**A. Sudintas**^{1}
,
**P. Paškevicius**^{2}
,
**L. Patašiene**^{3}
,
**L. Ragulskis**^{4}

^{1, 2, 3}Kaunas University of Technology, Kaunas, Lithuania

^{4}Vytautas Magnus University, Kaunas, Lithuania

^{1}Corresponding author

Journal of Measurements in Engineering, Vol. 3, Issue 2, 2015, p. 48-53.

Received 20 March 2015; received in revised form 10 June 2015; accepted 23 June 2015; published 30 June 2015

Copyright © 2015 JVE International Ltd.

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Measurement of eigenfrequency using a dynamically directed vibroexciter is investigated. Lyapunov indicators are calculated for various values of parameters of the system. It is shown that when the frequency of excitation coincides with the eigenfrequency there is a positive Lyapunov indicator. Thus chaotic motion takes place and causes uncertainty in the measurement of eigenfrequency.

**Keywords:** uncertainty of measurement, vibroexciter, dynamical directivity, eigenfrequency, Lyapunov indicator, chaos.

Measurement of eigenfrequency using a dynamically directed vibroexciter is investigated. Lyapunov indicators are calculated for various values of parameters of the system. Calculation of Lyapunov indicators is based on the material presented in [1, 2].

It is shown that when the frequency of excitation coincides with the eigenfrequency there is a positive Lyapunov indicator. Thus chaotic motion takes place and causes uncertainty in the measurement of eigenfrequency. Related problems of chaotic motion are investigated in [3-5].

The system is described by the following equation:

(1)

$\left[M\right]\left\{\ddot{\delta}\right\}+\left[C\right]\left\{\dot{\delta}\right\}+\left[K\right]\left\{\delta \right\}=\left\{F\right\},$
where the upper dot denotes differentiation with respect to time and:

(2)

$\left[M\right]=\left[\begin{array}{cc}1+\mu & \mu z\mathrm{c}\mathrm{o}\mathrm{s}\alpha \\ \mu z\mathrm{c}\mathrm{o}\mathrm{s}\alpha & \mu {z}^{2}\end{array}\right],$
(3)

$\left[C\right]=\left[\begin{array}{cc}{h}_{y}& 2\mu \dot{z}\mathrm{c}\mathrm{o}\mathrm{s}\alpha -\mu z\dot{\alpha}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \\ 0& {h}_{\alpha}+2\mu z\dot{z}\end{array}\right],$
(4)

$\left[K\right]=\left[\begin{array}{ll}{p}_{y}^{2}& 0\\ 0& 0\end{array}\right],$
(5)

$\left\{F\right\}=\left\{\begin{array}{c}-\mu \ddot{z}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \\ -\mu zg\mathrm{c}\mathrm{o}\mathrm{s}\alpha \end{array}\right\},$
(6)

$z=A\mathrm{c}\mathrm{o}\mathrm{s}\omega t,$
(7)

$\left\{\delta \right\}=\left\{\begin{array}{l}y\\ \alpha \end{array}\right\},$
where $y$ and $\alpha $ are the generalized coordinates, $A$ is the amplitude of excitation, $\omega $ is the frequency of excitation, $\mu $ is the inertial parameter of the system, ${h}_{y}$ and ${h}_{\alpha}$ are the damping parameters of the system, ${p}_{y}$ is the stiffness parameter of the system, *g* is the acceleration of gravity, $t$ is time.

The following initial conditions are assumed:

(8)

$\left\{\delta \left(0\right)\right\}=\left\{\begin{array}{l}0\\ \frac{\pi}{4}\end{array}\right\},$
(9)

$\left\{\dot{\delta}\left(0\right)\right\}=\left\{\begin{array}{l}0\\ 0\end{array}\right\}.$
The linearized system is described by the following equation:

(10)

$\left[\stackrel{-}{M}\right]\left\{\ddot{\stackrel{-}{\delta}}\right\}+\left[\stackrel{-}{C}\right]\left\{\dot{\stackrel{-}{\delta}}\right\}+\left[\stackrel{-}{K}\right]\left\{\stackrel{-}{\delta}\right\}=\left\{\stackrel{-}{F}\right\},$
where:

(11)

$\left[\stackrel{-}{M}\right]=\left[\begin{array}{cc}1+\mu & \mu z\mathrm{c}\mathrm{o}\mathrm{s}\alpha \\ \mu z\mathrm{c}\mathrm{o}\mathrm{s}\alpha & \mu {z}^{2}\end{array}\right],$
(12)

$\left[\stackrel{-}{C}\right]=\left[\begin{array}{cc}{h}_{y}& 2\mu \dot{z}\mathrm{c}\mathrm{o}\mathrm{s}\alpha -2\mu z\dot{\alpha}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \\ 0& {h}_{\alpha}+2\mu z\dot{z}\end{array}\right],$
(13)

$\left[\stackrel{-}{K}\right]=\left[\begin{array}{cc}{p}_{y}^{2}& -\mu z\ddot{\alpha}\mathrm{s}\mathrm{i}\mathrm{n}\alpha -2\mu \dot{z}\dot{\alpha}\mathrm{s}\mathrm{i}\mathrm{n}\alpha -\mu z{\dot{\alpha}}^{2}\mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mu \ddot{z}\mathrm{c}\mathrm{o}\mathrm{s}\alpha \\ 0& -\mu z\ddot{y}\mathrm{s}\mathrm{i}\mathrm{n}\alpha -\mu zg\mathrm{s}\mathrm{i}\mathrm{n}\alpha \end{array}\right],$
(14)

$\left\{\stackrel{-}{F}\right\}=\left\{\begin{array}{l}0\\ 0\end{array}\right\},$
(15)

$\left\{\stackrel{-}{\delta}\right\}=\left\{\begin{array}{l}dy\\ d\alpha \end{array}\right\},$
where $d$ denotes differential.

The following four initial conditions are assumed:

(16)

$\left[\begin{array}{llll}\left\{{\stackrel{-}{\delta}}^{\left(1\right)}\left(0\right)\right\}& \left\{{\stackrel{-}{\delta}}^{\left(2\right)}\left(0\right)\right\}& \left\{{\stackrel{-}{\delta}}^{\left(3\right)}\left(0\right)\right\}& \left\{{\stackrel{-}{\delta}}^{\left(4\right)}\left(0\right)\right\}\end{array}\right]=\left[\begin{array}{llll}\left\{\begin{array}{l}1\\ 0\end{array}\right\}& \left\{\begin{array}{l}0\\ 1\end{array}\right\}& \left\{\begin{array}{l}0\\ 0\end{array}\right\}& \left\{\begin{array}{l}0\\ 0\end{array}\right\}\end{array}\right],$
(17)

$\left[\begin{array}{llll}\left\{{\dot{\stackrel{-}{\delta}}}^{\left(1\right)}\left(0\right)\right\}& \left\{{\dot{\stackrel{-}{\delta}}}^{\left(2\right)}\left(0\right)\right\}& \left\{{\dot{\stackrel{-}{\delta}}}^{\left(3\right)}\left(0\right)\right\}& \left\{{\dot{\stackrel{-}{\delta}}}^{\left(4\right)}\left(0\right)\right\}\end{array}\right]=\left[\begin{array}{llll}\left\{\begin{array}{l}0\\ 0\end{array}\right\}& \left\{\begin{array}{l}0\\ 0\end{array}\right\}& \left\{\begin{array}{l}1\\ 0\end{array}\right\}& \left\{\begin{array}{l}0\\ 1\end{array}\right\}\end{array}\right],$
where number in the superscript denotes which initial condition is represented.

In the step $i$ for Lyapunov indicator $j=$ 1, 2, 3, 4 the following calculations are performed:

(18)

${s}_{k}={\left\{{\stackrel{-}{\delta}}^{\left(j\right)}\right\}}^{T}\left\{{\stackrel{-}{\delta}}^{\left(k\right)}\right\}+{\left\{{\dot{\stackrel{-}{\delta}}}^{\left(j\right)}\right\}}^{T}\left\{{\dot{\stackrel{-}{\delta}}}^{\left(k\right)}\right\},\mathrm{}\mathrm{}\mathrm{}k=1,\dots ,j-1,$
(19)

$\left\{\begin{array}{l}\left\{{\stackrel{-}{\delta}}^{\left(j\right)}\right\}:=\left\{{\stackrel{-}{\delta}}^{\left(j\right)}\right\}-\sum _{k=1}^{j-1}{s}_{k}\left\{{\stackrel{-}{\delta}}^{\left(k\right)}\right\},\\ \left\{{\dot{\stackrel{-}{\delta}}}^{\left(j\right)}\right\}:=\left\{{\dot{\stackrel{-}{\delta}}}^{\left(j\right)}\right\}-\sum _{k=1}^{j-1}{s}_{k}\left\{{\dot{\stackrel{-}{\delta}}}^{\left(k\right)}\right\},\end{array}\right.$
(20)

${d}_{i}^{\left(j\right)}=\sqrt{{\left\{{\stackrel{-}{\delta}}^{\left(j\right)}\right\}}^{T}\left\{{\stackrel{-}{\delta}}^{\left(j\right)}\right\}+{\left\{{\dot{\stackrel{-}{\delta}}}^{\left(j\right)}\right\}}^{T}\left\{{\dot{\stackrel{-}{\delta}}}^{\left(j\right)}\right\}},$
(21)

$\left\{\begin{array}{l}\left\{{\stackrel{-}{\delta}}^{\left(j\right)}\right\}:=\frac{1}{{d}_{i}^{\left(j\right)}}\left\{{\stackrel{-}{\delta}}^{\left(j\right)}\right\},\\ \left\{{\dot{\stackrel{-}{\delta}}}^{\left(j\right)}\right\}:=\frac{1}{{d}_{i}^{\left(j\right)}}\left\{{\dot{\stackrel{-}{\delta}}}^{\left(j\right)}\right\},\end{array}\right.$
where $:=$ denotes the assignment operator.

Lyapunov indicator is determined from:

(22)

${\lambda}_{j}T=\sum _{i=1}^{m}\mathrm{l}\mathrm{n}{d}_{i}^{\left(j\right)},$
where ${\lambda}_{j}$ is the Lyapunov indicator, $T$ is time of integration, $m$ is the number of time steps.

The following parameters of the investigated system are assumed: $A=$ 0.01, $\mu =$ 0.3, ${h}_{y}=$ 0.2, ${p}_{y}=$ 20*π*, ${h}_{\alpha}=$ 0.2, $g=$ 9.81.

**Fig. 1.**
${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$ as functions of time when $\omega =$ 10$\pi $

**Fig. 2.**
${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$ as functions of time when $\omega =$ 15$\pi $

Lyapunov indicators as functions of time when $\omega =$ 10$\pi $ are presented in Fig. 1 and when $\omega =$ 15$\pi $ are presented in Fig. 2.

From the presented results it is seen that all Lyapunov indicators are negative. Thus there is no chaotic motion and measurements can be performed.

Lyapunov indicators as functions of time when $\omega =$ 25$\pi $ are presented in Fig. 3, when $\omega =$ 30$\pi $ are presented in Fig. 4 and when $\omega =$ 40$\pi $ are presented in Fig. 5.

**Fig. 3.**
${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$ as functions of time when $\omega =$ 25$\pi $

**Fig. 4.**
${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$ as functions of time when $\omega =$ 30$\pi $

**Fig. 5.**
${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$ as functions of time when $\omega =$ 40$\pi $

From the presented results it is seen that all Lyapunov indicators are negative. Thus there is no chaotic motion and measurements can be performed.

Lyapunov indicators as functions of time when $\omega =$ 20$\pi $ are presented in Fig. 6.

**Fig. 6.**
${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$ as functions of time when $\omega =$ 20$\pi $

From the presented results it is seen that the first Lyapunov indicator is positive. Thus there is chaotic motion and it causes uncertainty in the measurements.

Lyapunov indicators as functions of frequency of excitation in the interval from $\omega =$ 15$\pi $ to $\omega =$ 30$\pi $ are presented in Fig. 7.

From the presented results the interval where the first Lyapunov indicator is positive is approximately seen. In this interval there is chaotic motion and it causes uncertainty in the measurements.

**Fig. 7.**
${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$ as functions of frequency of excitation

Measurement of eigenfrequency using a dynamically directed vibroexciter is investigated. Lyapunov indicators are calculated for various values of parameters of the system. Low frequency excitations and high frequency excitations are investigated. From the presented results it is seen that all Lyapunov indicators in both cases are negative. Thus there is no chaotic motion and measurements can be performed.

It is shown that when the frequency of excitation coincides with the eigenfrequency there is a positive Lyapunov indicator. Thus chaotic motion takes place and it causes uncertainty in the measurement of eigenfrequency. This fact reduces the precision of the measurement procedure.

Lyapunov indicators as functions of frequency of excitation are obtained. From the presented results the frequency interval where the first Lyapunov indicator is positive is approximately seen. In this interval there is chaotic motion and it causes uncertainty in the measurements of eigenfrequency.

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