ġ(t) ти x(t) k v(t) ➤f(t) Figure 1: A model of piezosensors Figure 2: A load cell to measure force 2. A start-up company aims to develop a load cell that measures forces varying quickly with respect to time. An initial design consists of a linear spring and a dashpot in series; see Fig. 2. The spring has a spring constant k, and the dashpot has a damping coefficient c. When a force f(t) is applied, the displacement (t) is measured. By measuring the displacement z(t), 1 the company hopes to reconstruct the force f(t). The transfer function from f(t) to x(t) is given as H(s) cs+k cks (2) Answer the following questions (a) What is the frequency response function G(w) from fƒ(t) to x(t)? (b) Derive the magnitude |G(w)| and phase LG(w). Each should be a function of the driving frequency w. Identify a cutoff frequency we, above and below which |G(w)| behaves very differently. Derive asymptotic behavior of |G(w) when w> we (c) Sketch the Bode plot of the magnitude |G(w). Show the cutoff frequency and asymptotes in the Bode plot. (d) Based on the Bode plot you obtain in part (c), answer the following questions. i. What is the frequency range for which the load cell has a constant gain (i.e., sensi- tivity) to reconstruct the force f(t)? In other words, what is the bandwidth of this load cell? ii. What can you do to increase the bandwidth? Please justify your answer. iii. Explain why the current design in Fig. 2 cannot measure a static force. (e) The CEO of the company wants to modify the design in Fig. 2 so that it can measure a static force. If this can be done, the company will sell the load cell as a scale to measure weights. Engineer X receives the assignment from the CEO. To meet the assignment, what options does Engineer X have in modifying the design? Explain the rationale. [Hint: resigning from the job is not an option!]
ġ(t) ти x(t) k v(t) ➤f(t) Figure 1: A model of piezosensors Figure 2: A load cell to measure force 2. A start-up company aims to develop a load cell that measures forces varying quickly with respect to time. An initial design consists of a linear spring and a dashpot in series; see Fig. 2. The spring has a spring constant k, and the dashpot has a damping coefficient c. When a force f(t) is applied, the displacement (t) is measured. By measuring the displacement z(t), 1 the company hopes to reconstruct the force f(t). The transfer function from f(t) to x(t) is given as H(s) cs+k cks (2) Answer the following questions (a) What is the frequency response function G(w) from fƒ(t) to x(t)? (b) Derive the magnitude |G(w)| and phase LG(w). Each should be a function of the driving frequency w. Identify a cutoff frequency we, above and below which |G(w)| behaves very differently. Derive asymptotic behavior of |G(w) when w> we (c) Sketch the Bode plot of the magnitude |G(w). Show the cutoff frequency and asymptotes in the Bode plot. (d) Based on the Bode plot you obtain in part (c), answer the following questions. i. What is the frequency range for which the load cell has a constant gain (i.e., sensi- tivity) to reconstruct the force f(t)? In other words, what is the bandwidth of this load cell? ii. What can you do to increase the bandwidth? Please justify your answer. iii. Explain why the current design in Fig. 2 cannot measure a static force. (e) The CEO of the company wants to modify the design in Fig. 2 so that it can measure a static force. If this can be done, the company will sell the load cell as a scale to measure weights. Engineer X receives the assignment from the CEO. To meet the assignment, what options does Engineer X have in modifying the design? Explain the rationale. [Hint: resigning from the job is not an option!]
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
Related questions
Question
![ġ(t)
ти
x(t)
k
v(t)
➤f(t)
Figure 1: A model of piezosensors
Figure 2: A load cell to measure force
2. A start-up company aims to develop a load cell that measures forces varying quickly with
respect to time. An initial design consists of a linear spring and a dashpot in series; see Fig. 2.
The spring has a spring constant k, and the dashpot has a damping coefficient c. When a
force f(t) is applied, the displacement (t) is measured. By measuring the displacement z(t),
1
the company hopes to reconstruct the force f(t). The transfer function from f(t) to x(t) is
given as
H(s)
cs+k
cks
(2)
Answer the following questions
(a) What is the frequency response function G(w) from fƒ(t) to x(t)?
(b) Derive the magnitude |G(w)| and phase LG(w). Each should be a function of the driving
frequency w. Identify a cutoff frequency we, above and below which |G(w)| behaves very
differently. Derive asymptotic behavior of |G(w) when w<we and w >> we
(c) Sketch the Bode plot of the magnitude |G(w). Show the cutoff frequency and asymptotes
in the Bode plot.
(d) Based on the Bode plot you obtain in part (c), answer the following questions.
i. What is the frequency range for which the load cell has a constant gain (i.e., sensi-
tivity) to reconstruct the force f(t)? In other words, what is the bandwidth of this
load cell?
ii. What can you do to increase the bandwidth? Please justify your answer.
iii. Explain why the current design in Fig. 2 cannot measure a static force.
(e) The CEO of the company wants to modify the design in Fig. 2 so that it can measure a
static force. If this can be done, the company will sell the load cell as a scale to measure
weights. Engineer X receives the assignment from the CEO. To meet the assignment,
what options does Engineer X have in modifying the design? Explain the rationale.
[Hint: resigning from the job is not an option!]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1720cd77-2294-45a6-8b70-43447dc02920%2F7823cd28-3925-4127-9521-4924340034e7%2F9crlqdm_processed.png&w=3840&q=75)
Transcribed Image Text:ġ(t)
ти
x(t)
k
v(t)
➤f(t)
Figure 1: A model of piezosensors
Figure 2: A load cell to measure force
2. A start-up company aims to develop a load cell that measures forces varying quickly with
respect to time. An initial design consists of a linear spring and a dashpot in series; see Fig. 2.
The spring has a spring constant k, and the dashpot has a damping coefficient c. When a
force f(t) is applied, the displacement (t) is measured. By measuring the displacement z(t),
1
the company hopes to reconstruct the force f(t). The transfer function from f(t) to x(t) is
given as
H(s)
cs+k
cks
(2)
Answer the following questions
(a) What is the frequency response function G(w) from fƒ(t) to x(t)?
(b) Derive the magnitude |G(w)| and phase LG(w). Each should be a function of the driving
frequency w. Identify a cutoff frequency we, above and below which |G(w)| behaves very
differently. Derive asymptotic behavior of |G(w) when w<we and w >> we
(c) Sketch the Bode plot of the magnitude |G(w). Show the cutoff frequency and asymptotes
in the Bode plot.
(d) Based on the Bode plot you obtain in part (c), answer the following questions.
i. What is the frequency range for which the load cell has a constant gain (i.e., sensi-
tivity) to reconstruct the force f(t)? In other words, what is the bandwidth of this
load cell?
ii. What can you do to increase the bandwidth? Please justify your answer.
iii. Explain why the current design in Fig. 2 cannot measure a static force.
(e) The CEO of the company wants to modify the design in Fig. 2 so that it can measure a
static force. If this can be done, the company will sell the load cell as a scale to measure
weights. Engineer X receives the assignment from the CEO. To meet the assignment,
what options does Engineer X have in modifying the design? Explain the rationale.
[Hint: resigning from the job is not an option!]
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