3. A small cantilever beam is shown in figure (2). Its response is dominated by a resonant mode having an equivalent mass meq = .002 kg and an equivalent stiffness keq = 270 N/m. The beam has an initial deflection at its end with zero initial velocity. Determine the equivalent dashpot constant ceq so that its response decays as fast as possible. w xx) Figure 2: A cantilever beam. 00000. x(1)

You have to design a structure of a thin rectangular beam supporting a lamp with a mass of 20 kg as shown in Figure 1. The beam is fixed on the wall with modulus of elasticity of 200 GPa, length, I = 1.5 m, thickness, t= 2.3 cm and width, w = 3.2 cm. a) Derive the general solution to determine the displacement of the lamp for time t. b) Evaluate the maximum acceleration in vertical direction measured at the lamp if a gust wind initially excited the beam with initial displacement x = 0.1 cm and initial velocity v = 0 m/s. Ignore the mass of the beam. c) The procurement manager offers another option for the material of the beam. The material has a modulus of elasticity of 100 GPa. Discuss the impact of changing to this material in terms of vibration response. Lamp Figure 1 Wall

An object with a mass of 0.5 kg is released at a distance of 2 m, the spring coefficient k = 100 N / m and the object is in the same direction after impact. If it is known to rebound on the road throughout;  a) Find the maximum compression and maximum spring force in the spring. b) Find the rebound height if e = 0.5. (Assume that the spring behaves linearly during deformation, when the object leaves the spring, the spring regains its initial length (L) and energy is conserved during compression.) ① The object is 2 m above the arc and is at rest. ② The object is about to hit the spring. ③ It is the moment when the object compresses the spring as much as. ④ It is the moment when the spring is most compressed, the object is at rest at this moment. ⑤ It is the moment when the object loses contact with the spring. ⑥ The object is at the top h of the spring as a result of becomes immobile at distance.

4. The space needle observation tower can be considered as a cantilever beam that is fixed at the ground (Figure 3). Find an idealized spring-mass system equation of the observation tower and determine the natural frequency. [Hints: For the study of transverse vibration, the top mass can be considered a point mass, and the supporting structure (beam) can be approximated as a spring to obtain the single-degree-of-freedom model. The transverse deflection of the beam, due to a load F is given by FL/3EI] Figure 3. Space Needle observation tower in Seattle, Washington.