6. Beats. Consider a free undamped oscillator with mass m = 1 and stiffness k = 3025, which satisfies the differential equation x"(t) + 3025x(t) = 0. The natural frequency is wo = √k/m = 55 and the general solution is xe(t) = c₁ cos(55t) + C2 sin(55t). Now suppose we subject this oscillator to a periodic external force with amplitude 500 and frequency 45: x"(t) + 3025x(t) = 500 cos(45t). (a) Find a particular solution of the form xp(t) = A cos(45t) + B sin(45t). (b) Find the general solution r(t) = xc(t) + xp(t). (c) Find the unique solution r(t) with initial conditions x(0) = 0 and x'(0) = 0. (1) n (0) 100 [TT TT 11
6. Beats. Consider a free undamped oscillator with mass m = 1 and stiffness k = 3025, which satisfies the differential equation x"(t) + 3025x(t) = 0. The natural frequency is wo = √k/m = 55 and the general solution is xe(t) = c₁ cos(55t) + C2 sin(55t). Now suppose we subject this oscillator to a periodic external force with amplitude 500 and frequency 45: x"(t) + 3025x(t) = 500 cos(45t). (a) Find a particular solution of the form xp(t) = A cos(45t) + B sin(45t). (b) Find the general solution r(t) = xc(t) + xp(t). (c) Find the unique solution r(t) with initial conditions x(0) = 0 and x'(0) = 0. (1) n (0) 100 [TT TT 11
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Plz solve all parts asap will definitely upvote
![3025,
6. Beats. Consider a free undamped oscillator with mass m = 1 and stiffness k =
which satisfies the differential equation
x"(t) + 3025x(t) = 0.
The natural frequency is wo
√k/m = 55 and the general solution is e(t) = C₁ cos(55t) +
C₂ sin(55t). Now suppose we subject this oscillator to a periodic external force with amplitude
500 and frequency 45:
x"(t) + 3025x(t)
500 cos(45t).
(a) Find a particular solution of the form xp(t) = A cos(45t) + B sin(45t).
(b) Find the general solution r(t) = xc(t) + xp(t).
(c) Find the unique solution r(t) with initial conditions (0) = 0 and x'(0) = 0.
(d) Express your solution in the form x(t) = C sin(at) sin(ßt). [Hint: Use the trig identities
-
cos(a - b) = cos a cos 3 + sin a sin 3,
cos(a + B) = cos a cos - sin a sin 3,
cos(a + 3) = 2 sin a sin 6.]
cos(a - b)
(e) Plot your solution r(t) for t between 0 and 3π/5. [Use a computer.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96191c65-e923-461d-9658-eb756d234e0f%2Ff416834b-b118-472e-9659-9bebf089aeb3%2Fe9cgd9i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3025,
6. Beats. Consider a free undamped oscillator with mass m = 1 and stiffness k =
which satisfies the differential equation
x"(t) + 3025x(t) = 0.
The natural frequency is wo
√k/m = 55 and the general solution is e(t) = C₁ cos(55t) +
C₂ sin(55t). Now suppose we subject this oscillator to a periodic external force with amplitude
500 and frequency 45:
x"(t) + 3025x(t)
500 cos(45t).
(a) Find a particular solution of the form xp(t) = A cos(45t) + B sin(45t).
(b) Find the general solution r(t) = xc(t) + xp(t).
(c) Find the unique solution r(t) with initial conditions (0) = 0 and x'(0) = 0.
(d) Express your solution in the form x(t) = C sin(at) sin(ßt). [Hint: Use the trig identities
-
cos(a - b) = cos a cos 3 + sin a sin 3,
cos(a + B) = cos a cos - sin a sin 3,
cos(a + 3) = 2 sin a sin 6.]
cos(a - b)
(e) Plot your solution r(t) for t between 0 and 3π/5. [Use a computer.]
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