Hello You already know how to deal with simple structures such as cables or beams. In this video, we will investigate the case of more complex structures submitted to dynamic loadings! A first category of complex structure are the regular structures involving a smooth displacement field. They may be considered as generalized 1DOF oscillators. It means that we may consider the 1DOF equation of motion in Z(t), at the top of the structure, but incorporating generalized mass (m)tilde, stiffness (k)tilde and loading (p)tilde. Such generalized variables account for the spatial variations of the displacement in the structure, in a very simple way! If we consider a more complex structure such as a basketball hoop, we must account for the deformation of two beams. We may simplify the problem by considering that the mass m represents a player and is concentrated at point C! This case is more complex than the previous ones since we have to determine the forces along both beams! If we consider the top beam , it is rather simple since the normal force T=F_x and the shear force N=F_y. The moment only depends on F_y with a lever arm L-x. If we consider the left post , T=F_y and N=-F_x. The moment depends on both F_y, constant lever arm L, as well as on F_x with a lever arm H-y. We may then write the displacements of the mass u_x and u_y with respect to the forces F_x and F_y. K^-1 is the inverse of the stiffness matrix and may be identified as follows. The shear terms have been neglected and the compression terms may also be neglected. We thus keep the only bending terms depending on the product EI. Let us consider a young basketball player jumping at the hoop! The equation of motion gives a first term proportional to the acceleration vector with a concentrated mass. The second term is proportional to the displacement vector with the stiffness matrix determined previously. And the right hand side term only has a vertical component exerted by the young boy! The eigenmodes may be determined by writing that the determinant of K minus omega^2M should be 0! The eigenmodes of the system are as follows. The first eigenvector is phi_1={1 , -(1+sqrt(2))} and the second eigenvector is phi_2={1 , -(1-sqrt(2))}. The displacement vector u is finally q_1(t) phi_1 + q_2(t) phi_2. If we now consider a more complex structure - a building with 3 stiff floors. The bending stiffness of each wall is k_1 at the top, k_2 at the middle and k_3 at the bottom. Since the floors are very stiff, the horizontal displacements at each level will be the same for each wall. We may thus combine the stiffnesses to get an equivalent 3 DOFs system. The stiffnesses will now be 3k_1, 3k_2 and 3k_3. What about the stiffness matrix of the whole structure? Its expression involves the stiffness 3k_1 only at point 1. Conversely, it combines the stiffness of the upper and lower levels at point 2. We thus get 3k_1+3k_2. Similarly , we get 3k_2+3k_3 at point 3. If concentrated masses are assumed, the mass matrix is simply diagonal. Finally, we simply have to solve our favourite eigenvalue problem: det(K-omega^2 M)=0. Let us choose the values of the stiffnesses and masses. k_1 = 200 kN/m, k_2 = 400 kN/m and k_3 = 600 kN/m. Remember that each value corresponds to a single wall. m_1 = 1t, m_2 = 1.5t and m_3 = 2t. The eigenfrequencies are then obtained omega_1 = 14.5 rad/s, omega_2 = 31 rad/s, omega_3 = 46 rad/s. The first eigenmode corresponds to a simple bending mode. The second eigenmode leads to a vibration node between point 1 and 2. The third eigenmode leads to two vibration nodes between point 1 and 2 and between point 2 and 3. Finally, the three components displacement vector may be determined as the sum of q_i(t) phi_i on the three modes. Complex system but simple calculation! For more complex systems a finite element model is generally needed. However the basic principles are the same! This is a FEM model of the Normandy bridge. As you can see , the vibrations of this very complex structure may be approximated through a large number of 3D beams! In some cases, the structure is not easy to model using beams only. For this building mock-up, another strategy is needed! Here are some of the 2D and 3D modes of the structure. For the 2D modes at the top, solid finite elements are used. The first eigenfrequency is 7.75 Hz and the second eigenfrequency is 32.8 Hz. For the 3D modes, shell elements are considered. They are a generalization of beams in 3 dimensions. For the first 3D mode, the eigenfrequency is 7.65 Hz and for the second 3D mode, the eigenfrequency is 33.2 Hz. There is a slight difference between the 2D and 3D frequencies since the finite element models are different! Let us now summarize this video! We have detailed the dynamic analysis for a basketball hoop involving two beams only. For more complex models, a discrete, or finite element, model is needed! We may also consider the vibration and eigenmodes of an entire aircraft combining beam elements and shell elements. As you can see, some of the modes are nice but probably not so comfortable!