Mechanical Vibrations Misc Topics Base Excitation & Seismic Peter Avitabile Mechanical Engineering Department University of Massachusetts Lowell 1 Dr. Peter Avitabile
Seismic events are often moele with an input applie as a base excitation Dr. Peter Avitabile
With the motion of the base enote as y an the motion of the mass relative to the intertial reference frame as x, the ifferential equation of motion becomes Substitute m & x k(x into the equations to give = (3.5.1) z = y) c(x& x m & z + cz& + y) & The equation is assume to be in stanar form with F/m equal to the negative of the acceleration kz y = & y (3.5.) (3.5.3) 3 Dr. Peter Avitabile
Consier the SDOF system with base excitation The excitation shown is that of the 1940 N-S El Centro earthquake (commonly use) 4 Dr. Peter Avitabile
The response of the SDOF system (or MDOF system if esire) can be compute for any given MCK system. Obviously the response will be ictate by the natural frequency of the structural system. Since the majority of the energy of a seismic event is well below 0 Hz, often times a system can be analyze statically if the natural frequency of the structure is above 33 Hz. When this is the case then the static forces are approximate by F=ma for ease of computation. 5 Dr. Peter Avitabile
The response of the SDOF system with 1 Hz 150 El Centro Earthquake - N-S Acceleration - 1940 100 Acceleration Input(inch/sec ) 50 0-50 8 6 Seismic Response - 1 Hz - 10% Dam -100 4-150 0 4 6 8 10 1 14 16 18 0 time(sec) isplacement(inch) 0 - -4 6 Dr. Peter Avitabile
The response of the SDOF system with >33 Hz 150 El Centro Earthquake - N-S Acceleration - 1940 100 Acceleration Input(inch/sec ) 50 0-50 0.15 0.1 Seismic Response - Freq > 33 Hz -100 0.05 0-150 0 4 6 8 10 1 14 16 18 0 time(sec) isplacement(inch) -0.05-0.1-0.15 7 Dr. Peter Avitabile
Seismic Response - Pseuo Response Analysis The response of the SDOF system is epenent upon its natural frequency an amping. Obviously there are an infinite number of combinations that exist an the response of each SDOF system in each environment must be etermine in this case. This is an extremely time consuming analysis that must be performe for all equipment use in builings an structures that are prone to seismic environments. An alternate approach is typically use as iscusse next. 8 Dr. Peter Avitabile
Seismic Response - Pseuo Response Analysis Consier a builing that is subjecte to a seismic isturbance. EQUIPMENT ROOF M X K C FLOOR M X 1 K C TWO STORY BUILDING ANALYTICAL REPRESENTATION 9 Dr. Peter Avitabile
Seismic Response - Pseuo Response Analysis A coarse moel of the builing is generate to represent the gross overall weight an effective builing stiffness charateristics. The base motion is use as input to etermine the amplification an filtering that occurs to the groun motion at various levels in the builing (ie, VTB1_4). This moifie input is then use for each level of the builing to etermine the pseuo-isplacement, pseuo-velocity an pseuo-acceleration that various SDOF systems will be subjecte to when the groun excitation is applie. 10 Dr. Peter Avitabile
Seismic Response - Pseuo Response Analysis Using this approach, the actual equipment at each level is not specifically moele. The effective response of a variety of assume SDOF systems with various frequencies an ampings are compute to etermine the response in the builing. In this way the equipment manufacturer is provie response spectrums that are use to esign their particular equipment epening on the location in the builing. 11 Dr. Peter Avitabile
With no amping, the relative response is compute using the convolution (Duhamel) integral as 1 z(t) = && y( τ)sin ωn ω an when consiering amping as n t 0 ( t τ) τ (4..5) z(t) 1 = && y( τ)e ω t 0 ςω n (t τ) sin ω ( t τ) τ 1 Dr. Peter Avitabile
This equation is typically use for shock loaing consierations as well as for seismic applications. Typically, for earthquake analysis, the velocity spectra is use extensively. Differentiating z(t) & 1 = && y( τ)e ω t 0 ζω n (t τ) [ ζω sin ω ( t τ) + ω cosω ( t τ) ] τ n 13 Dr. Peter Avitabile
This equation can be written as z(t) & = where e ζω n t 1 ζ P + Q sin ( ω φ) t P = && y( τ)e 0 t Q = && y( τ)e 0 ςωnt ςωnt cosω sin ω ττ ττ φ = tan 1 ( ) P 1 ζ + Qζ Pζ Q 1 ζ 14 Dr. Peter Avitabile
It is important to realize that in shock spectrum analysis, only the maximum response is compute. This implies that there coul be many ifferent excitation shock spectrums that cause the same response. It is this feature that that makes shock response spectrum analysis so attractive. 15 Dr. Peter Avitabile
The spectrums can be evelope an then use for esign even though the actual loaing may be ifferent, the same response is achieve. Its limitation is in fatigue analysis where the energy associate with ifferent frequencies may cause ifferent failures to occur even though the max response is the same. 16 Dr. Peter Avitabile
The maximum velocity spectrum can be written as S v = z(t) & max = e ζω n 1 ζ t P + Q max an the corresponing isplacement an acceleration values are S S = z(t) = v max ω a max n v n S = & z(t) = ω S 17 Dr. Peter Avitabile
Seismic Response - Pseuo Response Analysis Conceptually the SDOF response is shown in the figure below p 1 f 1 p f p 3 f 3 m 1 m m 3 k 1 c 1 k c k 3 c 3 MODE 1 MODE MODE 3 18 Dr. Peter Avitabile
Seismic Response - Pseuo Response Analysis Typical Response Spectrum Typical Design Spectrum 19 Dr. Peter Avitabile