Dynamic Performance Investigation of Base Isolated Structures
By Ather K. Sharif
Definition of decibel scaling used
It is important that the signal of interest, in this case the response due to the train source is clearly above ambient levels
The magnitude of a level in decibels is ten times the logarithm to the base 10 of the ratio of power-like quantities, i.e.
where: L = level of power-like quantity
X = quantity under consideration
XO = reference quantity of the same kind
A difference in the levels of two like quantities X1 and X2 is described by the same formula because, by the rules of logarithms, the reference quantity is automatically divided out as follows:
Where Transmissibility (insertion loss or gain) is defined as the non-dimensional ratio of the response amplitude of a system to the excitation amplitude. The ratio may be one of forces, displacements, velocities or accelerations. A doubling of level causes a 6dB increase, and a tenfold change is equivalent to 20dB. A positive insertion loss indicates a disbenefit, and conversely a negative insertion loss indicates a benefit.
The word ‘disbenefit’ is not currently in the English dictionary, although is used to convey the obvious interpretation.
Abbreviations used in Figures
SDOF single degree of freedom
FE finite element
Trans transmissibility
crit critical damping ratio
ch channel
psd power spectral density (auto spectrum)
col column
TR test room
V vertical
L longitudinal
T tangential
List of Symbols
Chapter 2
h depth of source
VP P wave velocity
VR Rayleigh wave velocity
VS S wave velocity
Chapter 3
aw frequency weighted acceleration
t time
VDV Vibration Dose Value
Chapter 5
(t) function of time
fS spring force
fD damping force
fI inertial force
k spring constant
c damping coefficient
cc critical damping coefficient
m mass
p, po applied force
u displacement
ù du/dt (velocity)
ü d2u/dt2 (acceleration)
ur relative displacement
U absolute displacement
ug ugo ground displacement
peff effective load
Z arbitrary complex constant
s constant
w n undamped natural angular frequency
w D damped natural angular frequency
x damping ratio
Z1, Z2 constants
A constant
B constant
C vector amplitude
- vector amplitude
q phase angle
G1 G2 constants
- frequency ratio
- phase angle
y 1, y 2 phase angles
M Dynamic magnification factor
W work over cycle
t time
T transmissibility
h , h o loss factor
fn natural cyclic frequency
j mode number
w j angular frequency of mode j
L length of column
m integer
E Young’s modulus
- density
x j modal critical damping ratio
w j angular frequency of mode j
D w j frequency interval for mode j
- alpha, Rayleigh damping constant
- beta, Rayleigh damping constant
d m logarithmic decrement determined from waveform m cycles apart
w a frequency point above resonance
w b frequency point below resonance
w r resonance frequency
q a angle to point (a) above resonance
q b angle to point (b) below resonance
K constant
i ¶ -1
Re real
Im imaginary
Chapter 6
Gxx auto spectra of stationary random process x(t)
Gyy auto spectra of stationary random process y(t)
Gxy cross spectra between two stationary random processes
x(t) and y(t)
Ttotal Total transmissibility
Tdirect Direct transmissibility
x(t) function of time (t)
y(t) function of time (t)
h(t ) unit impulse response function
H(f) Fourier transform of impulse response function
Y(f) finite Fourier transform of y(t)
X(f) finite Fourier transform of x(t)
g xy2(f) coherence function
- standard deviation
m mean value of a random variable
Be effective bandwidth of spectral window
T record length
- number of statistical degrees of freedom
n number of adjacent spectral lines to the side of central value
e b bias error
Br half power point bandwidth at resonance
x critical damping ratio
fr resonance frequency
e r random error
To optimum averaging time
Bo optimum bandwidth
CT time resolution bias error coefficient
Chapter 7
Transmissibilities
D1 test block to un-loaded raft
D2 test block to raft loaded with un-isolated mass on ‘rigid’ blocks
D3 test block to un-isolated mass
D4 test block to isolated mass
D5 test block to raft loaded with isolated mass
D6 raft loaded with isolated mass to isolated mass
Si = Component area
a i = Component sound absorption coefficient
W = sound power (watts)
r c = characteristics impedance of air (407 mks rayls)
Sx = area of radiating surface (m2)
V = r.m.s velocity of vibration of surface (m/s)?
SWL = Sound Power Level (dB) ?
Wref = 10-12 watts
SPL =Sound Pressure Level (dB re20m Pa)?
S = total surface area of room (m2)
Lp = SPL dB re 20
Pa
La = rms vibration acceleration of floor (re: 10-6g)
f = frequency, either octave or 1/3rd octave
Lv = Vibration re 10-5m/s2
LAmaxf = A-weighted maximum SPL using fast time weighting
Lvmax = vibration velocity re 10-9 m/sec
Chapter 10
Appendix 3.1
aw frequency weighted acceleration
VDV Vibration Dose Value
eVDV estimated Vibration Dose Value
aw(rms) r.m.s frequency weighted acceleration
t event duration
n event number
N total number of events
r.m.s root mean square
r.m.q root mean quad
Appendix 8.1
F = force
mr = mass x radius of gyration
w = angular frequency