Demonstration of different beam models by simple experiment

Transcription

Demonstration of different beam models by
simple experiment
Anujit Khutia, Utsa Majumder, Sumanta Neogy (corresponding author) and Arghya Nandi
Mechanical Engineering Department, Jadavpur University, Calcutta-700 032, India
E-mail: [email protected]
Abstract Experimental validation of various beam models is presented. The validation could form part
of a course in strength of materials, mechanical vibration and finite element analysis of structures.
Natural frequency is used as the key parameter. Such a presentation will also acquaint students with
the various instruments used in a dynamics laboratory. However, it suggests that the finite element
method, as well as other analytical or numerical tools, merely model the original structure and so
reminds students of the importance of experiments.
Keywords Euler–Bernoulli beam; Timoshenko beam; Rayleigh beam; natural frequency experiment;
finite elements
Notation
E
I
w
wn
l
r
k
A
J
G
d
t
x
y
p(x)
b
Young’s modulus
moment of inertia
Frequency
nth natural frequency
length of the beam
linear density of the beam
shear deflection constant
cross-sectional area
polar moment of inertia
modulus of rigidity
height of the beam
time
x-coordinate along the axis of the beam
y-coordinate normal to the beam axis
intensity of transverse loading on a beam
non dimensional frequency parameter
Introduction
In structural mechanics, a ‘beam’ is defined as a structural member that is reasonably long compared with its lateral dimensions when suitably supported and subjected to transverse forces so applied as to induce bending in an axial plane [1].
Different models of beams based on their length–depth ratio are covered in a standard introductory course in structural mechanics. A course in mechanical vibration
typically follows a course in structural mechanics, and once again various beam
models are covered. Now, over and above the elastic forces, the inertia forces also
International Journal of Mechanical Engineering Education 34/1
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A. Khutia et al.
have to be considered. The topics are once again discussed when the student takes
a course in finite elements of structures, in which both static analysis and dynamic
analysis are addressed. Finally, the student undergoes training in experimental techniques for structural dynamics of machine dynamics. The present exercise may be
considered a fitting experiment in such a course, because it permits students to bridge
with appropriate experimentation the theoretical concepts they have studied earlier.
Natural frequency is the parameter used here for the purpose of experimentally
identifying various types of beam. The reason for such a choice is that it is simple
to measure and such an experiment can be carried out using instruments which are
affordable and widely available.
In essence, the present work can be viewed as a comprehensive exercise in the
accurate numerical and experimental determination of natural frequencies. In the
process students are acquainted with the various instruments used. If some importance is given to experimental methods, a conventional and theoretically biased
vibration course packed with mathematical treatments becomes more interesting and
useful for a budding mechanical engineer.
Finally, computer graphics have achieved such a level of polish and versatility as
to inspire great trust in the underlying analysis, a trust that may be unwarranted [2].
The present work will help remind the student of the importance of experiments in
engineering.
To achieve the objectives discussed above, the natural frequencies of free–free
beams have been determined by an impact test using an instrumented hammer, a
high-frequency accelerometer and a dynamic signal analyser. This is a standard technique for experimental determination of natural frequency, mode shape and
damping. In spite of the wide acceptance of impact testing, it has various finer elements which are still subjects of contemporary research. Bill Fladung discussed how
Windows can be used for impact testing [3]. Karsen and Little presented the advantages of impact testing [4]. Brown pointed out the drawbacks of impact testing [5].
Singal, Gorman and Forgues used impact testing to investigate the free vibration of
rectangular plates [6]. Mohammed, Khan and Ramamurti carried out impact testing
on square and rectangular plates to determine the damping ratios [7].
Theoritical background
The standard theoretical beam model along with their assumptions are discussed
briefly.
Euler–Bernoulli model
To determine the differential equation for lateral vibration of the beam, the forces
and moments on an elemental part of the beam are considered [8]. Fig. 1 shows the
free-body diagram of such an elemental length of beam. The governing differential
equation is:
EI
d4y
∂2 y
+ p( x, t ) + r 2 = 0
4
dx
∂t
(1)
Beam models
13
Fig. 1
Free-body diagram for the Euler–Bernoulli Model.
And the natural frequency of vibration for nth mode is:
w n = (b nl)
2
EI
rl 4
(2)
The values of (bnl) depend on the boundary conditions [8]. For a free–free beam:
( b1l ) = 4.73004
(b 2 l ) = 7.85320
Here each cross-section originally plane is assumed to be plane and normal to the
longitudinal fibres of the beam.
Timoshenko and Rayleigh beam models
In the Timoshenko model, rotary inertia and shear deformation effects are taken into
account. Fig. 2 shows the free-body diagram of an elemental length of a Timoshenko
beam model. The equation of motion for free vibration is [8]:
EI
∂4 y
∂2 y
EIr  ∂ 4 y
Jr ∂ 4 y
+ r 2 −J +
+
=0
4
2
2

∂x
∂t
kAG  ∂x ∂t
kAG ∂t 4
(3)
Here, if the shear terms are neglected the Raleigh model for the beam is obtained,
where the equation of motion is:
EI
∂4 y
∂2 y
∂4 y
+
r
−
J
=0
∂x 4
∂t 2
∂x 2 ∂t 2
(4)
With a Rayleigh beam, after deformation the plane sections remain plane and perpendicular to the neutral axis. In a deformed Timoshenko beam the plane sections
remain plane but are not perpendicular to the neutral axis.
14
A. Khutia et al.
Fig. 2
Free-body diagram for the Timoshenko and Rayleigh beam models.
Three-dimensional solid model
A beam can be defined as a one-dimensional structural member in flexure. Geometrically two of the dimensions should be much smaller than the length for a structure to qualify as a beam. The next general model is a plate or a shell, which is a
two-dimensional structural member. Here only one dimension needs to be small
compared with the other two. In this work, investigations have been carried out on
members of square cross-section. So, as the length decreases (keeping the crosssection area same), the structure is expected to transform to a three-dimensional
model and the intermediate plate model cannot be used. Solving the threedimensional model is an exercise in the theory of elasticity. Analytical methods
would be quite complicated. Liew et al. [9] have solved the natural frequency problem of a three-dimensional flat solid using the Rayleigh–Ritz method. However,
even this analysis is quite involved. As an alternative, the finite element technique
has been chosen for solving the three-dimensional model. The finite element mesh
has been sufficiently refined to ensure that the numerical results are accurate enough
to give a reliable representation of the three-dimensional model.
Experimental setup
The full experimental setup is shown in Fig. 3.
Specimen preparation
The beam structure chosen for this experiment has a square cross-section of
19 mm × 19 mm (0.75″ square). In order to study the nature of various beam models,
the length–thickness ratio needs to be varied. This is achieved by taking various
lengths of specimen while keeping the cross-section the same.
Beam models
15
Fig. 3
Photograph of the full experimental setup.
Choice of boundary conditions
For an experiment to determine natural frequency one can choose various
combinations of boundary conditions: simply supported, cantilever, free–free
clamped–clamped, clamped–hinged, etc. But as the experiment has to cover a wide
range of frequencies (approximately 400–20 000 Hz) it is very difficult to support
the beam in a jaw (clamped conditions) or keep it in a simply supported configuration using rollers. In particular, as the beam becomes shorter, its natural frequency
rises (i.e. it becomes more rigid) and so the flexibility of the beam becomes comparable to that of the supporting structure. This causes serious error in highfrequency measurements. Further, clamped and simple supports very often spoil the
repeatability of the experiment as they inject non-quantifiable errors due to the nonuniformity of clamping/tightening forces. To avoid such errors, Silva and Gomes
[10] employed a free–free boundary condition for accurate determination of the
natural frequency of a cracked beam in their study of the effects of a crack on natural
frequency.
Boundary condition simulation
A free–free boundary condition is one in which there are no geometrical boundaries
imposed on the structure. Clearly, this becomes a semi-definite system, having rigidInternational Journal of Mechanical Engineering Education 34/1
16
A. Khutia et al.
Fig. 4
Original type of support and equivalent spring mass system.
TABLE 1
Item
Impact hammer
Accelerometer
Amplifier
OR25 PC Pack II
List of instruments used
Manufacturer
Type
SI. No.
Kistler
Kistler
Kistler
Oros
9722A2000
8774A50
5114
–
C 125735
C 192514
C 160084
–
body modes of zero frequency. In finite element parlance, these ‘zero eigenvalues’
are sometimes avoided by having natural frequencies close to zero or much lower
than the flexural natural frequencies. To simulate this computationally, weak springs
are added conveniently to the semi-definite degrees of freedom. Experimentally the
same idea has been applied by placing either end of the beam on two soft pieces of
sponge. The experimental setup may be idealised as a spring mass system with two
degrees of freedom, as shown in Fig. 4. The system will have two different natural
frequencies corresponding to the rigid-body modes (considering motion in a single
transverse plane).
Sensor attachment
The present experiment involves measurement of high frequencies (of the order of
20 kHz). Direct bolting of the sensor to the beam is avoided, for convenience. The
sensor is attached to the beam using a lighter attachment, as shown in Fig. 5.
Details of instruments used
The instruments required for the experiment (Table 1) are affordable to any vibration laboratory. The price of an impact hammer with amplifier would be typically
US$2000, while an accelerometer and conditioner would cost around US$800. The
Beam models
17
(a)
(b)
Fig. 5
(a) Schematic diagram showing base fixed on the beam, and (b) photograph of
beam with sensor (top left of beam; impact hammer also shown).
amplifier for the hammer would not be required for natural frequency measurement
as FRF (Frequency Response Function) is not required – FFT (Fast Fourier Transform) is enough to perform the experiment. The dynamic signal analyser used is
more expensive (approximate price €10 000), but with the advent of PC-based instrumentation and data acquisition systems the response spectrum can be obtained using
a cheaper instrument (e.g. the price of ACD 216 from Picotech, UK, is of the order
of £750).
18
A. Khutia et al.
Results
Natural frequency for the Euler–Bernoulli model
Natural frequency can be calculated from equation 2 by using a scientific
calculator.
Natural frequency for the Rayleigh and Timoshenko models
Equations 3 and 4 can be solved using a fully analytical method, a semi-numerical
method or a fully numerical method like finite elements. Shames and Dym [11] have
provided an analytical solution for both Rayleigh and Timoshenko beams simply
supported at either end. Thomas, Wilson and Wilson [12] have presented elegant
ways to solve Timoshenko beam problems using finite elements.
Here, the finite element method (using beam elements) was used to solve
2
equation 3 and 4 numerically, by taking k = 0 and k = , respectively. A standard
3
finite elements package (ANSYS) was used. The elements taken were two-dimensional elastic beam elements with two nodes. Fig. 6 shows the mode shape for the
first and second natural frequencies for a Timoshenko beam. Rayleigh mode shapes
are similar.
Natural frequency for a three-dimensional solid model
To solve the three-dimensional solid model, ANSYS was again used. A 20-noded
brick element was used and a 2 × 2 × 2 integration rule was employed. To ensure
accurate results, the aspect ratio was kept equal to 2. A subspace solver was used
with a shift to calculate the eigen frequencies. Fig. 7 shows the mode shapes for the
first and the second natural frequencies of a typical case.
Natural frequency from experiment
The beam is hit by an impact hammer with a suitable tip to excite the range of frequency of interest. The real-time FFT of the response is then obtained. Two exam1
1
Y
Y
X
X
(a)
Fig. 6
(b)
Mode shape for the first (a) and second (b) natural frequencies for a Timoshenko
beam (I/d = 20).
Beam models
19
(a)
Fig. 7
(b)
Mode shape for the first (a) and second (b) natural frequencies for a threedimensional solid model beam (I/d = 5).
ples of the resulting spectra are shown in Fig. 8. The spectra show clearly the natural
frequencies of the system. All the peaks of spectrum correspond to different beam
bending modes. It needs to be noted that all the spectra have two peaks close to zero
hertz. Fig. 9 shows a close-up of the spectrum near the zero frequency to detail these
peaks. As discussed, these correspond to the rigid-body modes of vibration.
The ratio of the frequencies of the two-rigid-body modes as obtained theoretically
and experimentally are 1.732 and 1.483, respectively. Further, the spectrum peaks
appear rather blunt. The cause of such an observation is that the accelerometer under
consideration has a cut-off at a lower value of 20 Hz. A different accelerometer with
a lower cut-off would obviously improve the peaks as well as the experimental
results. At this point the teacher may point out the limitation of the probes used for
a certain measurement.
The above observation stresses the correctness of the measurement made. Further,
students may be given an attitude that, if they are unable to explain a certain experimented result, they should not necessarily start thinking the experiment was wrongly
performed but they should also learn to ponder over its proper explanation.
All the natural frequencies for the first and second bending modes obtained above
are tabulated in Tables 2 and Table 3, respectively.
The experimental results match the theoretical ones well. Frequencies, as
expected, increase as the l/d ratio decreases. Another point to note is that the experimentally obtained values lie below the theoretical frequency values in most cases –
a typical characteristic of experimental values. The reason for such a trend is that
experiments provide boundary conditions less stiff than those of the theoretical
models. Further, unavoidable material effects present in the structure of the steel bar
also reduce the experimental values. These results are shown graphically in Fig. 10.
The computationally obtained values of natural frequencies using the
20
A. Khutia et al.
(a)
(b)
Fig. 8
Spectrum of natural frequency obtained from experiment (a) for l/d = 7 and (b) for
l/d = 14.
Beam models
21
Fig. 9
Natural frequency of rigid-body modes of vibration.
TABLE 2
Natural frequency (first bending mode)
Theoretical frequency (Hz)
l/d ratio
5
6
7
8
9
10
11
12
14
15
20
25
Euler–Bernoulli
model
Rayleigh
model
Timoshenko
model
Three-dimensional
model
Experimental
frequency (Hz)
11 221.47
7 792.69
5 725.24
4 383.39
3 463.42
2 805.37
2 318.49
1 948.17
1 431.31
1 246.83
701.34
448.86
10 418.00
7 396.30
5 509.40
4 256.97
3 385.70
2 754.60
2 284.38
1 924.58
1 419.20
1 237.90
699.10
448.25
10 182.00
7 227.90
5 429.94
4 207.21
3 352.54
2 732.60
2 268.90
1 913.46
1 413.06
1 233.20
697.56
447.61
9962.50
7148.76
5365.30
4168.00
3327.46
2715.80
2257.33
1905.18
1408.52
1229.80
696.45
447.15
9675.0
7150.0
5337.5
4100.0
3262.5
2687.5
2193.75
1843.2
1362.5
1212.5
687.5
442.5
22
A. Khutia et al.
TABLE 3
Natural frequency (second bending mode)
Theoretical frequency (Hz)
l/d ratio
5
6
7
8
9
10
11
12
14
15
20
25
Euler–Bernoulli
model
Rayleigh
model
Timoshenko
model
Three-dimensional
model
Experimental
frequency (Hz)
30 932.41
21 480.84
15 781.84
12 082.97
9 547.04
7 733.10
6 390.99
5 370.21
3 945.46
3 436.93
1 933.28
1 237.30
26 588.00
19 250.68
14 531.66
11 333.11
9 072.58
7 419.70
6 176.46
5 218.96
3 864.70
3 376.20
1 915.40
1 230.80
25 038.00
18 284.62
13 912.11
10 923.56
8 793.75
7 224.60
6 036.51
5 116.31
3 806.23
3 330.90
1 900.20
1 224.20
23 346.00
17 357.03
13 364.19
10 580.42
8 568.58
7 071.00
5 928.43
5 038.14
3 762.52
3 297.33
1 889.10
1 219.70
–*
17 600.00
13 350.00
10 412.50
8 412.50
7 037.50
5 737.50
4 950.00
3 687.50
3 300.00
1 875.00
1 212.50
*Data out of range of OROS data acquisition system.
Euler–Bernoulli model, Rayleigh model, Timoshenko model and three-dimensional
solid model decrease progressively in that order. This is because the models become
more flexible and allow for additional inertias. For example, a Rayleigh quotient
overestimates the natural frequency of the structure due to the stiffening of the structure. For high values of the l/d ratio all the theoretically obtained curves converge.
The experimental results also merge with the computed results. As the beam
becomes shorter, however, the curves diverge. The experimental points fall well
below the corresponding Timoshenko points and also below the line representing
three-dimensional model (except for l/d = 6, second natural frequency).
However, it needs to be noted that the experimental curve, especially for the
second mode, does not follow the theoretical trend exactly. Those are two probable
causes of error:
• The sensor and its base are attached to one end and act as an extra mass (unless
the probe is placed at a node where the mode will be suppressed).
• At frequencies as high as 10 kHz, the calibration curve of the sensor shows about
±2% error [13].
The latter appears to have a large effect on the experimental trend obtained at l/d =
6, for the second natural frequency. A more elegant representation of the data may
be obtained if the non-dimensional frequency.
w model
2
× (b n l )
w Euler
is computed for all the cases and plotted with the non-dimensional length (l/d), as
in Fig. 11.
Beam models
23
(a)
(b)
Fig. 10 Plots of the experimental and theoretical results for the first (a) and second (b)
natural frequencies.
24
A. Khutia et al.
(a)
(b)
Fig. 11
Non-dimensional frequency as a function of non-dimensional length, for the first
natural frequency (a) and the second natural frequency (b).
Beam models
25
The percentage error for different beam models and experimental values (with the
simplest Euler model as the reference) are shown in Fig. 12. This representation
helps magnify deviation in the values obtained.
It is evident from Fig. 10 that the Rayleigh, Timoshenko, three-dimensional
models and experimental results for either of the natural frequencies show similar
variation. From an engineering viewpoint, an error margin of 5% may be considered good enough. If one is able to accept this degree of accuracy, it is clear from
Fig. 10 that for the first mode up to l/d = 8 the Euler–Bernoulli beam gives acceptable results, and for the second mode up to l/d = 14 provides similar accuracy.
Students can therefore clearly see the appropriateness of the rule of thumb that
the strength of a material for a Euler or thin beam is measured with a minimum
l/d = 10. But from the date it can be inferred clearly that if higher modes are taken
into consideration, for obtaining an accuracy of 5%, l/d must be much higher.
However, it would extremely difficult to conduct such an experiment at reasonable
cost, as the frequencies involved rise appreciably. Here the limitations of the thumb
rule may be pointed out to students.
Once the Euler–Bernoulli model has been shown to be unacceptable, it is vital to
point out the strength of the Timoshenko model. It is important to emphasise that,
up to l/d = 5 for both modes, the Timoshenko model predicts results within 6% of
those of the three-dimensional solid model. As with the Euler model, it is easy to
see that, for higher modes, Timoshenko models would diverge away from the solid
model. So, it becomes evident that, up to l/d = 5, an involved three-dimensional
analysis is not really warranted.
The need for a three-dimensional analysis arises if structures shorter than l/d = 5
or modes higher than the second one are to be considered. It is important to note
that, in mechanical and structural systems, a natural frequency of 17–18 kHz is
rare and so the need to go to higher modes will not be very common. To address
the analysis of structures with l/d < 5, it is important to point out that only a threedimensional analysis will give a more accurate result, but under such circumstances
the support mass itself becomes flexible (at least compared with the short structure)
and so the greater effort given to the accurate analysis will have limited practical
value.
Conclusion
In the theoretical models, no originality is claimed, but appropriate graphical representation helps to bring out the meaningfulness of the results. The stress is on the
choice of model. However, as a word of caution, the cut-off value of l/d depends on
the boundary conditions.
The authors feel that exact results of the type given are not commonly available
in the literature. From the engineering viewpoint, students will learn that an
engineering structure is not obliged to behave as a computer dictates [2]. In
other words, the world of analysis approximates the real world and not the reverse
[14].
26
A. Khutia et al.
(a)
(b)
Fig. 12 Percentage error (with the simplest Eules model as the reference) for different
beam models and experimental values for the first natural frequency (a) and the second
natural frequency (b).
Beam models
27
References
[1] S. P. Timoshenko and D. H. Young, Elements of Strength of Material (Affiliated East West Press,
New Delhi, 1980).
[2] R. D. Cook, D. S. Malkus and M. E. Plesha, Concepts and Applications of Finite Element Analysis (John Wiley, New York, 1989).
[3] B. Fladung, Windows used for impact testing, in 15th International Modal Analysis Conference
(1997), pp. 1662–1666.
[4] C. D. van Karsen and E. F. Little, ‘The strengths of impact testing’, in 15th International Modal
Analysis Conference (1997), pp. 1667–1671.
[5] D. Brown, ‘Weaknesses of impact testing’, in 15th International Modal Analysis Conference (1997),
pp. 1672–1676.
[6] R. K. Singal, D. J. Gorman and S. A. Forgues, ‘A comprehensive analytical solution for free vibration of rectangular plates with classical edge conditions: experimental verifications’, Experimental
Mechanics, 32(1) (1992), 21–23.
[7] D. R. A. Mohammad, N. U. Khan and V. Ramamurti, On the roll of rayleigh damping’, Journal of
Sound and Vibration, 185(2) (1995), 207–218.
[8] W. T. Thomson, Theory of Vibration with Applications (2nd edn) (Prentice Hall India, New Delhi,
1982).
[9] K. M. Liew, K. C. Hung and M. K. Lim, ‘Free vibration study on stress-free three dimensional
elastic solids’, Journal of Applied Mechanics, 62 (1995), 159–164.
[10] J. M. Silva and A. J. M. Gomes, ‘Experimental dynamic analysis of cracked free-free beams’, Experimental Mechanics 30 (1990), 20–25.
[11] I. H. Shames and C. L. Dym, Energy in Finite Element Methods in Structural Mechanics (New Age
International Publishers Ltd, Wiley Eastern Ltd, New Delhi, 1995).
[12] D. L. Thomas, J. M. Wilson and R. R. Wilson, ‘Timoshenko beam finite elements’, Journal of Sound
and Vibration, 31(3) (1973), 315–330.
[13] Instruction Manual, Ceramic Shear Accelerometers (Low Impedance Voltage Mode), edition 11/01,
Kistler.
[14] J. L. Meriam, Engineering Mechanics (John Wiley & Sons, New York, 1980).

Demonstration of different beam models by simple experiment

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