PSI - Issue 63

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

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Using the method of virtual work, the flexibility matrix [ δ ] was established, and subsequently the stiffness matrix [ k ] = [ δ ] -1 was determined for this model with two degrees of freedom. � � � � 0 0 ��� 42606.67 42606.67 � � � �� 6750000 � 5906250 � 5906250 6750000 � We are looking for the zero determinant of the matrix �� ��� �� ∙ � �� ; this gives us first and second natural frequencies. In this case, the quadratic equation is solved. We obtain  n 1 = 14.063 rad/s and  n 2 = 54.464 rad/s by solving a quadratic equation. Natural frequencies  1 = 14.078 rad/s and  2 = 56.311 rad/s were solved numerically using the given procedure by the bisection method. The differences between the values are 0.11% and 3.28%.

Fig. 3. Numerical solution of natural frequencies of simple beam using the zero determinant of a matrix.

The third frequency  3 = 126.5 rad/s was also determined using the presented procedure (see Fig. 3). Fig. 4 shows the first and second mode shapes obtained by the given procedure. In the first mode shape, the beam vibrates to the phase; in the second, it vibrates in the antiphase.

Fig. 4. First and second mode shapes of the simple beam.