Evaluation_Formulae

3.2.1.1 Evaluation Formulae

The axial deviation of the saw blade induced by asymmetric oscillation can be described by the following formula:

,(3.36)

or

,(3.37)

where x_0(r) – amplitude of forced oscillation as a function of the strained point from the saw blade centre [mm],

ω_s = 2.π.f_d – circular frequency of radially propagating wave,

ω_n = 2.π.f_n – circular frequency of the saw blade,

n – number of the saw blade revolutions,

t – time,

φ – phase constant,

k – number of nodal averages (k = 2, 3, 4, …),

f_d – dynamic frequency of the saw blade [Hz],

f_n – basic frequency of the saw blade oscillation [Hz].

Asymmetric waves can propagate in two mutually opposite directions, either rotationwise or counter-rotationwise. The velocity increase of these waves in the rotating system can be expressed by ratio that was observed as stable in relation to the saw blade (Siklienka, Svoreň, 1997). On the other hand, the absolute angular velocities in the stationary system (relative) are expressed by for the wave propagating counter-rotationwise and the absolute angular velocities are expressed by for the wave propagating rotationwise.

The frequencies of these waves can be calculated by the following expressions:

,(3.38)

and for:

,(3.39)

where f₁ – frequency of the wave propagating rotationwise,

f₂ – frequency of the wave propagating counter-rotationwise

The resonance state comes if the multiple of excitation frequency is equal to frequency f₁ eventually f₂, while the excitation frequency of the saw blade rotation is equal to frequency f_n.

On the basis of the above information, the dynamic inherent frequency of the saw blade is determined by the following expression:

,(3.40)

λ is a factor depending on both the oscillation shape and the clamping ratio. For non-modified saw blades it is λ = 1.5.

In consequence of the above mentioned cyclic axial loading, an excessive increase of the strain occurs on the bearing located closer to the saw blade, which is confirmed also by the maintenance practice.

To the negative consequences of saw blade resonance oscillation belongs the apparition of fatigue cracks in the gullet, noisy running, axially and radially excentric running of the saw blades resulting in wider cutting gap and, moreover, the electricity consumption is increased, too. Therefore, great effort is made to investigate these phenomena by applying new modern methods that make it possible to analyze the dynamic behaviour of saw blades. Propre application of the results obtained at construction of saw blades as well as at the cutting process can reduce the risk of damage and increase their durability.

If during the exposure time the object recorded on a hologram performs a harmonic motion, then during the reconstruction the average time value of the object wave appears. In one of the first works published on holographic interferometry Powell and Stetson introduced the method of vibration analysis (Powell, Stetson, 1965).

The brightest fringe in this pattern merges with the nodal area, i.e. with those places on the object surface that do not move during the oscillation. Besides the bright nodal fringes in the holographic image other fringes can be observed, each one corresponding to the geometric position of the points oscillating with equal amplitude.

An example of this process is the vibration of the disc whose surface regularly moves in the rhythm of sinusoidal oscillation round the equilibrium position. The imediate deviation of each point in the direction along with the z-axis is characterised by time function:

,(3.41)

where d_z – amplitude of mechanical oscillations,

⌊⌋ – angular frequency of oscillations.

Path difference ⌊⌋(x, t) of the light beam is given by expression:

,(3.42)

while for this difference the following expression is valid:

, (3.43)

where φ – angle of incidence of the light beam,

ψ – angle of reflection of the light beam (toward the observer).

By application of phase difference:

,(3.44)

the complex amplitude of the light wave will have the form .

For illumination intensity in the reconstructed image the following is valid:

,(3.45)

and furthermore:

,(3.46)

where J₀ is Bessell function of the first kind and zero order.

On the basis of the previous formulae we can derive the following equation for deviation d_z:

,(3.47)

where N_i() is i-th zero order of quadrate of Bessel function ,

⌊⌋ – wavelength of the used light.

Formula (3.47) enables us to determine – from measurable values ⌊⌋ , φ, ψ and from the table values Ni(J₀₂) (Vest, 1979) – the values d_z necessary for the analysis of reconstructed interferograms.