Maximum Displacement of Forced Vibration using Natural Frequency Calculator

✖Static Force is a force that keeps an object at rest.ⓘ Static Force [F_x]			+10% -10%
✖Damping Coefficient is a material property that indicates whether a material will bounce back or return energy to a system.ⓘ Damping Coefficient [c]			+10% -10%
✖The Angular Velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time.ⓘ Angular Velocity [ω]			+10% -10%
✖Stiffness of Spring is a measure of the resistance offered by an elastic body to deformation. every object in this universe has some stiffness.ⓘ Stiffness of Spring [k]			+10% -10%
✖Natural Circular Frequency is a scalar measure of rotation rate.ⓘ Natural Circular Frequency [ω_n]			+10% -10%

✖Total Displacement is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position.ⓘ Maximum Displacement of Forced Vibration using Natural Frequency [d_mass]

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Maximum Displacement of Forced Vibration using Natural Frequency

Formula

`"d"_{"mass"} = "F"_{"x"}/(sqrt(("c"*"ω"/"k")^2+(1-("ω"/"ω"_{"n"})^2)^2))`

Example

`"17.59301m"="20N"/(sqrt(("5Ns/m"*"10rad/s"/"60N/m")^2+(1-("10rad/s"/"21rad/s")^2)^2))`

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Download Frequency of Under Damped Forced Vibrations Formulas PDF

Maximum Displacement of Forced Vibration using Natural Frequency Solution

STEP 0: Pre-Calculation Summary

Formula Used

Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity/Stiffness of Spring)^2+(1-(Angular Velocity/Natural Circular Frequency)^2)^2))
d_mass = F_x/(sqrt((c*ω/k)^2+(1-(ω/ω_n)^2)^2))
This formula uses 1 Functions, 6 Variables

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

Total Displacement - (Measured in Meter) - Total Displacement is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position.
Static Force - (Measured in Newton) - Static Force is a force that keeps an object at rest.
Damping Coefficient - (Measured in Newton Second per Meter) - Damping Coefficient is a material property that indicates whether a material will bounce back or return energy to a system.
Angular Velocity - (Measured in Radian per Second) - The Angular Velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time.
Stiffness of Spring - (Measured in Newton per Meter) - Stiffness of Spring is a measure of the resistance offered by an elastic body to deformation. every object in this universe has some stiffness.
Natural Circular Frequency - (Measured in Radian per Second) - Natural Circular Frequency is a scalar measure of rotation rate.

STEP 1: Convert Input(s) to Base Unit

Static Force: 20 Newton --> 20 Newton No Conversion Required
Damping Coefficient: 5 Newton Second per Meter --> 5 Newton Second per Meter No Conversion Required
Angular Velocity: 10 Radian per Second --> 10 Radian per Second No Conversion Required
Stiffness of Spring: 60 Newton per Meter --> 60 Newton per Meter No Conversion Required
Natural Circular Frequency: 21 Radian per Second --> 21 Radian per Second No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

d_mass = F_x/(sqrt((c*ω/k)^2+(1-(ω/ω_n)^2)^2)) --> 20/(sqrt((5*10/60)^2+(1-(10/21)^2)^2))

Evaluating ... ...

d_mass = 17.5930102473354

STEP 3: Convert Result to Output's Unit

17.5930102473354 Meter --> No Conversion Required

FINAL ANSWER

17.5930102473354 ≈ 17.59301 Meter <-- Total Displacement

(Calculation completed in 00.004 seconds)

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Home » Physics » Theory of Machine » Vibrations » Longitudinal and Transverse Vibrations » Frequency of Under Damped Forced Vibrations » Maximum Displacement of Forced Vibration using Natural Frequency

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Created by Anshika Arya

National Institute Of Technology (NIT), Hamirpur

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Verified by Dipto Mandal

Indian Institute of Information Technology (IIIT), Guwahati

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< 15 Frequency of Under Damped Forced Vibrations Calculators

Total Displacement of Forced Vibrations

Go Total Displacement = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant)+(Static Force*cos(Angular Velocity*Time Period-Phase Constant))/(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))

Particular Integral

Go Particular Integral = (Static Force*cos(Angular Velocity*Time Period-Phase Constant))/(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))

Maximum Displacement of Forced Vibration using Natural Frequency

Go Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity/Stiffness of Spring)^2+(1-(Angular Velocity/Natural Circular Frequency)^2)^2))

Static Force using Maximum Displacement or Amplitude of Forced Vibration

Go Static Force = Total Displacement*(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))

Maximum Displacement of Forced Vibration

Go Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))

Phase Constant

Go Phase Constant = atan((Damping Coefficient*Angular Velocity)/(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2))

Damping Coefficient

Go Damping Coefficient = (tan(Phase Constant)*(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2))/Angular Velocity

Maximum Displacement of Forced Vibration at Resonance

Go Total Displacement = Deflection under Static Force*Stiffness of Spring/(Damping Coefficient*Natural Circular Frequency)

Maximum Displacement of Forced Vibration with Negligible Damping

Go Total Displacement = Static Force/(Mass suspended from Spring*(Natural Circular Frequency^2-Angular Velocity^2))

Static Force when Damping is Negligible

Go Static Force = Total Displacement*(Mass suspended from Spring*Natural Circular Frequency^2-Angular Velocity^2)

Complementary Function

Go Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant)

External Periodic Disturbing Force

Go External Periodic Disturbing Force = Static Force*cos(Angular Velocity*Time Period)

Deflection of System under Static Force

Go Deflection under Static Force = Static Force/Stiffness of Spring

Static Force

Go Static Force = Deflection under Static Force*Stiffness of Spring

Total Displacement of Forced Vibration given Particular Integral and Complementary Function

Go Total Displacement = Particular Integral+Complementary Function

Maximum Displacement of Forced Vibration using Natural Frequency Formula

What is undamped free vibration?

The simplest vibrations to analyze are undamped, free, one degree of freedom vibrations. "Undamped" means that there are no energy losses with movement (whether intentional, by adding dampers, or unintentional, through drag or friction). An undamped system will vibrate forever without any additional applied forces.

What is forced vibration?

Forced vibrations occur if a system is continuously driven by an external agency. A simple example is a child's swing that is pushed on each downswing. Of special interest are systems undergoing SHM and driven by sinusoidal forcing.

How to Calculate Maximum Displacement of Forced Vibration using Natural Frequency?

Maximum Displacement of Forced Vibration using Natural Frequency calculator uses Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity/Stiffness of Spring)^2+(1-(Angular Velocity/Natural Circular Frequency)^2)^2)) to calculate the Total Displacement, The Maximum Displacement of Forced Vibration using Natural Frequency formula implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object. Total Displacement is denoted by d_mass symbol.

How to calculate Maximum Displacement of Forced Vibration using Natural Frequency using this online calculator? To use this online calculator for Maximum Displacement of Forced Vibration using Natural Frequency, enter Static Force (F_x), Damping Coefficient (c), Angular Velocity (ω), Stiffness of Spring (k) & Natural Circular Frequency (ω_n) and hit the calculate button. Here is how the Maximum Displacement of Forced Vibration using Natural Frequency calculation can be explained with given input values -> 17.59301 = 20/(sqrt((5*10/60)^2+(1-(10/21)^2)^2)).

FAQ

What is Maximum Displacement of Forced Vibration using Natural Frequency?

The Maximum Displacement of Forced Vibration using Natural Frequency formula implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object and is represented as d_mass = F_x/(sqrt((c*ω/k)^2+(1-(ω/ω_n)^2)^2)) or Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity/Stiffness of Spring)^2+(1-(Angular Velocity/Natural Circular Frequency)^2)^2)). Static Force is a force that keeps an object at rest, Damping Coefficient is a material property that indicates whether a material will bounce back or return energy to a system, The Angular Velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time, Stiffness of Spring is a measure of the resistance offered by an elastic body to deformation. every object in this universe has some stiffness & Natural Circular Frequency is a scalar measure of rotation rate.

How to calculate Maximum Displacement of Forced Vibration using Natural Frequency?

The Maximum Displacement of Forced Vibration using Natural Frequency formula implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object is calculated using Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity/Stiffness of Spring)^2+(1-(Angular Velocity/Natural Circular Frequency)^2)^2)). To calculate Maximum Displacement of Forced Vibration using Natural Frequency, you need Static Force (F_x), Damping Coefficient (c), Angular Velocity (ω), Stiffness of Spring (k) & Natural Circular Frequency (ω_n). With our tool, you need to enter the respective value for Static Force, Damping Coefficient, Angular Velocity, Stiffness of Spring & Natural Circular Frequency 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 Total Displacement?

In this formula, Total Displacement uses Static Force, Damping Coefficient, Angular Velocity, Stiffness of Spring & Natural Circular Frequency. We can use 5 other way(s) to calculate the same, which is/are as follows -

Total Displacement = Deflection under Static Force*Stiffness of Spring/(Damping Coefficient*Natural Circular Frequency)
Total Displacement = Static Force/(Mass suspended from Spring*(Natural Circular Frequency^2-Angular Velocity^2))
Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))
Total Displacement = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant)+(Static Force*cos(Angular Velocity*Time Period-Phase Constant))/(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))
Total Displacement = Particular Integral+Complementary Function

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