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.