Question

A mass m = 1kg attached to a spring of rigidity k = 16 is set in motion using a force F (t) = 3 sin (4t). Assuming that there is no damping and zero initial conditions:

a) Find the corresponding complementary solution yc (t).

b) Find the corresponding particular solution yp (t).

c) Find the corresponding general solution y (t) taking into account the initial conditions.

d) In this problem, is the motion of the mass stable or unstable? Justify your answer.

Answer 1

love LAS gave Natural positi of spring 0 m t 01 =) -kx + ma" — m -Ang + F(t) 2011 Equilibrium mg=ks m=1kg, k= 16 N/m in motion External force flt)= 3 sin4t het a (t) denote the displacement of the mass from Equilibrium, By Newton's recond low: mal -K 18+ x) + mg + Fit) -WA- tkx= Flt) ("img (1) = ks) no + 16x= 3 sinat Initial conditions: 210) = 0, x 10) = 0 - (2) (0) x" + 164 = 3 Sintt → (02+16/4= 3 sinet auxiliary Equation: m²+16=0 to complementory solution yc (t)= cq los 4t+ca sinat, where Ca and ca are arb arbitrary constants. (b) Particular solutim y pltl=1 3 sin 4t (D2+16) > Yplt) = 3 puting D²= -16 02+16) D? +16=0 so =-3xt as 4t 1 sinat D²+ a2 -t Cosat Deď dt m= + 4 I Sinqt 2X4 - Yplt) = - 3t Con4t 2a here a= 8 ho yplt) = -3 (tlo 4t) 8 Scanned with CamScanner

colo (6) The general solutions y= Yelt) + Yplt) yıtl=C₂ sat+ C2 sim 4t-3 t wat Now y10)=0 (1 = y'it) = -44 sin4 t +462 47 4t - 44t Sin4 t +64t Now y'lo)=0 => 462 462= 3 6 3 X 1 = 0 =C2 = 3 8 32 T loy(t)= 3 Sin 4 t- 3 t sous 4 t) B 32 3. Sin 4t - at 3 tas 4t Ami ylt)= 3 Sin 4 t 32 (d) This is under damped oncillations due to presence of sine and conine torms in the solution, and lim y(t) to so it does not approach the Equilibrium position over time. Hence the motion of the mass in unstable. csscanned with CamScanner
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