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Total Displacement of Forced Vibrations Formula

GoTotal Displacement of Forced Vibration given Particular Integral and Complementary Function Formula

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Total displacement in forced vibrations is the sum of the steady-state displacement caused by the external force and any transient displacement. Check FAQs

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

d_tot - Total Displacement?A - Amplitude of Vibration?ω_d - Circular Damped Frequency?ϕ - Phase Constant?F_x - Static Force?ω - Angular Velocity?t_p - Time Period?c - Damping Coefficient?k - Stiffness of Spring?m - Mass suspended from Spring?

Total Displacement of Forced Vibrations Example

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Here is how the Total Displacement of Forced Vibrations equation looks like with Values.

Here is how the Total Displacement of Forced Vibrations equation looks like with Units.

Here is how the Total Displacement of Forced Vibrations equation looks like.

1.7146 = 5.25 \cdot \cos (6 - 55) + \frac{20 \cdot \cos (10 \cdot 1.2 - 55)}{\sqrt{{(5 \cdot 10)}^{2} - {(60 - 0.25 \cdot 10^{2})}^{2}}}

1.7146m = 5.25m \cdot \cos (6Hz - 55°) + \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}}}

1.7146 = 5.25 \cdot \cos (6 - 55) + \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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Total Displacement of Forced Vibrations Solution

Follow our step by step solution on how to calculate Total Displacement of Forced Vibrations?

FIRST Step Consider the formula

d tot = A \cdot \cos (ω d - ϕ) + \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

∴

d tot = 5.25m \cdot \cos (6Hz - 55°) + \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

∴

d tot = 5.25m \cdot \cos (6Hz - 0.9599rad) + \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

∴

d tot = 5.25 \cdot \cos (6 - 0.9599) + \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

∴

d tot = 1.71461194420038m

LAST Step Rounding Answer

∴

d tot = 1.7146m

Total Displacement of Forced Vibrations Formula Elements

Variables

Functions

Total Displacement

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

Symbol: d_tot

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Total Displacement

Amplitude of Vibration

Amplitude of Vibration is the maximum displacement of an object from its equilibrium position in a vibrational motion under external force.

Symbol: A

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Amplitude of Vibration

Circular Damped Frequency

Circular Damped Frequency is the frequency at which an under damped system vibrates when an external force is applied, resulting in oscillations.

Symbol: ω_d

Measurement: FrequencyUnit: Hz

Note: Value should be greater than 0.

Formulas to find Circular Damped Frequency

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

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

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 to find Total Displacement

Go Total Displacement of Forced Vibration given Particular Integral and Complementary Function

d tot = x 2 + x 1

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 Total Displacement of Forced Vibrations?

Total Displacement of Forced Vibrations evaluator 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 evaluate 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 evaluate Total Displacement of Forced Vibrations using this online evaluator? To use this online evaluator 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.

FAQs on Total Displacement of Forced Vibrations

What is the formula to find Total Displacement of Forced Vibrations?

The formula of Total Displacement of Forced Vibrations is expressed as 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)). Here is an example- 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)).

How to calculate Total Displacement of Forced Vibrations?

With 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) we can find Total Displacement of Forced Vibrations using the 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)). This formula also uses Cosine (cos), Square Root (sqrt) function(s).

What are the other ways to Calculate Total Displacement?

Here are the different ways to Calculate Total Displacement-

Total Displacement=Particular Integral+Complementary Function

Can the Total Displacement of Forced Vibrations be negative?

No, the Total Displacement of Forced Vibrations, measured in Length cannot be negative.

Which unit is used to measure Total Displacement of Forced Vibrations?

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

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Total Displacement of Forced Vibrations Formula

​GoTotal Displacement of Forced Vibration given Particular Integral and Complementary Function Formula

Total Displacement of Forced Vibrations Example

Total Displacement of Forced Vibrations Solution

Total Displacement of Forced Vibrations Formula Elements

Other Formulas to find Total Displacement

Other formulas in Frequency of Under Damped Forced Vibrations category

How to Evaluate Total Displacement of Forced Vibrations?

FAQs on Total Displacement of Forced Vibrations

Variable Name

GoTotal Displacement of Forced Vibration given Particular Integral and Complementary Function Formula