## Damping Ratio Solution

STEP 0: Pre-Calculation Summary
Formula Used
Damping Ratio = Actual Damping/Critical Damping
ζ = C/Cc
This formula uses 3 Variables
Variables Used
Damping Ratio - Damping Ratio in control system is defined as the ratio with which any signal gets decayed.
Actual Damping - Actual Damping in a control system refers to the level of damping present in the system as a result of all the physical and electrical components that make up the system.
Critical Damping - Critical Damping refers to the amount of damping required in a system to return to its equilibrium state as quickly as possible without overshooting.
STEP 1: Convert Input(s) to Base Unit
Actual Damping: 0.6 --> No Conversion Required
Critical Damping: 0.5 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
ζ = C/Cc --> 0.6/0.5
Evaluating ... ...
ζ = 1.2
STEP 3: Convert Result to Output's Unit
1.2 --> No Conversion Required
FINAL ANSWER
1.2 <-- Damping Ratio
(Calculation completed in 00.000 seconds)
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Indian Institute of Technology,Roorlee (IITR), Roorkee
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## < 10+ Fundamental Formulas Calculators

Bandwidth Frequency given Damping Ratio

## Bandwidth Frequency given Damping Ratio

Formula
"f"_{"b"} = "f"*((sqrt(1-(2*("ζ"^2))))+sqrt(("ζ"^4)-(4*("ζ"^2))+2))

Example
"54.96966Hz"="23Hz"*((sqrt(1-(2*("0.1"^2))))+sqrt(("0.1"^4)-(4*("0.1"^2))+2))

Calculator
LaTeX
Bandwidth Frequency = Frequency*((sqrt(1-(2*(Damping Ratio^2))))+sqrt((Damping Ratio^4)-(4*(Damping Ratio^2))+2))
Angle of Asymptotes

## Angle of Asymptotes

Formula
"ϕ"_{"k"} = (((2*"K")+1)*pi)/("P"-"Z")

Example
"951.4286°"=(((2*"18")+1)*pi)/("13"-"6")

Calculator
LaTeX
Angle of Asymptotes = (((2*Parameter for Root Locus)+1)*pi)/(Number of Poles-Number of Zeros)
Damping Ratio or Damping Factor

## Damping Ratio or Damping Factor

Formula
"ζ" = "c"/(2*sqrt("M"*"K"_{"spring"}))

Example
"0.188147"="16"/(2*sqrt("35.45kg"*"51N/m"))

Calculator
LaTeX
Damping Ratio = Damping Coefficient/(2*sqrt(Mass*Spring Constant))
Gain-bandwidth Product

## Gain-bandwidth Product

Formula
"G.B" = "modulus"("A"_{"M"})*"BW"

Example
"56.16Hz"="modulus"("0.78")*"72b/s"

Calculator
LaTeX
Gain-Bandwidth Product = modulus(Amplifier Gain in Mid-band)*Amplifier Bandwidth
Damped Natural Frequency

## Damped Natural Frequency

Formula
"ω"_{"d"} = "f"*(sqrt(1-("ζ")^2))

Example
"22.88471Hz"="23Hz"*(sqrt(1-("0.1")^2))

Calculator
LaTeX
Damped natural frequency = Frequency*(sqrt(1-(Damping Ratio)^2))
Resonant Peak

## Resonant Peak

Formula
"M"_{"r"} = 1/((2*"ζ")*sqrt(1-("ζ")^2))

Example
"5.025189"=1/((2*"0.1")*sqrt(1-("0.1")^2))

Calculator
LaTeX
Resonant Peak = 1/((2*Damping Ratio)*sqrt(1-(Damping Ratio)^2))
Resonant Frequency

## Resonant Frequency

Formula
"ω"_{"r"} = "f"*sqrt(1-2*("ζ")^2)

Example
"22.76884Hz"="23Hz"*sqrt(1-2*("0.1")^2)

Calculator
LaTeX
Resonant Frequency = Frequency*sqrt(1-2*(Damping Ratio)^2)
Number of Asymptotes

## Number of Asymptotes

Formula
"N"_{"a"} = "P"-"Z"

Example
"7"="13"-"6"

Calculator
LaTeX
Number of Asymptotes = Number of Poles-Number of Zeros
Transfer Function for Closed and Open Loop System

## Transfer Function for Closed and Open Loop System

Formula
"G"_{"s"} = "C"_{"s"}/"R"_{"s"}

Example
"0.458333"="22"/"48"

Calculator
LaTeX
Transfer Function = Output of System/Input of System
Closed-loop Gain

## Closed-loop Gain

Formula
"A"_{"f"} = 1/"β"

Example
"0.25"=1/"4"

Calculator
LaTeX
Gain-with-feedback = 1/Feedback Factor

## Damping Ratio Formula

Damping Ratio = Actual Damping/Critical Damping
ζ = C/Cc

## How the values of actual damping and critical damping affects the system?

If the actual damping ratio is greater than the critical damping ratio, the system is said to be overdamped, which means the system response is slow and it takes a long time to reach the equilibrium state after a disturbance. On the other hand, if the actual damping ratio is less than the critical damping ratio, the system is said to be underdamped and the system will have oscillations before reaching the equilibrium state.

## How is damping ratio used?

To characterize the amount of damping in a system a ratio called the damping ratio (also known as damping factor and % critical damping) is used. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. The formula for the damping ratio is used for the mass-spring-damper model.

## How to Calculate Damping Ratio?

Damping Ratio calculator uses Damping Ratio = Actual Damping/Critical Damping to calculate the Damping Ratio, Damping Ratio is a parameter, usually denoted by ζ (zeta) that characterizes the frequency response of a second-order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator. Damping Ratio is denoted by ζ symbol.

How to calculate Damping Ratio using this online calculator? To use this online calculator for Damping Ratio, enter Actual Damping (C) & Critical Damping (Cc) and hit the calculate button. Here is how the Damping Ratio calculation can be explained with given input values -> 1.2 = 0.6/0.5.

### FAQ

What is Damping Ratio?
Damping Ratio is a parameter, usually denoted by ζ (zeta) that characterizes the frequency response of a second-order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator and is represented as ζ = C/Cc or Damping Ratio = Actual Damping/Critical Damping. Actual Damping in a control system refers to the level of damping present in the system as a result of all the physical and electrical components that make up the system & Critical Damping refers to the amount of damping required in a system to return to its equilibrium state as quickly as possible without overshooting.
How to calculate Damping Ratio?
Damping Ratio is a parameter, usually denoted by ζ (zeta) that characterizes the frequency response of a second-order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator is calculated using Damping Ratio = Actual Damping/Critical Damping. To calculate Damping Ratio, you need Actual Damping (C) & Critical Damping (Cc). With our tool, you need to enter the respective value for Actual Damping & Critical Damping and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Damping Ratio?
In this formula, Damping Ratio uses Actual Damping & Critical Damping. We can use 2 other way(s) to calculate the same, which is/are as follows -
• Damping Ratio = Damping Coefficient/(2*sqrt(Mass*Spring Constant))
• Damping Ratio = -ln(Percentage Overshoot/100)/ sqrt((pi^2)+ln(Percentage Overshoot/100)^2)
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