After completing the design of a spring-damper system for a bridge, you are assigned your nexttask. The company you work for has been commissioned to design a new two-storey buildingon-campus at the university.The building in Figure 1 can be modelled as a two-DOF system, with point masses for the firstfloor and the roof. The mass of the columns can be assumed negligible compared to the mass ofthe floor/roof.m2m₁X2x1Figure 1: Schematic diagram of the proposed building designThe first floor is supported by four, cylindrical concrete columns of diameter d₁ = 0.5m. Theroof is also supported by four cylindrical columns but these were required to be of diameter d₂ =0.3 m as a consequence of weight considerations. The Young's Modulus of concrete can be takenas E = 25 GPa and the height of every column is 4m. The masses of the first floor and the roof arem₁ = 40000kg and m₂ = 32000kg, respectively. The bending stiffness of one column modelled3EIas a beam of circular cross-seciton can be expressed as k = where E is the Young's modulus,73.I is the second moment of area of beam's cross-section and I is the beam's length.Ek₂= 1863786.657Ek₁= 14381069.89W₁ = 7.114053 rad/sec with eigenvector a1[1, 7.62988]¹ andW2 = 20.34097 rad/sec with eigenvector a2[1, -0.16383]¹1. The site for the building is between two existing buildings. As a result of CFD modellingand experimental data, a nodal force vector off(t) =has been estimated due to the higher frequency eddies as a result of turbulence near groundlevel.6000 sin (27)2500 sin (1.4nt)Using direct matrix inversion (no modal analysis), calculate the particular solution of eachmass in the absence of damping.2. Your boss contacts you to share data they have recently obtained. It has become clear thatthe effect of damping, arising from the joints between columns and the floor/roof is notnegligible. The damping factor can be assumed to take the form of Rayleigh dampingx(0) =C=aK+BMwith a = 0.15 and ß = 2.0.(a) Perform a modal analysis to calculate the steady state motion of each mass.(b) Using the steady-state solution, evaluate the transient solution to obtain the total solu-tion.Initial conditions can be taken as8₁m and (0) =[8]ms-¹.(c) Is the effect of damping helpful to this design?(d) Describe whether the transient solution is important for the design of this building whenonly wind loading is considered.

As per the rule of the company, at a time, maximum three sub-parts can be answered. Solving first…

After completing the design of a spring-damper system for a bridge, you are assigned your next task. The company you work for has been commissioned to design a new two-storey building on-campus at the university. The building in Figure 1 can be modelled as a two-DOF system, with point masses for the first floor and the roof. The mass of the columns can be assumed negligible compared to the mass of the floor/roof. m₂ m₁ X2 x1 Figure 1: Schematic diagram of the proposed building design The first floor is supported by four, cylindrical concrete columns of diameter d₁ = 0.5m. The roof is also supported by four cylindrical columns but these were required to be of diameter d₂ = 0.3m as a consequence of weight considerations. The Young's Modulus of concrete can be taken as E = 25 GPa and the height of every column is 4 m. The masses of the first floor and the roof are m₁ = 40000kg and m₂ = 32000 kg, respectively. The bending stiffness of one column modelled 3EI as a beam of circular cross-seciton can be expressed as k=13, where E is the Young's modulus, I is the second moment of area of beam's cross-section and I is the beam's length. 14381069.89 Ek₂= 1863786.657 Eki W₁= 7.114053 rad/sec with eigenvector a1[1, 7.62988)¹ and W₂ = 20.34097 rad/sec with eigenvector a2[1, -0.16383]™ 1. The site for the building is between two existing buildings. As a result of CFD modelling and experimental data, a nodal force vector of f(1) = 6000 sin (271) 2500 sin (1.4) has been estimated due to the higher frequency eddies as a result of turbulence near ground level. Using direct matrix inversion (no modal analysis), calculate the particular solution of each mass in the absence of damping. 2. Your boss contacts you to share data they have recently obtained. It has become clear that the effect of damping, arising from the joints between columns and the floor/roof is not negligible. The damping factor can be assumed to take the form of Rayleigh damping C=aK+BM with a = 0.15 and ß = 2.0. (a) Perform a modal analysis to calculate the steady state motion of each mass. (b) Using the steady-state solution, evaluate the transient solution to obtain the total solu- tion. Initial conditions can be taken as x(0) = -8- m and x(0) = ms-¹. (c) Is the effect of damping helpful to this design? (d) Describe whether the transient solution is important for the design of this building when only wind loading is considered.