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

doi:10.1007/s10483-025-3197-8

摘要/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

中图分类号:

O313

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 37-62.

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[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 37-62.

图/表 28

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

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

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

$L / h$	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

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

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

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

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

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

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