Question: Given equation, describes a linear, second-order, constant-coefficient differential equation used to model a mechanical spring damper system.
(i) By expressing Eq., in the following form:
(d2y/dt2) + ((ωn/Q)(dy/dt)) + ωn2y(t) = (1/M)x(t)
determine the values of ωn and Q in terms of mass M, damping factor r, and the spring constant k.
(ii) The variable ωn denotes the natural frequency of the spring damper system. Show that the natural frequency ωn can be increased by increasing the value of the spring constant k or by decreasing the mass M.
(iii) Determine whether the system is (a) linear, (b) time-invariant, (c) memoryless,
(d) causal, (e) invertible, and (f) stable.