The Hancock Tower in Boston experienced large swaying due to winds, causing motion sickness in occupants of the upper floors. To reduce the oscillations, a "tuned mass damper" (TMD) was installed to act as a vibration absorber. We want to study the behavior of this system through a simplified 2-DOF model. The design concept behind a TMD is to reduce the amplitude of vibration of a primary structure by introducing a secondary spring and mass that can be tuned to a specific frequency. No damping is required for the TMD to reduce the amplitude of vibration of the primary structure. There is an accompanying video on Canvas that describes a similar TMD used for the Shanghai Tower and mentions the issue of the Hancock Tower - I suggest you watch this video before doing anything else. Hancock Tower parameters (some values are estimated): Height: 240 m (60 stories) Total floor area: 191,380 m Simplified rectangular cross-section: 100 m x 32 m Mass: Assuming 950 kg/m, total mass = 182E6 kg Observed sway period: 6 sec TMD Mass: m2 = 600 tons = 5.44E5 kg Max travel of TMD mass: +/-2 m relative to building Tall buildings are generally designed for 100 MPH wind loads. Since we are going to be performing an actual forced response analysis, we will assume the portion of the wind speed at any particular frequency is 15% of that overall value and also assume the force is uniformly distributed along the height of the building. Use a Ca value of 1.5 for the building's drag coefficient. While no damping is required in a TMD to be effective, all physical structures have some amount of damping. So, to be as realistic as possible, we will consider both the undamped system and then assume a small amount of damping is present in both the building (3 = 1%) and the TMD itself (we will study a range of values). j 1) To begin, use the parameters above to determine appropriate values for mi, k1, and f1. Set up an equivalent SDOF system for the building and determine the values to match the observed oscillations of the building alone before the TMD was installed. Treat the building as a uniform cantilever beam and use the displacement at the top as the DOF. Use this as the location for the equivalent force due to the wind as well. 2) Derive the EOM in matrix form for the 2-DOF system. Assume the TMD is located at the top of the building. 3) Determine the value of k that will drive the response of the building to zero at the building's original natural frequency.