1. Consider the mass-damper system shown in Fig. 1. Velocity v(t) of the mass takes the following form v(t) = − (1 − j)e³(2πt) + jej (4πt) 1 - ej(6πt) +c.c. 2 (1) where j√-1 and c.c. represents complex conjugates of the three complex terms in (1). Moreover, the velocity v(t) and time t are both in MKS units. Answer the following questions. (a) What is the fundamental frequency wo of the velocity v(t)? Please include the unit of wo. (b) Compare the velocity v(t) in (1) with the following complex Fourier series 8 v(t): = Σ Cejnwot n=-∞ where wo is the fundamental frequency. Calculate the magnitude and phase of In can be positive or negative. Obtain and plot the complex spectrum of v(t). (c) Integrate the velocity v(t) in (1) to obtain displacement x(t) as (2) Сп where x(t) = =- (1-1) ej(2πt) + jej (4nt). - 2πj 4πj 1 ~ēj (6πt) + c.c. 2 (6πj) (3) Express the displacement x(t) in the form of trigonometric functions (e.g., sine and cosine functions). (d) For the velocity v(t) shown in (1), does its Fourier transform exist? Explain why. (e) When a Fourier series converges to a periodic function f(t), i.e., 8 f(t) = A0+(An cos nwot + B sin nwot) n=1 what is minimized when more terms are retained in the Fourier series? (4)
1. Consider the mass-damper system shown in Fig. 1. Velocity v(t) of the mass takes the following form v(t) = − (1 − j)e³(2πt) + jej (4πt) 1 - ej(6πt) +c.c. 2 (1) where j√-1 and c.c. represents complex conjugates of the three complex terms in (1). Moreover, the velocity v(t) and time t are both in MKS units. Answer the following questions. (a) What is the fundamental frequency wo of the velocity v(t)? Please include the unit of wo. (b) Compare the velocity v(t) in (1) with the following complex Fourier series 8 v(t): = Σ Cejnwot n=-∞ where wo is the fundamental frequency. Calculate the magnitude and phase of In can be positive or negative. Obtain and plot the complex spectrum of v(t). (c) Integrate the velocity v(t) in (1) to obtain displacement x(t) as (2) Сп where x(t) = =- (1-1) ej(2πt) + jej (4nt). - 2πj 4πj 1 ~ēj (6πt) + c.c. 2 (6πj) (3) Express the displacement x(t) in the form of trigonometric functions (e.g., sine and cosine functions). (d) For the velocity v(t) shown in (1), does its Fourier transform exist? Explain why. (e) When a Fourier series converges to a periodic function f(t), i.e., 8 f(t) = A0+(An cos nwot + B sin nwot) n=1 what is minimized when more terms are retained in the Fourier series? (4)
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Transcribed Image Text:1. Consider the mass-damper system shown in Fig. 1. Velocity v(t) of the mass takes the
following form
v(t) = − (1 − j)e³(2πt) + jej (4πt)
1
-
ej(6πt)
+c.c.
2
(1)
where j√-1 and c.c. represents complex conjugates of the three complex terms in (1).
Moreover, the velocity v(t) and time t are both in MKS units. Answer the following questions.
(a) What is the fundamental frequency wo of the velocity v(t)? Please include the unit of
wo.
(b) Compare the velocity v(t) in (1) with the following complex Fourier series
8
v(t):
=
Σ Cejnwot
n=-∞
where wo is the fundamental frequency. Calculate the magnitude and phase of
In can be positive or negative. Obtain and plot the complex spectrum of v(t).
(c) Integrate the velocity v(t) in (1) to obtain displacement x(t) as
(2)
Сп
where
x(t) =
=-
(1-1) ej(2πt)
+
jej (4nt).
-
2πj
4πj
1
~ēj (6πt) + c.c.
2 (6πj)
(3)
Express the displacement x(t) in the form of trigonometric functions (e.g., sine and
cosine functions).
(d) For the velocity v(t) shown in (1), does its Fourier transform exist? Explain why.
(e) When a Fourier series converges to a periodic function f(t), i.e.,
8
f(t) = A0+(An cos nwot + B sin nwot)
n=1
what is minimized when more terms are retained in the Fourier series?
(4)
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