In the following problem we study a springboard vibrating system. The system is made of one springboard having a young modulus E = 7 GPa and a spring of stiffness K=10000 N/m placed at point B. The water under the springboard generates a viscous damping equivalent to c=800 N.s/m at point C. The cross sectional area of the springboard is a 5x30 cm rectangle, 5 cm being the thickness t of the springboard and 30 cm the width. The man stands on the tip C of the springboard to jump in a pool. Consider a one-degree of freedom system in this study. The generalized coordinate is to be the vertical deflection at point C. Consider: L1=50 cm and L2= 2.5 m

Part 1:  

a) Neglect the mass of the springboard in this question. Find the equivalent stiffness of the system. Hint: You may first find the reaction of the spring function of the weight of the man W using the superposition method. 

b) The man having a weight of 80 kg steps on the springboard with a downward velocity v=0.1m/s. Find the response of the springboard. Neglect the mass of the springboard in this question again and use keq =2.5 KN/m.

Part 2:

In this part we aim to design a critically damped system by adding a new damper at point C in parallel to the water. 

a) Find the constant of linear damping of the new damper. 

b) Find the response of the springboard in a critical damping case when x0=0.1 m and ????!=0 m/s. 

c) Design the extra damper to add in parallel to the water by finding the needed length L of the following viscous damper. This damper consists of 2 coaxial rigid cylinders, the inner one connected to point C and the outer one is fixed connected to the ground. A liquid having a viscosity µ=20 Pa.s fills the gap between the 2 cylinders. Consider t=2 mm and d= 10 mm.

Springboard L2 L1 Water Displacement L.