The Statistical Energy Analysis (SEA)

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## The Statistical Energy Analysis (SEA)

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**The Statistical Energy Analysis (SEA)**S E A by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**• Methods used for vibration problems: by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**• Methods used for vibration problems: -Usually we are dealing with models like FEM, (BEM) and analytical models which enable us to calculate for deterministic loads and defined model parameters deterministic responses. -Typically the calculated value is given in detail with respect to frequency, time and location. -However, the level of discretization of time/frequency and the geometric data has to be defined at the basis of theoretical considerations regarding wave-lengths, eigenmodes etc. -The following introductory example shows, that at higher frequencies the reliability of the result of calculation might be considerably reduced. by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**2. Introductory example: • room (25 m3) • limited by a steel plate • one of the boundary surfaces is excited by a harmonic load • 18 points in the room are considered by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**2. Introductory example: • The figure shows for the 18 points • in the room all measured transfer • functions between the harmonic • load and the sound pressure. • lt can clearly be seen, that at • higher frequencies the transfer • functions differ considerably. Level difference sound pressure - harmonic force frequency Hz Hz Wheel of a bike by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**2. Introductory example: - reason for the high differences: different contributions of single modes which are close together regarding their eigenfrequency. So e.g. in the centre of the room and a tonal excitation at 250 Hz, a difference of about 20 dB (factor 10) between the individual functions is observed. Level difference sound pressure - harmonic force frequency Hz Hz by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**2. Introductory example: - Even slight temperatur differences in the room, which practically cannot be eliminated, influence the positions of the Eigenfrequencies so that a detailed prediction cannot be given Level difference sound pressure - harmonic force frequency Hz Hz by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**2. Introductory example: • The air inside the room also shows • modes (starting at about 50 Hz) by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**2. Introductory example: • Possible Uncertainties of… • boundary conditions (e.g. clamped/free edge) • dynamic material properties (e.g. concrete: E ~ 30kN/mm^2) • masses of the materials (e.g. concrete: 25 kN/m^2) • damping • load distribution (e.g. position of the machine) • frequency of excitation (e.g. velocity of train) • ... by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**3. Historical example: In the early 1960s: -prediction of the vibrational response to rocket noise of satellite launch vecicles and their payloads -problem: the frequency range of significant response contained the natural frequencies of a multitude of higher order modes: -the Saturn launch vehicle possessed about 500.000 natural frequencies in the range 0 to 2000 Hz by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**4. Motivation for SEA: • The both examples above are leading to the insight that • at higher frequencies a method with less detailing has to be accepted. • -A detailed analysis at the basis of FEM approach (input at a point of • excitation, output at a point of observation) would lead to results which • are very sensitive to slight changes in the input parameters • (factor 10!). • -In order to obtain acceptable sensitivities of the results, but to describe • nevertheless the system response, we will give the results in an averaged • sense. by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**4. Motivation for SEA: by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**5. Deterministic approach: modal superposition mode shape (point of observation) system response contribution of the i.th mode by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**5. Deterministic approach: modal superposition influence of the geometry of excitation amplification functioninfluence of the frequency of excitation by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.1 Shift to energy • In the first step a shift from velocities to energy is carried out. • the mean kinetic energy is proportional to the mean square velocity mode shape (point of observation) contribution of the i.th mode by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.2 Averaging in the SEA • Now we increase the prediction accuracy by appropriate averaging • in several steps by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.3 Averaging over the points of observation („ Step 1“) • by this step the phase information gets lost by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.3 Averaging over the points of observation („ Step 1“) Orthogonality of modeshapes („Summing up the modal energy“) by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.4 Averaging over the points of excitation („ Step 2“) • By this averaging, the information about the shape of the individual • eigenmodes is eliminated and has no longer to be considered • This means: the modes don‘t have to be calculated! by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.4 Averaging over the points of excitation („ Step 2“) mean modal force modal mass by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.4 Averaging over the points of excitation („ Step 2“) force amplification function total mass no information about the modes necessary! by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.5 Averaging over the frequencies of excitation („ Step 3“) -To simplify the mean square velocity once again, we assume several similar modes N in a frequency band amplification by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.5 Averaging over the frequencies of excitation („ Step 3“) force total mass damping frequency band by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Energetic approach: 6.5 Averaging over the frequencies of excitation („ Step 3“) Energy within a certain frequency band: centre frequency force frequency band total mass damping by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**6. Mean input power -We are looking at one „sub-system“ (frequency band) -We assume a steady state vibration: „the mean input power, which is introduced during one cycle of vibration equals to the dissipated power due to damping“ (compare SDOF system). -mean input power in a frequency band: force frequency band total mass input power is independent from damping by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**7. Balance of power- hydrodynamic analogy Mean input power P Energy E in the sub-system Dissipated energy by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**7. Balance of power- hydrodynamic analogy -every sub-system is considered as a energy reservoir -The dissipated energy is proportional to the absolute dynamic energy E of the sub-system: damping by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**7. Balance of power- hydrodynamic analogy Expansion to coupled systems: -For every sub-system holds: by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**7. Balance of power- hydrodynamic analogy Expansion to coupled systems: -Energy flow between two sub-systems: modal energy coupling loss factor by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**8. Equations of the SEA The governing equations can be derived by considering: the loss of energy by damping the energy flow between every pair of sub-systems (coupling) by Michael Fischer JASS 2006 in St. Petersburg**damping**The Statistical Energy Analysis (SEA) 8. Equations of the SEA coupling by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**8. Equations of the SEA -Related to the different possible deflection patterns (e.g. bending, shear, torsional waves): each part of the structure might appear as various energy reservoirs and thus described by various governing equations. -FE: usually a high dicretization of the structure is necessary -SEA: based on calculation of global values computational costs are much smaller interactive planning by the engineer is possible by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**8. Conclusions and look into the future -Energy methods have a huge impact on the methodology of noise and vibration prediction -especially hybrid methods can carry out vibroacoustic investigations with a good confidence by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**8. Conclusions and look into the future -example: by Michael Fischer JASS 2006 in St. Petersburg**Rail-Impedance-Model RIM**The Statistical Energy Analysis (SEA) 8. Conclusions and look into the future by Michael Fischer JASS 2006 in St. Petersburg**The Statistical Energy Analysis (SEA)**Thank you for your attention! by Michael Fischer JASS 2006 in St. Petersburg