Vibrational tests of a system with two coupled beams and a distributed tip mass

We present the dynamical response of a nonlinear system designed for vibrational energy harvesting. The mechanical resonator is composed of two steel beams. The beams are coupled by a screw together at the ends with various distance element. Additionally, two piezoelectric patches are placed on their surface giving output powers on electric loads. In the article, we examine the output powers and dynamics of the system for variable frequency, different sizes of distance elements and several tip mass distributions.


Introduction
Recently, various nonlinear systems where proposed for vibration energy harvesting [1][2][3][4]. They guarantee broadband frequency energy converters important for variable vibration energy sources [5][6][7] comparing to linear devices [8]. The role of nonlinearities [5,6] is important non only in broadening of frequency in the main resonance region through inclination of the resonance curve but also in involving multiple solutions. Larger amplitude solutions were observed in the double well system resonators [6,9]. Following the concept of the inverted elastic pendulum which undergoes a buckling bifurcation of single to double equilibria under load of a tip mass [10,11], we propose the system of specific elasticity based on the coupled beams and a tip mass. In this system we observe buckling phenomenon for larger amplitudes of excitation.
The tested system is characterized by a simple construction. The two flexible beams are coupled with a spacer. In addition, the end of the system is loaded with a tip mass. The separation distance and the value and distribution of tip mass can be changed in particular measurements. To our knowledge, the beam arrangement with this geometry has not been studied before. The construction of the system creates conditions to excite high amplitude vibrations at various modes involving snap through and impact phenomena. Therefore, the main aim of the work is to examine the dynamics of this system in various configurations and at different levels of excitation. The dynamics of both beams is studied by measuring the average relative power generated in the piezoelectric elements.

Experiment configuration
The geometry of the investigated system is presented in Fig. 1 1. Two beams are fixed to the shaker base with a constant distance D. Figure 1. System geometry. P 3 , P 4 -top and bottom patches (we will use the same notation for power output on the corresponding electrical resistors R 3 and R 4 ). d -spacer thickness. Difference between normal and long masses is shown. The photos show the configurations of the long tip mass with two and four additional nuts screwed to its ends, marked as 2x2 and 2x4, respectively Both beams were coupled through a distance element with thickness d (Fig. 1). A piezoelectric patch was sticked to each beam. Their ends are screwed with a spacer of thickness d between them. The screw joining the ends of both beams is simultaneously the tip mass of the system. In the measurements, we use two tip masses: normal (short) and long, both with the same value m = 17.95 g. In addition, at the ends of the long mass we screw the pairs of nuts, obtaining new configurations of tip mass: 2x2 and 2x4, as it was shown in figure 1. Such determinations of the tip mass are related to the weight of a single nut, which equals to about 2 g The scheme of the measurement system is shown in Fig. 3. A series of measurements of the system were made at various distances d, tip masses and levels of excitation. There was no stimulation control in the measurements. Therefore, to minimize the effect of variable excitation on the measurement results, all measured values have been normalized. For this purpose, the following conversions were made. Firstly, the window was moved along the time axis with discrete values corresponding to the measuring points. The width of the window equals n points on time axis and the length of the shift step is ∆t s . Next, for each window was calculated the square of the standard deviation of the excitation σ 2 a which was expressed in (m/s 2 ) 2 . For both piezo voltages, the average normalized electrical power in the window was calculated according to the formula: where σ a is a standard deviation of excitation acceleration calculated for n steps.

Results
In order to gain an insight into the dynamics of the tested system, we present comparative results that illustrate the impact of changing the various parameters of the experiment on  (c) (d) Figure 3. Continuation indicate the nonlinear effects. The assumed measuring procedure of solution continuation leads to dynamical hysteresis.
To explore these differences we repeated experiments for various system parameters. The next figure (Fig. 4) shows how the power generated in the system is modify under the influence of a small change in the tip mass distribution.  In these measurements, we examine the system with d = 3.0 mm excited at the level of U E = 50 mV. Both presented results differ in tip mass configuration: (a) -2x2, (c) -2x4 (see definition in Fig. 1). Figures (b) and (d), for both cases, show the magnification of the resonance area near 60 Hz. A slight modification of the tip mass induces clear changes in the system vibrations. A new resonance occurs for a frequency of around 30 Hz. At higher frequencies, two effects can be noted for the resonance area of 60 Hz. Firstly, in both piezoelectric elements there is a large increase in normalized power, mainly for the sweeping towards lower frequencies (red). The second effect is the shift of resonance peaks towards the lower frequencies, which is definitely greater for the red lines -the direction of the sweep to the left (red). To make it easier to observe this phenomenon, the resonance area around 60Hz was shown in zoom in figures 4b and d.  Distance d is an important parameter, significantly affecting the dynamics of the system. Interestingly, due to the geometrical conditions (see Fig. 1), the distance d play some role in the contact appearance and loss during buckling of one of the beam. These impacting solution can give fairly large power for higher frequency [12].
The results of the system dynamics study for different values of this parameter are presented in Fig. 5. It shows the results obtained for highest excitation on the level of U E = 100mV. Figure 5 contains two three-dimensional plots showing the power charac- distributed, one can probably find the optimal distance d, for which the system's efficiency will be the highest.