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Lab sheet for Vibration Experiment EG238

创建日期：2024-04-29 16:57:02

所属分类：物理Physical

标签： EG238

Lab sheet for Vibration Experiment

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Lab sheet for Vibration Experiment

EG238/EG268/EGA229
1 Introduction
This laboratory session studies the first two vi-
bration modes of a cantilever beam. The beam is
excited by a small electrodynamic shaker, or set
in motion and observed performing free vibra-
tions. The vibration and force inputs are mea-
sured by means of a piezo-electric accelerome-
ter and force transducer respectively, which are
acquired digitally into the National Instruments
LABVIEW system.
2 Objectives
In addition to providing a practical perspective
on some of the theory taught in Dynamics 1
(EG260) and 2 (EG360), the exercise aims to cul-
tivate a critical and objective approach to com-
paring theory with experiments.
The aim of the experiment is to compare
three different methods to determine natural fre-
quency(s) of the beam, and in some case their
associated damping ratios. The methods are:
Amplitude decay test
Broadband frequency measurement
Stepped sine test
3 Theory
For more detail see the additional theory docu-
ment provided on blackboard, and of course re-
fer to your notes from EG-260 Dynamics 1 and
Mode ωi
√
L4ρA
EI
1 3.516
2 22.034
3 61.697
4 120.902
Table 1: Natural frequencies for the first four
vibration modes of a uniform cantilever beam
the many text books on the subject (for example
Inman (2001); Rao (2011); Meirovitch (2010)).
Note that all frequencies are expressed in radi-
ans per second unless stated otherwise.
Consider the situation shown in Figure 1,
where a cantilever beam, assumed to be mod-
elled as an Euler-Bernoulli beam. Because the
beam is a continuous structure, it has any num-
ber of natural frequencies, each associated with
a given pattern of vibration known as a mode
shape. Table 1 allows the calculation of the first
4 natural frequencies and mode shapes of a can-
tilever beam, and Figure 2 illustrates the mode
shapes. Note that as the mode gets higher, the
frequency increases and the shape gets more com-
plicated. Furthermore, higher mode shapes have
points where the amplitude is zero, known as
nodes.
3.1 Free damped vibration
If the beam is displaced then released, initially
its response will include a combination of all its
modes of vibration. However, this response will
decay due to damping, and it can be shown that
1
Lx1 x2
fx1 = F0 cos Ωt
wx2(t) = W cos(Ωt− φ)w(x, t)
Figure 1: General situation of a cantilever beam forced harmonically at position x1 with a vibration
response w(x, t)
higher modes of vibration will decay far more
quickly than the first. Therefore, after some
transient behaviour the free response will become
dominated by just the first mode, in a manner
that is analogous to a single degree of freedom
system. Therefore once the transients have de-
cayed, the acceleration signal of the tip (x2 in
Figure 1) will be given by:
ax2(t) = e
−ζ1ω1tA¯ cos(ωd1t− ψ1) (1)
Where ω1 is the first natural frequency of the
beam, ζ1 is the first modal damping ration and
ωd1 is the first damped natural frequency, given
by ωd1 = ω1
√
1− ζ21 1. A¯ is the initial ampli-
tude and ψ1 is the phase (we do not need to
know either of these for the present purpose).
This gives us a technique to measure the first
damped modal frequency and damping ratio; we
can ‘twang’ the beam, observe the vibration of
a point on the beam and apply the logarithmic
decrement method from SDOF theory to the re-
sulting time trace. Once the transient contribu-
tions of other modes have decayed, we use
ζ =
δ√
4pi2 + δ2
where δ =
1
n
ln
(
A¯0
A¯n
)
(2)
where A¯0 is the peak height of an arbitrarily cho-
sen vibration cycle, and A¯n is the peak height n
cycles later. The damped natural frequency can
be found from the time taken for n cycles.
1When damping is light i.e. ζi << 1, it is quite com-
mon to assume ωi = ωdi
3.2 Forced vibration
The steady state displacement response ampli-
tude measured at position x2 of a beam forced
harmonically at position x1 can be found by:∣∣∣∣WF0
∣∣∣∣ = |H(Ω)12| = ∑
i
Ui12√
(ω2i − Ω2)2 + (2ζiωiΩ)2
(3)
where Ui12 is a modal constant related to the dis-
placement of mode shapes at points x1 and x2
(and again, we do not need its value). H(Ω)12
is known as the receptance and is a type of Fre-
quency Response Function (FRF); it tells us the
ratio of displacement response amplitude to forc-
ing amplitude at any given frequency. The form
of equation (3) shows that when the structure is
forced near the ith natural frequency, the recep-
tance will become large, and the ith mode will
dominate the response. This gives us another
way of finding natural frequencies; we can excite
the structure at different frequencies in turn; fre-
quencies that give peaks in receptance are natu-
ral frequencies. Searching for natural frequencies
by exciting each frequency in turn is known as
stepped sine testing.
3.3 Broadband excitation and
peak picking
If we assume that the response is linear, the prin-
ciple of superposition applies. This means that
instead of testing each frequency in turn, we can
excite a range of frequencies simultaneously, ob-
tain the FRF in one short test and then use it
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
-0.5
0
0.5
1
Mode 1
Mode 2
Mode 3
Mode 4
Figure 2: The first 4 modeshapes of a cantilever beam.
to calculate the properties. This is known as
broadband testing. An easy way to do this is
is to send a random noise signal through the
shaker. The amplitude of each frequency com-
ponent in the time signals from the force trans-
ducer and accelerometer are extracted through a
discrete Fourier transform (see EG360). The sen-
sor returns acceleration, not displacement, but
a displacement amplitude by applying a factor
of 1/Ω2 to the acceleration amplitude 2. Then
equation (3) can be evaluated directly at each
frequency to give the receptance FRF, where the
resonant peaks show the damped natural fre-
quencies.
The damping ratio for each mode can also be
found, by noting that highly damped modes have
short, more rounded resonant peaks whereas
lightly damped modes have tall sharp peaks. To
do this we need to find the two frequencies near
any given peak in the receptance FRF for which
|H(ω)| = |H(ωd)| /
√
2 (4)
This is shown in Figure 3, where the two fre-
quencies where this is met are labelled ω1 and ω2
2To see why, just differentiate wx2(t) = W cos(Ωt−φ)
twice. In fact similar calculations can be performed on
an accelerance FRF, which uses acceleration amplitude
instead of displacement amplitude.
and are known as half power points. Once the
half power frequencies and the damped natural
frequency have been found for a given mode, the
damping ratio can be found by:
ζ=
ω2 − ω1
2ωd
(5)
3
Figure 3: Terms used in peak picking.
4 Lab Instructions
See the handout ‘Using the LabView VIs’ for
more detailed information on using the software.
4.1 Initial measurements
Use rulers and/or callipers to obtain measure-
ments of the length and cross sectional dimen-
sions of your beam.
4.2 Decay test
1. Open and run the VI ‘EC268 decay’.
2. Manually deflect the tip of the beam, then
release it and instantly click ‘Take Reading’
– this may need two people. Aim to get a
nice time trace of the tip vibration ampli-
tude slowly decaying.
3. Zoom in on a region of the acceleration
graph after the initial transients have died
away, record the heights of a series of peaks;
use at least 5 peaks. You will use this data
to estimate the natural frequency and damp-
ing ratio of the system using the formulas for
logarithmic decay.
4. Export the data from the graph you have
used to excel and save.
4.3 Broadband test
1. Open the VI ‘EG268 Broadband’.
2. After running the VI for a short while, click
on ‘Acquire response’. Make sure all fre-
quency graphs are zoomed to the approxi-
mate region 0-300Hz. There should be two
peaks visible in the ‘Accelerance’ graph (ig-
nore data below 10Hz, the sensors are hope-
less at these frequencies).
3. Note the frequencies where the two peaks
occur.
4. Repeat steps 2 and 3 ten times, each time
pasting the data from the graph ‘Acceler-
ance’ into a master spread sheet, which will
have columns: frequency, accelerance1, ac-
celerance2..accelerance10. Ensure that the
4
x-axis zoom remains constant during this
process, so that the same frequency range
is obtained each time.
4.4 Stepped sine test
1. Open and run the VI ‘EG268 sin’
2. With amplitude set to 0.1V, set frequency to
a value near to the second natural frequency
that you found in part 4.3. Record/compute
the data in Table 2, estimating the ampli-
tudes from the time series graphs.
3. Repeat Step 2 at various values in the range
of the 2nd peak in the broadband test. Tip:
start with a coarse frequency sweep, then
take more readings near the peak.
4. With the frequency set to your best estimate
of the 2nd natural frequency, let each team
member run a finger along the beam. There
will be a point along the beam at which
you feel no vibration; this is the node of
the modeshape. Measure how far this oc-
curs along the beam - take an estimate for
each team member.
5 Assignment
5.1 Predictions for natural fre-
quencies
Using your measurements from section 4.1,
1. Calculate the cross sectional area A, and the
second moment of area I = bd
3
12
of the beam3.
Using typical values for the mass density and
Young’s modulus of steel, calculate the lin-
ear mass density ρA and flexural stiffness EI
of the beam. (2 marks)
2. Using column two of Table 1, give predicted
values for the 1st and 2nd natural frequen-
cies of the beam. (1 mark)

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