Hello! You already know the theory of beam vibration. The dynamic features of a beam depend on the way the mass m and the stiffness , given in bending by the product EI, are distributed in the structure. In the Transamerica Pyramid in San Francisco, the mass and stiffness are distributed in a very complex way! In this experimental lab, we will observe the vibrations of a frame structure in order to determine its eigenfrequencies and eigenmodes. From the actual structure, we will consider a reduced scale experiment on a mock-up. It is a 3 storey building mock-up excited at its base by an actuator. Each floor is connected to four flexible beams. The motion is purely horizontal and we will thus consider three vibrating masses at levels 1, 2 and 3. The mock-up may be seen as a 3 DoFs system and the motion at each floor will be recorded by a laser vibrometer. The vibrometer allows to measure the motion of several points along the structure. Let us now determine experimentally the eigenmodes of this structure. The excitation at the base of the mock-up is sinusoidal and the motion at each storey and each frequency is recorded. At the top, you can see the sinusoidal motion measured by the vibrometer at 6.5Hz as well as the motion at 17Hz. Sweeping the frequency and considering the amplitude at the top divided by the amplitude at the floor, we get the experimental transfer function w.r.t frequency. Three eigenfrequencies are found: the first one at 7.5Hz, the second one around 24Hz and the third one at 35Hz. The vibration of the mock-up is governed by the vibration of its three floors. We thus evidenced that it is a 3DoF system! Let us now identify the modal shape of the structure at each eigenfrequency! For the first mode, the eigenfrequency is 7.5Hz. What is the corresponding mode shape? As shown by the experiment, the motion of the whole building is in the same direction! The only vibration node is at the bottom where the structure is clamped to its support. The motion of each floor being measured with the vibrometer, we easily get the first eigenmode of the equivalent 3DoFs system. We do not consider the motion along each beam but only the vibration at each floor! For the second mode, the eigenfrequency is 23.8Hz. What is the corresponding mode shape? As shown by the experiment v5, the motion of the top of the building is in the opposite direction to that of the bottom part! We also observe a vibration node in the top part of the structure. From the recorded motion at each floor, we get the second eigenmode of the equivalent 3DoFs system. The vibration node in the top part appears clearly in this video! For the third mode, the eigenfrequency is 35Hz. What is the corresponding mode shape? As shown by the experiment, the motions at the top and at the bottom of the building are in the same direction. That of the central part is in the opposite direction! We thus have two vibration nodes along the structure. From the recorded motion at each floor, we get the third eigenmode of the equivalent 3DoFs system. Two vibration nodes clearly appear along the structure! We now have the experimental eigenmodes of the mock-up. Is it possible to define a theoretical model for this structure? Since we measured the motion of the three vibrating masses at each floor level, we may think about a 3 DoFs model with the mass m of each floor concentrated at each level and the stiffness k of the four beams superimposed between each level. We thus have a stiffness of 4k between each floor. The expression of the stiffness matrix includes the stiffness of a single beam , 12EI/L, and the matrix is obtain by combining the influence of each beam. A factor 4 thus affects the whole matrix and a factor 2 the 1st and 2nd diagonal terms. They correspond to the floors which are linked to two beam groups: one below and one above! The masses being concentrated at the level of each floor, the mass matrix is simply diagonal! Let us now determine the theoretical eigenfrequencies from this simple model! The eigenfrequencies can be determined from this eigenvalue problem: det(K-omega^2 M)=0. Since we have a 3 DoF system, we get 3 eigenfrequencies. Let us determine the three eigenmodes! The first eigenmode is at 10.4Hz which is significantly different from the experiment. The second eigenmode is at 29Hz and we can notice the differences in the mode shapes between the model (in blue) and the experiment (in red). The third eigenmode is estimated at 42Hz. These comparisons show that the 3 DoF model does not perfectly match the experimental results. The frequencies are too high!! And we may wonder why?! Indeed it is only due to the fact that the mass of the beams is not included in the model! If we do so, it is possible to reduce the theoretical eigenfrequencies and to get much better comparisons with the experiments! The final experimental study considers the influence of a pendulum on the eigenmodes of the structure. As shown in this graph , the amplitudes of the higher modes, in dark blue, are significantly reduced due to the pendulum. We may thus use such a system as an antiseismic device! This solution has already been used for the Taipei tower and it is very efficient to reduce the effects of actual quakes!! Let us now summarize this video! We have detailed the experimental analysis of simple dynamic systems. We have measured the eigenfrequencies, the eigenmodes and performed comparisons with a simple 3DoF model. More complex structures can also be studied in the lab. The eigenmodes of this guitar soundboard are shown by sand grains perfectly aligning along the vibration nodes!! Nice modes for good vibes!!
