Examples of frame calculation for dynamic loading

Let us consider the application of the static method in the dynamics of structures
on examples of calculation of frames loaded by dynamic loads. Considering systems with a finite number of degrees of freedom, we will find extreme values of dynamic bending moments (MD), transverse (QD) and longitudinal forces (ND).

Calculation of a statically determinate frame

For the frame shown in (Fig. 6.13), it is necessary to determine the dynamic forces (MD,QD,ND) with the amplitude of the disturbing force P=2 kN, the frequency of the vibration load θ=0.5ω_min and masses m1=400 kg, m2=500 kg, where ωmin is the lowest frequency of self oscillations. Taking into account the assumptions, the position of the masses at any moment of time is determined by the following parameters: mass m₁ can move vertically and mass m₂ can move horizontally.
These masses have no other displacements and the system has two degrees of freedom. Let us determine the frequencies of self oscillations. Based on the equality (33), the equation of vibration for a system with two degrees of freedom is as follows:

Expanding the determinant we get

where

(41)

where δ₁₁,δ₁₂,δ₂₁,δ₂₂ – displacements of masses caused by static forces P=1, applied alternately at the points of mass location in the direction of displacement of these masses.
The diagrams of bending moments from forces P₁=1 and P₂=1 are shown in Fig. 6.14a, b.

Substituting the displacement values into equation (41), we have

where

The frequencies of self oscillations will be:

Circular frequency of vibration load

The diagram of bending moments from the static action of vibration load according to the
condition M^р_st= M1⋅P is shown in Fig. 6.15.

We find the maximum values of inertial forces according to condition (40):

(42)

Substituting into equation (42), we have:

Where

We obtain the MD diagram according to the condition

The diagrams M_iI_i and the final diagram of bending moments M_D are shown in Fig. (6.16).

The correctness of constructing the M_D diagram consists of checking the equilibrium of the system nodes. We check the equilibrium of nodes E and F (Fig. 6.17):