## Damping Ratio given Percentage Overshoot Solution

STEP 0: Pre-Calculation Summary
Formula Used
Damping Ratio = -ln(Percentage Overshoot/100)/ sqrt((pi^2)+ln(Percentage Overshoot/100)^2)
ζ = -ln(PO/100)/ sqrt((pi^2)+ln(PO/100)^2)
This formula uses 1 Constants, 2 Functions, 2 Variables
Constants Used
pi - Archimedes' constant Value Taken As 3.14159265358979323846264338327950288
Functions Used
ln - Natural logarithm function (base e), ln(Number)
sqrt - Squre root function, sqrt(Number)
Variables Used
Damping Ratio - Damping Ratio in control system is defined as the ratio with which any signal gets decayed.
Percentage Overshoot - Percentage Overshoot refers to an output exceeding its final, steady-state value.
STEP 1: Convert Input(s) to Base Unit
Percentage Overshoot: 70 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
ζ = -ln(PO/100)/ sqrt((pi^2)+ln(PO/100)^2) --> -ln(70/100)/ sqrt((pi^2)+ln(70/100)^2)
Evaluating ... ...
ζ = 0.112808450693581
STEP 3: Convert Result to Output's Unit
0.112808450693581 --> No Conversion Required
0.112808450693581 <-- 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 given Percentage Overshoot Formula

Damping Ratio = -ln(Percentage Overshoot/100)/ sqrt((pi^2)+ln(Percentage Overshoot/100)^2)
ζ = -ln(PO/100)/ sqrt((pi^2)+ln(PO/100)^2)

## 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 given Percentage Overshoot?

Damping Ratio given Percentage Overshoot calculator uses Damping Ratio = -ln(Percentage Overshoot/100)/ sqrt((pi^2)+ln(Percentage Overshoot/100)^2) to calculate the Damping Ratio, The Damping Ratio given Percentage Overshoot formula 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 given Percentage Overshoot using this online calculator? To use this online calculator for Damping Ratio given Percentage Overshoot, enter Percentage Overshoot (PO) and hit the calculate button. Here is how the Damping Ratio given Percentage Overshoot calculation can be explained with given input values -> 0.112808 = -ln(70/100)/ sqrt((pi^2)+ln(70/100)^2).

### FAQ

What is Damping Ratio given Percentage Overshoot?
The Damping Ratio given Percentage Overshoot formula 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 = -ln(PO/100)/ sqrt((pi^2)+ln(PO/100)^2) or Damping Ratio = -ln(Percentage Overshoot/100)/ sqrt((pi^2)+ln(Percentage Overshoot/100)^2). Percentage Overshoot refers to an output exceeding its final, steady-state value.
How to calculate Damping Ratio given Percentage Overshoot?
The Damping Ratio given Percentage Overshoot formula 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 = -ln(Percentage Overshoot/100)/ sqrt((pi^2)+ln(Percentage Overshoot/100)^2). To calculate Damping Ratio given Percentage Overshoot, you need Percentage Overshoot (PO). With our tool, you need to enter the respective value for Percentage Overshoot 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 Percentage Overshoot. 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 = Actual Damping/Critical Damping
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