PSI - Issue 63

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

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Fig. 1. Bisection method.

The bisection method is a root-finding method that applies to any continuous function f ( x ) for which one knows two values with opposite signs (see Fig. 1). The method consists of repeatedly bisecting the interval (Eq. (9)) defined by these values and then selecting the subinterval in which the function changes sign and therefore must contain a root. � � �� � � � 9 � Next, we use Eq. (8) to determine the mode shapes. Since the matrix �� � � �� ∙ � �� is singular, the solution continues by selecting any one value of the displacement vector { u ( t ) }. In this case, we give the last one value equal to one and count the other values needed to determine mode shape for the corresponding natural frequency ω n . 3. Examples 3.1. Simple beam The procedure was first applied to a simple concrete beam with 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 . For a simple beam with one degree of freedom, the first natural frequency can be determined from Eq. (10). � � �  � 10 � The mass m is taken as half of the mass of the beam. The stiffness k of the simple beam is � 48 ∙ / � . For values m = 6400 kg and k = 125000 MN/m, the first natural frequency was set  n = 13.975 rad/s. Natural frequency  1 = 14.078 rad/s was solved numerically using the given procedure. The difference between the values is 0.7%. Then, a model with two degrees of freedom (see Fig. 2) has also been created for the simple beam. In this case, the mass m is taken around a third of the mass of the beam.

Fig. 2. A model of the simple beam with two degrees of freedom.