Frequency Analysis of a rotating shaft

AIM: - To simulate the FEA frequency analysis on the rotating shaft.

PHYSICS: -

Vibration: - To and fro motion of a particle w.r.t to a fix point is is called vibration.

Amplititude: - The maximum didplacement of particle during vibration is called amplitude.

Resonance: - When the when the frequency of disturbiong force is equal to the natural frequency of system, this phenomena is called resonance. This results in larger amplitude. The system will oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies. Frequencies at which the response amplitude is relative maximum are also known as resonant frequencies or resonance frequencies of the system.

The increase in amplitude of oscillation of an electric or mechanical system exposed to a periodic force whose frequency is equal or very close to the natural undamped frequency of the system. Resonance phenomena occur with all types of vibrations or wave there is mechanical resonance, accuastic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave. Resonant systems can be used to generate vibrations of a specific frequency (e.g., musical instrumemtal), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters).

Resonance of a driven, damped harmonic oscillator: -

Consider a damped mass on a spring driven by a sinusoidal, externally applied force. Newton's 2nd law takes the form

m d 2 x d t 2 = F 0 sin ⁡ ( ω t ) − k x − c d x d t , {\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}},}

(1)

where m is the mass, x is the displacement of the mass from the equilibrium point, F₀ is the driving amplitude, ω is the driving angular frequency, k is the spring constant, and c is the viscous damping coefficient. This can be rewritten in the form

d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = F 0 m sin ⁡ ( ω t ) , {\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {F_{0}}{m}}\sin(\omega t),}

(2)

where

ω 0 = k m {\displaystyle \omega _{0}={\sqrt {\frac {k}{m}}}} is called the undamped angular frequency of the oscillator or the natural frequency

ζ = c 2 m k {\displaystyle \zeta ={\frac {c}{2{\sqrt {mk}}}}} is called the damping ratio.

Many sources also refer to ω₀ as the resonant frequency. However, as shown below, when analyzing oscillations of the displacement x(t), the resonant frequency is close to but not the same as ω₀. In general the resonant frequency is close to but not necessarily the same as the natural frequency. The general solution of Equation (2)is the sum of a trasient solution that depends on initial conditions and a stedy state solution that is independent of initial conditions and depends only on the driving amplitude F₀, driving frequency ω, undamped angular frequency ω₀, and the damping ratio ζ. The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution.

It is possible to write the steady-state solution for x(t) as a function proportional to the driving force with an induced phase change φ,

x ( t ) = F 0 m ( 2 ω ω 0 ζ ) 2 + ( ω 0 2 − ω 2 ) 2 sin ⁡ ( ω t + φ ) , {\displaystyle x(t)={\frac {F_{0}}{m{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}}\sin(\omega t+\varphi ),}

(3)

where

φ = arctan ⁡ ( 2 ω ω 0 ζ ω 2 − ω 0 2 ) + n π . {\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi .}

The phase value is usually taken to be between −180° and 0 so it represents a phase lag for both positive and negative values of the arctan argument.

Steady-state variation of amplitude with relative frequency ratio( ω / ω 0 {\displaystyle \omega /\omega _{0}} ) and damping ratio (ζ {\displaystyle \zeta } )

Resonance occurs when, at certain driving frequencies, the steady-state amplitude of x(t) is large compared to its amplitude at other driving frequencies. For the mass on a spring, resonance corresponds physically to the mass's oscillations having large displacements from the spring's equilibrium position at certain driving frequencies. Looking at the amplitude of x(t) as a function of the driving frequency ω, the amplitude is maximal at the driving frequency

ω r = ω 0 1 − 2 ζ 2 . {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}

ω_r is the resonant frequency for this system. Again, note that the resonant frequency does not equal the undamped angular frequency ω₀ of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including ω₀, but the maximum response is at the resonant frequency.

Also note that ω_r is only real and non-zero if ζ < 1 / 2 {\displaystyle \zeta <1/{\sqrt {2}}} , so this system can only resonate when the harmonic oscillator is significantly underdamped. For systems with a very small damping ratio and a driving frequency near the resonant frequency, the steady state oscillations can become very large.

Resonance of a pendulum

For other driven, damped harmonic oscillators whose equations of motion do not look exactly like the mass on a spring example, the resonant frequency remains

ω r = ω 0 1 − 2 ζ 2 , {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}},}

but the definitions of ω₀ and ζ change based on the physics of the system. For a pendulum of length l and small displacement angle θ, Equation (1) becomes

m l d 2 θ d t 2 = F 0 sin ⁡ ( ω t ) − m g θ − c l d θ d t {\displaystyle ml{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-mg\theta -cl{\frac {\mathrm {d} \theta }{\mathrm {d} t}}}

and therefore

ω 0 = g l , {\displaystyle \omega _{0}={\sqrt {\frac {g}{l}}},}

ζ = c 2 m l g . {\displaystyle \zeta ={\frac {c}{2m}}{\sqrt {\frac {l}{g}}}.}

Mode: – The mode is a statistical term that refers to the most frequently occurring number found in a set of numbers. The mode is found by collecting and organizing data in order to count the frequency of each result. The result with the highest count of occurrences is the mode of the set, also referred to as the modal value.

A mode shape is a specific pattern of vibration executed by a mechanical system at a specific frequency. Different mode shapes will be associated with different frequencies. The experimental technique of modal analysis discovers these mode shapes and the frequencies.

Modes are inherent properties of a structure and are determined by the material properties (mass, damping and stiffness) and boundary conditions of the structure. Each mode is defined by a natural (modal or resonant) frequency, modal damping and a mode shape (i.e. the so called Modal parameters).

whirling of shaft