The document Vibration Isolation Notes | EduRev is a part of the Mechanical Engineering Course Theory of Machines (TOM).

All you need of Mechanical Engineering at this link: Mechanical Engineering

**Vibrations Causing Unbalanced Forces in a Mechanical Running System**

**1. Rotating Unbalance**

(Rotors, whose C.G. is not coinciding with the axis of shaft).

At time, t

f_{un} in a particular direction,

f_{un} = (M_{Rotor} eω^{2})

where, f_{0} = (m_{Rotor} eω^{2}) — max volume of unbalanced force.

ω — force frequency or excitation.

**2. Reciprocating Unbalance: (in Piston-crank)**

f_{un} = m_{R}rω^{2} sinθ

[m_{R}—mass of Reciprocating Ports]

(mass of Piston + mass of crosshead + mass of connecting Rod)

Where, ω = forced frequency and m = machine mass (whole) which is under vibrations.

**3. Forced-Damped Systems (Perfect Reality)**

Forced vibrations of a damped spring mass system

This is the equation of forced-Damped System.

∴ The solution will be, x = c · f + P · I

After some time, CF = 0

After solving,

Amplitude of forced vibrations (A):

Hence, x = PI

∴ x = A sin(ωt - ϕ)

Where, A = Amplitude of steady state vibrations (independence of time) (forced vibrations)

Running system vibrations will never stop.

Every machine/mechanical running system must have one running life.

**4. Magnification factor (M.F.)**

∴ M.F. depends upon:**(i)** ω/ω_{n}**(ii)** ζ

Magnification Factor

**As Underdamping Increases**

⇒ ζ↓

⇒ MF ↑ ⇒ A↑

⇒ Running life decrease ↓**At Resonance**

[ω/ω_{n}= 1]

**Note:** At some time 't',

F = f_{0}sin[ωt]

x = Aω cos ωt

= Aω[sin{π/2 + (ωt - ϕ)}]

The Basic Equation was,

- Spring and damping force’s max values are perpendicular to each other.
- Inertia force lies exactly opposite (At 180°) To the spring force’s max value.

**6. Vibration Isolations**

It is used to isolate the ground from the vibrations of the Running machine so as to save other stationary m/cs from these vibration effects.

Vibration Isolation

F_{τ} = force transmitted to the ground

F_{τ} < < < < < F_{0}

∈ = F_{τ }/ F_{0} = Transmissivity

0 < ∈ < 1

∈ → 0 (for Belts)

Now, F_{τ} = Resultant of forces of spring force and damping force (max values)__Hence, ′∈′ depends upon:__**(i)** ω/ω_{n}**(ii) **ζ**Note:** If, w/w_{n} = 0 w/w_{n} = √2 ⇒ ∈ = 1

**As Underdamping**↑ ⇒ ζ↓

∈ will be constant, if w/w_{n}= √2**Vibration Isolation is Effective**

When ∈ < 1

⇒ (w/w_{n}> √2) → called "Effective Vibration-isolation-zone".**In Effective Vibration Isolation Zone:**

w/w_{n}> √2; Then At ζ = 0, e is minimum.

∴ No damping → is Best. (∵ = ∈ → 0 Best)

Hence, Damping is dangerous in this Zone.

⇒ Ass ↑ (in this zone). ∈↑ ⇒ f_{τ}↑

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!

94 videos|41 docs|28 tests