Complementary Function Calculator

✖Amplitude of Vibration is the maximum displacement of an object from its equilibrium position in a vibrational motion under external force.ⓘ Amplitude of Vibration [A]			+10% -10%
✖Circular Damped Frequency is the frequency at which an under damped system vibrates when an external force is applied, resulting in oscillations.ⓘ Circular Damped Frequency [ω_d]			+10% -10%
✖Phase Constant is a measure of the initial displacement or angle of an oscillating system in under damped forced vibrations, affecting its frequency response.ⓘ Phase Constant [ϕ]			+10% -10%

✖Complementary Function is a mathematical concept used to solve the differential equation of under damped forced vibrations, providing a complete solution.ⓘ Complementary Function [x₁]

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Complementary Function Solution

STEP 0: Pre-Calculation Summary

Formula Used

Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant)
x₁ = A*cos(ω_d-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) - Complementary Function is a mathematical concept used to solve the differential equation of under damped forced vibrations, providing a complete solution.
Amplitude of Vibration - (Measured in Meter) - Amplitude of Vibration is the maximum displacement of an object from its equilibrium position in a vibrational motion under external force.
Circular Damped Frequency - (Measured in Hertz) - Circular Damped Frequency is the frequency at which an under damped system vibrates when an external force is applied, resulting in oscillations.
Phase Constant - (Measured in Radian) - Phase Constant is a measure of the initial displacement or angle of an oscillating system in under damped forced vibrations, affecting its frequency response.

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: 55 Degree --> 0.959931088596701 Radian (Check conversion here)

STEP 2: Evaluate Formula

Substituting Input Values in Formula

x₁ = A*cos(ω_d-ϕ) --> 5.25*cos(6-0.959931088596701)

Evaluating ... ...

x₁ = 1.68969819244576

STEP 3: Convert Result to Output's Unit

1.68969819244576 Meter --> No Conversion Required

FINAL ANSWER

1.68969819244576 ≈ 1.689698 Meter <-- Complementary Function

(Calculation completed in 00.020 seconds)

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Created by Anshika Arya

National Institute Of Technology (NIT), Hamirpur

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< Frequency of Under Damped Forced Vibrations Calculators

Static Force using Maximum Displacement or Amplitude of Forced Vibration

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

Static Force when Damping is Negligible

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

Deflection of System under Static Force

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

Static Force

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

Complementary Function Formula

LaTeX Go

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

What is Forced Vibration?

Forced vibration occurs when a system is subjected to an external periodic force, causing it to oscillate at the frequency of the applied force rather than its natural frequency. This type of vibration can be seen in systems like machinery, where external influences, such as motors or seismic activity, induce motion. The response of the system depends on factors such as the amplitude of the applied force, damping characteristics, and the mass of the system. Unlike free vibrations, which occur without external influence, forced vibrations can lead to steady-state conditions where the system oscillates continuously at the driving frequency.

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, Complementary Function formula is defined as a mathematical representation of the oscillatory motion of a system under the influence of an external force, describing the frequency of under damped forced vibrations, where the system's natural frequency is affected by the damping force and the external force. 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 (ω_d) & Phase Constant (ϕ) and hit the calculate button. Here is how the Complementary Function calculation can be explained with given input values -> 1.689698 = 5.25*cos(6-0.959931088596701).

FAQ

What is Complementary Function?

Complementary Function formula is defined as a mathematical representation of the oscillatory motion of a system under the influence of an external force, describing the frequency of under damped forced vibrations, where the system's natural frequency is affected by the damping force and the external force and is represented as x₁ = A*cos(ω_d-ϕ) or Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant). Amplitude of Vibration is the maximum displacement of an object from its equilibrium position in a vibrational motion under external force, Circular Damped Frequency is the frequency at which an under damped system vibrates when an external force is applied, resulting in oscillations & Phase Constant is a measure of the initial displacement or angle of an oscillating system in under damped forced vibrations, affecting its frequency response.

How to calculate Complementary Function?

Complementary Function formula is defined as a mathematical representation of the oscillatory motion of a system under the influence of an external force, describing the frequency of under damped forced vibrations, where the system's natural frequency is affected by the damping force and the external force 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 (ω_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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