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A free vibration experiment was conducted on a cantilever beam and the acceleration signal was measured by an accelerometer at the end of the beam. Using the Matlab function "cumsum" and twice integrating the recorded acceleration signal, the displacement values were obtained and filtered with a 3th-order high pass butterworth filter. The maximum displacement value was found to be reasonable comparing to the value observed in the experiment. The Matlab function "ginput" was then used to manually measure the possible periods of motion and calculating the first natural frequency. The frequency content of the signal was calculated by discrete Fourier transformation and applying Matlab function "FFT". Then the first 4 natural frequencies were identified. Finally using the Euler-Bernoulli theory, a dynamic model of the beam was created and the first four natural frequencies were calculated and compared with the experimental results.
The objective of this experiment is to introduce the concept of resonance frequency (natural frequency) from both an experimental and theoretical point of view. It should be obtained through the following objectives.
- Numerical calculating of displacement from acceleration signal using the Matlab function cumsum.
- Using the Matlab function ginput to manually measure the periods of vibration on acceleration signal and identify the first natural frequency.
- Carry out a discrete Fourier transform to calculate the frequency content of displacement signal using Matlab function FFT and identify the first 4 natural frequencies.
- Create of a dynamic model of the beam using Euler-Bernoulli theory and Calculating of the first four natural frequencies.
Natural frequency is the key issue when studying the dynamic behaviour of mechanical systems. Resonance is the tendency of a driven system to oscillate at maximum amplitude at certain frequencies. Natural frequencies are the frequencies at which a system naturally vibrates once it has been set into motion i.e. free vibrations. In practice, resonance occurs close to the natural frequencies. A continuous structure (such as a cantilever beam) has an infinite number of natural frequencies. Knowledge of a structure natural frequency is essential for understanding dynamic behaviour of a structure in a certain environment. The easiest way to observe (or measure) the natural frequencies is to let a mechanical system vibrate freely. Applying a sudden impact, the mechanical system starts to vibrate at its natural frequencies with acceleration "a (t)" and damp down to zero with a typical exponential decay .
The velocity of vibration V (t) can be obtained by integrating "a (t)" and consequently the displacement is the result of integrating V (t) over time.
V (t) = âˆ«da(t)/dt S (t) = âˆ«dV(t)/dt
The first natural frequency has the lowest frequency and causes the largest displacements while the others cannot be easily distinguished due to the high accelerations and low amplitudes .
Period T is the duration of a single repetition of a cyclic motion. In other words period of vibrating is the distance between the two similar points on the graph of its motion. The number of motion cycles repeated in time unit is frequency (f or Ï‰) and can be determined by measuring T
f= 1/T cycles/second
Fourier transform is one of the most important tools when analyzing natural frequencies. It transforms the domain of the signals x (t) from time to frequency.
X (Ï‰) = âˆ« x (t) â‹… eâˆ’iÏ‰ t . dt
The maximum values of X (Ï‰) of a vibration signal x (t) are the natural frequencies.
Euler-Bernoulli theory describes the static and dynamic behavior of beams. Equation (1) is the differential equation of motion of beams which describes the dynamic behavior of the beam .
It is assumed that the weight of beam is ignorable
In free vibration distributed loads on the beam should be equal to zero, that is:
Solving the equation (1) results: w(x, t) = T (t). X ()
T (t) only describes the transient motion of the beam. X () is related to natural frequencies and mode shapes and defines in equation (2).
Constants to depend to boundary conditions of the beam .
The system set up for the experiment consists of a cantilever supported beam with an accelerometer installed at the end of it (Figure 1).
Figure 1. Cantilever beam, data acquisition unit and accelerometer
The beam has a rectangular uniform cross section of h=0.0095 m height, b=0.016 m width and a length of L=0.678 m. It is made of steel with beam E=196 GN/m2 and Ï=7900 Kg/m3.
The accelerometer is linear and can measure the vertical acceleration at the free end of the beam. Its output is in milivolt and it is calibrated before and the conversion coefficient value is equal to .
The accelerometer is connected to a data acquisition unit (Figure 2) which is itself connected to a laptop through a USB connector. The laptop can run the Matlab software.
Figure 2. The data acquisition unit
Procedure and Calculations
1. Experimental Data
The accelerometer was mounted at the end of the beam and connected to the data acquisition unit which itself connected to laptop through the USB-unit. After running Matlab and setting fs =10000 Hz and N=50000 the following syntax was used to acquire data in software.
>> volt = AcqNUpdates('dev2/ai0',-5,5,N,fs,-1);
Then by placing the fingertip at the end of the beam and Pressing it down around 5-10 mm and releasing it, the beam started to vibrate freely. The acceleration signal in milivolt was recorded for 5 seconds. The experiment was repeated for two times. Finally the accelerometers sensitivity in the documentation was checked.
Then the signal was converted to m/s2 by multiplying it by and plotted as a function of time by using Matlab. Figure 3 shows the acceleration in time domain.
Figure 3. Acceleration (m/s2) vs time(s)
The acceleration decreased continuously and finally the beam stops its vibration.
2. Calculation of the displacement
Displacement of the free end of the beam was calculated by twice integrating the acceleration values by applying the cumsum function of in Matlab. It was then plotted as it shown in figure 4.
Figure 4. Displacement (m) vs time(s)
The free end of the beam sweep a curve instead of a line and this can affect the experiment result and enter errors in acceleration values. Also some environmental noises might be measured as signal values. The effect of such noises must be decreased to have a pure accurate signal. This was done by creating a 3th-order highpass butterworth filter with a cutoff frequency at 3 Hz and filter the displacement signal using Matlab command below:
[b,a] = butter (3, 3/ (fs/2),'high');
xf = filter(b,a,x);
The displacement signal after filtration was plotted. Figure 5 shows the filtered signal. Consequently the maximum displacement was found to be about 1.5 mm.
Figure 5. Displacement (m) vs time (s) after using highpass filter
3. Resonance Frequency
Matlab ginput function can be used to manually measure the distance between the two picks in Figure 5. This is the period of the first natural frequency. However, for a better view, figure 5 was zoomed which is shown in figure 6. And the period was measured.
Figure 6. Scaled displacement (m) vs. time(s) after using high pass filter
T = 0.0323 s then f = 1/T = 30.9438 Hz
Other natural frequencies cannot be clarified from the graphs.
4. Frequency Analysis
Matlab FFT function was used to transforming the displacement signal from time domain to frequency domain. Then it was plotted in order to find the natural frequencies. Figure 6 shows the displacement signal versus frequency. The natural frequencies were determined directly from figure 7 as below:
f1= 31.20 Hz It is very close to f1 obtained from Resonance Frequency method in 2.3
f2= 205.52 Hz
f3= 422.78 Hz
f4= 562.80 Hz
Figure 7. Displacement signal in frequency domain
5. Analytical Model
In order to carry out a mathematical analysis for vibrations of a continuous structure, an analytical model should be created. This model is clarified for a cantilever beam by using Euler-Bernoulli theory. Applying the boundary conditions (3) of the cantilever beam to the equation (2) results a system of equations.
Vibration is only possible for the non-trivial solution. This is true if the determinant of the system of equations is zero which results the natural frequencies relation (4). (4)
m= Ï.v= Ï (h.b.L)
m=0 .814 Kg
The first four natural frequencies were calculated
Filtration of signal omits the noises and leads to better more accurate results. From the acceleration or displacement signals in time domain, only the first natural frequency can be determined. It is very difficult to determine the other natural frequencies from the time domain signals. Transforming the signal to frequency domain creates the possibility of simply determining a considerable number of the natural frequencies. So "FFT" is a necessity while determining the natural frequencies of a continuous structure.
There is considerable difference between the values of the natural frequencies obtained from the experiment and the analytical model. This difference comes from the errors enter the experimental results and also the assumption of governing the equation (4). It was assumed that the mass of the beam is small and can be ignored. So the model should be modified by taking the weight of the beam into account. Also the free end of the beam moves on a curve while vibrating and the accelerometer only measures the vertical acceleration which causes errors in measurement.