensionless natural frequencies Ω of FGP curved beams with the porosity distribution type 1 under the HH boundary condition (θ0=60°)"

$L / h$	Mode number	Present	Ref. [53]			Ref. [54]
$L / h$	Mode number	Present	CLT	FSDT	SIN	Solution I	Solution II
10	1	8.193 3	8.286 4	8.200 4	8.183 7	8.199 1	8.215 2
	2	35.377 6	37.237 1	35.823 0	35.563 0	35.745 1	35.814 4
	3	75.657 9	84.101 0	77.788 5	76.707 6	77.399 3	77.535 9
50	1	8.318 4	8.321 3	8.317 8	8.317 1	8.317 7	8.318 4
	2	37.752 1	37.834 1	37.769 7	37.756 9	37.765 8	37.769 0
	3	86.611 8	87.062 9	86.371 7	86.666 5	86.708 4	86.715 8

Table 4

Fig. 8

Table 5

Frequency results (Hz) for the structure shown in Fig. 9"

Boundary condition	$e 0$	Mode number
		1		2		3
		ANSYS	Present	ANSYS	Present	ANSYS	Present
CCCC	0.25	49.721 0	50.670 4	52.489 5	53.080 4	58.216 8	58.939 5
	0.50	46.962 3	47.946 1	49.906 4	50.485 2	55.062 7	55.766 7
	0.75	43.075 1	43.696 9	46.013 8	46.533 3	50.987 2	51.610 6
CFCF	0.25	15.862 1	16.330 9	38.518 7	39.101 2	51.698 2	52.349 0
	0.50	14.986 4	15.603 9	36.648 9	37.216 8	49.042 7	49.738 5
	0.75	14.125 3	14.887 5	34.125 6	34.917 7	45.062 9	45.700 6

Table 5

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Table 6

The dimensionless critical velocity and the maximum DMF for different cases investigated in Figs. 12 and 13"

Boundary condition	Result	Porosity distribution
		Type 1			Type 2			Type 3
		$e 0 = 0.25$	$e 0 = 0.50$	$e 0 = 0.75$	$e 0 = 0.25$	$e 0 = 0.50$	$e 0 = 0.75$	$e 0 = 0.25$	$e 0 = 0.50$	$e 0 = 0.75$
CCCC	$v ¯ critical$	0.700 0	0.680 0	0.660 0	0.690 0	0.650 0	0.590 0	0.700 0	0.660 0	0.600 0
	$DMF maximum$	1.339 5	1.619 7	2.055 5	1.373 1	1.730 5	2.392 1	1.366 6	1.727 6	2.460 6
	$η ¯$	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0
CFCF	$v ¯ critical$	0.230 0	0.220 0	0.210 0	0.220 0	0.210 0	0.190 0	0.220 0	0.210 0	0.190 0
	$DMF maximum$	7.162 5	8.573 5	10.712 3	7.412 5	9.404 9	13.245 3	7.362 3	9.310 7	13.256 0
	$η ¯$	1.000 0	1.000 0	1.000 0	1.000 0	1.000 0	1.000 0	1.000 0	1.000 0	1.000 0
CSCS	$v ¯ critical$	0.700 0	0.670 0	0.660 0	0.680 0	0.640 0	0.570 0	0.690 0	0.650 0	0.600 0
	$DMF maximum$	1.345 2	1.625 9	2.062 1	1.381 6	1.747 0	2.434 7	1.372 9	1.735 5	2.471 7
	$η ¯$	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0
SSSS	$v ¯ critical$	0.700 0	0.670 0	0.660 0	0.680 0	0.630 0	0.560 0	0.690 0	0.650 0	0.600 0
	$DMF maximum$	1.346 3	1.627 9	2.065 9	1.382 0	1.747 1	2.435 4	1.373 7	1.736 6	2.473 4
	$η ¯$	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0	0.505 0

Table 6

Fig. 14

Fig. 15

Table 7

The dimensionless critical velocity and the maximum DMF in different cases investigated in Fig. 15"

Result	Porosity distribution
	Type 1			Type 2			Type 3
	$N = 1$	$N = 3$	$N = 5$	$N = 1$	$N = 3$	$N = 5$	$N = 1$	$N = 3$	$N = 5$
$v ¯ critical$	1.020 0	0.780 0	0.640 0	1.010 0	0.740 0	0.610 0	0.950 0	0.720 0	0.600 0
$DMF maximum$	1.307 5	1.270 6	1.212 0	1.519 5	1.437 6	1.332 0	1.544 7	1.528 9	1.453 4
$η ¯$	0.510 0	0.505 0	0.505 0	0.510 0	0.505 0	0.505 0	0.510 0	0.505 0	0.505 0

Table 7

Fig. 16

Fig. 17

Table 8

The dimensionless critical velocity and the maximum DMF in different cases investigated in Fig. 17"

Result	Porosity distribution
	Type 1			Type 2			Type 3
	$θ 0 = 50 ∘$	$θ 0 = 100 ∘$	$θ 0 = 150 ∘$	$θ 0 = 50 ∘$	$θ 0 = 100 ∘$	$θ 0 = 150 ∘$	$θ 0 = 50 ∘$	$θ 0 = 100 ∘$	$θ 0 = 150 ∘$
$v ¯ critical$	0.770 0	0.560 0	0.310 0	0.720 0	0.460 0	0.240 0	0.700 0	0.510 0	0.280 0
$DMF maximum$	1.313 8	2.477 2	3.793 8	1.472 1	2.882 2	4.751 3	1.585 5	2.967 5	4.598 3
$η ¯$	0.525 0	0.530 0	0.535 0	0.525 0	0.515 0	0.505 0	0.525 0	0.530 0	0.535 0

Table 8

Fig. 18

Fig. 19

Table 9

The dimensionless critical velocity and the maximum DMF in different cases investigated in Fig. 19"

Result	Porosity distribution
	Type 1			Type 2			Type 3
	$R 2 R 1$ =1.5	$R 2 R 1 = 2.0$	$R 2 R 1 = 2.5$	$R 2 R 1 = 1.5$	$R 2 R 1 = 2.0$	$R 2 R 1 = 2.5$	$R 2 R 1 = 1.5$	$R 2 R 1 = 2.0$	$R 2 R 1 = 2.5$
$v ¯ critical$	0.740 0	0.480 0	0.370 0	0.690 0	0.440 0	0.340 0	0.680 0	0.450 0	0.340 0
$DMF maximum$	1.305 7	1.687 0	1.999 8	1.483 1	1.911 4	2.283 2	1.576 3	2.049 8	2.457 4
$η ¯$	0.505 0	0.535 0	0.530 0	0.515 0	0.530 0	0.530 0	0.520 0	0.530 0	0.530 0

Table 9

References 55

[1]	YUAN, W., LIAO, H., GAO, R., and LI, F. Vibration and sound transmission loss characteristics of porous foam functionally graded sandwich panels in thermal environment. Applied Mathematics and Mechanics (English Edition), 44(6), 897–916 (2023) https://doi.org/10.1007/s10483-023-3004-7
[2]	IZADI, M., ABEDI, M., and VALVO, P. S. Free vibration analysis of a functionally graded porous triangular plate with arbitrary shape and elastic boundary conditions using an isogeometric approach. Thin-Walled Structures, 205, 112422 (2024)
[3]	LIU, K., ZONG, S., LI, Y., WANG, Z., HU, Z., and WANG, Z. Structural response of the U-type corrugated core sandwich panel used in ship structures under the lateral quasi-static compression load. Marine Structures, 84, 103198 (2022)
[4]	CEN, Q., XING, Z., WANG, Q., LI, L., WANG, Z., WU, Z., and LIU, L. Molding simulation of airfoil foam sandwich structure and interference optimization of foam-core. Chinese Journal of Aeronautics, 37, 325–338 (2024)
[5]	ZHANG, P., SCHIAVONE, P., and QING, H. Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model. Applied Mathematics and Mechanics (English Edition), 44(12), 2071–2092 (2023) https://doi.org/10.1007/s10483-023-3059-9
[6]	WANG, Y. and SIGMUND, O. Topology optimization of multi-material active structures to reduce energy consumption and carbon footprint. Structural and Multidisciplinary Optimization, 67(1), 5 (2024)
[7]	CHANDRASHEKHARA, K. and BANGERA, K. M. Free vibration of composite beams using a refined shear flexible beam element. Computers & Structures, 43(4), 719–727 (1992)
[8]	MATSUNAGA, H. Free vibration and stability of thin elastic beams subjected to axial forces. Journal of Sound and Vibration, 191(5), 917–933 (1996)
[9]	CHEN, W. Q., LÜ, C. F., and BIAN Z. G. A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Applied Mathematical Modelling, 28(10), 877–890 (2004)
[10]	KISA, M. Free vibration analysis of a cantilever composite beam with multiple cracks. Composites Science and Technology, 64(9), 1391–1402 (2004)
[11]	SAYYAD, A. S. Comparison of various refined beam theories for the bending and free vibration analysis of thick beams. Applied and Computational Mechanics, 5(2), 217–230 (2011)
[12]	GIUNTA, G., BISCANI, F., BELOUETTAR, S., FERREIRA, A. J. M., and CARRERA, E. Free vibration analysis of composite beams via refined theories. Composites Part B: Engineering, 44(1), 540–552 (2013)
[13]	XIAO, Z., ZHANG, R., and DAI, H. Dynamic characteristics analysis of variable cross-section beam under thermal vibration environment. Structures, 61, 105941 (2024)
[14]	MORENO-GARCÍA, P., DOS SANTOS, J. V. A., and LOPES, H. A review and study on Ritz method admissible functions with emphasis on buckling and free vibration of isotropic and anisotropic beams and plates. Archives of Computational Methods in Engineering, 25, 785–815 (2018)
[15]	CANALES, F. G. and MANTARI, J. L. Free vibration of thick isotropic and laminated beams with arbitrary boundary conditions via unified formulation and Ritz method. Applied Mathematical Modelling, 61, 693–708 (2018)
[16]	NGUYEN, N. D. and NGUYEN, T. D. Chebyshev polynomial-based Ritz method for thermal buckling and free vibration behaviors of metal foam beams. Applied Mathematics and Mechanics (English Edition), 45(5), 891–910 (2024) https://doi.org/10.1007/s10483-024-3116-5
[17]	JAFARI-TALOOKOLAEI, R. A., ATTAR, M., VALVO, P. S., LOTFINEJAD-JALALI, F., GHASEMI SHIRSAVAR, S. F., and SAADATMORAD, M. Flapwise and chordwise free vibration analysis of a rotating laminated composite beam. Composite Structures, 292, 115694 (2022)
[18]	KRISHNAN, A. and SURESH, Y. J. A simple cubic linear element for static and free vibration analyses of curved beams. Computers & Structures, 68(5), 473–489 (1998)
[19]	KANG, B., RIEDEL, C. H., and TAN, C. A. Free vibration analysis of planar curved beams by wave propagation. Journal of Sound and Vibration, 260(1), 19–44 (2003)
[20]	BOZHEVOLNAYA, E. and SUN, J. Q. Free vibration analysis of curved sandwich beams. Journal of Sandwich Structures & Materials, 6(1), 47–73 (2004)
[21]	ZHU, Z. H. and MEGUID, S. A. Vibration analysis of a new curved beam element. Journal of Sound and Vibration, 309(1-2), 86–95 (2008)
[22]	HAJIANMALEKI, M. and QATU, M. S. Static and vibration analyses of thick, generally laminated deep curved beams with different boundary conditions. Composites Part B: Engineering, 43(4), 1767–1775 (2012)
[23]	SADEGHPOUR, E., SADIGHI, M., and OHADI, A. Free vibration analysis of a debonded curved sandwich beam. European Journal of Mechanics-A/Solids, 57, 71–84 (2016)
[24]	JAFARI-TALOOKOLAEI, R. A., ABEDI, M., and HAJIANMALEKI, M. Vibration characteristics of generally laminated composite curved beams with single through-the-width delamination. Composite Structures, 138, 172–183 (2016)
[25]	KHODABAKHSHPOUR-BARIKI, S., JAFARI-TALOOKOLAEI, R. A., ATTAR, M., and EYVAZIAN, A. Free vibration analysis of composite curved beams with stepped cross-section. Structures, 33, 4828–4842 (2021)
[26]	CORRÊA, R. M., ARNDT, M., and MACHADO, R. D. Free in-plane vibration analysis of curved beams by the generalized/extended finite element method. European Journal of Mechanics-A/Solids, 88, 104244 (2021)
[27]	SAYYAD, A. S. and AVHAD, P. V. A new higher order shear and normal deformation theory for the free vibration analysis of sandwich curved beams. Composite Structures, 280, 114948 (2022)
[28]	KITIPORNCHAI, S., CHEN, D., and YANG, J. Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Materials & Design, 116, 656–665 (2017)
[29]	CHEN, D., YANG, J., and KITIPORNCHAI, S. Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams. Composites Science and Technology, 142, 235–245 (2017)
[30]	NGUYEN, N. D., NGUYEN, T. N., NGUYEN, T. K., and VO, T. P. A new two-variable shear deformation theory for bending, free vibration and buckling analysis of functionally graded porous beams. Composite Structures, 282, 115095 (2022)
[31]	CHEN, D., REZAEI, S., ROSENDAHL, P. L., XU, B. X., and SCHNEIDER J. Multiscale modelling of functionally graded porous beams: buckling and vibration analyses. Engineering Structures, 266, 114568 (2022)
[32]	SU, X., HU, T., ZHANG, W., KANG, H., CONG, Y., and YUAN, Q. Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions. Applied Mathematics and Mechanics (English Edition), 45(6), 983–1000 (2024) https://doi.org/10.1007/s10483-024-3125-8
[33]	CHEN, C., LI, D., ZHOU, X., and ZHOU, L. Thermal vibration analysis of functionally graded graphene platelets-reinforced porous beams using the transfer function method. Engineering Structures, 284, 115963 (2023)
[34]	NGUYEN, N. D., NGUYEN, T. N., NGUYEN, T. K., and VO, T. P. A Legendre-Ritz solution for bending, buckling and free vibration behaviours of porous beams resting on the elastic foundation. Structures, 50, 1934–1950 (2023)
[35]	LEI, Y. L., GAO, K., WANG, X., and YANG, J. Dynamic behaviors of single- and multi-span functionally graded porous beams with flexible boundary constraints. Applied Mathematical Modelling, 83, 754–776 (2020)
[36]	NOORI, A. R., ASLAN, T. A., and TEMEL, B. Dynamic analysis of functionally graded porous beams using complementary functions method in the Laplace domain. Composite Structures, 256, 113094 (2021)
[37]	CHEN, D., YANG, J., SCHNEIDER, J., KITIPORNCHAI, S., and ZHANG, L. Impact response of inclined self-weighted functionally graded porous beams reinforced by graphene platelets. Thin-Walled Structures, 179, 109501 (2022)
[38]	TURAN, M., YAYLACI, E. U., and YAYLACI, M. Free vibration and buckling of functionally graded porous beams using analytical, finite element, and artificial neural network methods. Archive of Applied Mechanics, 93, 1351–1372 (2023)
[39]	ZHAO, J., WANG, Q., DENG, X., CHOE, K., XIE, F., and SHUAI, C. A modified series solution for free vibration analyses of moderately thick functionally graded porous (FGP) deep curved and straight beams. Composites Part B: Engineering, 165, 155–166 (2019)
[40]	ZHOU, Z., CHEN, M., and XIE, K. Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams. Applied Mathematics and Mechanics (English Edition), 41(4), 567–586 (2020) https://doi.org/10.1007/s10483-020-2594-7
[41]	PHAM, Q. H., TRAN, V. K., and NGUYEN, P. C. Hygro-thermal vibration of bidirectional functionally graded porous curved beams on variable elastic foundation using generalized finite element method. Case Studies in Thermal Engineering, 40, 102478 (2022)
[42]	SAYYAD, A. S., AVHAD, P. V., and HADJI, L. On the static deformation and frequency analysis of functionally graded porous circular beams. Forces in Mechanics, 7, 100093 (2022)
[43]	DENG, H., CHEN, K. D., CHENG, W., and ZHAO, S. G. Vibration and buckling analysis of double-functionally graded Timoshenko beam system on Winkler-Pasternak elastic foundation. Composite Structures, 160, 152–168 (2017)
[44]	HAN, F., DAN, D. H., and DENG, Z. C. A dynamic stiffness-based modal analysis method for a double-beam system with elastic supports. Mechanical Systems and Signal Processing, 146, 106978 (2021)
[45]	CHEN, B., LIN, B., LI, Y., and TANG, H. Exact solutions of steady-state dynamic responses of a laminated composite double-beam system interconnected by a viscoelastic layer in hygrothermal environments. Composite Structures, 268, 113939 (2021)
[46]	ZHAO, X., MENG, S. Y., ZHU, W. D., ZHU, Y. L., and LI, Y. H. A closed-form solution of forced vibration of a double-curved-beam system by means of the Green's function method. Journal of Sound and Vibration, 561, 117812 (2023)
[47]	WATTANASAKULPONG, N., CHAIKITTIRATANA, A., and PORNPEERAKEAT, S. Chebyshev collocation approach for vibration analysis of functionally graded porous beams based on third-order shear deformation theory. Acta Mechanica Sinica, 34(6), 1124–1135 (2018)
[48]	ZHAO, J., WANG, Q., DENG, X., CHOE, K., ZHONG, R., and SHUAI, C. Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions. Composites Part B: Engineering, 168, 106–120 (2019)
[49]	JAFARI-TALOOKOLAEI, R. A., KARGARNOVIN, M. H., and AHMADIAN, M. T. Dynamic response of a delaminated composite beam with general lay-ups based on the first-order shear deformation theory. Composites Part B: Engineering, 55, 65–78 (2013)
[50]	WANG, Y. and SIGMUND, O. Multi-material topology optimization for maximizing structural stability under thermo-mechanical loading. Computer Methods in Applied Mechanics and Engineering, 407, 115938 (2023)
[51]	REDDY, J. N. Mechanics of Laminated Composite Plates and Shells; Theory and Analysis, 2nd ed., CRC Press, Boca Raton, 362–364 (2003)
[52]	CHEN, D., YANG, J., and KITIPORNCHAI, S. Free and forced vibrations of shear deformable functionally graded porous beams. International Journal of Mechanical Sciences, 108-109, 14–22 (2016)
[53]	POLIT, O., ANANT, C., ANIRUDH, B., and GANAPATHI, M. Functionally graded graphene reinforced porous nanocomposite curved beams: bending and elastic stability using a higher-order model with thickness stretch effect. Composites Part B: Engineering, 166, 310–327 (2019)
[54]	HOSSEINI, S. A. H. and RAHMANI, O. Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model. Applied Physics A, 122, 1–11 (2016)
[55]	WU, D., LIU, A., HUANG, Y., HUANG, Y., PI, Y., and GAO, W. Dynamic analysis of functionally graded porous structures through finite element analysis. Engineering Structures, 165, 287–301 (2018)