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Complementary Function Formula

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Complementary Function is a mathematical concept used to solve the differential equation of under damped forced vibrations, providing a complete solution. Check FAQs

x 1 = A \cdot \cos (ω d - ϕ)

x₁ - Complementary Function?A - Amplitude of Vibration?ω_d - Circular Damped Frequency?ϕ - Phase Constant?

Complementary Function Example

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With units

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Here is how the Complementary Function equation looks like with Values.

Here is how the Complementary Function equation looks like with Units.

Here is how the Complementary Function equation looks like.

1.6897 = 5.25 \cdot \cos (6 - 55)

1.6897m = 5.25m \cdot \cos (6Hz - 55°)

1.6897 = 5.25 \cdot \cos (6 - 55)

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Complementary Function Solution

Follow our step by step solution on how to calculate Complementary Function?

FIRST Step Consider the formula

x 1 = A \cdot \cos (ω d - ϕ)

Next Step Substitute values of Variables

∴

x 1 = 5.25m \cdot \cos (6Hz - 55°)

Next Step Convert Units

∴

x 1 = 5.25m \cdot \cos (6Hz - 0.9599rad)

Next Step Prepare to Evaluate

∴

x 1 = 5.25 \cdot \cos (6 - 0.9599)

Next Step Evaluate

∴

x 1 = 1.68969819244576m

LAST Step Rounding Answer

∴

x 1 = 1.6897m

Complementary Function Formula Elements

Variables

Functions

Complementary Function

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

Symbol: x₁

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Complementary Function

Amplitude of Vibration

Amplitude of Vibration is the maximum displacement of an object from its equilibrium position in a vibrational motion under external force.

Symbol: A

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Amplitude of Vibration

Circular Damped Frequency

Circular Damped Frequency is the frequency at which an under damped system vibrates when an external force is applied, resulting in oscillations.

Symbol: ω_d

Measurement: FrequencyUnit: Hz

Note: Value should be greater than 0.

Formulas to find Circular Damped Frequency

Phase Constant

Phase Constant is a measure of the initial displacement or angle of an oscillating system in under damped forced vibrations, affecting its frequency response.

Symbol: ϕ

Measurement: AngleUnit: °

Note: Value should be greater than 0.

Formulas to find Phase Constant

cos

Cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse of the triangle.

Syntax: cos(Angle)

Other formulas in Frequency of Under Damped Forced Vibrations category

Go Static Force using Maximum Displacement or Amplitude of Forced Vibration

F x = d max \cdot (\sqrt{{(c \cdot ω)}^{2} - {(k - m \cdot ω^{2})}^{2}})

Go Static Force when Damping is Negligible

F x = d max \cdot (m) \cdot ({ω nat}^{2} - ω^{2})

Go Deflection of System under Static Force

x o = \frac{F x}{k}

Go Static Force

F x = x o \cdot k

How to Evaluate Complementary Function?

Complementary Function evaluator uses Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant) to evaluate 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 evaluate Complementary Function using this online evaluator? To use this online evaluator for Complementary Function, enter Amplitude of Vibration (A), Circular Damped Frequency (ω_d) & Phase Constant (ϕ) and hit the calculate button.

FAQs on Complementary Function

What is the formula to find Complementary Function?

The formula of Complementary Function is expressed as Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant). Here is an example- 1.689698 = 5.25*cos(6-0.959931088596701).

How to calculate Complementary Function?

With Amplitude of Vibration (A), Circular Damped Frequency (ω_d) & Phase Constant (ϕ) we can find Complementary Function using the formula - Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant). This formula also uses Cosine (cos) function(s).

Can the Complementary Function be negative?

No, the Complementary Function, measured in Length cannot be negative.

Which unit is used to measure Complementary Function?

Complementary Function is usually measured using the Meter[m] for Length. Millimeter[m], Kilometer[m], Decimeter[m] are the few other units in which Complementary Function can be measured.

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Complementary Function Formula

Complementary Function Example

Complementary Function Solution

Complementary Function Formula Elements

Other formulas in Frequency of Under Damped Forced Vibrations category

How to Evaluate Complementary Function?

FAQs on Complementary Function

Variable Name