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Resonance Testing
This is series of forced vibration tests to discover experimentally
the natural frequencies and normal modes of an engineering
structure or machine. The results are often used to check out the
original mathematical model to see if it was accurate and
representative of the real structure.
1. The structure is supported in an appropriate way and excited
by electromagnetic or hydraulic or mechanical excitors – or
some other way.
2. The displacement is monitored at one on more places on the
structure. For small damping (the usual situation), the natural
frequencies are given by maximum amplitudes, or from a
vector plot as in one-degree of freedom systems.
1ω 2ω 3ω 4ω
o
o
F
zIm
o
o
F
zRe
ω Excitation frequency
Resonance at maximum spacing
for equal frequency increments
Amplitude
3. At each natural frequency, the normal mode shape is
measured using transducers (i.e. accelerometers etc.).
Suitable care is taken with the excitation, the measured mode
can approximate closely to an undamped normal mode.
4. To check the overall accuracy of the measurements, an
orthogonality “check” is often made. This uses the measured
modes in conjunction with the mass matrix estimated for the
mathematical model of the structure under test. Thus, if mR is
the measured modal matrix, and eM is the estimated mass
matrix, then me
T
m RMR will not necessarily be diagonal.
The nearer to diagonal it is, the greater the confidence in the
measured modes and estimated mass. In practice, if all off-
diagonal terms ⊗⊗⊗ < jjiiij mmm 1.0
this is regarded as acceptable as an orthogonality check. If so,
the diagonal terms are taken as the true generalised masses,
and the off-diagonal are put equal to zero.
Aircraft Dynamic Testing
- Rigid body (R/B) modes
- Symmetric and Asymmetric modes
- Resonance Testing
- Flight Flutter Testing
- Aeroelastic Model Testing
R/B Modes
Consider cantilever beam, 1st bending normal mode.
K.E. = [ ]∫s
dxtxzxm0
2
1 ),()(2
1& and if )()(),( 111 tqxftxz = , this becomes
[ ] 2
11
0
2
112
1
2
1qMdxqfm
s
&& =∫ where mass dgeneralise1 ≡M
P.E. = 2
11
0
2
2
1
2
2
1)(
2
1qKdx
dx
zdxEI
s
=
∫ where stiffness dgeneralise1 ≡K
For:
the generalised mass and stiffness are 12M and 12K and the
natural frequency 1
11
M
K=ω is unchanged.
Now consider “free-free” case
Introduce a R/B mode like
z
s
x
Symmetrical mode Asymmetrical mode
K.E. = ( ) dxzzm BR
s
s
2
/12
1&& +∫
+
−
( )∫+
−
++=s
s
BRBR dxzzzzm2
//1
2
1 22
1&&&&
P.E. = (P.E.) elastic
Note: Zero coupling between symmetric elastic bending mode
and asymmetric R/B mode (and vice-versa) since
( ) 0/1 =∫+
−
dxzzm
s
s
BR&&
Example: let m(x)= 1, uniform mass distribution
1
2
1 )( qs
xxz
= (parabolic bending mode)
2/ qz BR = (rigid body heave mode)
No R/B mode:
K.E. 2
1
2
1
4
5
2
2
1
2
1qsqdx
s
xs
s
&& =
= ∫
+
−
P.E. 2
1
2
1
2
13 5
2
2
14
2
1qsq
s
EIω== , say,
giving (natural frequency)2 = 2
1ω =4
10
s
EI.
Datum Symmetric (heave, if beam is
a/c wing)
Asymmetric (roll)
with R/B mode:
K.E.
++=
+
= ∫
+
−
2
221
2
1
2
21
2
23
4
5
2
2
1
2
1
qsqqsqs
dxqqs
xs
s
&&&&
&&
From Lagrange:
[ ] [ ]
=
=
00
05/2
23/2
3/25/2 2
1ωsKsM
From characteristic equation: (soln t
oeqqλ= )
05
4
45
16
5
4
9
4
5
4
0
23
23
2
5
2
5
2
2
1
242
1
24
22
22
1
2
2
=+=+
−
=+
ωλλωλλ
λλ
λωλ
s
1
22
2
3or0 ωλλ ±==
From the 2nd
equation of motion:
023
221 =+ qq &&&&
3
1
1
2
1
2 −==∴q
q
q
q
&&
&&
We could do a similar calculation in A/S mode and find
Resonance testing (carried out on the ground)
Objective - to check accuracy of theoretical model
- to provide basis for flight flutter testing
Node
A/S mode shape:
Again, natural frequency would be > 1ω
and a zero frequency for the R/B mode
2/3
1/3
Node
Symmetric mode
shape:
Note that natural frequency has
increased to 3/2 × original
Theoretical models are like stick model below
Alternatively FEM model representation is employed which
requires many more degrees-of-freedom analysis hence increased
computational time.
http://www.swri.edu/3pubs/BROCHURE/d18/AirStruc/StructuralAnalysis.pdf
Resonance testing provides:
- Natural frequencies
- Mode shapes – more difficult to measure and require large
number of accelerometers and data acquisition system
lumped masses including inertia properties
Stiffness EI, GJ etc distributions
for all branches
Equations of motion with damping and forcing
Equations transformed into normal modes are like:
[ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] { })(tFRqRKRqRMRTTT
=+&& (zero damping)
or
[ ] { } [ ] { } { })(***tFqKqM =+&&
where [ ]∗M and [ ]*K are diagonal. (So equations are uncoupled like
1 d.o.f.)
Consider forcing – assume we have a single force )(tjφ at point j.
[ ] { }
=
0
)(
0
)(
11
tzz
zz
tFRjiji
j
T
φ
M
MMM
LL
LMMM
LL
=
=
M
M
M
M
)(
)(
)(
)(
*
*
11
tF
tF
tz
tz
ijij
jj
φ
φ
i.e. the amount of forcing in a mode depends on the displacement
in that mode at the forcing point.
Thus, forcing at a node will produce zero generalised force in that
mode – and hence zero response. That is why in resonance tests,
multiple exciters are sometime used to ensure good responses in
higher-order modes.
Damping
We can introduce the damping (estimated or measured) into the
mathematical model directly into the normal coordinates, as a
damping ratio.
ith
mode
mode 1
In engineering, structural damping ratio usually small, so can
ignore coupling effects ( 02.0=ζ or less).
i.e. for ith normal coordinate
)(***
,
*tFqkqcqm iiiicritiiii =++ &&& ζ
where ***
, 2 iicriti mkc =
Example
Obtain equations in n coordinates and introduce 2% critical
damping. Find amplitudes of each n coordinate at 1st natural
(undamped) frequency for damped system.
Do the usual to obtain [ ]
−=
5.01
11R and the two natural frequencies
m
k=1ω and
m
k
2
52 =ω
and hence
m
2m
x2 x1
k
k
2k
F(t) [ ]
[ ]
−
−=
=
31
12
20
01
kK
mM
[ ] [ ][ ]
=
5.10
03mRMR
T ; [ ] [ ][ ]
=
75.30
03kRKR
T
{ } [ ] { } )(5.0
1)(
1
0
5.01
11)(and)(
1
0)( tFtFtFRtFtF
T
−=
−=∴
=
Equations (undamped) are: - in n-coordinates
)(5.0
1
75.30
03
5.10
03
2
1
2
1tF
q
qk
q
qm
−=
+
&&
&&
2% critical damping is:
=
×
×
37.20
0304.0
75.35.120
033202.0 mk
km
km
Equations become:
)(312.03 111 tFkqqmkqm =++ &&&
)(5.075.3095.05.1 222 tFkqqmkqm −=++ &&&
Now put ti
oeFtFω=)( where
m
k== 1ωω
1st equation
( ) ti
o
ti
o eFeqkimkm 11
11
2
1 312.03ωωωω =++−
k
Fi
ki
Fq oo
o 33.812.0
1 −== (phase lagging by 90o)
2nd
equation
( ) ti
o
tieFeqkimkm 11 5.075.3095.05.1 201
2
1
ωωωω −=++−
leading to
)0094.0222.0(2 ik
Fq o
o +−= (almost 180o out of phase with phase angle
2.4o)
Flight Flutter Testing
Like a resonance test in the air. Objective is to obtain modal
frequencies and dampings. A reduction of a modal damping
towards zero implies possible flutter.
Wing displacements { } [ ]{ }qfz = and following a disturbance
{ } { } { } )( ti
o
t
o eqeqq ωµλ ±==
where iµ and iω are plotted for the modes of interest at ever
increasing EAS, usually covering a range of Mach numbers and
altitudes and payloads (different fuel cases or weapon
configurations).
In the air, structure may be excited by a number of methods
- steady forced excitation (may be “swept” and signal into
aileron/elevator/rudder jack or vibrating mass)
V (E.A.S)
V (E.A.S)
- µ
ω
Note that all values at zero speeds
correspond to resonance test values
Flutter ?
- impulse – stick taps
rudder kicks
bonkers
Sensors are required to measure response
e.g. – accelerometers
- strain gauges
- displacement transducers (or control surfaces)
Hopefully, there is reasonable correlation with flutter
calculations to provide confidence in increasing airspeed for the
next flutter clearance test. (Remember, the aerodynamic
modelling has least experimental back-up).
Aeroelastic model testing
A dynamically scaled model is constructed. Resonance tests to
check mass/stiffness. Wind tunnel tests (possibly in pressurised
W/T) provide check on flutter calculations.
Recommended