A direct approach for the calculation of the natural frequencies and vibration mode shapes of a perfectly clamped-free beam with additional stepwise eccentric distributed masses is developed, along with its corresponding equations. Firstly there is contrived influence of a mass, located on a given position along the beam, upon the modal energies, via an energy analysis method. Secondly, the mass participation coefficient is defined as a function of the mass location and the bending vibration mode number. The proposed coefficient is employed to deduce the mathematical relation for the frequencies of beams with supplementary eccentric loads, generally available for any boundary conditions. The accuracy of the obtained mathematical relation was examined in comparison with the numerical simulation and experimental results for a cantilever beam. For this aim, several finite element models have been developed, individualized by the disturbance extent and the mass increase or decrease. Also, one real system was tested. The comparisons between the analytically achieved results and those obtained from experiments proved the accuracy of the developed mathematical relation.
During operation, engineering structures are subjected to loads, often produced by supplementary masses. These structures can suffer various damage types, including corrosion due to the interaction with the environment, which produce alterations of the geometry and the mechanical/physical properties [
Development of damage detection methods has drawn the interest of numerous researchers [
Herein, the cases of mass increase and decrease are equally treated, for different beam segment lengths, affected by these mass distortions. The idea was to find a reliable equation, which is able to indicate the natural frequencies, valid for beams irrespective of their support type and for any mass change scenario.
In prior research devoted to the analysis of beams with open cracks (see [
The aim of this section is to introduce a mathematical relation contrived by the authors, which predicts the frequency changes due to mass variation. In order to illustrate the case, a cantilever beam is used. In this example, the geometrical asymmetry assures an unequivocal link between the mass disturbance, defined by position and intensity, and the frequency changes. The analyzed prismatic beam of steel, illustrated in Figure
Cantilever beam model.
The natural frequency
The beam subjected to a supplementary load, induced by a mass
Cantilever beam with additional mass and its continuous and finite element models.
For finding the influence of mass changes upon the beam dynamics, the cantilever beam is considered as continuous and without mass. A mass
Dynamically equivalent systems having the mass located at different positions.
The kinetic energy calculated for the tip mass
If the kinetic energies in the two cases are equal, the same natural frequency for them both is obtained. The dependency between the two masses
Note that, for the cantilever beam, the largest displacements are always achieved at the free end. Thus,
The uniformly distributed inertial mass, loading on the cantilever beam in mode two, and the mass participation in the kinetic energy of each slice.
Consequently, the total equivalent mass, for the
Therefore, the frequency can be expressed with regard to the equivalent mass, as
The contribution to the total kinetic energy can be derived from (
The mass participation in the total kinetic energy of the uniformly distributed loads for 100 slices placed along the beam.
Now, the beam with a uniform cross section loaded with supplementary mass on a beam segment of length
The equivalent mass contribution of the segment subjected to a mass increase is
The total equivalent specific mass, contributing to the kinetic energy, is found by summarizing the equivalent specific masses of this segment and the other two segments, which are not subjected to any additional mass. These last two are
Figure
The mass participation in the total kinetic energy for 100 slices placed along the beam: (a) in case of supplementary mass; (b) in case of mass loss.
In the case of a supposed uniform stiffness, the natural frequencies for the beam with increased mass between points
Obviously, the natural frequencies decrease due to an additional mass. In the case of mass loss, that is,
From (
If the beam is subjected to the action of more masses, the frequencies can be similarly derived, but a particular approach is imposed for each specific segment. This means that in (
Equation (
In practical applications, such as damage detection or vibration control, the frequency deviation caused by the mass alteration is relevant. It is
The relative frequency shift, achieved by dividing the frequency change due to mass alteration to that of the uniform beam, becomes
Because the contrived relations, expressing the frequency of the nonuniformly loaded beam, the frequency shift, and the relative frequency shift of transverse modes, contain just terms dependable on the support type (e.g., mode shapes and wave numbers), the relations are generally valid. The single requirement is the involvement of these features obtained for the right boundary conditions. In the next sections, the obtained mathematical relations are examined as opposed to numerical simulations and experimental results in the case of a cantilever beam.
To prove the correctness of the contrived mathematical relations, by means of the finite element analysis (using ANSYS software), a series of simulations were performed. The initial investigations have taken into consideration the uniform beam, having a length
Afterwards, beams presenting nonuniform mass distributions were considered. The particular cases are individualized by the disturbance position, extent, and intensity. Regarding the disturbance extent, two typical cases are defined: (a) deviation from the uniform mass distribution present on large segments and (b) slim regions affected by the altered mass load. It has to be noted that the mass increase or decrease is modeled by a changed mass density of the respective segment. The parameters of the regions subjected to mass alteration are comprehensively presented in Tables
Parameters reflecting the segments affected by mass alteration, large regions.
Scenario | Mass density |
Left limit |
Right limit |
Extent |
---|---|---|---|---|
A | 4000 | 300 | 400 | 100 |
B | 10000 | 300 | 400 | 100 |
C | 12000 | 600 | 650 | 50 |
D | 2000 | 700 | 750 | 50 |
E | 14000 | 500 | 550 | 50 |
F | 3000 | 800 | 850 | 50 |
Parameters reflecting the segments affected by mass alteration, slim regions.
Scenario | Mass density |
Left limit |
Right limit |
Extent |
---|---|---|---|---|
G | 4000 | 100 | 110 | 10 |
H | 10000 | 100 | 110 | 10 |
I | 12000 | 150 | 160 | 10 |
J | 2000 | 150 | 160 | 10 |
K | 14000 | 200 | 210 | 10 |
L | 3000 | 200 | 210 | 10 |
The first three natural frequencies for the transverse vibration modes are found using the
Natural frequencies analytically derived for wide affected region scenarios.
Mode |
Uniform beam | Beam with stepped load | |
---|---|---|---|
|
Scenario |
|
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1 | 4.07690 | A | 4.08977 |
B | 4.06976 | ||
C | 4.02542 | ||
D | 4.20082 | ||
E | 4.03397 | ||
F | 4.23059 | ||
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2 | 25.54951 | A | 26.52667 |
B | 25.04867 | ||
C | 25.17304 | ||
D | 25.65486 | ||
E | 24.63272 | ||
F | 25.60767 | ||
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3 | 71.53939 | A | 75.09158 |
B | 69.76202 | ||
C | 70.40146 | ||
D | 73.74612 | ||
E | 71.42629 | ||
F | 71.87511 |
Natural frequencies analytically derived for concentrated mass loads.
Mode |
Uniform beam | Beam with stepped load | |
---|---|---|---|
|
Scenario |
|
|
1 | 4.07690 | G | 4.07691 |
H | 4.07689 | ||
I | 4.07683 | ||
J | 4.07699 | ||
K | 4.07661 | ||
L | 4.07713 | ||
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2 | 25.54951 | G | 25.55209 |
H | 25.54805 | ||
I | 25.53872 | ||
J | 25.56472 | ||
K | 25.51040 | ||
L | 25.58046 | ||
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3 | 71.53939 | G | 71.58216 |
H | 71.51493 | ||
I | 71.39000 | ||
J | 71.75063 | ||
K | 71.11268 | ||
L | 71.88067 |
Natural frequencies found by FEA for large affected region scenarios.
Mode |
Uniform beam | Beam with stepped load | |
---|---|---|---|
|
Scenario |
|
|
1 | 4.08986 | A | 4.10330 |
B | 4.08236 | ||
C | 4.03839 | ||
D | 4.21423 | ||
E | 4.04695 | ||
F | 4.24427 | ||
|
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2 | 25.62655 | A | 26.62145 |
B | 25.11124 | ||
C | 25.25335 | ||
D | 25.73864 | ||
E | 24.718923 | ||
F | 25.68827 | ||
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3 | 71.75447 | A | 75.53598 |
B | 70.08363 | ||
C | 70.67144 | ||
D | 74.09155 | ||
E | 71.65492 |
Natural frequencies found by FEA for concentrated mass loads.
Mode |
Uniform beam | Beam with stepped load | |
---|---|---|---|
|
Scenario |
|
|
1 | 4.08986 | G | 4.08963 |
H | 4.08963 | ||
I | 4.08961 | ||
J | 4.08979 | ||
K | 4.08947 | ||
L | 4.08993 | ||
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2 | 25.62655 | G | 25.62894 |
H | 25.62506 | ||
I | 25.61647 | ||
J | 25.64190 | ||
K | 25.58856 | ||
L | 25.65777 | ||
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3 | 71.75447 | G | 71.79798 |
H | 71.73294 | ||
I | 71.60898 | ||
J | 71.96386 | ||
K | 71.32889 | ||
L | 72.09396 |
The relative frequency shifts are derived and compared to have a better overview on the results. Tables
The relative frequency shifts for large affected region scenarios.
Mode |
Scenario |
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1 | A |
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B |
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C |
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D |
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E |
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F |
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2 | A |
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B |
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C |
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D |
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E |
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F |
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3 | A |
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B |
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C |
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D |
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E |
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The relative frequency shifts for the regions affected by concentrated loads.
Mode |
Scenario |
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1 | G |
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H |
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I |
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J |
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K |
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L |
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2 | G |
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H |
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I |
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J |
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K |
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L |
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3 | G |
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H |
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I |
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J |
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K |
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L |
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Taking a look at Tables
The relative frequency shifts analytically derived (left bars for each mode) and evaluated by FEA (bars positioned at right) for the mass increase on an extended region.
The relative frequency shifts analytically derived (left bars for each mode) and evaluated by FEA (bars positioned at right) for the mass decrease on an extended region.
The relative frequency shifts analytically derived (left bars for each mode) and evaluated by FEA (bars positioned at right) for the mass increase on a narrow segment.
The relative frequency shifts analytically derived (left bars for each mode) and evaluated by FEA (bars positioned at right) for the mass decrease on a narrow segment.
The relative frequency shifts (RFSh) have got a special attention. They clearly characterize any nonuniformity in the beam’s mass distribution, indicating at the same time the location and level of supplementary load. For the same position, the RFSh differ by amplitudes, which are in a relation defined by the additional load level (e.g., ratio between the regular density and the density achieved in the region, where the distortion is present). Thus, by normalizing the relative frequency shifts a sequence of values is attained, constituting a pattern that indicates the region in which the mass nonuniformity is present. These patterns are in direct relation to the square of the mode shape value at the affected region location and they can be analytically defined if the boundary conditions are known.
The experimental tests have been designed to complete the numerical analysis, in order to prove the correctness of the theoretical findings. These were carried out on a cantilever beam similar to that chosen for the FEA. Here, the first seven natural frequencies of the weak axis bending modes, for the uniform beam and the beam subjected to an additional mass (see scenario E), are targeted. To assure the clamped end with a proper fixing, the beam was clamped in a vise, as presented in Figure
The experimental setup: (a) stand overview; (b) detailed view of the region with added masses.
The acquisition system consists of a laptop, a compact chassis (NI cDAQ-9172) with NI 9234 signal acquisition modules, and a Kistler 8772 accelerometer that is fixed close to the beam’s free end. Since a high precision in evaluating the natural frequencies is required, due to the expected small frequency changes, the implementation of a properly developed virtual instrument in the
A noncontact acoustic excitation was employed to produce controlled input. The acoustic excitation system chain consists of virtual free software, acoustic amplifier with frequency equalizer, and low-frequency speaker. The
Additional masses are distributed between points
The measured natural frequencies
Natural frequencies of the uniform beam.
Mode |
|
|
|
---|---|---|---|
1 | 4.07690 | 4.08986 | 4.045 |
2 | 25.54951 | 25.62655 | 25.956 |
3 | 71.53939 | 71.75447 | 72.654 |
4 | 140.18865 | 140.62745 | 142.615 |
5 | 231.74189 | 232.52003 | 235.801 |
6 | 346.18225 | 347.45183 | 352.505 |
7 | 483.51075 | 485.45783 | 491.164 |
Table
Natural frequencies of the beam loaded by a stepped mass (scenario E).
Mode |
|
|
|
---|---|---|---|
1 | 4.03397 | 4.046953 | 3.988 |
2 | 24.63272 | 24.71892 | 25.044 |
3 | 71.42629 | 71.65493 | 72.528 |
4 | 135.42063 | 136.1956 | 138.036 |
5 | 230.28665 | 231.3189 | 234.551 |
6 | 335.93814 | 338.4368 | 343.192 |
7 | 477.90911 | 480.9907 | 485.748 |
The relative frequency shifts are calculated and compared to accomplish a deeper analysis. For the three approaches, the results are indicated in Table
Relative frequency shifts.
Mode |
|
|
|
---|---|---|---|
1 | 1.0528 | 1.04913 | 1.409147 |
2 | 3.5882 | 3.54176 | 3.513638 |
3 | 0.1575 | 0.13873 | 0.173425 |
4 | 3.370280 | 3.15148 | 3.210742 |
5 | 0.597020 | 0.51657 | 0.530108 |
6 | 2.928860 | 2.59462 | 2.641948 |
7 | 1.095870 | 0.92019 | 1.102687 |
Relative frequency shifts for scenario E.
The results presented in Figures
In this paper a mathematical relation for the prediction of frequency changes due to alteration of mass is proposed. This relation associates the effect of a local mass on the natural frequencies with the square of the mode shape corresponding to that location. Since the relation has as input parameters the position, the mass alteration, and the beam mode shape, it becomes easily adaptable to any kind of boundary conditions. The accuracy of the achieved mathematical relation has been examined by numerical simulations and experiments, in the case of a cantilever beam, and the results confirmed its robustness and reliability.
The relative frequency shifts derived from this relation provide a simple way to describe the vibrational behavior of beams with nonuniform distributed mass. Thus, it is possible to create a database containing numerous scenarios afterwards employable to find the region subjected to mass distortion and derive its significance.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work has been funded by the Sectoral Operational Programme Human Resources Development 2007–2013 of the Ministry of European Funds through the Financial Agreement POSDRU/159/1.5/S/132395.