PSI - Issue 63

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

36

Nachum and Altus (2007) deal with the study of the natural frequencies and mode shapes of non-homogeneous (deterministic and stochastic) rods and beams. The solution is based on the functional perturbation method. The frequencies and mode shapes are considered functionals of the non-homogeneous properties. The work of authors Silva and Nicoletti (2017) focuses on the placement of natural frequencies of beams to desired frequency regions. More specifically, they investigated the effects of combining mode shapes to shape a beam to change its natural frequencies, both numerically and experimentally. In the experimental work, published by Brøns and Thomsen (2019), a scanning laser Doppler vibrometer is used to measure mode shapes and natural frequencies of beam bending vibrations above the critical frequency predicted by Timoshenko theory. Above the critical frequency, the mode shapes are intricate, so to quantitatively compare theoretical and experimental mode shapes, the modal assurance criterion is applied. The analytical analysis of the in-plane vibrations of the beam curved in different mode shapes is presented in Nicoletti (2020). The author focused on the slender beams, which present interesting dynamic characteristics when they are curved according to a given shape. When the beam is curved according to one of its mode shapes, the natural frequency associated with that mode tends to increase significantly without affecting the other natural frequencies. A finite element formulation is proposed for the natural vibration analysis of pipes conveying fluids in a paper presented by Attia et al. (2024). The study develops a robust mathematical procedure that combines the real and imaginary components of the eigenvectors to form physically attainable (i.e., real) mode shapes. This paper deals with numerical solutions of natural frequencies and mode shapes. The method of stiffness constants is used. The procedure can be used for any planar bar construction, for which the stiffness matrix and mass matrix are determined. For the numerical solution of the zero determinate of a matrix, the bisection method is used. The presented procedure was programmed using the commonly available spreadsheet editor Microsoft Excel (using the macro programming language Visual Basic for Applications). 2. Solution procedure This procedure describes the procedure for determining the natural frequencies and mode shapes of the planar bar construction. Construction is characterized by cross-section characteristics like area A , moment of inertia I , and material characteristics like modules of elasticity E , density of material ρ m . When solving, the construction is divided into elements. Each element is identified by its length ∆ s i . First, we need to determine the stiffness matrix [ k] and the mass matrix [ m] for a given structure. Then the natural frequencies and mode shapes are solved. 2.1. Stiffness matrix and mass matrix The stiffness matrix [ k] is obtained by localizing the global stiffness matrices of the elements [ k i ] into which the structure is divided. Matrix localization is based on code numbers, where three code numbers are assigned to each point. The global stiffness matrix [ k i ] in Eq. (2) of the i -th element is obtained by the transformation of the local stiffness matrix �� � ∗ � in Eq. (1). � ∗ � � ⎢ ⎢ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ � � � � � 0 0 � � � � � � 0 0 0 � � � � �� � � � � � � �� � � 0 � � � � � �� � � � � � � �� � � 0 � � � � �� � � � � � � � � 0 � � � �� � � � � � � � � � � � � � � 0 0 � � � � � 0 0 0 � � � � � �� � � � � � �� � � 0 � � � � �� � � � � � �� � � 0 � � � � �� � � � � � � � � 0 � � � �� � � � � � � � � ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ � 1 �

� � � �� � �� ∗ ��� � � �

� 2 �