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Time Response in Overdamped Case Formula

GoTime Response in Undamped Case Formula

GoTime Response of Critically Damped System Formula

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Time Response for Second Order System is defined as the response of a second-order system towards any applied input. Check FAQs

C t = 1 - (\frac{e^{- (ζ over - (\sqrt{({ζ over}^{2}) - 1})) \cdot (ω n \cdot T)}}{2 \cdot \sqrt{({ζ over}^{2}) - 1} \cdot (ζ over - \sqrt{({ζ over}^{2}) - 1})})

C_t - Time Response for Second Order System?ζ_over - Overdamping Ratio?ω_n - Natural Frequency of Oscillation?T - Time Period for Oscillations?

Time Response in Overdamped Case Example

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Here is how the Time Response in Overdamped Case equation looks like with Values.

Here is how the Time Response in Overdamped Case equation looks like with Units.

Here is how the Time Response in Overdamped Case equation looks like.

0.8075 = 1 - (\frac{e^{- (1.12 - (\sqrt{({1.12}^{2}) - 1})) \cdot (23 \cdot 0.15)}}{2 \cdot \sqrt{({1.12}^{2}) - 1} \cdot (1.12 - \sqrt{({1.12}^{2}) - 1})})

0.8075 = 1 - (\frac{e^{- (1.12 - (\sqrt{({1.12}^{2}) - 1})) \cdot (23Hz \cdot 0.15s)}}{2 \cdot \sqrt{({1.12}^{2}) - 1} \cdot (1.12 - \sqrt{({1.12}^{2}) - 1})})

0.8075 = 1 - (\frac{e^{- (1.12 - (\sqrt{({1.12}^{2}) - 1})) \cdot (23 \cdot 0.15)}}{2 \cdot \sqrt{({1.12}^{2}) - 1} \cdot (1.12 - \sqrt{({1.12}^{2}) - 1})})

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Time Response in Overdamped Case Solution

Follow our step by step solution on how to calculate Time Response in Overdamped Case?

FIRST Step Consider the formula

C t = 1 - (\frac{e^{- (ζ over - (\sqrt{({ζ over}^{2}) - 1})) \cdot (ω n \cdot T)}}{2 \cdot \sqrt{({ζ over}^{2}) - 1} \cdot (ζ over - \sqrt{({ζ over}^{2}) - 1})})

Next Step Substitute values of Variables

∴

C t = 1 - (\frac{e^{- (1.12 - (\sqrt{({1.12}^{2}) - 1})) \cdot (23Hz \cdot 0.15s)}}{2 \cdot \sqrt{({1.12}^{2}) - 1} \cdot (1.12 - \sqrt{({1.12}^{2}) - 1})})

Next Step Prepare to Evaluate

∴

C t = 1 - (\frac{e^{- (1.12 - (\sqrt{({1.12}^{2}) - 1})) \cdot (23 \cdot 0.15)}}{2 \cdot \sqrt{({1.12}^{2}) - 1} \cdot (1.12 - \sqrt{({1.12}^{2}) - 1})})

Next Step Evaluate

∴

C t = 0.807466086195714

LAST Step Rounding Answer

∴

C t = 0.8075

Time Response in Overdamped Case Formula Elements

Variables

Functions

Time Response for Second Order System

Time Response for Second Order System is defined as the response of a second-order system towards any applied input.

Symbol: C_t

Measurement: NAUnit: Unitless

Note: Value should be greater than 0.

Formulas to find Time Response for Second Order System

Overdamping Ratio

Overdamping Ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance.

Symbol: ζ_over

Measurement: NAUnit: Unitless

Note: Value should be greater than 1.

Formulas to find Overdamping Ratio

Natural Frequency of Oscillation

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.

Symbol: ω_n

Measurement: FrequencyUnit: Hz

Note: Value should be greater than 0.

Formulas to find Natural Frequency of Oscillation

Time Period for Oscillations

Time Period for Oscillations is the time taken by a complete cycle of the wave to pass a particular interval.

Symbol: T

Measurement: TimeUnit: s

Note: Value should be greater than 0.

Formulas to find Time Period for Oscillations

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.

Syntax: sqrt(Number)

Other Formulas to find Time Response for Second Order System

Go Time Response in Undamped Case

C t = 1 - \cos (ω n \cdot T)

Go Time Response of Critically Damped System

C t = 1 - e^{- ω n \cdot T} - (e^{- ω n \cdot T} \cdot ω n \cdot T)

Other formulas in Second Order System category

Go Bandwidth Frequency given Damping Ratio

f b = ω n \cdot (\sqrt{1 - (2 \cdot ζ^{2})} + \sqrt{ζ^{4} - (4 \cdot ζ^{2}) + 2})

Go Delay Time

t d = \frac{1 + (0.7 \cdot ζ)}{ω n}

Go First Peak Overshoot

M o = e^{- \frac{π \cdot ζ}{\sqrt{1 - ζ^{2}}}}

Go First Peak Undershoot

M u = e^{- \frac{2 \cdot ζ \cdot π}{\sqrt{1 - ζ^{2}}}}

How to Evaluate Time Response in Overdamped Case?

Time Response in Overdamped Case evaluator uses 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)))) to evaluate the Time Response for Second Order System, Time Response in Overdamped Case occurs when the damping factor/damping ratio is more than 1 during the process of damping. Time Response for Second Order System is denoted by C_t symbol.

How to evaluate Time Response in Overdamped Case using this online evaluator? To use this online evaluator for Time Response in Overdamped Case, enter Overdamping Ratio (ζ_over), Natural Frequency of Oscillation (ω_n) & Time Period for Oscillations (T) and hit the calculate button.

FAQs on Time Response in Overdamped Case

What is the formula to find Time Response in Overdamped Case?

The formula of Time Response in Overdamped Case is expressed as 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)))). Here is an example- 0.807466 = 1-(e^(-(1.12-(sqrt((1.12^2)-1)))*(23*0.15))/(2*sqrt((1.12^2)-1)*(1.12-sqrt((1.12^2)-1)))).

How to calculate Time Response in Overdamped Case?

With Overdamping Ratio (ζ_over), Natural Frequency of Oscillation (ω_n) & Time Period for Oscillations (T) we can find Time Response in Overdamped Case using the formula - 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)))). This formula also uses Square Root (sqrt) function(s).

What are the other ways to Calculate Time Response for Second Order System?

Here are the different ways to Calculate Time Response for Second Order System-

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

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Time Response in Overdamped Case Formula

​GoTime Response in Undamped Case Formula

​GoTime Response of Critically Damped System Formula

Time Response in Overdamped Case Example

Time Response in Overdamped Case Solution

Time Response in Overdamped Case Formula Elements

Other Formulas to find Time Response for Second Order System

Other formulas in Second Order System category

How to Evaluate Time Response in Overdamped Case?

FAQs on Time Response in Overdamped Case

Variable Name

GoTime Response in Undamped Case Formula

GoTime Response of Critically Damped System Formula