Appendix A List of Abbreviations

44 Pages·1999·0.49 MB·English

Appendix A
List of Abbreviations
The abbreviations used in this thesis are listed below.
B&K Bru¨el and Kjær, a manufacturer of instrumentation
DOF Degrees of Freedom
DSP Digital Signal Processor
FEA Finite Element Analysis
FEM Finite Element Modelling
FIR Finite Impulse Response filter
FX-LMS filtered-x Least Means Squares control algorithm
GPIB General Purpose Interface Bus
IIR Infinite Impulse Response filter
HP High pass filter
KE Kinetic Energy
LMS Least Means Squares control algorithm
LP Low pass filter
x-LMS filtered-x Least Means Squares control algorithm
289
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Appendix B
Definition of Power Transmission for
Tonal Excitation
A harmonic force driving a structure may be described in complex form. The instan-
taneous measure of power P is given by (Skudrzyk 1968)
P = fv = (cid:7)(fˆ)(cid:7)(vˆ) (cid:17)= (cid:7)(fˆvˆ) (B.1)
where fˆand vˆ denote the complex force and velocity respectively, f and v denote the
real part of force and velocity respectively and (cid:7)(x) is the real part of the bracketed
expression. Force and velocity can be expressed as real and imaginary components.
The force and velocity can be expressed as
(cid:25) (cid:26)
1
f = fˆ+fˆ∗ (B.2)
2
1
v = (vˆ+vˆ∗) (B.3)
2
(cid:1) (cid:2)
1
⇒ fv = (cid:7)(fˆvˆ)+(cid:7)(fˆvˆ∗) (B.4)
2
where fˆ∗ and vˆ∗ are the complex conjugates of the force and velocity respectively. By
integrating the power over time and dividing by the time duration, a time averaged
291
292 AppendixB DefinitionofPowerTransmissionforTonalExcitation
measure of power is obtained. The force and velocity can be written as
f = F expjωt+φ (B.5)
v = V expjωt+θ (B.6)
where exp is the exponential function, ω is the driving frequency, φ and θ are phase
angle delays. By substituting these equations into equation (B.4) the expression for
power becomes
1
P = FV [cos(2ωt+φ+θ)+cos(φ−θ)] (B.7)
2
The first cosine term in Eq. (B.7) is an oscillatory function at twice the frequency of
the driving force. The second cosine term is a constant term which is dependent on
the relative phase angle between the force and velocity. The time averaged vibrational
power transmission is calculated by integrating Eq. (B.7) with respect to time over
one period and dividing by the duration. The time average of the first cosine term in
equation (B.7) equates to zero, which leaves
1
(cid:20)fv(cid:21) = FV cos(φ−θ) (B.8)
t
2
which can be shown to be equal to
1
(cid:20)fv(cid:21) = (cid:7)(fv∗) (B.9)
t
2
1
= (cid:7)(f∗v) (B.10)
2
Equation (B.9) describes the time averaged power transmission as half the real part of
the force times the complex conjugate of the velocity. Equation (B.9) only applies for
tonal noise problems; it does not apply for random noise.
Appendix C
Beam Model
The equations that are used in Pan et al. (1993) are described in this Appendix. The
model is described here because the equations are directly referred to in section 5.2 to
show how the matrices have to be reduced in dimensions in order to solve the system
of equations.
C.1 List of Symbols
The terms used in this model are:
a,b ,b ,c quadratic equation coefficients.
1 2
ci imaginary part of c
C Ith modal contribution of the mass of the lower mount.
I(cid:1),I
D displacement vector of rigid body.
0
Dt,Db displacement vector of upper and lower mounting.
E Young’s modulus of the beam.
F ,F ,F the forces acting on the rigid mass.
x y z
Fc,Fc,Fc the control forces applied by the actuator.
x y z
G ,··· ,G , matrices used in the construction of the quadratic equation
1 4
I moment of inertia of the beam.
b
I ,I ,I moments of inertia of the rigid body.
x y z
293
294 AppendixC BeamModel
(cid:8)(x) imaginary part of x.
k spring stiffness in the ith direction.
i
K six dimensional stiffness matrix of isolator.
L length of the beam.
b
m modal mass of the beam for the Ith mode.
I
m mass of the rigid body
0
M ,M ,M the moments acting on the rigid mass.
x y z
Mc,Mc,Mc the control moments applied by the actuator.
x y z
q vector containing real elements of the real and imaginary parts of
c
the control force.
Q primary excitation force vector on rigid body.
0
Qt,Qb force vector active on the upper and lower mount.
Q active isolator control force vector.
c
Rt coupling matrix between primary excitation force vector and the
force on the upper mount.
Rb mode shape matrix at the lower mount of the beam.
(cid:7)(x) the real part of x.
S cross sectional area of the beam.
w complex modal amplitude.
p
x X location of the isolator on the beam.
b
x ,y ,z the point where the rigid mass is connected to the isolator.
t t t
x ,y ,z the three translational displacements of the top mass.
0 0 0
Z impedance of the rigid body.
0
Z impedance of the beam.
p
α quadratic equation matrix coefficient.
β quadratic equation matrix coefficient.
χ contribution of the beam to the beam impedance.
I
η damping loss factor in the ith direction.
i
C.2 TheBeamModel 295
η Ith modal loss factor.
I
ω frequency of analysis.
ω Ith resonance frequency of the beam.
I
ρ density of the beam.
θ ,θ ,θ the three angular rotational displacements of the top mass.
x0 y0 z0
C.2 The Beam Model
The primary excitation of the rigid body Q is a 6 dimensional force vector which can
0
be expressed as
(cid:15) (cid:16)
T
Q = F F F M M M (C.1)
0 x y z x y z
where F and M refer to the force and moments in the subscript directions respectively.
The spring produces an equal and opposite force on each mass given by:
Qt = −Qb = K(Db −Dt) (C.2)
where Qt, Qb, Dt, Db are the 6 dimensional force and displacement vectors of the
top and bottom masses respectively. K is the complex diagonal stiffness matrix of
the isolator mount which can be expressed as k (1+jη ), where i = 1···6, η are the
i i i
√
damping loss factors and j = −1.
The displacement of the top mass is given by
t t T
D = [R ] D (C.3)
0
where [Rt]T is the transpose of the coupling matrix between moments and translational
296 AppendixC BeamModel
displacements and is defined as follows
 
 1 0 0 0 0 0 
 
 
 0 1 0 0 0 0 
 
 
 0 0 1 0 0 0 
t  
R =   (C.4)
 0 −zt yt 1 0 0 
 
 
 z 0 −x 0 1 0 
t t
 
−y x 0 0 0 1
t t
where (x , y , z ) is the point where the rigid mass is connected to the isolator.
t t t
D is the complex displacement vector of the centre of gravity of the top mass
0
defined as
(cid:15) (cid:16)
T
D = x y z θ θ θ (C.5)
0 0 0 0 x0 y0 z0
where x ,y ,z are the three linear displacements and θ ,θ ,θ are the three angular
0 0 0 x0 y0 z0
rotational displacements.
A sufficient number of modes must be included in the modelling of the beam to
accurately describe the dynamics. The displacement of the beam is considered to be
the sum of 1,··· ,P modes and can be described by
b b T
D = [R ] w (C.6)
p
where [Rb]T is the transpose of the mode shape functions for each degree of freedom
of the beam and is defined as
 (cid:25) (cid:26) (cid:25) (cid:26) (cid:25) (cid:26) 
− πh cos πxb , 0, sin πxb , 0, − π cos πxb , 0
 2L L L L L 
 b (cid:25) b (cid:26) (cid:25) b (cid:26) b (cid:25) b (cid:26) 
 −2πh cos 2πxb , 0, sin 2πxb , 0, −2π cos 2πxb , 0 
Rb =  2Lb Lb Lb Lb Lb  (C.7)
 . . . . . . 
 .. .. .. .. .. .. 
 (cid:25) (cid:26) (cid:25) (cid:26) (cid:25) (cid:26) 
−Pπh cos Pπxb , 0, sin Pπxb , 0, −2π cos Pπxb , 0
2L L L L L
b b b b b
C.2 TheBeamModel 297
and w is the modal complex amplitude and defined as:
p
(cid:15) (cid:16)
T
w = w , w , ··· w (C.8)
p 1 2 P
The equations of motion of the rigid body and beam are
Z w = RbQb −RbQ (C.9)
p p c
t t t
Z D = Q +R Q R Q (C.10)
0 0 0 c
where Z is the impedance of the rigid body defined as
0
 
 m0 0 0 0 0 0 
 
 
 0 m 0 0 0 0 
0
 
 
 0 0 m 0 0 0 
Z0 = −ω2 0  (C.11)
 0 0 0 Ix 0 0 
 
 
 0 0 0 0 I 0 
y
 
0 0 0 0 0 I
z
where m is the mass of the rigid body, I ,I ,I are the moments of inertia about the
0 x y z
centre of gravity and ω is the driving frequency of Q .
0
Z is the impedance of the beam defined as
p
 
χ −C −C ··· −C
 1 11 12 1P 
 
 
−C χ −C ··· −C
 21 2 22 2P 
Z =   (C.12)
p  ... ... ... ... 
 
C −C ··· χ −C
P1 P2 P PP
where the contribution of the Ith beam mode is given by
(cid:22) (cid:23)
χ = m ω2 +jη ω2 −ω2 (C.13)
I I I I I
298 AppendixC BeamModel
1
m = ρSL (C.14)
I b
2
+
I2π2 EI
ω = b (radians/sec) (C.15)
I L2 ρS
b
and the inertia of the lower mount is taken into account by the term
Iπx I(cid:3)πx
C = mbω2sin sin (C.16)
I(cid:1),I L L
b b
The active isolator control force vector is given by
(cid:15) (cid:16)
T
Q = Fc Fc Fc Mc Mc Mc (C.17)
c x y z x y z
where the superscript c represent the control actuator.
Equations (C.9) and (C.10) can be compactly expressed in matrix form as
    
 A11 A12  D0   Q0 +RtQc 
   =   (C.18)
A A w −RbQc
21 22 p
where the coefficients of the matrix A are
t t T
A = Z +R K[R ] (C.19)
11 0
A = −RtK[Rb]T (C.20)
12
A = −RbK[Rt]T (C.21)
21
b b T
A = Z +R K[R ] (C.22)
22 p
The power into the beam is defined by
ω (cid:20) (cid:21)
Power = (cid:8) [Db]H(Qb +Q ) (C.23)
c
2