Comparison of Natural Frequencies Values of Circular Saw Blade Determined by Different Methods

In most cases high cutting speed is used when cutting wood by circular saw. This results in oscillating of circular saw blade, which may lead to destroying the tool, machine or hurting the operator. The aim of this paper is to show that it is possible to apply any of the methods used in the research as an equivalent method for obtaining the values of natural frequencies of circular saw blade. The article deals with three methods for obtaining the values of natural frequencies. The fi rst method is modal analysis, the second method is the determination of values measured experimentally and the last method calculates the values by Bessel functions. A circular saw blade was used with the diameter of 350 mm and 36 teeth on the blade.


INTRODUCTION 1. UVOD
Circular saw dynamic features such as workpiece characteristics, circular saw blade accuracy, and static and dynamic properties of the tool infl uence the accu-racy of sawing, surface roughness, operating noise level, tool life, etc.
As the circular saw blade is the most common device for cutting wood and wood based materials in the wood industry, it is important to deal with the problem of oscillation of circular saw blade.Vibrations are DRVNA INDUSTRIJA 66 (2) 123-128 (2015) emitted during the cutting process and when the critical rotational speed is reached, the circular saw blade becomes unstable.So the reduction of the amplitude of oscillations is essential for improving the parameters such as surface quality, precision of cutting, increase of the yield and longer tool life, but also for reducing noise.Therefore, it is necessary to fi nd solutions to eliminate these adverse effects.
There are many methods for determining natural frequencies (the next step is to calculate critical rotational speed).The values calculated using various equations do not give an idea of how the circular saw blade behaves.For better understanding, a software program was used in this paper that displays the deformation of the circular saw blade as a result of oscillation.The results of circular saw blade static modal analysis (for nodal diameter n=4, and nodal circles m=0) is shown in Figure 1.

MATERIJAL I METODE
In practice, one of the most important questions is how to evaluate the critical rotational speed of circu-lar saw blade.To solve this problem, it is necessary to determine the values of natural frequencies.The parameters of the circular saw blade used in this paper are shown in Table 1.(1) Here, the bending stiffness of the blade is: (2) Where: h -thickness of circular saw blade / debljina lista kružne pile, mm E -modulus of elasticity / modul elastičnosti, Nm -2 (E=2.1× 10 11 ) n -Poisson ratio / Poissonov omjer (n=0.33).
The differential equation is solved by separation of variables.The resulting function can be written in the form: (3) Where: n -number of nodal diameters / broj čvornih promjera.
Function R(k,r) is solved by Bessel functions. (4) Where: (5) r -density / gustoća, kgm -3 (r=7800 kgm -3 ) J n -the Bessel function of the fi rst kind of order n / Besselova funkcija prve vrste reda n N n -the Bessel function of the second kind of order n / Besselova funkcija druge vrste reda n After the theoretical solution, the formula obtained for natural frequency was: The resulting values of modal analysis are infl uenced by various parameters (material, type of model mesh, number of elements, etc).Five modal analyses were made with different element sizes.In the modal analysis Shell 5, shell elements of the model were used with the maximum size of 5 mm.In the modal analysis Shell 10, shell elements of the model were used with the maximum size of 10 mm and in the modal analysis Shell 20, the maximum element size was 20 mm.The density of circular saw blade mesh for modal analysis Shell 10 is shown in Fig. 2.

Experimental measurements 2.3. Eksperimentalna mjerenja
Solid circular saw blade, whose parameters are presented above in Table 1, was used for experimental measurements of natural frequencies.The method (harmonic test) was presented in more detail in the paper of Siklienka and Svoreň (1997).
Natural frequencies of the non-rotating circular saw blade f (n=0) were measured for n = 1, 2, 3, 4 on an experimental stand in the Technical University in Zvolen (Fig. 3).

RESULTATI I DISKUSIJA
In the next paragraph, mathematical and experimental results are presented in the same sequence as in paragraph Material and Methods.The calculated values of natural frequencies determined by the Bessel function are shown in Table 2. Leopold and Münz (1992) showed that the type (shell/solid) and number of elements had no signifi cant infl uence on calculation accuracy of natural frequencies of circular saw blade.Our results confi rm the same (see the values in Table 3 and 4).
Table 3 and 4 show two very similar frequency values for n=1.These values of natural frequency of cosine and sine components were called split mode by Yu and Mote (1987).
When comparing values in different columns (Shell 5) and (Shell 20), i.e. for maximum and minimum density of mesh, the difference is practically neg-  ligible (see Fig. 4), but time of computing for Shell 20 is shorter than for Shell 5.
The modal analysis with solid elements was made on a circular saw blade with the element size of 10 mm (Solid 10) and 20 mm (Solid 20).The calcu-lated results are shown in Table 4, and graphically in Figure 5.
The values of natural frequencies, which were determined by experimental measuring, (using the apparatus in Fig. 3), are shown in Table 5.The criteria for  As shown in Figure 6, the values of natural frequencies for n = 1 are the same for B.F. and Shell 5.
Values are relatively identical for n = 2, and for n = 3 the values determined by modal analysis are pretty lower than the values determined by other two methods, which are identical.For n = 4, the differences between values are relatively high.It could be said that the values determined by modal analysis are in accordance with the values determined by other methods used in the paper.

ZAKLJUČCI
On the basis of the experiments, it could be stated that: 1. Application of fi nite element method with various types of elements (shell/solid) and dimensions (5 / 10 / 20) resulted in approximate data of natural frequencies for shape modes n = 1, 2, 3, 4.This corresponds to the results of other authors cited in references.2. The values of natural frequencies determined by all three methods are acceptable.Considering the difference in the values of natural frequencies, for n=1, between values determined by experimental method and with the use of Bessel functions, and between values determined by experimental method and by fi nite element methods, it seems that the mathematical apparatus is not able to solve exactly the state of thin plate (saw disc is thin plate).3. The good conformity of the three results shows that manufacturers could use the available software that can simulate relatively exactly the behaviour of the studied phenomena -in our case natural frequencies and other parameters in the real operating conditions.The use of software will save the time and money of manufacturers for production of circular saw blades and their testing.

Figure 5
Figure 5Values of natural frequencies determined by modal analysis of circular saw blade with solid elements Slika 5. Vrijednosti vlastitih frekvencija dobivenih modalnom analizom kružne pile s volumenskim elementima

Table 2
Calculation values of natural frequencies determined by Bessel functions (B.F.)