Solve the model for a driven spring/mass system with damping d²x dt² where m = 1/2, ß = 1, k = 5, and the driving function f is the meander function given below with amplitude 10, and a = π. f(t) m Ox(t) = 2(²-=-= dx + B + kx = f(t), x(0) = 0, x'(0) = 0, dt 10 -10 CO 0 x(t) = 2(4- •* sin(3t) ——* cos(3t) + 4 (-1)^[1 - e - n=1 - e* sin(3t) - O x(t) = 2(1 - * cos(3t) — e - Ox(t) = 2(1 x(t) = 2(1-e* cos(3t) -t ) = 2(1 - e* cos(3t) - 2a * sin(3t)) + 4 -t 3a 00 * cos(3t) + 4 (-1)^[1 - e(t - 2n) sin(3(t - 2nn)) - e(t - 2nr) cos(3(t - 2n)]2(t-2nit) n=1 -t 00 sin(3t)) + 4 (-1) [1- — - - n=1 sin(3t)) 4a 00 n=1 e-(t-nit) sin(3 (t-nit)) - e-(-) cos(3(tn)) (t - nπ) (-1)" [1 -¯(t - m²) cos(3{(t− n)) - e(t - mm) sin(3(t - mi))](t – nr) -e - e-(t-2nn) cos(3(t - 2nπ))--(-2mm) sin(3(t-2nn)) (t-2nn) 3
Solve the model for a driven spring/mass system with damping d²x dt² where m = 1/2, ß = 1, k = 5, and the driving function f is the meander function given below with amplitude 10, and a = π. f(t) m Ox(t) = 2(²-=-= dx + B + kx = f(t), x(0) = 0, x'(0) = 0, dt 10 -10 CO 0 x(t) = 2(4- •* sin(3t) ——* cos(3t) + 4 (-1)^[1 - e - n=1 - e* sin(3t) - O x(t) = 2(1 - * cos(3t) — e - Ox(t) = 2(1 x(t) = 2(1-e* cos(3t) -t ) = 2(1 - e* cos(3t) - 2a * sin(3t)) + 4 -t 3a 00 * cos(3t) + 4 (-1)^[1 - e(t - 2n) sin(3(t - 2nn)) - e(t - 2nr) cos(3(t - 2n)]2(t-2nit) n=1 -t 00 sin(3t)) + 4 (-1) [1- — - - n=1 sin(3t)) 4a 00 n=1 e-(t-nit) sin(3 (t-nit)) - e-(-) cos(3(tn)) (t - nπ) (-1)" [1 -¯(t - m²) cos(3{(t− n)) - e(t - mm) sin(3(t - mi))](t – nr) -e - e-(t-2nn) cos(3(t - 2nπ))--(-2mm) sin(3(t-2nn)) (t-2nn) 3
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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Question
![Solve the model for a driven spring/mass system with damping
d²x
dx
+ ß + kx = f(t), x(0) = 0, x'(0) = 0,
वहर
dt
where m = 1/2, ß = 1, k = 5, and the driving function f is the meander function given below with amplitude 10, and a = 1.
f(t)
m
Ox(t)
Ox(t) = 2
= 2(²-1/² -
x(t)
=
10
-10
00
- - .¯* sin(3t) — —¯* cos(3t) + 4
O x(t) = 2(1 - ²* cos(3t) —
e
-
Ox(t) = 2(1
2a
= 2(1-e* cos(3t) -
(-1)″ [1 -
-e* e-(²-
-t
) = 2(1 - e* cos(3t) -
00
= 2(²-=-=
- e* sin(3t) --* cos(3t) + 4 (-1)^[1 - e(t = 2n) sin(3(t - 2nn)) - e(t - 2nr) cos(3(t - 2n)]2(t - 2nit)
n=1
3a
* sin(3t)) + 4
n=1
-t
e* sin(3t))
4a
00
n=1
e-(t-nt) sin(3(t - nm))--(-) cos(3(t - nn)) (t-nit)
(-1)" [1 - ¯(t−m²) cos(3(t - nit)) - e-(tr) sin(3(t - mi))]2(t − n)
00
* sin(3t)) + 4 (-1) [1- e-(t-2nn) cos(3 (t - 2nπ)) - e-(-2mm) sin(3(t-2nπ)) (t-2nn)
3
n=1
X](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b5d6349-72f7-48e1-ae0c-329bb96c39f5%2F0a60d0e2-f8cd-4f11-b821-b7731ef900f1%2F0nw1ff_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Solve the model for a driven spring/mass system with damping
d²x
dx
+ ß + kx = f(t), x(0) = 0, x'(0) = 0,
वहर
dt
where m = 1/2, ß = 1, k = 5, and the driving function f is the meander function given below with amplitude 10, and a = 1.
f(t)
m
Ox(t)
Ox(t) = 2
= 2(²-1/² -
x(t)
=
10
-10
00
- - .¯* sin(3t) — —¯* cos(3t) + 4
O x(t) = 2(1 - ²* cos(3t) —
e
-
Ox(t) = 2(1
2a
= 2(1-e* cos(3t) -
(-1)″ [1 -
-e* e-(²-
-t
) = 2(1 - e* cos(3t) -
00
= 2(²-=-=
- e* sin(3t) --* cos(3t) + 4 (-1)^[1 - e(t = 2n) sin(3(t - 2nn)) - e(t - 2nr) cos(3(t - 2n)]2(t - 2nit)
n=1
3a
* sin(3t)) + 4
n=1
-t
e* sin(3t))
4a
00
n=1
e-(t-nt) sin(3(t - nm))--(-) cos(3(t - nn)) (t-nit)
(-1)" [1 - ¯(t−m²) cos(3(t - nit)) - e-(tr) sin(3(t - mi))]2(t − n)
00
* sin(3t)) + 4 (-1) [1- e-(t-2nn) cos(3 (t - 2nπ)) - e-(-2mm) sin(3(t-2nπ)) (t-2nn)
3
n=1
X
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