1. The vibration of a dental drill is governed by the following differential equation +0.1+ (106+ k)x = u(t) (1) where u(t) is the input force drilling on a tooth and x(t) is the output vibration of the drill. In addition, k is a constant stiffness parameter, which depends on the bearings used in the dental drill. (a) Determine the transfer function H(s). (b) Assume that k = 2×106. If u(t) = cos 1000t, what is the magnitude and phase of H(s)? (c) A dentist's nightmare is to have the drill shaking violently while drilling a tooth. Assume that the dentist drill is operating at 1000 Hz (or 6280 rad/s). What stiffness parameter k do you want to avoid? Otherwise, you will never sell the dental drill. (d) Due to design constraints (e.g., costs, dimensions, vendors, and fatigue life), you choose bearings with k = 2 × 106. Originally, the dental drill is designed to operate at 350 Hz (or 2200 rad/s), but you want to reduce the vibration further. Give two alternatives to make your design better. Explain why.