PSI - Issue 63

Lenka Koubova / Procedia Structural Integrity 63 (2024) 35–42

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3.2. Parabolic arc This procedure can be used for any planar bar construction with mass matrix [ m ] and stiffness matrix [ k ]. It was also used to solve the parabolic arc. The shape of the parabolic arc describes Eq. (11). ݖ ൌ ସ௙௫ మ ௟ మ ሺ 11 ሻ

Fig. 5. Parabolic arc.

The arc was divided into 16 elements, ∆ x = 1 m. The arc is supported by the joint support on both sides. A computational model with 47 degrees of freedom was created. The parabolic arc has the same parameters as a simple beam, i.e., length l = 16 m, rectangular cross-section with parameters h = 0.4 m, b = 1 m, modulus of elasticity E = 20 GPa, and density of material ρ m = 2000 kg/m 3 . The height of the arc is f = 8 m. In this paper, the first six natural frequencies and mode shapes are presented.

Table 1. Natural frequencies.

Natural frequency ω i [rad/s]

Frequency order i

1 2 3 4 5 6

18.23 50.15

95.70 152.59 224.13 309.10

Fig. 6. Numerical solution of natural frequencies of parabolic arc using the zero determinant of a matrix.