**R. Maskeliūnas**^{1}
,
**K. Ragulskis**^{2}
,
**P. Paškevičius**^{3}
,
**L. Patašienė**^{4}
,
**L. Ragulskis**^{5}

^{1}Vilnius Gediminas Technical University, Vilnius, Lithuania

^{2, 3, 4}Kaunas University of Technology, Kaunas, Lithuania

^{5}Vytautas Magnus University, Kaunas, Lithuania

^{1}Corresponding author

Journal of Measurements in Engineering, Vol. 4, Issue 1, 2016, p. 15-22.

Received 27 October 2015; received in revised form 5 January 2016; accepted 17 January 2016; published 31 March 2016

Copyright © 2016 JVE International Ltd.

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In the paper plane vibrations of an elastic circular structure are analyzed. Investigations of a two dimensional problem are performed. It is assumed that vibrations are taking place. Several first eigenmodes are investigated. Results obtained by using the experimental procedure of stroboscopic geometric moiré are analyzed. By applying the superimposed moiré technique, the results are investigated. Different numbers of gaps are used and provide results of different quality.

**Keywords:** circular structure, elastic structure, plane vibrations, eigenmode, stroboscopic moiré, geometric moiré, superimposed moiré, experimental results.

In this paper plane vibrations of an elastic circular structure are analyzed. Investigations of a two dimensional problem are performed. It is assumed that vibrations are taking place. Several first eigenmodes are investigated. Results obtained by using the experimental procedure of stroboscopic geometric moiré are analyzed. By applying the superimposed moiré technique, the results are investigated. Different numbers of gaps are used and provide results of different quality.

This paper continues the analysis of superimposed moiré techniques and of vibrations of circular structures. Similar problems are analyzed in [1-10] and in a number of other similar papers.

Angular moiré lines are represented as:

(1)

${I}_{\theta}=\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\left(\frac{\pi}{{\lambda}_{\theta}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\frac{y}{x}\right),$
where $x$ and $y$ are the coordinates of a point of a two dimensional structure in the status of equilibrium, ${\lambda}_{\theta}$ determines the width of angular moiré lines, ${I}_{\theta}$ is the intensity of the image painted on the surface of the structure.

Radial moiré lines are represented as:

(2)

${I}_{r}=\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\left(\frac{\pi}{{\lambda}_{r}}\sqrt{{x}^{2}+{y}^{2}}\right),$
where ${\lambda}_{r}$ determines the width of radial moiré lines, ${I}_{r}$ is the intensity of the image painted on the surface of the structure.

Superimposed moiré lines are represented as:

(3)

$I=\mathrm{m}\mathrm{a}\mathrm{x}\left({I}_{\theta},{I}_{r}\right),$
where $I$ is the intensity of the image painted on the surface of the structure.

In the earlier papers the special function of the following type was introduced:

(4)

$F\left(i,\lambda ,x\right)=\left\{\begin{array}{l}0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{}\frac{1}{2}+\left(1+i\right)j\frac{x}{\lambda}\frac{1}{2}+i+\left(1+i\right)j,\mathrm{}\mathrm{}\mathrm{}j=0,\pm 1,\pm 2,\dots ,\\ \mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\frac{\pi}{\lambda}x,\mathrm{}\mathrm{}\mathrm{}\text{elsewhere},\end{array}\right.$
where $i=\mathrm{}$0, 1, 2,… determines the gap between the moiré lines, $\lambda $ determines the width of moiré lines.

Angular moiré lines with gaps are represented as:

(5)

${I}_{\theta}=F\left(i,{\lambda}_{\theta},\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\frac{y}{x}\right).$
Radial moiré lines with gaps are represented as:

(6)

${I}_{r}=F\left(i,{\lambda}_{r},\sqrt{{x}^{2}+{y}^{2}}\right).$
Investigations of a circular elastic structure are performed. It is assumed that the internal radius of the circular structure is fixed. Stroboscopic geometric moiré images for the angular and radial directions of fringes for the eighth eigenmode are shown.

When there are no gaps (that is the gap width is $i=$ 0) the images are represented in Fig. 1. When there is a single gap (that is the gap width is $i=$ 1) the images are represented in Fig. 2. When there is a double gap (that is the gap width is $i=$ 2) the images are represented in Fig. 3.

The eighth eigenmode is investigated. Results are obtained by using superimposed stroboscopic geometric moiré.

When there are no gaps (that is the gap width is $i=$ 0) the images are represented in Fig. 4. When there is a single gap (that is the gap width is $i=$ 1) the images are represented in Fig. 5. When there is a double gap (that is the gap width is $i=$ 2) the images are represented in Fig. 6.

**Fig. 1.**
Problem without gaps

a) Angular fringes

b) Radial fringes

The obtained results of investigations indicate that it is a great problem to perform the interpretation of the image when the problem is without gaps. And when there are greater gaps the number of obtained moiré fringes decreases. This has a negative effect to the accuracy of the performed measurements. From the obtained results it is possible to provide recommendations for the choice of the number of gaps. Acceptable results when $i=$ 1 are obtained.

**Fig. 2.**
Problem with a single gap

a) Angular fringes

b) Radial fringes

**Fig. 3.**
Problem with a double gap

a) Angular fringes

b) Radial fringes

**Fig. 4.**
Problem without gaps

**Fig. 5.**
Problem with a single gap

**Fig. 6.**
Problem with a double gap

This investigation is based on the use of the one dimensional model. In the previous paper when performing the analysis by using the one dimensional model the assumption was made that the intensity of lines changes smoothly according to the law of cosine squared. But in many engineering applications the lines are black and white. Thus here the one dimensional model corresponding to the problem when the lines are black and white is investigated.

The following special function is introduced:

(7)

$\stackrel{-}{F}\left(i,\lambda ,x\right)=\left\{\begin{array}{l}1,\mathrm{}\mathrm{}\mathrm{}F\left(i,\lambda ,x\right)\frac{1}{2},\\ 0,\mathrm{}\mathrm{}\mathrm{}\text{elsewhere}.\end{array}\right.$
For the case when the displacements are negative moiré lines are:

(8)

${I}_{1}=\stackrel{-}{F}\left(i,\lambda ,x-{u}_{-}\right),$
where ${u}_{-}$ is the displacement, ${I}_{1}$ is the intensity of the image. Thus:

(9)

${u}_{-}=-kx,$
where $k$ is a constant.

For the case when the displacements are positive moiré lines are:

(10)

${I}_{2}=\stackrel{-}{F}\left(i,\lambda ,x-{u}_{+}\right),$
where ${u}_{+}$ is the displacement, ${I}_{2}$ is the intensity of the image. Thus:

(11)

${u}_{+}=kx.$
The stroboscopic image has the usual representation of intensity:

(12)

${I}_{s}=\frac{{I}_{1}+{I}_{2}}{2}.$
In the performed investigation parameters have the values: $\lambda =\text{0.8}$ and $k=\text{0.1}$.

${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=\text{0}$ are graphically shown in Fig. 7.

**Fig. 7.**
${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=$ 0: 13 maximums in the envelope of the stroboscopic image

**Fig. 8.**
${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=$ 1: 7 maximums in the envelope of the stroboscopic image

**Fig. 9.**
${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=$ 2: 5 maximums in the envelope of the stroboscopic image

**Fig. 10.**
${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=$ 3: 4 maximums in the envelope of the stroboscopic image

**Fig. 11.**
${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=$ 4: 2 maximums in the envelope of the stroboscopic image

**Fig. 12.**
${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=$ 5: 2 maximums in the envelope of the stroboscopic image (distance between the observed maximums is greater than when $i=$ 4)

${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=\text{1}$ are graphically shown in Fig. 8.

${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=\text{2}$ are graphically shown in Fig. 9.

${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=\text{3}$ are graphically shown in Fig. 10.

${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=\text{4}$ are graphically shown in Fig. 11.

*I*
_{1}, *I*_{2} and *I*_{s} for $i=5$ are graphically shown in Fig. 12.

Thus one can see from the presented results that when the gap increases the intervals between the maximums of the envelope of final investigated images also increase. On the basis of this fact it is concluded that one can interpret the displacements by using moiré images that have gaps.

In the performed investigation based on the method of superimposed moiré both moiré images in the analyzed problem of a circular elastic structure in two dimensions performing plane vibrations are represented simultaneously.

The obtained results of investigations indicate that it is a great problem to perform the interpretation of the image when the problem is without gaps. And when bigger gaps are used accuracy of measurement decreases. From the obtained results it is possible to provide recommendations for the choice of the number of gaps. Acceptable results when $i=$ 1 are obtained.

Theoretical investigation of the one dimensional problem when the lines are black and white is performed. This is true in a number of engineering applications. From the obtained graphical relationships, it is concluded that one can interpret the displacements by using moiré images that have gaps.

The obtained results are used for the analysis and measurement of vibrations of mechanical devices, especially when high precision is required.

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