Question

3. Consider a system consisting of a Ikg mass connected to a spring with spring co -ION/. and a friction coefficient of 6N/m, subject to an external force of 15 cos 22 newtons). (a) (5 points) Write down the general solution to this system. myll thy't ny=0 "y"+by't loy=0 F(+)= Iscoszt m 2 tlom +10 - (0) (3 points) Label the steady state response and the transient part of the son above (0) (3 points) is the 3 points) Is the mass-spring system underdamped, critically damped, or over- damped? Explain. ing b-16 K=10 (5 points) Rewrite the steady state response in amplitude-phase form.

Answer 1

m=1kg, k=100 0 kolo N b = 6 MS/M F = 15 cos at 19) ega of motion P er of motion: restoring force clamping face criven force = -kx = -box = ircosat then en A = -ku - bdx + 15 4032 - + Godx + 10% = 151686) (a) + egy + box = 15669 2t 7 + 6 +0 =D T: +6436-40 = -321 - then, X = Ge Cost) + sen Let now = A Coszt + Bximat = -3 Asin 2x + 28c0bgt = -4A Coszt 4-46 Sint x Put u bault, (-4AC02 - ue sin 2 t) + 6-2 Arinzt +26t) +10 A02t + Brint) = 15LO2+ + vin 24- + Coszt (-4A +12B +10A) + 6n2+1-4G-12A +106) = 15102 tornat > A 10 6A +126 = 15 Get-12A + B = -66 +12=15 965 - B = 3/3 A= 31 so Xp = 2t + Smit x = xp+Xc {cosat XIE) = Ge sint & & sinat - cost the

16) Xp = steady state response Xe a transient response - (1) here k h = 10 4m2 - 4 = undee clamped . 4m2