mechanical vibrations misc topics base excitation &...

1 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.457 Mechanical Vibrations - Misc Topics

Mechanical VibrationsMisc Topics

Base Excitation & Seismic

Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell


Seismic Response - Support Motion

Seismic events are often modeled with an inputapplied as a ‘base excitation’



With the motion of the base denoted as ‘y’ andthe motion of the mass relative to the intertialreference frame as ‘x’, the differential equationof motion becomes

Substituteinto the equations to give

The equation is assumed to be in standard formwith F/m equal to the negative of the acceleration

)yx(c)yx(kxm &&&& −−−−= (3.5.1)yxz −=

(3.5.2)

(3.5.3)ykzzczm &&&&& −=++



Consider the SDOF system with base excitation

The excitation shown is that of the 1940 N-SEl Centro earthquake (commonly used)



The response of the SDOF system (or MDOFsystem if desired) can be computed for any givenMCK system. Obviously the response will bedictated by the natural frequency of thestructural system.Since the majority of the energy of a seismicevent is well below 20 Hz, often times a systemcan be analyzed statically if the natural frequencyof the structure is above 33 Hz.When this is the case then the static forces areapproximated by F=ma for ease of computation.



The response of the SDOF system with 1 Hz

0 2 4 6 8 10 12 14 16 18 20-150

-100

-50

0

50

100

150El Centro Earthquake - N-S Acceleration - 1940

time(s ec)

Acc

eler

atio

n In

put(i

nch/

sec2

)

-4

-2

0

2

4

6

8S eis mic Res pons e - 1 Hz - 10% Dam

disp

lace

men

t(inc

h)



The response of the SDOF system with >33 Hz

0 2 4 6 8 10 12 14 16 18 20-150

-100

-50

0

50

100

150El Centro Earthquake - N-S Acceleration - 1940

time(s ec)

Acc

eler

atio

n In

put(i

nch/

sec2

)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15S eis mic Res pons e - Freq > 33 Hz

disp

lace

men

t(inc

h)


Seismic Response - Pseudo Response Analysis

The response of the SDOF system is dependentupon its natural frequency and damping. Obviouslythere are an infinite number of combinations thatexist and the response of each SDOF system ineach environment must be determined in this case.This is an extremely time consuming analysis thatmust be performed for all equipment used inbuildings and structures that are prone to seismicenvironments.An alternate approach is typically used asdiscussed next.



Consider a building that is subjected to a seismicdisturbance.

K

C

M

K

C

M X

X 1

2 ROOF

FLOOR

TWO STORY BUILDING ANALYTICAL REPRESENTATION

��

��

��

��

��

��

EQUIPMENT

��

��

��

��

��



A coarse model of the building is generated torepresent the gross overall weight and effectivebuilding stiffness charateristics. The base motionis used as input to determine the amplification andfiltering that occurs to the ground motion atvarious levels in the building (ie, VTB1_4).This modified input is then used for each level ofthe building to determine the pseudo-displacement,pseudo-velocity and pseudo-acceleration thatvarious SDOF systems will be subjected to whenthe ground excitation is applied.



Using this approach, the actual equipment at eachlevel is not specifically modeled. The effectiveresponse of a variety of “assumed” SDOF systemswith various frequencies and dampings arecomputed to determine the response in thebuilding.In this way the equipment manufacturer isprovided “response spectrums” that are used todesign their particular equipment depending on thelocation in the building.



With no damping, the relative response iscomputed using the convolution (Duhamel) integralas

and when considering damping as

(4.2.5)( ) ττ−ωτω

−= ∫ dtsin)(y1)t(z

t

0n

n&&

( ) ττ−ωτω

−= ∫ τ−ςω− dtsine)(y1)t(z

t

0d

)t(

d

n&&



This equation is typically used for shock loadingconsiderations as well as for seismic applications.

Typically, for earthquake analysis, the velocityspectra is used extensively. Differentiating

( ) ( )[ ] ττ−ωω+τ−ωζω−τω

= ∫ τ−ζω− dtcostsine)(y1)t(z

t

0dddn

)t(

d

n&&&



This equation can be written as

where

( )φ−ω+ζ−

=ζω−

d22

2

tsinQP

1e)t(z

n

&

ττωτ= ∫ ςω− dcose)(yPt

0d

tn&&

ττωτ= ∫ ςω− dsine)(yQt

0d

tn&&

( )2

21

1QPQ1Ptanζ−−ζ

ζ+ζ−−=φ −



It is important to realize that in shock spectrumanalysis, only the maximum response is computed.

This implies that there could be many differentexcitation shock spectrums that cause the sameresponse.

It is this feature that that makes shock responsespectrum analysis so attractive.



The spectrums can be developed and then used fordesign even though the actual loading may bedifferent, the same response is achieved.

Its limitation is in fatigue analysis where theenergy associated with different frequencies maycause different failures to occur even though themax response is the same.



The maximum velocity spectrum can be written as

and the corresponding displacement andacceleration values are

max

222

t

maxv QP1e)t(zS

n

+ζ−

==ζω−

&

nv

maxdS)t(zS ω== vnmaxa S)t(zS ω== &&


m

k

1

1 c 1

m p

3

3

c 3

m

k

2

2 c 2

f 3

k 3

p 2 f 2 f 1 p 1

MODE 1 MODE 2 MODE 3


Conceptually the SDOF response is shown in the figure below



Typical Response Spectrum Typical Design Spectrum

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