PSI - Issue 63

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

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Like the stiffness matrix, the mass matrix [ m ] is determined by localizing the global mass matrices of the elements [ m i ] into which the structure is divided. The global mass matrix [ m i ] in Eq. (4) of the i -th element is obtained by the transformation of the local mass matrix � � ∗ � in Eq. (3). � ∗ ��� � ∙ � ∙ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ � � 0 0 � � 0 0 0 � � � � � ��∙ � � ��� 0 � � � ��∙ � � ��� 0 � ��∙ � � ��� � � � ��� 0 � ��∙ � � ��� � � � � ��� � � 0 0 � � 0 0 0 � � � � ��∙ � � ��� 0 � � � � ��∙ � � ��� 0 ��∙ � � ��� � � � � ��� 0 ��∙ � � ��� � � � ��� ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ � 3 � � 4 � An angle ψ i is needed to determine the transformation matrix [ T i ] in Eq. (5). This angle indicates the rotation of the element i from the horizontal axis. � � �� ⎢ ⎢ ⎣ ⎢ ⎢ ⎡ � � 0 0 0 0 � � � 0 0 0 0 0 0 1 0 0 0 0 0 0 � � 0 0 0 0 � � � 0 0 0 0 0 0 1 ⎥ ⎥ ⎦ ⎥ ⎥ ⎤ � 5 � 2.2. Natural frequencies and mode shapes The method of stiffness constants is used to determine natural frequencies and mode shapes in this case. The solved construction is replaced by a computational model, i.e., a system with n degrees of freedom. The equations of motion can be written (in matrix notation): � ���� ��� � � � ��� ��� ��� 0 � . � 6 � In Eq. (6), { u ( t ) } is the displacement vector. Its second derivation ��� � � � determines the acceleration vector. It is assumed, that the construction vibrates in the harmonic motion. Then equations of motion can be rewritten into Eq. (7), and further they can be modified into Eq. (8). �� �� ∙ � ��� ��� � � � ��� ��� ��� 0 � � 7 � �� ��� �� ∙ � ���� ��� ��� 0 � � 8 � We are looking for the zero determinant of the matrix �� ��� �� ∙ � �� ; this gives us the natural frequencies ω n . The determination of the zero determinant is numerical in this solution. The values of natural frequency are chosen with the given difference, and the determinant is calculated. In the section where the sign of the determinant changes, the value of the natural frequency is found. The bisection method is used (Moheuddin et al. (2019)). � � � � � �� ∗ �� � � �