Total Displacement of Forced Vibrations 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%
✖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%
✖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%

✖Total displacement in forced vibrations is the sum of the steady-state displacement caused by the external force and any transient displacement.ⓘ Total Displacement of Forced Vibrations [d_tot]

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Total Displacement of Forced Vibrations Solution

STEP 0: Pre-Calculation Summary

Formula Used

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))
d_tot = A*cos(ω_d-ϕ)+(F_x*cos(ω*t_p-ϕ))/(sqrt((c*ω)^2-(k-m*ω^2)^2))
This formula uses 2 Functions, 10 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

Total Displacement - (Measured in Meter) - Total displacement in forced vibrations is the sum of the steady-state displacement caused by the external force and any transient displacement.
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.
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.
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

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)
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
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

d_tot = A*cos(ω_d-ϕ)+(F_x*cos(ω*t_p-ϕ))/(sqrt((c*ω)^2-(k-m*ω^2)^2)) --> 5.25*cos(6-0.959931088596701)+(20*cos(10*1.2-0.959931088596701))/(sqrt((5*10)^2-(60-0.25*10^2)^2))

Evaluating ... ...

d_tot = 1.71461194420038

STEP 3: Convert Result to Output's Unit

1.71461194420038 Meter --> No Conversion Required

FINAL ANSWER

1.71461194420038 ≈ 1.714612 Meter <-- Total Displacement

(Calculation completed in 00.020 seconds)

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Home » Physics » Theory of Machine » Vibrations » Longitudinal and Transverse Vibrations » Mechanical » Frequency of Under Damped Forced Vibrations » Total Displacement of Forced Vibrations

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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

Total Displacement of Forced Vibrations Formula

LaTeX Go

What is Damped?

Damped refers to the reduction or attenuation of oscillations in a system due to energy loss over time. This energy loss can occur from various factors, such as friction, air resistance, or internal material properties. In damped systems, the amplitude of vibrations decreases as energy dissipates, leading to a gradual settling of the system toward equilibrium. Damping can be classified into different types, including underdamped, critically damped, and overdamped, each affecting the system's response to disturbances.

How to Calculate Total Displacement of Forced Vibrations?

Total Displacement of Forced Vibrations calculator uses 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)) to calculate the Total Displacement, Total Displacement of Forced Vibrations formula is defined as a measure of the total movement of an object undergoing forced vibrations, taking into account the amplitude, frequency, and phase shift of the vibrations, as well as the damping and stiffness of the system. Total Displacement is denoted by d_tot symbol.

How to calculate Total Displacement of Forced Vibrations using this online calculator? To use this online calculator for Total Displacement of Forced Vibrations, enter Amplitude of Vibration (A), Circular Damped Frequency (ω_d), Phase Constant (ϕ), Static Force (F_x), Angular Velocity (ω), Time Period (t_p), Damping Coefficient (c), Stiffness of Spring (k) & Mass suspended from Spring (m) and hit the calculate button. Here is how the Total Displacement of Forced Vibrations calculation can be explained with given input values -> 1.714612 = 5.25*cos(6-0.959931088596701)+(20*cos(10*1.2-0.959931088596701))/(sqrt((5*10)^2-(60-0.25*10^2)^2)).

FAQ

What is Total Displacement of Forced Vibrations?

Total Displacement of Forced Vibrations formula is defined as a measure of the total movement of an object undergoing forced vibrations, taking into account the amplitude, frequency, and phase shift of the vibrations, as well as the damping and stiffness of the system and is represented as d_tot = A*cos(ω_d-ϕ)+(F_x*cos(ω*t_p-ϕ))/(sqrt((c*ω)^2-(k-m*ω^2)^2)) or 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)). 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, 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, 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 Total Displacement of Forced Vibrations?

How many ways are there to calculate Total Displacement?

In this formula, Total Displacement uses Amplitude of Vibration, Circular Damped Frequency, Phase Constant, Static Force, Angular Velocity, Time Period, Damping Coefficient, Stiffness of Spring & Mass suspended from Spring. We can use 1 other way(s) to calculate the same, which is/are as follows -

Total Displacement = Particular Integral+Complementary Function

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