Question

PROBLEMS 89 3-30. A system composed of a mass of S kg and an elastic member having a modulus of 45 N/m is less than critically damped. When the mass is givén an initial displacement and released from rest, the overshoot (the displacement attained past the equilibrium position) is 25% Determine the dam ping factor and the damping constant. 3-31. A mass-spring system is critically damped. The spring constant is 3 Ib/in. and the body weighs 11.58 lb. (a) The system is given an initial displacement x,-2 in and initial velocity x::-30 in/sec. Determine (a) the overshoot. (b) If xo-2 in, and Xo ,--20 in./sec, obtain the overshoot. 3-32. A large gun with supporting base weighs 2500 lb, and has a recoil system composed of spring k 32 000 lb/ft and a viscous shock absorber (dasbpot) for which damping is critical. The recoil distance is 3 ft. Determine (a) the initial recoil velocity, (b) tbe time to return to a position 0.1 in. from the initial position, and (c) displacement at t 0.5 sec 3-33. An artillery piece and integral supporting base weigh 1800 lb and have a recoil spring for which k 24000 lb/ft. When fired, the system recoils 40 in. A dash pot having a critical damping constant is then engaged on the return stroke oDetermine (a) the initial velocity of recoil, (b) the damping constant, and (c) 3-34. For the case of Coulomb damping, by equating the loss of potential energy for the position of the gun and base at 0.5 sec of the return stroke a half cycle to the work done by the constant friction force, show that the loss of amplitude is 2Fa/k. Designate the initial amplitude as xo and a half cycle later as xo.s 3-35. A mass-spring system is subjected to an undetermined damping condition. The vibrating body weighs 47 lb and the spring modulus is 30 Ibin. The amplitudes of successive cycles were measured as 1.64, 1.59, 154, 149, 1.44,, in. (a) Define the type and magnitude of the damping force and (b) determine the frequency of the damped oscillation. 9 3-36. A system composed of mass of 57.61 kg and a spring having a modulus of 9100 N/m is subjected to a Coulomb damping force of 6.825 N. If the initial amphi- tude is 4 cm, determine (a) the amplitude at the end of 8 cycles and (b) the frequency of the damped oscillation. 3-37. A simple structure exhibits hysteresis damping The period of the vibration is 0.30 sec and the amplitude at the ninth cycle is 0.9309 times that at the first cycle. Determine the hysteresis damping constant b and wnte the equation of motion for this case

Please answer 3-34 and 3-35. Please provide all steps so I can follow along.

Answer 1

1 Equillibinium positiorn 2.34 าท mSurface with constant damping force Fa At 1 Cycle time or LxTime period Applying Lano of conservation of Energy b Change in kE oF block Change in PE of spring Energy dissipated in coulom b damping Work done against damping Dampingförce x Displacemer Loss of amplitude ( Response curve Ko Amplitute χ(t)

as Mass (m) 47lb 21-3188 Ig Spring modulus (K) = 30 lb/in= 525380s Logarithmic Decreement() NIm 640 026 I 1.44 ohere Damping factor Damping Coefficient (C): 2m5Bn 1ーー15253-805 = 15.698 rad/sec m v21.3188 C= 2x 213188x 4.138 x10-3 x 15.698 = 2.77 Ns/m Frequency of Damped oscillation UWI)-49n-b2. = 15-6981|-(4138 x10?) 15 697 radllsec Since the damping-Factor るis very small, hence there is very litHe difference between natural Frequency damped frequency