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3. For the following, choose the best description of the system from the following: Simple Harmonic Motion (SHM) Overdamped (OD) Underdamped (UD) Critically

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3. For the following, choose the best description of the system from the following:
Simple Harmonic Motion (SHM) Overdamped (OD) Underdamped (UD) Critically
Damped (CD) Beating (B) Resonant (R) Steady-State plus Transient (S ST) 3. y” + 43/ = 0
b. y” + (1.8)2y = cos(2t)
c. y” + 4)! = cos(2t)
d. y” + y’ + y = 0
e. y” + y’ + y = c0505)
f. y”+2y’+y=0
4. The motion of a force mass-spring system is described by the following IVP: u” + 9n = cos(3t), u(0) = 0, u’(0) = 0
(a) Explain why you expect resonance to occur. (b) Solve this WP and sketch the graph of the solution. 5. The motion of a force mass-spring system is described by the following IVP:
u" + (2.8)211 = cos(3t), 11(0) 2 0, u’(0) = 0
Explain why you expect the beats phenomenon to occur. (3) Explain why you expect resonance to occur. (b) Solve this IVP and write your solution in the form A sin(at) sinfltt).
(c) Determine the length of the beats and the period of the oscillation. 6. A mass m = 1 is attached to a sprin g with constant k = 2 and damping constant 1/. Determine the
value of 1: so that the motion is critically damped. 7. The position function of a mass-spring system satisfies the differential equation
mx” + yx’ + kx : cos(mt), 13(0) : 0, x’(0) : 0. Assumem = 1 and k = 9. [fy at 0, the amplitude of the forced oscillation is given by C = J;— (9—m2)2 +1,sz
Assume y = 1. Differentiate C to find the value of a; at which practical resonance occurs.
Determine the corresponding value of C. V. Laplace Transform 1. Use the definition of the Laplace transform to find 17(5) = .I {f (15)] for the following functions. (a)f<t)={03 33::
(we): 2′ 2:: 0, t<2
5e "gt, 2:: l
“W :{22 i2?

 
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