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Maximum Displacement of Forced Vibration using Natural Frequency Formula

GoMaximum Displacement of Forced Vibration at Resonance Formula

GoMaximum Displacement of Forced Vibration with Negligible Damping Formula

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Maximum displacement refers to the largest distance a vibrating system moves from its equilibrium position during oscillation. Check FAQs

d max = \frac{x}{\sqrt{\frac{(c^{2}) \cdot (ω^{2})}{k^{2}}} + {(1 - (\frac{ω^{2}}{{ω n}^{2}}))}^{2}}

d_max - Maximum Displacement?x - Deflection?c - Damping Coefficient?ω - Angular Velocity?k - Stiffness of Spring?ω_n - Natural Circular Frequency?

Maximum Displacement of Forced Vibration using Natural Frequency Example

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Here is how the Maximum Displacement of Forced Vibration using Natural Frequency equation looks like with Values.

Here is how the Maximum Displacement of Forced Vibration using Natural Frequency equation looks like with Units.

Here is how the Maximum Displacement of Forced Vibration using Natural Frequency equation looks like.

0.5615 = \frac{0.993}{\sqrt{\frac{(5^{2}) \cdot (10^{2})}{60^{2}}} + {(1 - (\frac{10^{2}}{{7.13}^{2}}))}^{2}}

0.5615m = \frac{0.993m}{\sqrt{\frac{({5Ns/m}^{2}) \cdot ({10rad/s}^{2})}{{60N/m}^{2}}} + {(1 - (\frac{{10rad/s}^{2}}{{7.13rad/s}^{2}}))}^{2}}

0.5615 = \frac{0.993}{\sqrt{\frac{(5^{2}) \cdot (10^{2})}{60^{2}}} + {(1 - (\frac{10^{2}}{{7.13}^{2}}))}^{2}}

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Maximum Displacement of Forced Vibration using Natural Frequency Solution

Follow our step by step solution on how to calculate Maximum Displacement of Forced Vibration using Natural Frequency?

FIRST Step Consider the formula

d max = \frac{x}{\sqrt{\frac{(c^{2}) \cdot (ω^{2})}{k^{2}}} + {(1 - (\frac{ω^{2}}{{ω n}^{2}}))}^{2}}

Next Step Substitute values of Variables

∴

d max = \frac{0.993m}{\sqrt{\frac{({5Ns/m}^{2}) \cdot ({10rad/s}^{2})}{{60N/m}^{2}}} + {(1 - (\frac{{10rad/s}^{2}}{{7.13rad/s}^{2}}))}^{2}}

Next Step Prepare to Evaluate

∴

d max = \frac{0.993}{\sqrt{\frac{(5^{2}) \cdot (10^{2})}{60^{2}}} + {(1 - (\frac{10^{2}}{{7.13}^{2}}))}^{2}}

Next Step Evaluate

∴

d max = 0.561471335970737m

LAST Step Rounding Answer

∴

d max = 0.5615m

Maximum Displacement of Forced Vibration using Natural Frequency Formula Elements

Variables

Functions

Maximum Displacement

Maximum displacement refers to the largest distance a vibrating system moves from its equilibrium position during oscillation.

Symbol: d_max

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Maximum Displacement

Deflection

Deflection refers to the displacement of a structural element or object under load. It measures how much a point moves from its original position due to applied forces.

Symbol: x

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Deflection

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

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

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

Natural Circular Frequency

Natural Circular Frequency is the frequency at which a system tends to oscillate in the absence of any external force.

Symbol: ω_n

Measurement: Angular VelocityUnit: rad/s

Note: Value should be greater than 0.

Formulas to find Natural Circular Frequency

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 to find Maximum Displacement

Go Maximum Displacement of Forced Vibration at Resonance

d max = x o \cdot \frac{k}{c \cdot ω n}

Go Maximum Displacement of Forced Vibration with Negligible Damping

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

Go Maximum Displacement of Forced Vibration

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

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 Maximum Displacement of Forced Vibration using Natural Frequency?

Maximum Displacement of Forced Vibration using Natural Frequency evaluator uses Maximum Displacement = (Deflection)/(sqrt(((Damping Coefficient^2)*(Angular Velocity^2))/(Stiffness of Spring^2))+(1-((Angular Velocity^2)/(Natural Circular Frequency^2)))^2) to evaluate the Maximum Displacement, Maximum Displacement of Forced Vibration using Natural Frequency formula is defined as the maximum amplitude of an object's oscillation when subjected to an external force, influenced by the natural frequency of the system, and is a critical parameter in understanding the behavior of underdamped forced vibrations. Maximum Displacement is denoted by d_max symbol.

How to evaluate Maximum Displacement of Forced Vibration using Natural Frequency using this online evaluator? To use this online evaluator for Maximum Displacement of Forced Vibration using Natural Frequency, enter Deflection (x), Damping Coefficient (c), Angular Velocity (ω), Stiffness of Spring (k) & Natural Circular Frequency (ω_n) and hit the calculate button.

FAQs on Maximum Displacement of Forced Vibration using Natural Frequency

What is the formula to find Maximum Displacement of Forced Vibration using Natural Frequency?

The formula of Maximum Displacement of Forced Vibration using Natural Frequency is expressed as Maximum Displacement = (Deflection)/(sqrt(((Damping Coefficient^2)*(Angular Velocity^2))/(Stiffness of Spring^2))+(1-((Angular Velocity^2)/(Natural Circular Frequency^2)))^2). Here is an example- 0.561471 = (0.993)/(sqrt(((5^2)*(10^2))/(60^2))+(1-((10^2)/(7.13^2)))^2).

How to calculate Maximum Displacement of Forced Vibration using Natural Frequency?

With Deflection (x), Damping Coefficient (c), Angular Velocity (ω), Stiffness of Spring (k) & Natural Circular Frequency (ω_n) we can find Maximum Displacement of Forced Vibration using Natural Frequency using the formula - Maximum Displacement = (Deflection)/(sqrt(((Damping Coefficient^2)*(Angular Velocity^2))/(Stiffness of Spring^2))+(1-((Angular Velocity^2)/(Natural Circular Frequency^2)))^2). This formula also uses Square Root (sqrt) function(s).

What are the other ways to Calculate Maximum Displacement?

Here are the different ways to Calculate Maximum Displacement-

Maximum Displacement=Deflection under Static Force*Stiffness of Spring/(Damping Coefficient*Natural Circular Frequency)
Maximum Displacement=Static Force/(Mass suspended from Spring*(Natural Frequency^2-Angular Velocity^2))
Maximum Displacement=Static Force/(sqrt((Damping Coefficient*Angular Velocity)^2-(Stiffness of Spring-Mass suspended from Spring*Angular Velocity^2)^2))

Can the Maximum Displacement of Forced Vibration using Natural Frequency be negative?

No, the Maximum Displacement of Forced Vibration using Natural Frequency, measured in Length cannot be negative.

Which unit is used to measure Maximum Displacement of Forced Vibration using Natural Frequency?

Maximum Displacement of Forced Vibration using Natural Frequency is usually measured using the Meter[m] for Length. Millimeter[m], Kilometer[m], Decimeter[m] are the few other units in which Maximum Displacement of Forced Vibration using Natural Frequency can be measured.

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Maximum Displacement of Forced Vibration using Natural Frequency Formula

​GoMaximum Displacement of Forced Vibration at Resonance Formula

​GoMaximum Displacement of Forced Vibration with Negligible Damping Formula

Maximum Displacement of Forced Vibration using Natural Frequency Example

Maximum Displacement of Forced Vibration using Natural Frequency Solution

Maximum Displacement of Forced Vibration using Natural Frequency Formula Elements

Other Formulas to find Maximum Displacement

Other formulas in Frequency of Under Damped Forced Vibrations category

How to Evaluate Maximum Displacement of Forced Vibration using Natural Frequency?

FAQs on Maximum Displacement of Forced Vibration using Natural Frequency

Variable Name

GoMaximum Displacement of Forced Vibration at Resonance Formula

GoMaximum Displacement of Forced Vibration with Negligible Damping Formula