Mechanical Vibrations

Name: Mechanical Vibrations
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Revisionato il 18/05/2026

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Appunti di Mechanical Vibrations basati su appunti personali del publisher presi alle lezioni del prof. Pellicano dell’università degli Studi di Modena e Reggio Emilia - Unimore, …

Esame Mechanical Vibrations

Facoltà Interfacoltà

Dal corso del Prof. Pellicano Francesco

Università Università degli Studi di Modena e Reggio Emilia

A.A. 2017-2018

72 pagine

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Estratto del documento

Mechanical Vibrations

Degrees of Freedom = the number of independent variables needed to completely describe the system dynamics.

NVH: Noise, Vibrations and Harshness: Vehicle comfortability

Base: a body vibrates when there is an oscillation

The diapason produces exactly a vibration of 440 Hz; it is used to calibrate

The effective value of a vibration is related to its energy and maybe to its destructive potential (Interquake)

Sources => where the vibration comes from, where the vibration energies

Factors: Path => How vibration is transferred

Receivers => Quantity of sound, vibration get the receiver.

1 DOF Systems

Generally the vibration is induced by a perturbation from the equilibrium position; it consists in an oscillation around the equilibrium position.

In the simplest case of a mass connected to a spring the equilibrium position is given by the spring's elongation Δ=mg/k

If a force is applied to the mass that moves from equilibrium position and starts to oscillate subjected to the elastic force Fe = -kx, considering the D'Alembert approach we get:

m&xuml; - k(x+ Δ) + mg = 0

m&xuml; + kx = 0

This is a linear ode of second order, we have to define the Cauchy's problem in order to get a solution =>

m&xuml; + kx=0
x(0)=x₀; ẋ(0) = ẋ₀

Initial position and initial velocity must be known

The same equation can be written as:

&xuml; + ω_n²x = 0

Defining ω_n = √k/m _{Natural circular frequency [rad/s]}

The general solution is given by a linear combination of sines and cosines functions:

x(t) = A sin ω_nt + B cos ω_nt

where A = x₀/ω_n; B = ẋ₀ can be calculated imposing the initial conditions.

Mechanical Vibrations

Degrees of Freedom - the number of independent variables needed to completely describe the system dynamics.

NVH: Noise, Vibrations, and Harshness: Vehicle confortability.

Bare: A body vibrates when there is an oscillation.

The diapason produces exactly a vibration of 440Hz, it is used to calibrate.

The effective value of a vibration is related to its energy and maybe to its destructive potential (in earthquake).

Source -> Where the vibration comes from; where the vibration originates.
Factors -> Path -> How vibration is transferred.
Receiver -> Quantity of sound, vibration gets the receiver.

1 DOF Systems

Generally the vibration is induced by a perturbation from the equilibrium position; it consists in an oscillation around the equilibrium position.

In the simplest case of a mass connected to a spring the equilibrium position is given by the spring's elongation. If a force is applied to the mass that moves from equilibrium position and starts to oscillate subjected to the elastic force F_e = -kx as considering the D'Alembert approach we get: -mx¨ - k(x+∆) + mg = 0

This is a linear ode of second order, we have to define the Cauchy's problem in order to get a solution ->

Initial position and initial velocity must be known

The same equation can be written as: x¨ + w_n²x = 0

The general solution is given by a linear combination of sines and cosines functions: x(t) = A sin w_nt + B cos w_nt where A = x₀/w_n; B = ẋ₀ can be calculated imposing the initial conditions.

Damper: a system which dissipates energy when stretched.

If u(t) is the relative position of the piston then forces of the damper can be formally written as:

F = F(du) = dF/du ⋅ u̇ + … = Cu̇ C=Damping coefficient

Consider again the system formed by a mass connected to the ground by a linear spring and a damper (see figure above), the equation is:

{ mü + Cu̇ + ku=0

Unforced system

u(0)=u₀ u₀=0

We search the solution of the equation using exponential functions (because the exponential function is an eigen-function with respect to the derivative operator, if we remark it derive another identical equation the solution is an exponential equation)

u(t) = A e^αt

If I derive u(t) and substituting in the equation we get (neglecting A)

m α² e^{α t} + Cα e^{α t} + k e^{α t = 0 ⇒ α² + C/m ⋅ α + k/m = 0}

⇒ α² + 2 ε w_n α + w_n² = 0

when: w_n = √(k/m)

Natural Circular frequency

ε = C/2 ⋅ √(km)

Damping Ratio

Resolving the 2^nd grade equation above w

Anteprima

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