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))
dmass = A*cos(ωd-ϕ)+(Fx*cos(ω*tp-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 is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position.
Amplitude of Vibration - (Measured in Meter) - Amplitude of Vibration is the greatest distance that a wave, especially a sound or radio wave, moves up and down.
Circular Damped Frequency - (Measured in Hertz) - Circular Damped Frequency refers to the angular displacement per unit time.
Phase Constant - (Measured in Radian) - Phase Constant tells you how displaced a wave is from equilibrium or zero position.
Static Force - (Measured in Newton) - Static Force is a force that keeps an object at rest.
Angular Velocity - (Measured in Radian per Second) - The Angular Velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time.
Time Period - (Measured in Second) - Time Period is the time taken by a complete cycle of the wave to pass a point.
Damping Coefficient - (Measured in Newton Second per Meter) - Damping Coefficient is a material property that indicates whether a material will bounce back or return energy to a system.
Stiffness of Spring - (Measured in Newton per Meter) - Stiffness of Spring is a measure of the resistance offered by an elastic body to deformation. every object in this universe has some stiffness.
Mass suspended from Spring - (Measured in Kilogram) - A Mass suspended from Spring is defined as the quantitative measure of inertia, a fundamental property of all matter.
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: 45 Degree --> 0.785398163397301 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
dmass = A*cos(ωd-ϕ)+(Fx*cos(ω*tp-ϕ))/(sqrt((c*ω)^2-(k-m*ω^2)^2)) --> 5.25*cos(6-0.785398163397301)+(20*cos(10*1.2-0.785398163397301))/(sqrt((5*10)^2-(60-0.25*10^2)^2))
Evaluating ... ...
dmass = 2.64887464500036
STEP 3: Convert Result to Output's Unit
2.64887464500036 Meter --> No Conversion Required
FINAL ANSWER
2.64887464500036 2.648875 Meter <-- Total Displacement
(Calculation completed in 00.020 seconds)

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National Institute Of Technology (NIT), Hamirpur
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15 Frequency of Under Damped Forced Vibrations Calculators

Total Displacement of Forced Vibrations
Go 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))
Particular Integral
Go 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))
Maximum Displacement of Forced Vibration using Natural Frequency
Go Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity/Stiffness of Spring)^2+(1-(Angular Velocity/Natural Circular Frequency)^2)^2))
Static Force using Maximum Displacement or Amplitude of Forced Vibration
Go Static Force = Total Displacement*(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))
Maximum Displacement of Forced Vibration
Go Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))
Phase Constant
Go Phase Constant = atan((Damping Coefficient*Angular Velocity)/(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2))
Damping Coefficient
Go Damping Coefficient = (tan(Phase Constant)*(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2))/Angular Velocity
Maximum Displacement of Forced Vibration at Resonance
Go Total Displacement = Deflection under Static Force*Stiffness of Spring/(Damping Coefficient*Natural Circular Frequency)
Maximum Displacement of Forced Vibration with Negligible Damping
Go Total Displacement = Static Force/(Mass suspended from Spring*(Natural Circular Frequency^2-Angular Velocity^2))
Static Force when Damping is Negligible
Go Static Force = Total Displacement*(Mass suspended from Spring*Natural Circular Frequency^2-Angular Velocity^2)
Complementary Function
Go Complementary Function = Amplitude of Vibration*cos(Circular Damped Frequency-Phase Constant)
External Periodic Disturbing Force
Go External Periodic Disturbing Force = Static Force*cos(Angular Velocity*Time Period)
Deflection of System under Static Force
Go Deflection under Static Force = Static Force/Stiffness of Spring
Static Force
Go Static Force = Deflection under Static Force*Stiffness of Spring
Total Displacement of Forced Vibration given Particular Integral and Complementary Function
Go Total Displacement = Particular Integral+Complementary Function

Total Displacement of Forced Vibrations Formula

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))
dmass = A*cos(ωd-ϕ)+(Fx*cos(ω*tp-ϕ))/(sqrt((c*ω)^2-(k-m*ω^2)^2))

What is undamped free vibration?

The simplest vibrations to analyze are undamped, free, one degree of freedom vibrations. "Undamped" means that there are no energy losses with movement (whether intentional, by adding dampers, or unintentional, through drag or friction). An undamped system will vibrate forever without any additional applied forces.

What is forced vibration?

Forced vibrations occur if a system is continuously driven by an external agency. A simple example is a child's swing that is pushed on each downswing. Of special interest are systems undergoing SHM and driven by sinusoidal forcing.

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, The Total displacement of forced vibrations formula implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object. Total Displacement is denoted by dmass 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 d), Phase Constant (ϕ), Static Force (Fx), Angular Velocity (ω), Time Period (tp), 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 -> 2.648875 = 5.25*cos(6-0.785398163397301)+(20*cos(10*1.2-0.785398163397301))/(sqrt((5*10)^2-(60-0.25*10^2)^2)).

FAQ

What is Total Displacement of Forced Vibrations?
The Total displacement of forced vibrations formula implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object and is represented as dmass = A*cos(ωd-ϕ)+(Fx*cos(ω*tp-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 greatest distance that a wave, especially a sound or radio wave, moves up and down, Circular Damped Frequency refers to the angular displacement per unit time, Phase Constant tells you how displaced a wave is from equilibrium or zero position, Static Force is a force that keeps an object at rest, The Angular Velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time, Time Period is the time taken by a complete cycle of the wave to pass a point, Damping Coefficient is a material property that indicates whether a material will bounce back or return energy to a system, Stiffness of Spring is a measure of the resistance offered by an elastic body to deformation. every object in this universe has some stiffness & A Mass suspended from Spring is defined as the quantitative measure of inertia, a fundamental property of all matter.
How to calculate Total Displacement of Forced Vibrations?
The Total displacement of forced vibrations formula implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object is calculated using 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 Total Displacement of Forced Vibrations, you need Amplitude of Vibration (A), Circular Damped Frequency d), Phase Constant (ϕ), Static Force (Fx), Angular Velocity (ω), Time Period (tp), Damping Coefficient (c), Stiffness of Spring (k) & Mass suspended from Spring (m). With our tool, you need to enter the respective value for Amplitude of Vibration, Circular Damped Frequency, Phase Constant, Static Force, Angular Velocity, Time Period, Damping Coefficient, Stiffness of Spring & Mass suspended from Spring and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
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 5 other way(s) to calculate the same, which is/are as follows -
  • Total Displacement = Deflection under Static Force*Stiffness of Spring/(Damping Coefficient*Natural Circular Frequency)
  • Total Displacement = Static Force/(Mass suspended from Spring*(Natural Circular Frequency^2-Angular Velocity^2))
  • Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity/Stiffness of Spring)^2+(1-(Angular Velocity/Natural Circular Frequency)^2)^2))
  • Total Displacement = Static Force/(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))
  • Total Displacement = Particular Integral+Complementary Function
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