Analysis using measurements

The measurements will be performed using the approach mentioned at the beginning of this chapter. Then the modes are obtained using a method that could be considered a variation on a method calledquadrature picking. More

information about quadrature picking can be found in [3]. The adaptions made to this method are taking the average of all measurement locations and using the coherence spectra (which also have been averaged over all points) as weighting functions to exclude unreliable data.

Grid

The first step in the modal analysis is to make a grid on which the measure- ments will be done. This grid was marked on the floor with masking tape. Figure 3.19 shows a sketch of the floor and the locations of the nodes. The followingx- and y-coordinates have been used:

~

x = n110, 250, 400, 550, 810, 1070, 1220, 1370 , 1510o[cm] (3.2)

~

y = n50 , 500, 775, 1000, 1250, 1620o[cm] (3.3)

Fig. 3.19. Sketch of the floor, with measurement locations indicated

During the non-collocated measurements the acceleration sensor was placed at [x, y] = [400,775] cm. This point is also highlighted in figure 3.19. The reason for placing the sensor there is that the lower modes are not expected to have a node point at this location. Placing the sensor too close to the wall will make it more difficult to detect resonances, whereas placing it too close to the center of the room makes it impossible to detect the modes that have a node point there.

Once the sensor was positioned at the right spot, a SpecTest stiffness mea- surement was done for each measuring location by exciting the floor there and saving the data file as ..KANTOOR3_XXXX_YYYY.mat, where XXXX

and YYYY denote the four-digit x- and y-coordinate of the point actuated, respectively.

Processing of the data

At a resonance frequency, the phase shift of the transfer function of the compliance (Eq. 3.4) should be either +90◦ or−90◦

H(f) = z(f)

F(f) (3.4)

When the phase ofH(f) is (close to) +90◦ or−90◦ it holds that:

k=H(f)k

k<H(f)k is a large number (3.5)

In modal analysis it is common to denote the transfer from input degree-of- freedomito output degree-of-freedom j as:

Hout,in(f) =Hj,i(f) (3.6) For the modal analysis done here, the floor’s response has been measured at a fixed location, so the output degree-of-freedom will always be the same one. Therefore, an alternative notation will be used for the transfer functions:

Hx,y(f) (3.7)

will denote the transfer function from the excitation force, applied at location [x, y], to the floor displacement.

By considering only one measurement location it is possible to overlook some resonance frequencies, as a measurement location may coincide with a node of a particular mode of vibration. When taking the entire floor into account by summing all the transfers this problem is solved. Mathematically, this means that: Nx X x=1 Ny X y=1 k=Hx,y(f)k Nx X x=1 Ny X y=1 k<Hx,y(f)k

with:

Nx: The number of x-coordinates (in this case: 9) Ny: The number of y-coordinates (in this case: 6)

x: The index corresponding to the x-coordinate (x= 1,2, . . . , Nx) y: The index corresponding to the y-coordinate (y= 1,2, . . . , Ny)

Now there is still one problem to overcome: at certain (especially high) frequencies, unreliable results may be obtained as the measurement data is subject to noise or other disturbances. The coherence spectra, which are also available from the measurements, are a measure for how reliable results at a certain frequency are, and can be used as a weighting function to filter out the data that’s too noisy. Therefore, Eq. 3.8 will be multiplied with the average coherence of all the measurement locations:

( 1 Nx Ny) Nx X x=1 Ny X y=1 (Cohx,y(f)) Nx X x=1 Ny X y=1 k=Hx,y(f)k Nx X x=1 Ny X y=1 k<Hx,y(f)k (3.9)

Now we can state that Eq. 3.9 must have a large iff is a resonance frequency.

As a check, the amplitude plot (also summed over all measurement locations) will be examined at a presumed resonance frequency to see if it shows a peak there.

Using the script which can be found in appendix C.5 , Eq. 3.9 has been plotted, together with the summed absolute values. The peaks of Eq. 3.9 were detected and their frequencies were highlighted. These peaks should correspond to resonance frequencies. The plot is shown in figure 3.20.

Fig. 3.20. Possible resonance frequencies

When zooming in on the frequency range of interest, this plot looks as shown in figure 3.21.

Fig. 3.21. Possible resonance frequencies between 0 and 50 Hz

To check if the found peaks indeed correspond to resonance frequencies, a script has been run to plot the real and imaginary part of the transfer func- tion for these frequencies. This script can be found in appendix C.6. At a resonance, the imaginary parts should be large, and have a shape that indi- cates a mode of vibration, whereas the real part should be relatively small. This may seem redundant, as this is also how the peaks were found, but visualizing the imaginary part (and thus the shape of the mode of vibration) is very helpful to assess the found frequencies.

Six natural frequencies were verified, possible higher ones may require more measurement locations (i.e., a higher resolution) to be verified. The found

modes of vibrations take place at the following frequencies: fres,1 = 7 Hz fres,2 = 10 Hz fres,3 = 14.25 Hz fres,4 = 16.5 Hz fres,5 = 24 Hz fres,6 = 42.25 Hz

The corresponding mode shapes will be shown next. Please note that the edges of the surface- and contour plots do not coincide with the physical walls of the building: these outer walls are represented by the black rectan- gle plotted around the surface- or countour plot. The figures have also been plotted using the script in appendix C.6.

In hindsight it would have been better to include extra measuring points along the walls: this would make the surface plots more insightful, and pro- vides information about boundary conditions, like how the floor is connected to the walls. Nevertheless, the measurements

First mode: 7 Hz

The first mode of vibration has a resonance frequency of 7 Hz and is shown in figure 3.22. It has the shape one would expect for the lowest mode of a plate-like structure that is fixed along the edges, such as this floor: it has one anti-node in the center.

Fig. 3.22. First mode: 7 Hz

Second mode: 10 Hz

The second mode of vibration has a resonance frequency of 10 Hz and is shown in figure 3.23. It has a near-constant deflection along the x-direction and a sine-shaped deflection along the y-direction, with one node in the center, and two anti-nodes at one quarter and three quarters of the length which move in anti-phase.

Fig. 3.23. Second mode: 10 Hz

Third mode: 14.25 Hz

The third mode of vibration has a resonance frequency of 14.25 Hz and is shown in figure 3.24. Its shape is somewhat similar to the shape of the second mode, but it has an additional node and anti-node. The peak in the absolute value plot around [x, y] = [550,775] cm indicates that there is some local resonance at this frequency. This could be due to a heating tube or something else suspended from the floor at this point that has a resonance close to 14.25 Hz, but there may be a different explanation for the peak.

Fig. 3.24. Third mode: 14.25 Hz

Fourth mode: 16.5 Hz

The fourth mode of vibration has a resonance frequency of 16.5 Hz and is shown in figure 3.25. It is basically the second mode shape, but with the x- and y-direction interchanged.

Fig. 3.25. Fourth mode: 16.5 Hz

Fifth mode: 24 Hz

The fifth mode of vibration has a resonance frequency of 24 Hz and is shown in figure 3.26. It has four anti-nodes: The upper left, upper right, and lower anti-node in the absolute value contour plot move in the same phase, and the upper middle one moves in anti-phase with the others.

Fig. 3.26. Fifth mode: 24 Hz

Sixth mode: 42.25 Hz

The sixth mode of vibration has a resonance frequency of 42.25 Hz and is shown in figure 3.27. It appears to have nine anti-nodes. The absolute value plot shows a ’column’ of three anti-nodes in the middle that are clearly visible. There are three more anti-nodes partially visible along the left edge of the plot and three along the right edge.

Fig. 3.27. Sixth mode: 42.25 Hz