Complementary Function Solution

STEP 0: Pre-Calculation Summary
Formula Used
Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant)
x1 = A*cos(ωd-ϕ)
This formula uses 1 Functions, 4 Variables
Functions Used
cos - Cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse of the triangle., cos(Angle)
Variables Used
Complementary Function - (Measured in Meter) - The Complementary Function is a part of the solution of the differential equation.
Amplitude of Vibration - (Measured in Meter) - Amplitude of Vibration is the greatest distance that a wave, especially a sound or radio wave, moves up and down.
Circular Damped Frequency - (Measured in Hertz) - Circular Damped Frequency refers to the angular displacement per unit time.
Phase Constant - (Measured in Radian) - Phase Constant tells you how displaced a wave is from equilibrium or zero position.
STEP 1: Convert Input(s) to Base Unit
Amplitude of Vibration: 5.25 Meter --> 5.25 Meter No Conversion Required
Circular Damped Frequency: 6 Hertz --> 6 Hertz No Conversion Required
Phase Constant: 45 Degree --> 0.785398163397301 Radian (Check conversion here)
STEP 2: Evaluate Formula
Substituting Input Values in Formula
x1 = A*cos(ωd-ϕ) --> 5.25*cos(6-0.785398163397301)
Evaluating ... ...
x1 = 2.52717321800662
STEP 3: Convert Result to Output's Unit
2.52717321800662 Meter --> No Conversion Required
FINAL ANSWER
2.52717321800662 2.527173 Meter <-- Complementary Function
(Calculation completed in 00.020 seconds)

Credits

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

Complementary Function Formula

Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant)
x1 = A*cos(ωd-ϕ)

Why do we need forced vibration?

The vibration of a moving vehicle is forced vibration, because the vehicle's engine, springs, the road, etc., continue to make it vibrate. Forced vibration is when an alternating force or motion is applied to a mechanical system, for example when a washing machine shakes due to an imbalance.

How to Calculate Complementary Function?

Complementary Function calculator uses Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant) to calculate the Complementary Function, The Complementary function formula is defined as a part of the solution for the differential equation of the under-damped forced vibrations. Complementary Function is denoted by x1 symbol.

How to calculate Complementary Function using this online calculator? To use this online calculator for Complementary Function, enter Amplitude of Vibration (A), Circular Damped Frequency d) & Phase Constant (ϕ) and hit the calculate button. Here is how the Complementary Function calculation can be explained with given input values -> 2.527173 = 5.25*cos(6-0.785398163397301).

FAQ

What is Complementary Function?
The Complementary function formula is defined as a part of the solution for the differential equation of the under-damped forced vibrations and is represented as x1 = A*cos(ωd-ϕ) or Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant). Amplitude of Vibration is the greatest distance that a wave, especially a sound or radio wave, moves up and down, Circular Damped Frequency refers to the angular displacement per unit time & Phase Constant tells you how displaced a wave is from equilibrium or zero position.
How to calculate Complementary Function?
The Complementary function formula is defined as a part of the solution for the differential equation of the under-damped forced vibrations is calculated using Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant). To calculate Complementary Function, you need Amplitude of Vibration (A), Circular Damped Frequency d) & Phase Constant (ϕ). With our tool, you need to enter the respective value for Amplitude of Vibration, Circular Damped Frequency & Phase Constant and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
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