According to Eq. (21), the amplitude of forced steady periodic oscillations for the system mx" + cx' + kx Fo cos wt is given by //
According to Eq. (21), the amplitude of forced steady periodic oscillations for the system mx" + cx' + kx Fo cos wt is given by //
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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KVL and KCL
KVL stands for Kirchhoff voltage law. KVL states that the total voltage drops around the loop in any closed electric circuit is equal to the sum of total voltage drop in the same closed loop.
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![**Steady Periodic Solution:**
The steady periodic solution is expressed as:
\[ x_{sp}(t) = \frac{\sqrt{E_0^2 + F_0^2}}{\sqrt{(k - m \omega^2)^2 + (c \omega)^2}} \cos(\omega t - \alpha - \beta), \]
where \(\alpha\) is defined in Eq. (22) and \(\beta = \tan^{-1}(F_0/E_0)\).
**Suggestion:** Add the steady periodic solutions separately corresponding to \(E_0 \cos \omega t\) and \(F_0 \sin \omega t\) (see Problem 25).
**27. Amplitude of Forced Steady Periodic Oscillations:**
According to Eq. (21), the amplitude of forced steady periodic oscillations for the system \(mx'' + cx' + kx = F_0 \cos \omega t\) is:
\[ C(\omega) = \frac{F_0}{\sqrt{(k - m \omega^2)^2 + (c \omega)^2}}. \]
- **(a)** If \(c \geq c_{cr}/\sqrt{2}\), where \(c_{cr} = \sqrt{4km}\), show that \(C\) steadily decreases as \(\omega\) increases.
- **(b)** If \(c < c_{cr}/\sqrt{2}\), show that \(C\) attains a maximum value (practical resonance) when:
\[
\omega = \omega_m = \sqrt{\frac{k}{m} - \frac{c^2}{2m^2}} < \omega_0 = \sqrt{\frac{k}{m}}.
\]
**28. Cart-with-Flywheel Example:**
As indicated by the cart-with-flywheel example discussed in this section, an unbalanced rotating machine part typically results in a force having amplitude proportional to the square of the frequency, \(\dots\) Show that the am](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03106fa3-330c-422f-ad56-a6718a5c3a13%2F4c96f016-1644-4481-9b90-0ec659a83bf9%2Ffhgw0rn.jpeg&w=3840&q=75)
Transcribed Image Text:**Steady Periodic Solution:**
The steady periodic solution is expressed as:
\[ x_{sp}(t) = \frac{\sqrt{E_0^2 + F_0^2}}{\sqrt{(k - m \omega^2)^2 + (c \omega)^2}} \cos(\omega t - \alpha - \beta), \]
where \(\alpha\) is defined in Eq. (22) and \(\beta = \tan^{-1}(F_0/E_0)\).
**Suggestion:** Add the steady periodic solutions separately corresponding to \(E_0 \cos \omega t\) and \(F_0 \sin \omega t\) (see Problem 25).
**27. Amplitude of Forced Steady Periodic Oscillations:**
According to Eq. (21), the amplitude of forced steady periodic oscillations for the system \(mx'' + cx' + kx = F_0 \cos \omega t\) is:
\[ C(\omega) = \frac{F_0}{\sqrt{(k - m \omega^2)^2 + (c \omega)^2}}. \]
- **(a)** If \(c \geq c_{cr}/\sqrt{2}\), where \(c_{cr} = \sqrt{4km}\), show that \(C\) steadily decreases as \(\omega\) increases.
- **(b)** If \(c < c_{cr}/\sqrt{2}\), show that \(C\) attains a maximum value (practical resonance) when:
\[
\omega = \omega_m = \sqrt{\frac{k}{m} - \frac{c^2}{2m^2}} < \omega_0 = \sqrt{\frac{k}{m}}.
\]
**28. Cart-with-Flywheel Example:**
As indicated by the cart-with-flywheel example discussed in this section, an unbalanced rotating machine part typically results in a force having amplitude proportional to the square of the frequency, \(\dots\) Show that the am
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