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Particular Integral Formula

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Particular Integral is the integral of a function that is used to find the particular solution of a differential equation in under damped forced vibrations. Check FAQs

x 2 = \frac{F x \cdot \cos (ω \cdot t p - ϕ)}{\sqrt{{(c \cdot ω)}^{2} - {(k - m \cdot ω^{2})}^{2}}}

x₂ - Particular Integral?F_x - Static Force?ω - Angular Velocity?t_p - Time Period?ϕ - Phase Constant?c - Damping Coefficient?k - Stiffness of Spring?m - Mass suspended from Spring?

Particular Integral Example

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With units

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Here is how the Particular Integral equation looks like with Values.

Here is how the Particular Integral equation looks like with Units.

Here is how the Particular Integral equation looks like.

0.0249 = \frac{20 \cdot \cos (10 \cdot 1.2 - 55)}{\sqrt{{(5 \cdot 10)}^{2} - {(60 - 0.25 \cdot 10^{2})}^{2}}}

0.0249m = \frac{20N \cdot \cos (10rad/s \cdot 1.2s - 55°)}{\sqrt{{(5Ns/m \cdot 10rad/s)}^{2} - {(60N/m - 0.25kg \cdot {10rad/s}^{2})}^{2}}}

0.0249 = \frac{20 \cdot \cos (10 \cdot 1.2 - 55)}{\sqrt{{(5 \cdot 10)}^{2} - {(60 - 0.25 \cdot 10^{2})}^{2}}}

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Particular Integral Solution

Follow our step by step solution on how to calculate Particular Integral?

FIRST Step Consider the formula

x 2 = \frac{F x \cdot \cos (ω \cdot t p - ϕ)}{\sqrt{{(c \cdot ω)}^{2} - {(k - m \cdot ω^{2})}^{2}}}

Next Step Substitute values of Variables

∴

x 2 = \frac{20N \cdot \cos (10rad/s \cdot 1.2s - 55°)}{\sqrt{{(5Ns/m \cdot 10rad/s)}^{2} - {(60N/m - 0.25kg \cdot {10rad/s}^{2})}^{2}}}

Next Step Convert Units

∴

x 2 = \frac{20N \cdot \cos (10rad/s \cdot 1.2s - 0.9599rad)}{\sqrt{{(5Ns/m \cdot 10rad/s)}^{2} - {(60N/m - 0.25kg \cdot {10rad/s}^{2})}^{2}}}

Next Step Prepare to Evaluate

∴

x 2 = \frac{20 \cdot \cos (10 \cdot 1.2 - 0.9599)}{\sqrt{{(5 \cdot 10)}^{2} - {(60 - 0.25 \cdot 10^{2})}^{2}}}

Next Step Evaluate

∴

x 2 = 0.0249137517546169m

LAST Step Rounding Answer

∴

x 2 = 0.0249m

Particular Integral Formula Elements

Variables

Functions

Particular Integral

Particular Integral is the integral of a function that is used to find the particular solution of a differential equation in under damped forced vibrations.

Symbol: x₂

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Particular Integral

Static Force

Static Force is the constant force applied to an object undergoing under damped forced vibrations, affecting its frequency of oscillations.

Symbol: F_x

Measurement: ForceUnit: N

Note: Value should be greater than 0.

Formulas to find Static Force

Angular Velocity

Angular velocity is the rate of change of angular displacement over time, describing how fast an object rotates around a point or axis.

Symbol: ω

Measurement: Angular VelocityUnit: rad/s

Note: Value should be greater than 0.

Formulas to find Angular Velocity

Time Period

Time Period is the duration of one cycle of oscillation in under damped forced vibrations, where the system oscillates about a mean position.

Symbol: t_p

Measurement: TimeUnit: s

Note: Value should be greater than 0.

Formulas to find Time Period

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

Damping Coefficient

Damping Coefficient is a measure of the rate of decay of oscillations in a system under the influence of an external force.

Symbol: c

Measurement: Damping CoefficientUnit: Ns/m

Note: Value should be greater than 0.

Formulas to find Damping Coefficient

Stiffness of Spring

The stiffness of spring is a measure of its resistance to deformation when a force is applied, it quantifies how much the spring compresses or extends in response to a given load.

Symbol: k

Measurement: Surface TensionUnit: N/m

Note: Value should be greater than 0.

Formulas to find Stiffness of Spring

Mass suspended from Spring

The mass suspended from spring refers to the object attached to a spring that causes the spring to stretch or compress.

Symbol: m

Measurement: WeightUnit: kg

Note: Value should be greater than 0.

Formulas to find Mass suspended from Spring

cos

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

Syntax: cos(Angle)

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 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})

How to Evaluate Particular Integral?

Particular Integral evaluator uses 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)) to evaluate the Particular Integral, Particular Integral formula is defined as a mathematical expression that represents the response of an underdamped system to an external force, providing the amplitude and phase of the resulting vibration in terms of the system's natural frequency, damping ratio, and forcing frequency. Particular Integral is denoted by x₂ symbol.

How to evaluate Particular Integral using this online evaluator? To use this online evaluator for Particular Integral, enter Static Force (F_x), Angular Velocity (ω), Time Period (t_p), Phase Constant (ϕ), Damping Coefficient (c), Stiffness of Spring (k) & Mass suspended from Spring (m) and hit the calculate button.

FAQs on Particular Integral

What is the formula to find Particular Integral?

The formula of Particular Integral is expressed as 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)). Here is an example- 0.024914 = (20*cos(10*1.2-0.959931088596701))/(sqrt((5*10)^2-(60-0.25*10^2)^2)).

How to calculate Particular Integral?

With Static Force (F_x), Angular Velocity (ω), Time Period (t_p), Phase Constant (ϕ), Damping Coefficient (c), Stiffness of Spring (k) & Mass suspended from Spring (m) we can find Particular Integral using the formula - 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)). This formula also uses Cosine (cos), Square Root (sqrt) function(s).

Can the Particular Integral be negative?

No, the Particular Integral, measured in Length cannot be negative.

Which unit is used to measure Particular Integral?

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

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Particular Integral Formula

Particular Integral Example

Particular Integral Solution

Particular Integral Formula Elements

Other formulas in Frequency of Under Damped Forced Vibrations category

How to Evaluate Particular Integral?

FAQs on Particular Integral

Variable Name