. As shown in the spring-damping system, where xt) is the input displacement, x.(1) is the output displacement, k is the spring stiffness, and c is the damping coefficient. (1)Find the transfer function of the system; (2) Calculate the amplitude-frequency characteristics characteristics of the system. and phase-frequency

A force of 20 newton stretches a spring 1 meter. A 5 kg mass is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 10 times the instantaneous velocity.   1) Let x denote the downward displacement of the mass from its equilibrium position. [Note that x>0 when the mass is below the equilibrium position. ] Assume the mass is initially released from rest from a point 3 meters above the equilibrium position. Write the differential equation and the initial conditions for the function x(t) 2)  Solve the initial value problem that you wrote above. 3)Find the exact time at which the mass passes through the equilibrium position for the first time heading downward. (Do not approximate.) 4)Find the exact time at which the mass reaches the lowest position. The "lowest position" means the largest value of x

Question 3 (a) A system is critically damped. Derive an expression for the time at which the dispacement achieves a maximum or minimum. What is the time that maximum displacement amplitude occurs for the following: For an aircraft landing, the natural frequency can be taken as 2.5 rad/s, initial displacement is 0 mm and sink rate, i.e. initial velocity is -50mm/s. For the result, roughy sketch the response of the system for the initial conditions. (b) The seat of a helicopter with a pilot weighs 1000N and is found to have a static deflection of 10mm under self-weight. The vibration of the rotor is transmitted to the base of the seat as harmonic motion with a frequency of 4Hz and an amplitude of 0.2mm. What are the levels of vibration felt by the pilot including velocity and acceleration?

The natural frequency of a system is determined to be 15 Hz ignoring the effects of damping. Calculate the damped natural frequency of the system assuming a damping ratio of 0.7.

The equation of motion of a certain mass-spring-damper system is 5x'' + cx' +10x = f(t). Suppose that f(t) = Fsin(wt). Define the magnitude ratio as M = X/F. Determine the natural frequency w_n, the peak frequency w_p, and the peak magnitude ratio M_p for damping ratio = 0.1 and the damping ratio = 0.3

Sketch frequency response of the system from part (a) in the form of Bode plot as accurately as you can.

A mass of 10kg oscillates at the end of a spring of stiffness, 5N/m. If the damping in the system is such that it gives a damping ratio of 0.45, determine. i) The critical damping. ii) The value of the damping coefficient. iii) The damped natural frequency. iv) The equation for the displacement if the initial velocity is 0.025m/s, and the initial displacement is 0 m.

Given a SDOF of spring-mass-coulomb damping, with mas of 0.1kg, stiffness of 10N s/m and a damping force of 0.1N is excited by a harmonic force of 1N amplitude at 1hz. Assuming steady-state condition, calculate the amplitude of vibrationn of the mass. How long the vibration will continu if the excitation is removed?

4. A mass attached to a spring hanging from the ceiling is pulled down 10 cm from its rest position and released. The amplitude decreases by 5% each second and the spring oscillates 3 times each second. Find the damping constant c rounded to 4 decimal places. Then find a function D(t) that gives the distance of the mass from equilibrium at t seconds after the mass was released.

Find the free-vibration response of a spring-mass system subject to Coulomb damping for the following initial conditions x(0) = 0.5 m, *(0) = 0. Data: m= 10kg, k = 200 N/m, µ = 0.5. Q1 . ...