Problem 3: A vibration absorber, as a tuned damper system including a main system and an absorber, is shown in the figure below. The main system is attached to the ground or a surface by a spring and a damper, and the absorber is attached to the main system by a spring only. The main system is subject to a harmonic force F() = cos(@-t) . The purpose of the absorber is to reduce or eliminate the vibrational amplitude of the main system. This type of problem often occurs when there is a need to reduce or eliminate the seismic effect on civil structures. For the given vibration absorber system design, the normalized amplitude y of the main system can be calculated as y= (7.1) where r is the ratio of absorber's mass to main system's mass, ( is the damping ratio of the main system, /) is the ratio of the nature frequency of the main system to the harmonic force frequency, and f: is the ratio of the nature frequency of the absorber to the harmonic force frequency. In this case study, r and ( are set as constants with 7=0.01 and (=0.01, whereas /| and are considered to be random variables that follow normal distributions, with #1~N (1, 0.0257) and #>~N (1, 0.025%), respectively. v It is considered as system failure when the normalized amplitude y reaches beyond a critical value of yo=28, thereby the limit state equation can be accordingly defined as G(5,, 5,) = (5,, 5,)28. (a) Plot the 3D surface of the normalized amplitude y with respect to ; and /%, and the contour of the limit state function G(f;, f;) = 0 within the design space of 0.9