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Review

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2 TEST A 1.What is the single most important property of vibrating structure? 2. What happens when structure is vibrating in resonance? 3.What controls the amplitude if excitation frequency equals the resonant frequency? 4.What is modal damping? 5.Under what circumstance damping needs to be defined accurately? 6.How are natural frequencies “spaced out”? 7. What is the relation between the eigenvalue, the natural circular frequency, frequency and period?

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3 1.What is a rigid body mode (RBM)? 2.What happens to natural frequencies if you have RBM? 3.How is mass affecting natural frequencies? 4.How is stiffness affecting natural frequencies? 5.How are tensile and compressive stresses affecting natural frequencies? 6.What is minimized by lower modes? 7.What is maximized by lower modes ? 8.List some possible candidates for pre-stress analysis. TEST B

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4 1.What modal analysis can be used for? 2. What are modeling considerations for Modal Analysis? 3.When are higher modes important? 4.What are possible problems with modeling supports in modal analysis? 5. What assumptions are made in the Modal Superposition Method? 6.Does the Modal Superposition Method reduce the cost of analysis, why? 7. What is the alternative to the Modal Superposition Method? TEST C

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5 1.What does dynamic response of a system depend on? 2.What problems are best solved in time domain? 3.What problems are best solved in frequency domain? 4.What is deterministic forcing function? 5.What is random forcing function? 6.Where does the term “spectral” came from? 7. What is the Power Spectral Density function, what units does it use? TEST D

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6 1.Why is the Acceleration Power Spectral Density most often used to characterize random mechanical vibrations? 2. What is the base excitation and force excitation? 3. What are mass participation factors, how are they used? 4.How do you interpret results of Random Vibrations analysis? 5.What is the “RMS stress” ? 6.What is the mean value of displacements and stresses in vibration analysis? TEST E

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7 1.What is the Response Spectrum Method? 2.What is the correspondence between arbitrary set of oscillators in the Response Spectrum Method and the analyzed structure? 3.What is the Response Spectrum Curve 4. Why is the modal analysis a pre-requisite to the Response Spectrum Method analysis? 5.Why modes need to be combined in the Response Spectrum Method? 6.What are methods for modal combinations? 7.What is usual cut-off frequency in seismic analysis? 8.How many modes are usually considered in seismic analysis? 9.How does “common sense” apply to dynamic analysis? TEST F

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8 APPENDIX

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10 A plot of PSD amplitude versus frequency gives us a way to characterize this complex random process in a visual format. See Figure 1. When we look at a complex random vibration recording, shown in Figure 2, we may wonder how to evaluate its constantly changing acceleration amplitude. The answer is to determine the average value of all the amplitudes within a given frequency range. Although acceleration amplitude at a given frequency does constantly change its average value tends to remain relatively constant. This is not intuitively obvious. When one observes the motion of a vibration system which is reproducing a PSD spectrum, its order and predictability are not at all apparent. The process looks literally random but in fact the process strictly obeys the limits defined in the PSD plot. This powerful characterization of the random process gives us a tool to easily reproduce spectra with our vibration testing systems. Why it is called PSD? Taken at face value, the term Power Spectral Density seems to imply that it has something to do with the distribution of power over a spectrum or frequency range. In vibration testing, this is not the case. Power Spectral Density specifications applied to vibration testing would be more aptly named Acceleration Spectral Density since we are measuring and subjecting our test items to varying accelerations. Power spectral density derives its name from its initial use to study the effects of random variations of the power absorbed in an electrical circuit. The underlying theory however, is fully applicable to other engineering analyses including acoustics and mechanical vibration. The use of this nomenclature in the vibration field has a long history and for this reason we don't suggest changing the term from PSD to ASD. DEFINITION OF POWER SPECTRAL DENSITY (PSD) -1

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11 What's g2/Hz ? The vertical axis of a PSD plot is scaled in the units of g2/Hz, very strange units indeed! What physical significance do these units have? By comparison, a common unit in engineering usage is lb./in2 which is a unit of pressure and is easily recognizable as such. g2/Hz does not make any sense at first glance since it is not recognizable as a physical quantity. In fact it is a combination unit where g2 is proportional to the physical quantity being measured and Hz (frequency) is used as a normalizing factor. To understand the unit g2/Hz consider how an instrument, which is used to analyze random vibration, functions. It sections the measured acceleration versus time signal into many small frequency segments called bins. Each bin is only a few hertz wide, that is it only contains information on the accelerations associated with a narrow bandwidth of frequencies. Using a simplified example, shown in Figure 3, if our instrument has a total bandwidth of 100 Hz, we might have 5 bins. Then each bin would be 20 hertz wide meaning it only has a 20 hertz wide view of the signal. The first bin would contain frequencies from 0 to 19 Hz, the second bin would contain frequencies from 20 to 39 Hz and so on. Each bin is referred to by its center frequency. For example, the first bin is called the 10 Hz bin. Consider a situation in which the same random vibration is analyzed with two different bin widths. The analysis using the larger bin width would yield larger average amplitudes since its bins cover a wider frequency range. To avoid this we divide the acceleration amplitudes of each bin by its bandwidth. In our example above, we would divide by 20 Hz. This is a normalizing process which is used to minimize errors that would occur if one analysis used a different bin bandwidth than another analysis. To obtain the required resolution, real PSD analyzers typically have hundreds of bins with fractional band-widths. Per PSD theory, the signal coming into the bin is squared and then averaged with the prior samples. The resulting quantity is the average value over the bandwidth of the bin. It is aptly called the mean square acceleration and has the units of g2. It is then divided by the bandwidth of the bin and this gives rise to the peculiar unit g2/Hz called Power Spectral Density, or just PSD. DEFINITION OF POWER SPECTRAL DENSITY (PSD) - 2

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12 The measurement of random processes is statistical in nature; all the concepts and definitions of statistics apply. A key concept is the fact that many random processes have so called Gaussian amplitude probability distributions. This means the positive and negative peaks in a random signal, when accumulated, will distribute themselves in the familiar bell shaped curve centered about a mean value (see Fig. 1). The Frequency of Occurrence graph in Figure 4 is a histogram of the acceleration vs. time random signal measured for 2 seconds. It defines a number of amplitude bins and the number of times the random acceleration peaks reached those levels. The histogram indicates that this random signal indeed has a Gaussian probability distribution. The mean value of a Gaussian probability curve is defined as the standard deviation, or sigma value, of the distribution. The mean, or sigma amplitude, in a Gaussian distribution will occur approximately 68.3% of the time. In our example, the positive sigma value lies between 0 and 1g, negative sigma lies between 0 and -1g. In other words, most of the random signal acceleration peaks will be contained between -1 and +1 g and will occur 68.3% of the time. Larger peaks will occur less frequently and will be limited to about +/-3g. Be aware that the Gaussian probability distribution does not indicate the random signal frequency content — that is the job of the PSD analysis. If we compute the area under a PSD plot and then take the square root, we have, in effect, computed the mean, or sigma value, of the signal. Remember the axes units of the PSD plot are g2/Hz vs. frequency in Hz. The g2 value is the mean squared value of the accelerations contained in the frequency bins. Computing the area of the PSD plot results in units of g2 This value represents the mean squared acceleration over the entire frequency range of the PSD analysis. The square root of this value is simply the mean value. We more commonly refer to this quantity as the rms g level of the PSD plot. In the Gaussian distribution analysis, we are concerned with amplitude distribution and its mean value. In PSD analysis, on the other hand, we are concerned with amplitude distribution with frequency and its associated mean value. The means from each analysis are computed differently but, in fact, have the same value. With a Gaussian distribution, we can make the assumption that the peak amplitudes in a PSD plot will be no more than about 3 times the rms g or sigma level. The fact that the rms g level dictates amplitudes has some interesting implications when we interpret the PSD plot. DEFINITION OF POWER SPECTRAL DENSITY (PSD) - 3

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13 Figure 4 DEFINITION OF POWER SPECTRAL DENSITY (PSD) - 4

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14 DEFINITION OF POWER SPECTRAL DENSITY (PSD) - 5 To get a better understanding of how grms levels and PSD's relate to each other, consider the PSD plot shown in Figure 5. Here the PSD value is constant and equal to 1.0 g2/Hz over a 100 Hz frequency range. The grms level of the plot is(1g2/Hz*100 Hz)0.5=10 grms, hence, we know the peak g levels in the signal will be around 3 sigma or 30g. Since the PSD value is flat, the energy level of the signal is evenly distributed over the frequency range. We have a group of sine waves ranging in frequency from 10 Hz to 110 Hz. occurring simultaneously. 3 sigma peaks occur as the result of random additions of these sine waves distributed over this frequency range. Now consider Figure 6. This plot has two levels of PSD over the same frequency range as Figure 1. The grms level of this plot is (1.333g2/Hz*50 + 0.67g2/Hz*50)0.5 = 10 grms. Since the grms level is the same, and we have Gaussian distribution, the peak g levels are again around 30 g. Since both plots have the same grms level, do they describe the same random signal? Certainly not as evidenced by the differences in PSD amplitudes. So what is happening here? In Figure 6, the distribution of vibration amplitudes is roughly twice as high in the region between 10 and 60 Hz as it is between 60 and 110 Hz. This means the average amplitudes of the sine waves in the lower region are approximately twice as large as the sine waves in the upper region. Their random sums, however, still add up to the same 3 sigma peak levels. In Figure 5, the average sine wave amplitudes are even over the whole frequency range.

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15 Figure 5Figure 6 DEFINITION OF POWER SPECTRAL DENSITY (PSD) - 6 Source www.lansmont.com/.www.lansmont.com/

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