Peak Time given Damping Ratio Calculator

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Second Order System

Fundamental Parameters

✖The natural frequency of oscillation refers to the frequency at which a physical system or structure will oscillate or vibrate when it is disturbed from its equilibrium position.ⓘ Natural Frequency of Oscillation [ω_n]			+10% -10%
✖Damping Ratio in control system is defined as the ratio with which any signal gets decayed.ⓘ Damping Ratio [ζ]			+10% -10%

✖Peak time is simply the time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot.ⓘ Peak Time given Damping Ratio [t_p]

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Formula

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Peak Time given Damping Ratio

Formula

`"t"_{"p"} = pi/("ω"_{"n"}*sqrt(1-"ζ"^2))`

Example

`"0.137279s"=pi/("23Hz"*sqrt(1-("0.1")^2))`

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Peak Time given Damping Ratio Solution

STEP 0: Pre-Calculation Summary

Formula Used

Peak Time = pi/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2))
t_p = pi/(ω_n*sqrt(1-ζ^2))
This formula uses 1 Constants, 1 Functions, 3 Variables

Constants Used

pi - Archimedes' constant Value Taken As 3.14159265358979323846264338327950288

Functions Used

sqrt - A square root function is a function that takes a non-negative number as an input and returns the square root of the given input number., sqrt(Number)

Variables Used

Peak Time - (Measured in Second) - Peak time is simply the time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot.
Natural Frequency of Oscillation - (Measured in Hertz) - The natural frequency of oscillation refers to the frequency at which a physical system or structure will oscillate or vibrate when it is disturbed from its equilibrium position.
Damping Ratio - Damping Ratio in control system is defined as the ratio with which any signal gets decayed.

STEP 1: Convert Input(s) to Base Unit

Natural Frequency of Oscillation: 23 Hertz --> 23 Hertz No Conversion Required
Damping Ratio: 0.1 --> No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

t_p = pi/(ω_n*sqrt(1-ζ^2)) --> pi/(23*sqrt(1-0.1^2))

Evaluating ... ...

t_p = 0.137279105086882

STEP 3: Convert Result to Output's Unit

0.137279105086882 Second --> No Conversion Required

FINAL ANSWER

0.137279105086882 ≈ 0.137279 Second <-- Peak Time

(Calculation completed in 00.004 seconds)

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Created by Aman Dhussawat

GURU TEGH BAHADUR INSTITUTE OF TECHNOLOGY (GTBIT), NEW DELHI

Aman Dhussawat has created this Calculator and 50+ more calculators!

Verified by Parminder Singh

Chandigarh University (CU), Punjab

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< 17 Second Order System Calculators

Time Response in Overdamped Case

Go Time Response for Second Order System = 1-(e^(-(Overdamping Ratio-(sqrt((Overdamping Ratio^2)-1)))*(Natural Frequency of Oscillation*Time Period for Oscillations))/(2*sqrt((Overdamping Ratio^2)-1)*(Overdamping Ratio-sqrt((Overdamping Ratio^2)-1))))

Time Response of Critically Damped System

Go Time Response for Second Order System = 1-e^(-Natural Frequency of Oscillation*Time Period for Oscillations)-(e^(-Natural Frequency of Oscillation*Time Period for Oscillations)*Natural Frequency of Oscillation*Time Period for Oscillations)

Bandwidth Frequency given Damping Ratio

Go Bandwidth Frequency = Natural Frequency of Oscillation*(sqrt(1-(2*Damping Ratio^2))+sqrt(Damping Ratio^4-(4*Damping Ratio^2)+2))

Rise Time given Damping Ratio

Go Rise Time = (pi-(Phase Shift*pi/180))/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2))

Time Response in Undamped Case

Go Time Response for Second Order System = 1-cos(Natural Frequency of Oscillation*Time Period for Oscillations)

Peak Time given Damping Ratio

Go Peak Time = pi/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2))

First Peak Undershoot

Go Peak Undershoot = e^(-(2*Damping Ratio*pi)/(sqrt(1-Damping Ratio^2)))

First Peak Overshoot

Go Peak Overshoot = e^(-(pi*Damping Ratio)/(sqrt(1-Damping Ratio^2)))

Number of Oscillations

Go Number of Oscillations = (Setting Time*Damped Natural Frequency)/(2*pi)

Time of Peak Overshoot in Second Order System

Go Time of Peak Overshoot = ((2*Kth Value-1)*pi)/Damped Natural Frequency

Rise Time given Damped Natural Frequency

Go Rise Time = (pi-Phase Shift)/Damped Natural Frequency

Delay Time

Go Delay Time = (1+(0.7*Damping Ratio))/Natural Frequency of Oscillation

Time Period of Oscillations

Go Time Period for Oscillations = (2*pi)/Damped Natural Frequency

Setting Time when Tolerance is 2 Percent

Go Setting Time = 4/(Damping Ratio*Damped Natural Frequency)

Setting Time when Tolerance is 5 Percent

Go Setting Time = 3/(Damping Ratio*Damped Natural Frequency)

Peak Time

Go Peak Time = pi/Damped Natural Frequency

Rise Time given Delay Time

Go Rise Time = 1.5*Delay Time

< 16 Second Order System Calculators

Time Response in Overdamped Case

Time Response of Critically Damped System

Rise Time given Damping Ratio

Go Rise Time = (pi-(Phase Shift*pi/180))/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2))

Time Response in Undamped Case

Go Time Response for Second Order System = 1-cos(Natural Frequency of Oscillation*Time Period for Oscillations)

Peak Time given Damping Ratio

Go Peak Time = pi/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2))

First Peak Undershoot

Go Peak Undershoot = e^(-(2*Damping Ratio*pi)/(sqrt(1-Damping Ratio^2)))

First Peak Overshoot

Go Peak Overshoot = e^(-(pi*Damping Ratio)/(sqrt(1-Damping Ratio^2)))

Number of Oscillations

Go Number of Oscillations = (Setting Time*Damped Natural Frequency)/(2*pi)

Time of Peak Overshoot in Second Order System

Go Time of Peak Overshoot = ((2*Kth Value-1)*pi)/Damped Natural Frequency

Rise Time given Damped Natural Frequency

Go Rise Time = (pi-Phase Shift)/Damped Natural Frequency

Delay Time

Go Delay Time = (1+(0.7*Damping Ratio))/Natural Frequency of Oscillation

Time Period of Oscillations

Go Time Period for Oscillations = (2*pi)/Damped Natural Frequency

Setting Time when Tolerance is 2 Percent

Go Setting Time = 4/(Damping Ratio*Damped Natural Frequency)

Setting Time when Tolerance is 5 Percent

Go Setting Time = 3/(Damping Ratio*Damped Natural Frequency)

Peak Time

Go Peak Time = pi/Damped Natural Frequency

Rise Time given Delay Time

Go Rise Time = 1.5*Delay Time

< 25 Control System Design Calculators

Time Response in Overdamped Case

Time Response of Critically Damped System

Bandwidth Frequency given Damping Ratio

Go Bandwidth Frequency = Natural Frequency of Oscillation*(sqrt(1-(2*Damping Ratio^2))+sqrt(Damping Ratio^4-(4*Damping Ratio^2)+2))

Rise Time given Damping Ratio

Go Rise Time = (pi-(Phase Shift*pi/180))/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2))

Percentage Overshoot

Go Percentage Overshoot = 100*(e^((-Damping Ratio*pi)/(sqrt(1-(Damping Ratio^2)))))

Time Response in Undamped Case

Go Time Response for Second Order System = 1-cos(Natural Frequency of Oscillation*Time Period for Oscillations)

Peak Time given Damping Ratio

Go Peak Time = pi/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2))

First Peak Undershoot

Go Peak Undershoot = e^(-(2*Damping Ratio*pi)/(sqrt(1-Damping Ratio^2)))

First Peak Overshoot

Go Peak Overshoot = e^(-(pi*Damping Ratio)/(sqrt(1-Damping Ratio^2)))

Gain-Bandwidth Product

Go Gain-Bandwidth Product = modulus(Amplifier Gain in Mid Band)*Amplifier Bandwidth

Resonant Frequency

Go Resonant Frequency = Natural Frequency of Oscillation*sqrt(1-2*Damping Ratio^2)

Number of Oscillations

Go Number of Oscillations = (Setting Time*Damped Natural Frequency)/(2*pi)

Time of Peak Overshoot in Second Order System

Go Time of Peak Overshoot = ((2*Kth Value-1)*pi)/Damped Natural Frequency

Rise Time given Damped Natural Frequency

Go Rise Time = (pi-Phase Shift)/Damped Natural Frequency

Steady State Error for Type Zero System

Go Steady State Error = Coefficient Value/(1+Position of Error Constant)

Delay Time

Go Delay Time = (1+(0.7*Damping Ratio))/Natural Frequency of Oscillation

Steady State Error for Type 2 System

Go Steady State Error = Coefficient Value/Acceleration Error Constant

Time Period of Oscillations

Go Time Period for Oscillations = (2*pi)/Damped Natural Frequency

Steady State Error for Type 1 System

Go Steady State Error = Coefficient Value/Velocity Error Constant

Setting Time when Tolerance is 2 Percent

Go Setting Time = 4/(Damping Ratio*Damped Natural Frequency)

Setting Time when Tolerance is 5 Percent

Go Setting Time = 3/(Damping Ratio*Damped Natural Frequency)

Number of Asymptotes

Go Number of Asymptotes = Number of Poles-Number of Zeroes

Peak Time

Go Peak Time = pi/Damped Natural Frequency

Q-Factor

Go Q Factor = 1/(2*Damping Ratio)

Rise Time given Delay Time

Go Rise Time = 1.5*Delay Time

Peak Time given Damping Ratio Formula

Peak Time = pi/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2))
t_p = pi/(ω_n*sqrt(1-ζ^2))

What does damping ratio depend on?

The energy dissipation is caused by material damping which basically depends on three factors: amplitude of stress, number of cycles and geometry. In the case of non-homogeneous stress distribution the geometry of the structure influences the vibration damping.

How to Calculate Peak Time given Damping Ratio?

Peak Time given Damping Ratio calculator uses Peak Time = pi/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2)) to calculate the Peak Time, The Peak time given damping ratio formula is defined as the time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot the response curve reaches to its maximum value. Peak Time is denoted by t_p symbol.

How to calculate Peak Time given Damping Ratio using this online calculator? To use this online calculator for Peak Time given Damping Ratio, enter Natural Frequency of Oscillation (ω_n) & Damping Ratio (ζ) and hit the calculate button. Here is how the Peak Time given Damping Ratio calculation can be explained with given input values -> 0.137279 = pi/(23*sqrt(1-0.1^2)).

FAQ

What is Peak Time given Damping Ratio?

The Peak time given damping ratio formula is defined as the time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot the response curve reaches to its maximum value and is represented as t_p = pi/(ω_n*sqrt(1-ζ^2)) or Peak Time = pi/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2)). The natural frequency of oscillation refers to the frequency at which a physical system or structure will oscillate or vibrate when it is disturbed from its equilibrium position & Damping Ratio in control system is defined as the ratio with which any signal gets decayed.

How to calculate Peak Time given Damping Ratio?

The Peak time given damping ratio formula is defined as the time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot the response curve reaches to its maximum value is calculated using Peak Time = pi/(Natural Frequency of Oscillation*sqrt(1-Damping Ratio^2)). To calculate Peak Time given Damping Ratio, you need Natural Frequency of Oscillation (ω_n) & Damping Ratio (ζ). With our tool, you need to enter the respective value for Natural Frequency of Oscillation & Damping Ratio 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 Peak Time?

In this formula, Peak Time uses Natural Frequency of Oscillation & Damping Ratio. We can use 3 other way(s) to calculate the same, which is/are as follows -

Peak Time = pi/Damped Natural Frequency
Peak Time = pi/Damped Natural Frequency
Peak Time = pi/Damped Natural Frequency

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