/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 161 A damped system has the followin... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 161

A damped system has the following parameters: \(m=2 \mathrm{~kg}, c=3 \mathrm{~N}-\mathrm{s} / \mathrm{m},\) and \(k=40 \mathrm{~N} / \mathrm{m} .\) Determine the natural frequency, damping ratio, and the type of response of the system in free vibration. Find the amount of damping to be added or subtracted to make the system critically damped.

Short Answer

Expert verified
The natural frequency of the system is \(2\sqrt{5}\, \text{rad/s}\) and the damping ratio is \(\frac{3\sqrt{5}}{20}\) which indicates an underdamped system. To make the system critically damped, the damping coefficient needs to be increased by \((4\sqrt{5}-3)\, \text{N路s/m}\).

Step by step solution

01

Determine the natural frequency of the undamped system

To calculate the natural frequency, we will use the formula: \[蠅_n = \sqrt{\frac{k}{m}}\] where: - \(蠅_n\) is the natural frequency - \(k\) is the stiffness - \(m\) is the mass Given: - \(m = 2 \,\text{kg}\) - \(k = 40 \,\text{N/m}\) Using these values in the formula, we get: \[ 蠅_n = \sqrt{\frac{40\,\text{N/m}}{2\,\text{kg}}} \]
02

Calculate the natural frequency

Calculating our value for the natural frequency: \[蠅_n = \sqrt{20}\] \[蠅_n = \sqrt{4 \times 5}\] \[蠅_n = 2 \sqrt{5}\, \text{rad/s}\]
03

Determine the damping ratio

To calculate the damping ratio, we will use the formula: \[ 尉 = \frac{c}{2 \sqrt{m k}} \] where: - \(尉\) is the damping ratio - \(c\) is the damping coefficient - \(m\) is the mass - \(k\) is the stiffness Given: - \(c = 3\, \text{N路s/m}\) - \(m = 2\, \text{kg}\) - \(k = 40\, \text{N/m}\) Using these values in the formula, we get: \[ 尉 = \frac{3\, \text{N路s/m}}{2 \sqrt{(2\, \text{kg})(40\, \text{N/m})}} \]
04

Calculate the damping ratio

Calculating our value for the damping ratio: \[ 尉 = \frac{3}{2 (2 \sqrt{5})} \] \[ 尉 = \frac{3}{4 \sqrt{5}} \] \[ 尉 = \frac{3 \sqrt{5}}{20} \]
05

Determine the type of response

Based on the damping ratio calculated, we can determine the type of response: - If \(尉 < 1\), the system is underdamped - If \(尉 = 1\), the system is critically damped - If \(尉 > 1\), the system is overdamped Since our calculated damping ratio \(尉 = \frac{3 \sqrt{5}}{20} < 1\) (approximately 0.335), the system is underdamped.
06

Find the amount of damping to be added or subtracted for critical damping

For the system to be critically damped, the damping ratio (\(尉\)) should equal 1. We will use the formula for the damping ratio to find the critical damping coefficient: \[ 尉 = \frac{c_{\text{critical}}}{2 \sqrt{m k}} \] Rearranging the formula for \(c_{\text{critical}}\): \[ c_{\text{critical}} = 2 \sqrt{m k} \] Given: - \(m = 2\, \text{kg}\) - \(k = 40\, \text{N/m}\) Using these values in the formula, we get: \[ c_{\text{critical}} = 2 \sqrt{(2\, \text{kg})(40\, \text{N/m})} \] \[c_{\text{critical}} = 4 \sqrt{5}\, \text{N路s/m}\] Now we can determine the amount of damping to be added or subtracted: \[螖c = c_{\text{critical}} - c\] \[螖c = 4 \sqrt{5}\, \text{N路s/m} - 3 \,\text{N路s/m}\]
07

Calculate the amount of damping to be added or subtracted for critical damping

Calculating the value of 螖c: \[螖c = (\, 4 \sqrt{5}-3) \text{N路s/m}\] Hence, the damping coefficient needs to be increased by \((4 \sqrt{5} - 3)\, \text{N路s/m}\) to make the system critically damped.

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Most popular questions from this chapter

An automobile is found to have a natural frequency of \(20 \mathrm{rad} / \mathrm{s}\) without passengers and 17.32 \(\mathrm{rad} / \mathrm{s}\) with passengers of mass \(500 \mathrm{~kg}\). Find the mass and stiffness of the automobile by treating it as a single-degree-of-freedom system.

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