Análisis del texto

This chapter is an experimental study of the dynamics of a steel cantilever beam with a motion limiting constraint on one side, subject to harmonic forcing. Impacts between the beam and the constraint can occur for a range of forcing frequency values. This results in vibro-impact motion of the beam. For systems which are linear away from the constraint, such as the beam system vibrating with small am plitude displacements, the nonlinearity in the system is induced by the non­ smooth nature of the impact.

The experimental apparatus used for this work has been used in the past by Bishop et ai and a dynamical study of the system was presented in (Bishop et al.

1996). The present setup has the addition of a specially constructed impact load cell to measure the force applied to the stop by the beam at each contact. The load cell was constructed using strain gauges mounted on a thin wall aluminium tube, such th at the longitudinal displacement of the tube is measured (as strain) and then related to the force of impact. This technique has similarities with the sensing block method (Chuman et al. 1997), for measuring an impact force using strain gauges mounted on a ‘block’.

Here attention is focused on interpretation of the dynamics of the beam using only an experimentally recorded signal from the tip of the impact stop. Interspike

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Impact

load cell A djustable im pact stop Displacem ent transducer Electrom agnetic exciter Beam

F ig u r e 4.1: A schem atic diagram of the experim ental a p p aratu s.

intervals are used to investigate the periodicity of the system. The coefficient of restitution ruie (see section 2.3.1) is used to model the vibro-impact behaviour and to compare numericaf predictions with experimental results. By measuring the duration of the impacts an assessment is made over the suitability of the instantaneous impact assumption used in the numerical modelling.

4.1

Experim ental setup

A schematic representation of the experimental setup used for this study is shown in figure 4.1. The impacting oscillator is a steel beam held vertically. One end of the beam is clamped to a heavy metal base, while the other is free to move laterally. Two steel walls are fixed at the metal base, one at each side of the beam. These are used to hold an adjustable impact stop, an electromagnetic exciter and a capacitative displacement transducer.

The forcing exciter was placed close to the node of the second lateral mode of vibration of the beam (see appendix B for estimation of this position). This is a small distance from the top of the beam and ensures minimal excitation of the second mode by forcing (Thompson et al. 1994).

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Depending on the position of the impact stop and the forcing frequency, the beam would perform vibro-impact motion. For these tests the impact stop was placed near the tip of the beam, with a pre-set gap between the beam and the tip of the im pact stop, referred to as the stop distance. This distance was fixed at a value corresponding approximately to a -0.092 volts reading in the displacement transducer. The transducer behaves linearly in the range of the oscillation ampli­ tude of the beam and 1 Volt corresponds to 10mm. Therefore, the stop distance was approximately 0.9mm.

For the particular configuration of stop and the beam used, vibro-impact mo­ tion occurs for forcing frequencies, / , in the range 19.9 < / < 24.5 Hz. T = 1 / / is the period of forcing. Close to the tip of the adjustable stop an impact load cell is mounted. This is used to record the strain exerted at the tip of the stop while the beam is in contact with it. The cell is capable of detecting longitudinal impacts with forces as high as 1 Newton.

The output from the load cell was recorded using a strain gauge monitor, linked to a National Instruments LabPC-l- data acquisition board. The board was also providing the analogue voltage signal required to drive the electromagnetic exciter through a direct link with the exciter. The data acquisition board was controlled by a PC running Labview 4.0 software. This allowed control of the frequency and waveform of the forcing, both of which could be programmed on the computer. In our experiments forcing has sinusoidal amplitude variation.

A detailed description of the experimental apparatus is given in appendix A.

4.2

R ecording o f data

In this section we describe the techniques used to record the impulse spike data from the load cell. The voltage signal, 6(r), where r is time, from the strain gauge monitor was digitally sampled and recorded using a personal computer. The maximum sampling rate i?, possible to achieve using this configuration was

R = 60000 samples/second. Figure 4.2a shows part of a time series recorded using

C H A P T E R 4. E X P E R I M E N T A L ANALYSI S T HR O U G H I MPULS E S P I K E R E C O R D I N G S 64 0.035 0.03 0.025 0.02 o > 0.015 2 0.01 0.005 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Time (seconds) _(a)

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 -0.005 --- '--- '--- '--- '--- '--- '--- '--- '--- '--- 0.034 0.0342 0.0344 0.0346 0.0348 0.035 0.0352 0.0354 0.0356 0.0358 0.036 Time (seconds) _(b)

F ig u r e 4.2: Tim e series of a vibro-im pact m otion showing response of im pact load cell, 6 (r), as strain in volts using a sam ple ra te of 60000 sam ples/second, (a) 5000 sam ples (b) 120 sam ple close up of im pulse spike, individual sam ples shown as diam onds. (Figure produced by D .J. Wagg)

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d ata from a mechanical experiment has been shown in (Pollard 1977).

At this rate of sampling, recording N = 5000 samples corresponds to 0.08 seconds of data. The sample contains one impulse spike, the remaining data be­ ing noise generated in the electronic circuitry used for instrum entation, and from external disturbance/ vibration of the system.