Particular Integral Calculator

✖Static Force is the constant force applied to an object undergoing under damped forced vibrations, affecting its frequency of oscillations.ⓘ Static Force [F_x]			+10% -10%
✖Angular velocity is the rate of change of angular displacement over time, describing how fast an object rotates around a point or axis.ⓘ Angular Velocity [ω]			+10% -10%
✖Time Period is the duration of one cycle of oscillation in under damped forced vibrations, where the system oscillates about a mean position.ⓘ Time Period [t_p]			+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%
✖Damping Coefficient is a measure of the rate of decay of oscillations in a system under the influence of an external force.ⓘ Damping Coefficient [c]			+10% -10%
✖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.ⓘ Stiffness of Spring [k]			+10% -10%
✖The mass suspended from spring refers to the object attached to a spring that causes the spring to stretch or compress.ⓘ Mass suspended from Spring [m]			+10% -10%

✖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.ⓘ Particular Integral [x₂]

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Particular Integral Solution

STEP 0: Pre-Calculation Summary

Formula Used

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))
x₂ = (F_x*cos(ω*t_p-ϕ))/(sqrt((c*ω)^2-(k-m*ω^2)^2))
This formula uses 2 Functions, 8 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)
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

Particular Integral - (Measured in Meter) - 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.
Static Force - (Measured in Newton) - Static Force is the constant force applied to an object undergoing under damped forced vibrations, affecting its frequency of oscillations.
Angular Velocity - (Measured in Radian per Second) - Angular velocity is the rate of change of angular displacement over time, describing how fast an object rotates around a point or axis.
Time Period - (Measured in Second) - Time Period is the duration of one cycle of oscillation in under damped forced vibrations, where the system oscillates about a mean position.
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.
Damping Coefficient - (Measured in Newton Second per Meter) - Damping Coefficient is a measure of the rate of decay of oscillations in a system under the influence of an external force.
Stiffness of Spring - (Measured in Newton per Meter) - 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.
Mass suspended from Spring - (Measured in Kilogram) - The mass suspended from spring refers to the object attached to a spring that causes the spring to stretch or compress.

STEP 1: Convert Input(s) to Base Unit

Static Force: 20 Newton --> 20 Newton No Conversion Required
Angular Velocity: 10 Radian per Second --> 10 Radian per Second No Conversion Required
Time Period: 1.2 Second --> 1.2 Second No Conversion Required
Phase Constant: 55 Degree --> 0.959931088596701 Radian (Check conversion here)
Damping Coefficient: 5 Newton Second per Meter --> 5 Newton Second per Meter No Conversion Required
Stiffness of Spring: 60 Newton per Meter --> 60 Newton per Meter No Conversion Required
Mass suspended from Spring: 0.25 Kilogram --> 0.25 Kilogram No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

x₂ = (F_x*cos(ω*t_p-ϕ))/(sqrt((c*ω)^2-(k-m*ω^2)^2)) --> (20*cos(10*1.2-0.959931088596701))/(sqrt((5*10)^2-(60-0.25*10^2)^2))

Evaluating ... ...

x₂ = 0.0249137517546169

STEP 3: Convert Result to Output's Unit

0.0249137517546169 Meter --> No Conversion Required

FINAL ANSWER

0.0249137517546169 ≈ 0.024914 Meter <-- Particular Integral

(Calculation completed in 00.004 seconds)

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Home » Physics » Theory of Machine » Vibrations » Longitudinal and Transverse Vibrations » Mechanical » Frequency of Under Damped Forced Vibrations » Particular Integral

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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

Particular Integral Formula

LaTeX Go

What is Particular Integral?

The particular integral is a specific solution to a non-homogeneous differential equation that addresses the external forces or inputs acting on a system. It complements the complementary function, which represents the system's natural response without external influences. Methods like undetermined coefficients or variation of parameters are often used to find the particular integral. The complete solution of the differential equation is the sum of the particular integral and the complementary function.

How to Calculate Particular Integral?

Particular Integral calculator 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 calculate 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 calculate Particular Integral using this online calculator? To use this online calculator 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. Here is how the Particular Integral calculation can be explained with given input values -> 0.024914 = (20*cos(10*1.2-0.959931088596701))/(sqrt((5*10)^2-(60-0.25*10^2)^2)).

FAQ

What is 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 and is represented as x₂ = (F_x*cos(ω*t_p-ϕ))/(sqrt((c*ω)^2-(k-m*ω^2)^2)) or 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)). Static Force is the constant force applied to an object undergoing under damped forced vibrations, affecting its frequency of oscillations, Angular velocity is the rate of change of angular displacement over time, describing how fast an object rotates around a point or axis, Time Period is the duration of one cycle of oscillation in under damped forced vibrations, where the system oscillates about a mean position, Phase Constant is a measure of the initial displacement or angle of an oscillating system in under damped forced vibrations, affecting its frequency response, Damping Coefficient is a measure of the rate of decay of oscillations in a system under the influence of an external force, 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 & The mass suspended from spring refers to the object attached to a spring that causes the spring to stretch or compress.

How to calculate Particular Integral?

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