Forced Harmonic Vibration Homework Help, Assignment Help, Online Tutor, Tutoring

Submit Assignment

* *

Payment Online Assignment Help

Metallurgical Engineering

Homework Help

sfsf

Forced Harmonic Vibration

Consider a system excited by a harmonic force F₁ sin ωt expressed by the matrix equation

Since the system is undamped, the solution can be assumed as

Substituting equation (ii) in equation (i), one obtains

Hence

Hence

(i) Two vectors x₁ and x₂ are normal if



(ii) Two vectors x₁ and x₂ are orthogonal if

   .

(iii) If x₁ and x₂ normal and orthogonal, they are called orthonormal, in that case

where is the Kronecker delta, defined by

Tuned Vibration Absorber

Consider a vibrating system of mass m₁, stiffness k₁, subjected to a force F sin ω_t. As studied in case of forced vibration of single-degree of freedom system, the system will have a steady state response given by

which will be maximum when ω = ω_n. Now to absorb this vibration, one may add a secondary spring and mass system as shown in figure below

The equation of motion for this system can be given by

As we know for steady state vibration, the system will vibrate with a frequency of the external excitation; we can assume the solution to be

Substituting Equation (3) in equation (2) one may write

Using Cramer's rule one may write

where

Now

Here λ₁ and λ₂ are the roots of the characteristic equation |Z(ω)| = 0. One may note that these roots are the normal mode frequency for this two-degrees of freedom system. These free-vibration frequencies can be given by

From equation (6), it is clear that,

Hence, if a system called the primary system with a stiffness k₁ mass m₁ is subjected to an exciting force or base motion to vibrate, it is possible to completely eliminate the vibration of the primary system by suitably designing an attached spring-mass system (secondary system) with stiffness k₂ and mass m₂ such that the natural frequency of the secondary system coincide with the exciting frequency.

This is the principle of dynamic vibration absorber.