Mechanical System Dynamics Proff. Corradi, Melzi and Zasso A.Y. 2021 – 2022 Assignment 2 In-plane Finite Element Model of a Transmission Tower A Ground wire Insulator string C B Conductor D E F G O1 O2 Figure 2 – Transmission Tower: structural scheme Figure 1 – Transmission Tower: physical system The system shown in Figure 1 is a Transmission Tower. Figure 2 represents a simplified in-plane scheme of the structure, suitable for the FE discretization. A simplified way to account for the dynamic interaction with the conductors consists in six lamped masses elastically suspended. Assume that the beams converging in any junctions are welded together. Point O1 and O2 are clamped to the ground. All dimensions in the drawing are in meters. The following tables summarize the main properties of the structure. green colour blue colour Beam sections of the tower m [kg/m] EA [N] 35.0 5.0e8 15.0 5.0e7 EJ [Nm2] 2.5e7 1.5e7 Conductor (suspended mass) ๐! 4e4 N/m ๐! 50 Ns/m ๐! 3000 kg Structural damping is defined according to the proportional damping assumption: [C] = α[M] + β[K], with α = 0.5 s-1 and β = 9.0e-5 s. For the structure shown in Figure 2, work out the following items and prepare a short report summarizing the solution obtained. 1. Define a FE model of the structure in the 0 – 15 Hz frequency range (check that for each element ๐, !",$ ≥ 1.5). " %&' 2. Compute the structure’s natural frequencies and vibration modes up to 9th mode. Plot the mode shapes with the indication of the associated natural frequencies. 3. Calculate the frequency response functions which relate the input force (horizontal force in node A) to following outputs. a) Horizontal displacement and horizontal acceleration of node D. Plot the corresponding magnitude and phase diagrams. b) Vertical displacement and vertical acceleration of node D. Plot the corresponding magnitude and phase diagrams. c) Shear force, bending moment and axial force evaluated in the section with node G (midpoint of the beam element). Plot the corresponding magnitude and phase diagrams. Assume the input force to vary in the 0 – 15 Hz frequency range and set the frequency resolution to 0.01 Hz. 4. Using the modal superposition approach and considering the structure’s modes below 10 Hz, calculate the frequency response functions which relate the same input force of question 3 (horizontal force applied in node A) to the horizontal and vertical displacement of node D, and the horizontal and vertical acceleration of node D. Plot the corresponding magnitude and phase diagrams superimposed to those obtained in question 3a and 3b. Point out the differences and comment the results. 5. Calculate the frequency response functions which relate the base excitation (imposed in-phase horizontal displacement in O1 and O2) to the horizontal displacement and acceleration of node A. Assume the input displacement to vary in the 0 – 15 Hz frequency range and set the frequency resolution to 0.01 Hz. Plot the corresponding magnitude and phase diagrams. 6. Optional question Galloping is a large-amplitude vibration of conductors associated with ice accretion over their surface. Ice formations make the conductor section similar to an aerofoil; therefore, unstable oscillations may arise due to the combination of vertical and torsional motion of the cables. Assume that, due to galloping, the conductor in node C vibrates with the following law: ๐ด๐๐๐ (Ω๐ก + ๐ฝ) where A = 0.1 m, f = 0.5 Hz (Ω = 2πf) and b is a random phase. Assume that the conductor in node B vibrates with the same law of the corresponding right-side one, with a phase shift computed as: ๐ ๐ฃ where d is the horizontal distance between the conductors and v = 10 m/s the wind speed. ๐=Ω a) Plot the time history (time length 5 s with resolution of 0.01 s) of the vertical and horizontal reaction forces at the ground points O1 and O2. b) Determine the amplitude of the shear force, bending moment and axial force along the EF beam element, in correspondence of node E.
