Structural Dynamics Concepts

Example: For the given frame, determine the internal forces from the vibration load. Units of measurement: m, kN (not shown in the diagrams). P(t) = 4 sinθt; q(t) = 2 sinθt.

Solution:

1. We perform a kinematic analysis and verify that the system is statically determinate and
   geometrically unchangeable. Dynamic degree of freedom n=1.
2. We construct a unit diagram M₁ from F₁=1.

3. We determine the displacement:

4. Calculate the frequency:

5. We write down the canonical equation for finding the forces of inertia:

6. We determine the main coefficients of the canonical equation:

at θ=0,8ω:

at θ=1,2ω:

7. We construct the load diagram Мp and determine the free terms of the equation:

8. We solve the canonical equation and determine the inertial forces:

at θ=0,8ω:

at θ=1,2ω:

We construct a diagram of dynamic bending moments separately for each frequency.

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