## What is Damping Ratio : Derivation & Its Cases

- Frank
- May 25, 2022
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Damping is the power on or to prevent or reduce its oscillation in an oscillatory system. So, in a physical system, the generation of damping can be done through the process that dissolves the stored energy within the oscillation. The best examples are resistance within electronic oscillators, viscous drag within mechanical systems, light absorption as well as scattering in optical oscillators. In other oscillation systems, damping doesn’t depend on loss of energy and that can be significant within bikes and biological systems. This article discusses an overview of the **damping ratio **and its derivation.

### What is Damping Ratio?

A **damping ratio definition** is a dimensionless measure used to describe how oscillations within a system can decompose once a disturbance occurs is known as the damping ratio. The behavior of oscillatory can be exhibited by many systems once they are worried about their location of stationary equilibrium. A mass balanced from a coil, once it is pulled and released then it bounces up & down. On every bounce, this system tries to return its balance location, however, overshoots it.

Sometimes, some losses moist the system & causes the oscillations to slowly decompose within amplitude to zero otherwise attenuate. So, this is the** significance of the damping ratio**. The system parameter like damping ratio is used to describe how quickly the oscillations decompose from one bounce to another. The **damping ratio symbol** is zeta (ζ), that can change from undamped like ζ = 0, underdamped like ζ < 1, critically damped like ζ = 1 & overdamped like ζ > 1.

The performance of oscillating systems is frequently used a different engineering fields like control, chemical, mechanical, structural & electrical. The physical amount that is fluctuating will change very much & could be the influence of a large building in the breeze otherwise the speed of the motor, but a normalized, otherwise non-dimensionalized approach can be suitable to describe common features of behavior.

### Damping Ratio in Control System

The damped harmonic oscillations within a mechanical system are very simple to understand through a spring-mass-damper system.

A spring-mass-damper system with an SDOF (single degree of freedom) mainly includes a spring, a mass & a damper. Motion can be defined through simply one independent coordinate namely time. In this spring system, ‘m’ signifies the moving mass, ‘k’ signifies the constant of spring & ‘c’ is the coefficient of damping.

Here, the constant of spring signifies the power utilizes through the spring once it is condensed for the length of a unit. The coefficient of damping is the power used through the damper once the mass goes with a unit speed.

The mass moves freely through one axis, however, the mass moves at any time and its motion can be opposed through the spring as well as the damper. In the above diagram, imagine that the mass goes down at a certain distance.

It reduces the spring to shift the damper through the same distance. The spring in the above system stores as well as releases energy in a single cycle. The damper simply absorbs energy & doesn’t discharge it reverse to the mass.

### Derivation

For this system, the equation is known a second order and normal differential equation. The **damping ratio formula in control system** is,

**d ^{2}x/dt^{2}+ 2 ζω_{0}dx/dt+ ω^{2}_{0}x = 0**

Here,

**ω0 = √k/m**

In radians, it is also called natural frequency

**ζ = C/2√mk**

The above equation is the **damping ratio formula** in the control system. The normal frequency is the system’s oscillation frequency if it is troubled like hit or tapped from a break.

#### Cases of Oscillation

Based on the quantity of damping there, a spring-mass system will exhibit different behaviors of oscillatory.

When the spring-mass system is entirely lossless, then mass would swing imprecisely through every bounce of equivalent height to the final. So, this hypothetical case is known as undamped.

If this system includes high losses, for instance, if an experiment like spring-mass were performed within a viscous liquid, the mass can gradually come back to its break location without ever exceeding. So, this case is known as overdamped.

Usually, the mass is inclined to overshoot its initial location, and after that, it returns, overshoots again. So with every overshoot, some amount of energy within this system can be dissipated & the oscillations will die to zero. So, this case is known as underdamped.

In between the cases of overdamped as well as underdamped, there exists a specific range of damping where the spring system will simply fail to overshoot & will not create a single oscillation. So this case is known as critical damping. The major difference between the two critical & overdamping is, in the critical type, the system comebacks to a stable location within the least amount of time.

Thus, this is all about an overview of the damping ratio and **how to find the damping ratio in the control system**. It is one kind of parameter with dimensionless that describes how an oscillating otherwise vibrating body comes to relax. If there is no damping, then an oscillating system will never approach to relax. However, that doesn’t occur in nature. Each Oscillating system approaches to relax or balance location after a fixed point of time. Amplitude slowly reduces through time & approaches zero. Here is a question for you, how do you decrease the damping ratio?