University:
California State University, Northridge
Course:
MATH 280 | Applied Differential Equations
Academic year:

2013
Views:

120

Pages:

2
Author:

litauiskzzn

Student Name: Student I.D.: Question 1. [24 points] After a mass weighing 10 pounds is attached to a 5-foot spring, the spring measures 7 feet. This mass is removed and replaced with another mass that weighs 8 pounds. The entire system is placed in a medium that offers a damping force that is numerically equal to the instantaneous velocity. (a)[10 points] Find the equation of motion if the mass is initially released from a point 12 foot below the equilibrium position with a downward velocity of 1 ft/s. Solution: The equation is 14 x 00 + x 0 + 5x = 0, x(0) = 12 , and x 0 (0) = 1 =⇒ x = e −2t ( 12 cos 4t + 21 sin 4t ). (b)[4 points] Express the equation of motion in the form given in p x(t ) = Ae −λt sin( ω2 − λ2 t + φ). Solution: x = p1 e −2t 2 sin(4t + π4 ). (c)[6 points] Find the times at which the mass passes through the equilibrium position heading downward. π Solution: x = p1 sin(4t + π4 ) = 0, and x 0 > 0 =⇒ 4t + π4 = 2kπ, k is an positive integer. =⇒ t = k2 − 16 ,k = 2 1, 2, 3, . . . . (d)[4 points] Graph the equation of motion. Solution: 1 Question 2. [24 points] A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 5 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (a)[16 points] Find the equation of motion if the mass is driven by an external force equal to f (t ) = 12 cos 2t + 3 sin 2t . Solution: If x 00 + 2x 0 + 5x = 12 cos 2t + 3 sin 2t , x(0) = 1 and x 0 (0) = 5, then x c = e −t (c 1 cos 2t + c 2 sin 2t ) and x p = 3 sin 2t , so that the equation of motion is x = e −t cos 2t + 3 sin 2t . (b)[4 points] Graph the transient and steady-state solutions on the same coordinate axes. Solution: (c)[4 points] Graph the equation of the motion. Solution: 2