Question

A mass of 5 kg is suspended from a spring of stiffness 46 kN/m. A dashpot is fitted between the mass and the support with a damping ratio of 0.3. Calculate the critical damping coefficient in Ns/m.
(A) 939
(B) 949
(C) 959
(D) 969

Answer 1

Hint:To solve this type of question we use the following formula for the coefficient of critical damping. Critical damping is defined as the approach of the moving system returning to its original position without undergoing oscillation.

Formula used:
${C_c} = 2m\sqrt {\dfrac{k}{m}} $
Where,
${C_c}$ =coefficient of critical damping.
 k = spring stiffness constant or force constant N/m
 m = mass of the object in kg

Complete step-by-step answer:
Let us have a quick approach of the system towards zero amplitude is denoted by${C_c}$. The damping coefficient is directly proportional to the spring stiffness or force constant (k) and inversely proportional to the mass of the object i.e. (m).
We are interested to find out the coefficient of damping.
 Let us first write the information given in the question.
Given,
\[m = 5kg\],
\[k = 46{\text{ KN/m}}\] \[ = {\text{ }}46,000{\text{ N/}}m\]
Damping ratio=0.3
Now, let us use the formula of coefficient of critical damping ${C_c} = 2m\sqrt {\dfrac{k}{m}} $and substitute the values in it.
On substituting the values we get,
${C_c} = 2 \times 5\sqrt {\dfrac{{46000}}{5}} $
Let us simplify the expression for better understanding
${C_c} = 2 \times 5\sqrt {9200} $
$ = 10 \times 95.92$
$ = 959.2N/m$
Hence, option (C) $959N/m$ is the correct option.

Additional information:
*Damping is an influence on a system (by the system itself or from outside) that causes reduction, restriction, or prevention in its oscillation. Damping dissipates the energy stored in the oscillations.
*The damping ratio is a system parameter that describes how rapidly oscillations decay from one bound to another. Depending on its values there are three types of damping.
*If the damping ratio is zero then the system is called undamped.
*If the damping ratio is equal to one then the system is called critically damped and if greater than one it is called overdamped.
*Similarly, when the damping ratio is less than one system is called underdamped.

Note:
*Between overdamped and underdamped cases, there is a certain level of damping at which the system will just fail to make a single oscillation. This is called critical damping.
*In critical damping, the system returns to equilibrium in minimum time.
*The coefficient of damping i.e. ${C_c}$ can be also calculated using direct and simplified equation which is,
\[{C_{critical}} = 2\sqrt {\left( {km} \right)} \]
Where,
\[{C_{critical}}\] = coefficient of damping
K= spring stiffness or force constant \[N/m\]
M=mass of object in kg