Consider a damped undriven oscillator whose equation of motion is given by *=-w²x-yż Assume that the system is critically damped, such that y=2wo. When this is the case, a general solution is given by r(t) = (A+Bt)et/2 where A and B are constants. ) Express the general solution as a complex function. [Hint: This is a trick question.] 3) Assume the oscillator is initially at rest at the origin. At t = 0, it is given an impulse T (i.e., a udden blow), causing it to launch with initial positive velocity vo. Determine specific expression or both x(t) and v(t) in terms of known values. C) Determine both the time when the velocity is zero and the time when the velocity starts to ncrease (i.e., when the acceleration changes sign). ) Extending from parts b and c, sketch both x(t) and v(t), making sure to clearly indicate the elevant values (e.g., the maximum displacement, etc...). [Hint: The velocity is zero when the
Consider a damped undriven oscillator whose equation of motion is given by *=-w²x-yż Assume that the system is critically damped, such that y=2wo. When this is the case, a general solution is given by r(t) = (A+Bt)et/2 where A and B are constants. ) Express the general solution as a complex function. [Hint: This is a trick question.] 3) Assume the oscillator is initially at rest at the origin. At t = 0, it is given an impulse T (i.e., a udden blow), causing it to launch with initial positive velocity vo. Determine specific expression or both x(t) and v(t) in terms of known values. C) Determine both the time when the velocity is zero and the time when the velocity starts to ncrease (i.e., when the acceleration changes sign). ) Extending from parts b and c, sketch both x(t) and v(t), making sure to clearly indicate the elevant values (e.g., the maximum displacement, etc...). [Hint: The velocity is zero when the
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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![Consider a damped undriven oscillator whose equation of motion is given by
* = -w²x-yi
Assume that the system is critically damped, such that r =
solution is given by
r(t) = (A + Bt)ert/2
2wo. When this is the case, a general
where A and B are constants.
A) Express the general solution as a complex function. [Hint: This is a trick question.]
B) Assume the oscillator is initially at rest at the origin. At t = 0, it is given an impulse T (i.e., a
sudden blow), causing it to launch with initial positive velocity vo. Determine specific expression
for both x(t) and v(t) in terms of known values.
C) Determine both the time when the velocity is zero and the time when the velocity starts to
increase (i.e., when the acceleration changes sign).
D) Extending from parts b and c, sketch both x(t) and v(t), making sure to clearly indicate the
relevant values (e.g., the maximum displacement, etc...). [Hint: The velocity is zero when the
object is at the maximum displacement.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8334bf9f-036f-42c6-8278-821b6fa1dd3e%2F20a812a7-8d96-4850-aa0e-0c3351d2f27d%2Fcq2w65d_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a damped undriven oscillator whose equation of motion is given by
* = -w²x-yi
Assume that the system is critically damped, such that r =
solution is given by
r(t) = (A + Bt)ert/2
2wo. When this is the case, a general
where A and B are constants.
A) Express the general solution as a complex function. [Hint: This is a trick question.]
B) Assume the oscillator is initially at rest at the origin. At t = 0, it is given an impulse T (i.e., a
sudden blow), causing it to launch with initial positive velocity vo. Determine specific expression
for both x(t) and v(t) in terms of known values.
C) Determine both the time when the velocity is zero and the time when the velocity starts to
increase (i.e., when the acceleration changes sign).
D) Extending from parts b and c, sketch both x(t) and v(t), making sure to clearly indicate the
relevant values (e.g., the maximum displacement, etc...). [Hint: The velocity is zero when the
object is at the maximum displacement.
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