Evaluation of the Damping Produced by the Motion of a Permanent Magnet inside of a Conductive Pipe
##plugins.themes.bootstrap3.article.main##
In this study, a method to evaluate the electromagnetic damping effect produced by the movement of a permanent magnet inside of a conductive pipe, is proposed. Neither the residual flux density of the magnet, nor the electric conductivity of the pipe is required, but instead, only the magnet height, the pipe length, and the falling time of the magnet inside the tube, are necessary to compute the damping coefficient. Accordingly, cylindrical neodymium magnets of close residual flux densities, but various diameters and heights, are tested against copper pipes of various thicknesses. Using the falling time measured for each magnetpipe combination, the corresponding damping coefficient is computed, and the influence of various geometrical parameters on the damping coefficient is clarified. Clearance between the magnet and pipe is identified as essential to describe this damping phenomenon.
Downloads
Introduction
It is common knowledge that a magnet moving inside of a conductive pipe produces a variable magnetic field, which in turn gives rise to eddy currents into the pipe wall, and hence, oppositely to the velocity vector of the magnet, an electromagnetic force is generated. This force tends to stop the magnet, although there is no direct contact with the pipe, and thus, it appears to be resistant in nature. Applications of this phenomenon, such as magnetic dampers [1]–[3] and breaks [4], [5], magnetic levitation based energy harvesters [6], etc., are designed based on the fact that the electromagnetic force proportionally varies to the magnet speed [7], [8]. Since this force is similar to a viscous damping force, the constant of proportionality is called damping coefficient, and it can be experimentally determined either from the falling motion [7]–[14], or from the reciprocating motion [1] of the magnet inside of the conductive pipe. Thus, damping coefficient can be estimated either from the measured falling time, which is considerably longer than the time elapsed for descent in an insulative pipe of the same length [7]–[14], or from the area of the forcedisplacement hysteresis loop [1].
Earlier computation methods of the damping coefficient have been developed for a cylindrical magnet, uniformly magnetized and modeled as a magnetic dipole, falling inside of a thinwalled conductive pipe [7]–[10]. Then, in order to account for the experimentally noted nonlinear dependence of damping on the pipe thickness, expression of the damping coefficient was revised for pipes of finite thickness [7], [10]. Besides, the dipole approximation, holding for thin magnets, was then revised for magnets of finite thickness [8], [15].
However, all computational methods proposed to estimate the damping coefficient require precise data for the residual flux density of the magnet, and the electric conductivity of the pipe. On one hand, values of the flux density are slightly different on the northpole and the southpole surfaces of the magnet. In fact, flux density depends on the measurement point selected on the surface. Thus, highclass equipment, using a threedimensional probe attached to a Gaussmeter, able to scan the entire surface, is required to measure the flux density distribution on the surface, and to determine the maximal, minimal, and average values of the flux density. On the other hand, some discrepancy between the actual electric conductivity of the pipe, and the catalog value of the corresponding conductive alloy, is to be expected [10].
Thus, due to the unreliable information on the residual flux density of the magnet, and the electric conductivity of the pipe, the accuracy of the predicted damping coefficient remains uncertain. Moreover, although from an applicative standpoint, clearance between the magnet and pipe seems to considerably influence the damping phenomenon, pertinent data is unavailable in the literature.
In this paper, damping coefficient is computed, without using the values of the residual flux density and electrical conductivity, but employing data on the magnet height, pipe length, and falling time of the magnet inside the tube. A number of 12 cylindrical neodymium permanent magnets, having close residual flux densities, but various dimensions, were tested against 6 types of copper pipes, having the same inner diameter and length, but various thicknesses. Influence of the magnet diameter for fixed magnet height, influence of the magnet height for fixed magnet diameter, influence of the pipe thickness for fixed pipe inner diameter and length, and influence of the clearance between the magnet and pipe, on the damping coefficient, is discussed.
Computation Method of the Damping Coefficient
Fig. 1 illustrates a schematic view of the magnet falling inside of a conductive pipe. Axial coordinate z, attached to the fixed pipe, has the point of origin on the upper end of the tube. Geometrical parameters are denoted as follows: d is the magnet diameter, h is the magnet height, L is the pipe length, D is the inner diameter of the pipe, H is the thickness of the pipe wall, and C/2 is the radial clearance between the magnet and pipe. Initial and final positions of the magnet are shown using the dashdoubledot lines.
Forces acting on the magnet during its falling motion are the weight mg, and the damping or levitation force F_{m} = cv (see Fig. 1). By applying Newton’s 2^{nd} Law, the movement equation of the magnet can be written as: (1)mdvdt=mg−Fm=mg−cv,
where m is the magnet mass, g = 9.8 m/s^{2} is the gravitational acceleration, t is the time, v is the magnet velocity, and c is the damping coefficient. Integrating (1) under the following initial conditions: (2)z(t=0)=0;v(t=0)=0
variation against time of the axial coordinate z (t), velocity v (t), and acceleration a (t) can be expressed as: (3){z(t)=gc¯2[c¯t−1+exp(−c¯t)]v(t)=dzdt(t)=gc¯[1−exp(−c¯t)],a(t)=d2zdt2(t)=g⋅exp(−c¯t)
where c¯=c/m is the ratio of the damping coefficient to the magnet mass, i.e., the specific damping coefficient.
Falling time T is the time elapsed until the magnet arrives at the final position, z = L – h, during its descent inside the pipe. Therefore, from (3), the axial coordinate, velocity, and acceleration of the magnet corresponding to its final position can be written as: (4){z(T)=L−h=gc¯2[c¯T−1+exp(−c¯T)]v(T)=gc¯[1−exp(−c¯T)]a(T)=g⋅exp(−c¯T)
Note that the 1^{st} expression of (4) leads to the following transcendent equation in c¯: (5)(L−h)c¯2=g[c¯T−1+exp(−c¯T)],
which can be solved numerically for a given pipe length and magnet height by inputting only the measured falling time. However, since the exponential term exp(−c¯T)≅ 0 can be neglected, as will be later proved in Section IV, (5) can be fairly approximated to the following quadratic equation: (6)L−hgc¯2−T⋅c¯+1=0.
Solving (6) and selecting the appropriate larger solution, one obtains the specific damping coefficient as follows: (7)c¯=g2(L−h)[T+T2−4L−hg].
Test Rig and Experimental Procedure
Pipes used in the Falling Tests
Falling tests of the magnet were performed using 6 types of conductive pipes, having the same inner diameter and length, but different wall thicknesses (see Table I). Thus, all pipes, denoted as P1–P6, were made in copper alloy C1220, and had a length of L = 1,000 mm, and an inner diameter of D = 12 mm. Electric conductivity of the copper alloy C1220 was of 49.3 MS/m, i.e., 85% of the value accepted by the International Annealed Copper Standard.
Pipe type  P1  P2  P3  P4  P5  P6 

Thickness, H (mm)  1  1.5  2  3  4  5 
Magnets used in the Falling Tests
Falling tests were carried out by employing 12 types of cylindrical neodymium magnets, having close residual flux densities, but various dimensions. Physical properties of these permanent magnets, denoted as M1–M12, i.e., the diameter d and height h, as well as the catalog values for the residual flux density B and mass m, are presented in Table II.
Magnettype  Diameter,d (mm)  Height, h (mm)  Flux density, B (mT)  Mass, m (g) 

M1  11.5  6  405.4  4.67 
M2  11.5  7  434.0  5.45 
M3  11.5  8  456.1  6.23 
M4  11.5  9  473.3  7.01 
M5  11.5  10  486.9  7.79 
M6  11.5  11  497.7  8.56 
M7  7  10  495.2  2.88 
M8  8  10  496.0  3.76 
M9  9  10  494.2  4.77 
M10  10  10  491.1  5.89 
M11  10.5  10  491.5  6.49 
M12  11  10  491.1  7.12 
Two types of falling tests have been performed:
 1) Falling tests for fixed magnet diameter, to a value of d = 11.5 mm, in which case the magnet height h was varied from 6 mm to 11 mm, with an increment of 1 mm (see magnets M1–M6 in Table II),
 2) Falling tests at fixed magnet height, to a value of h = 10 mm, in which case the magnet diameter was varied from 7 mm to 11.5 mm (see magnets M7–M12 and M5 in Table II).
Test Rig
Fig. 2 shows a schematic view of the test rig used in the falling tests of the magnet inside of the conductive tube. Pipe was fixed in a vertical position on a clamp support, and its verticality was checked by using a spirit level.
Since neodymium is a brittle material, the magnet can be easily broken during the falling tests by the collision with the metallic base of the clamp support at the bottom of the pipe. For this reason, a shockabsorbing sheet was used to cover the support base. Moreover, in order to evacuate the air during the falling tests of the magnet, a small gap of 1 mm thickness was opened between the pipe and the shockabsorbing sheet. The effect of the small deformation of the shock absorbing sheet, caused by the colliding magnet, and the influence of the small gap for air evacuation can be neglected relative to the falling length L – h, which was used in the theoretical model of Section II.
In order to hold the tested magnet at its initial position before commencing the falling test, a thin plastic plate and a small support magnet were used. For opposite polarities of the tested magnet and support magnet at the interface with the plastic plate, a small attractive force can be created to sustain the weight of the tested magnet. By removing the thin plastic plate together with the support magnet, this attractive force ceases, allowing the free falling of the tested magnet inside of the conductive pipe. In this way, the falling test can be materialized in concordance with the theoretical model presented in Section II.
Experimental Procedure
Falling tests of the magnet inside the conductive pipe were conducted, as follows:
1) Using the thin plastic plate and the support magnet, the tested magnet was placed inside the pipe, at its initial position, in such a way that its longitudinal axis was aligned with the longitudinal axis of the pipe.
2) Falling time was measured by using a stopwatch, started at the releasing instant, and stopped at the collision moment of the tested magnet. Note that the collision instant was identified simply by perceiving the collision noise.
3) In order to achieve reliable experimental results, for each combination of magnets and pipes, the falling tests were performed for 10 times, and the averaged value of the falling time was recorded. Using (7), the specific damping coefficient was computed, and then, the damping coefficient c=c¯⋅m was obtained.
Results and Discussions
Results obtained for all neodymium magnets M1–M12, i.e., the experimentally found mean value of the falling time T, as well as the computed values of the specific damping coefficient c¯ and damping coefficient c, are illustrated for all the copper pipes P1–P6 in Tables III–VIII, respectively.
Magnettype  Mean fallingtime, T (s)  Specific damping coefficient, c¯ (s^{−1})  Dampingcoefficient,c (Ns/m) 

M1  20.87  205.694  0.961 
M2  20.91  206.344  1.125 
M3  20.14  198.919  1.239 
M4  19.58  193.572  1.357 
M5  18.55  183.536  1.430 
M6  19.05  188.719  1.615 
M7  6.47  63.924  0.184 
M8  10.05  99.375  0.374 
M9  10.30  101.871  0.486 
M10  12.56  124.254  0.732 
M11  15.53  153.689  0.997 
M12  17.19  170.064  1.211 
Magnet type  Mean falling time, T (s)  Specific damping coefficient, c¯ (s^{−1})  Damping coefficient,c (Ns/m) 

M1  28.71  282.991  1.322 
M2  28.86  284.786  1.552 
M3  28.04  276.965  1.725 
M4  27.50  271.880  1.906 
M5  25.88  256.110  1.995 
M6  26.41  261.671  2.240 
M7  8.14  80.470  0.232 
M8  11.53  114.049  0.429 
M9  12.67  125.371  0.598 
M10  16.55  163.755  0.965 
M11  21.03  208.120  1.351 
M12  23.44  232.004  1.652 
Magnet type  Mean falling time, T (s)  Specific damping coefficient, c¯ (s^{−1})  Damping coefficient,c (Ns/m) 

M1  34.05  335.684  1.568 
M2  33.64  331.977  1.809 
M3  33.33  329.211  2.051 
M4  32.71  323.486  2.268 
M5  31.18  308.600  2.404 
M6  31.97  316.751  2.711 
M7  9.37  92.634  0.267 
M8  13.76  136.147  0.512 
M9  15.30  151.379  0.722 
M10  20.05  198.463  1.169 
M11  25.06  248.036  1.610 
M12  27.80  275.160  1.959 
Magnet type  Mean falling time, T (s)  Specific damping coefficient, c¯ (s^{−1})  Damping coefficient,c (Ns/m) 

M1  39.38  388.258  1.813 
M2  40.52  399.903  2.179 
M3  40.70  402.037  2.505 
M4  40.01  395.633  2.773 
M5  38.29  378.991  2.952 
M6  39.58  392.208  3.357 
M7  11.31  111.894  0.322 
M8  17.07  168.928  0.635 
M9  19.02  188.195  0.898 
M10  24.82  245.658  1.447 
M11  31.19  308.746  2.004 
M12  34.24  338.915  2.413 
Magnet type  Mean falling time, T (s)  Specific damping coefficient, c¯ (s^{−1})  Damping coefficient,c (Ns/m) 

M1  42.82  422.143  1.971 
M2  44.21  436.262  2.378 
M3  44.42  438.833  2.734 
M4  44.14  436.456  3.060 
M5  42.82  423.822  3.302 
M6  44.65  442.394  3.787 
M7  13.02  128.856  0.371 
M8  19.28  190.825  0.718 
M9  21.55  213.283  1.017 
M10  28.07  277.828  1.636 
M11  35.42  350.618  2.276 
M12  39.39  389.850  2.776 
Magnet type  Mean falling time, T (s)  Specific damping coefficient, c¯ (s^{−1})  Damping coefficient,c (Ns/m) 

M1  45.50  448.527  2.095 
M2  46.89  462.723  2.522 
M3  46.89  463.179  2.886 
M4  46.72  461.966  3.238 
M5  45.20  447.459  3.486 
M6  47.01  465.780  3.987 
M7  13.88  137.371  0.396 
M8  20.28  200.734  0.755 
M9  22.65  224.133  1.069 
M10  29.40  291.016  1.714 
M11  37.21  368.356  2.391 
M12  40.73  403.202  2.871 
Based on data shown by Tables III–VIII, it becomes now possible to evaluate the exponential term exp(−c¯T) of (5).
Except for the combinations of M7 with pipes P1 and P2, in which case exp(−c¯T)≅2×10−180 and 3×10−285, for all other combinations of magnets and tubes, exp(−c¯T)=0. Thus, evaluation method of the specific damping coefficient, based on (6) and (7), appears to be appropriate.
Moreover, by substituting exp(−c¯T)=0 in the 2^{nd} and 3^{rd} expressions of (4), velocity and acceleration of the magnet, at its final position inside the pipe, can be rewritten as: (8)v(T)=gc¯=mgc;a(T)=0
Analyzing the results shown by Tables III–VIII, and the magnet dimensions shown by Table II, one observes that the damping coefficient increases at augmentation of the height h (see M1–M6) and diameter d (see M7–M12) of the magnet, and also at augmentation of the pipe thickness H. Damping coefficient c ranges from the minimal value of 0.184 Ns/m, obtained for the combination of M7 with P1, to a maximal value of 3.987 Ns/m, found for the combination of M6 with P6. A detailed analysis of the effect of various geometrical parameters on the damping coefficient is given below.
Based on data from Table II, Fig. 3 shows the variation of the magnet mass m versus the magnet diameter d, for a fixed value of the magnet height, h = 10 mm, and versus the magnet height, for a fixed magnet diameter, d = 11.5 mm. As expected, magnet mass depends linearly on the magnet height, but parabolically on the magnet diameter.
Based on data from Table II, Fig. 4 presents the variation of the magnetic residual flux density B versus the magnet diameter d, for a fixed value of the magnet height, h = 10 mm, and versus the magnet height, for a fixed magnet diameter, d = 11.5 mm. On one hand, since for the magnets of the same height, h = 10 mm, the magnetic flux density varied from 486.9 mT, for M5, to 496.0 mT, for M8, these magnets can be regarded as of almost constant flux density, B = 491.45 mT, with a fluctuation of ±4.55 mT, i.e., with a fluctuation of only ±0.93%. On the other hand, for the magnets of the same diameter, d = 11.5 mm, the residual flux density B nonlinearly varied versus the magnet height h.
Fig. 5 illustrates the variation of the damping coefficient c versus the magnet diameter d, determined for fixed values of the magnet height, h = 10 mm, and pipe inner diameter, D = 12 mm, but for various values of the pipe thickness H = 1, 1.5, 2, 3, 4, and 5 mm (see P1–P6, Table I). Fig. 5 shows a nonlinear augmentation of the damping coefficient c against the magnet diameter d. Since these results are obtained at almost constant flux density, they appear to intrinsically reflect the influence of the magnet diameter.
Fig. 6 presents the variation of the damping coefficient c versus the magnet height h, determined for fixed values of the magnet diameter, d = 11.5 mm, and pipe inner diameter, D = 12 mm, but various values of the pipe thickness H = 1, 1.5, 2, 3, 4, and 5 mm (see P1–P6, Table I). Fig. 6 shows an almost linear augmentation of the damping coefficient c versus the magnet diameter, under the nonlinear fluctuation of the residual flux density, as displayed by Fig. 4.
Fig. 7 illustrates the variation of the damping coefficient c versus the pipe thickness H, determined for fixed values of the magnet height, h = 10 mm, and pipe inner diameter, D = 12 mm, but for various values of the magnet diameters d = 7, 8, 9, 10, 10.5, 11, and 11.5 mm (see M5 and M7–M12, Table II). Fig. 7 presents a nonlinear augmentation of the damping coefficient c versus the pipe thickness H. Again, since these results are obtained at almost constant residual flux density, they appear to genuinely reflect a saturationlike dependence of the damping coefficient on the pipe thickness.
Fig. 8 displays the variation of the damping coefficient c versus pipe thickness H, determined for fixed values of the magnet diameter, d = 11.5 mm, and pipe inner diameter, D = 12 mm, but various values of the magnet height h = 6, 7, 8, 9, 10, and 11 mm (see M1–M6, Table II).
Fig. 8 shows also a nonlinear, saturationlike pattern of augmentation for the damping coefficient c versus the pipe thickness H, under the nonlinear fluctuation of the residual flux density, as displayed by Fig. 4.
Fig. 9 illustrates the variation of the damping coefficient c versus the diametral clearance between the pipe and magnet, C = D – d, determined for fixed values of the magnet height, h = 10 mm, and pipe inner diameter, D = 12 mm, but for various values of the pipe thickness H = 1, 1.5, 2, 3, 4, and 5 mm (see P1–P6, Table I). Fig. 9 shows a nonlinear reduction of the damping coefficient c at augmentation of the gap C, which can be explained by the weakening of the electromagnetic interaction between the pipe and magnet. Once again, since these results are obtained at almost constant residual flux density of the magnet, they seem to accurately express the influence of the clearance. Due to the fact that, for all the mechanical systems involving the movement of a body inside of an interstice, clearance is the most important geometrical parameter to properly describe the movement, results shown by Fig. 9 can be used to adequately design the damper, associated to this electromagnetic damping effect.
Conclusion
In this work, the electromagnetic damping phenomenon, appearing during the movement of a cylindrical neodymium permanent magnet inside of a conductive copper pipe, was investigated, and the associated damping coefficient was evaluated using only the magnet height, the pipe length, and the falling time of the magnet inside the tube.
Under a procedure compliant with the theoretical model, two types of falling experiments were performed for various dimensions of the magnet and the pipe:
A) Tests with magnets of the same height, but variable diameter, in which the residual flux density of the magnet was almost constant. In these tests, the damping coefficient displayed the following intrinsic variation patterns versus the geometrical parameters:
 1) Nonlinear, concavelike augmentation pattern against the magnet diameter,
 2) Nonlinear, saturationlike augmentation pattern versus the pipe thickness,
 3) Nonlinear, concavelike reduction pattern against the clearance between the magnet and pipe.
B) Tests with magnets of the same diameter, but variable height, in which the residual flux density of the magnet nonlinearly varied versus the magnet height. In these tests, the damping coefficient showed the following dependences on the geometrical parameters:
 4) Linear augmentation versus the magnet diameter,
 5) Nonlinear, saturationlike augmentation pattern versus the pipe thickness.
References

Bae JS, Hwang JH, Park JS, Kwag DG. Modeling and experiments on eddy current damping caused by a permanent magnet in a conductive tube. J Mech Sci Tech. 2009;23:3024–35.
DOI  Google Scholar
1

Mobley FF, Tossman BE, Fountain GH. The attitude control and determination systems of the SAS—A satellite. APL Tech Dig. 1971;19:24–34.
Google Scholar
2

Weinberger MR. Drag force of an eddy current damper. IEEE Trans Aerosp Electron Syst. 1977;AES13(2):197–200.
DOI  Google Scholar
3

Anantha Krishna GL, Sathish Kumar KM. Evaluation of brake parameters in copper discs of various thicknesses and speeds using neodymiumironboron magnets. Matec Web Conf. 2018;114:01003.1–12.
DOI  Google Scholar
4

Ireson G, Twidle J. Magnetic braking revisited: activities for the undergraduate laboratory. European J of Physics. 2008;29:745–51.
DOI  Google Scholar
5

dos Santos Soares MP, Ferreira JAF, Simoes JAO, Pascoal R, Torrao J, Xue X, et al. Magnetic levitationbased electromagnetic energy harvesting: a semianalytical nonlinear model for energy transduction. Sci Rep. 2016;6:18579.1–9.
DOI  Google Scholar
6

Donoso G, Ladera CL, Martin P. Magnet fall inside a conductive pipe: motion and the role of the pipe wall thickness. Eur J Phys. 2009;30:855–69.
DOI  Google Scholar
7

Sahil J, Prita R, Ramachandran M. Investigation of a magnet falling through a copper tube. IOP Conf Ser: mater Sci Eng. 2020;810:12042.1–12.
DOI  Google Scholar
8

Bistafa SR. On the derivation of the terminal velocity for the falling magnet from dimensional analysis. Revista Brasileira de Ensino de Fisica. 2012;34(2):2101.1–4.
DOI  Google Scholar
9

Iniguez J, Raposo V, HernandezLopez A, Flores AG, Zazo M. Study of the conductivity of a metallic tube by analysing the damped fall of a magnet. Eur J Phys. 2004;25:593–604.
DOI  Google Scholar
10

Irvine B, Kemnetz M, Gangopadhyaya A, Ruubel T. Magnet traveling through a conducting pipe: a variation on the analytical approach. Am J Phys. 2014;82(4):273–9.
DOI  Google Scholar
11

Thottoli AK, Fayis M, Mohamed TC, Amjad T, Shameem PT, Mishab M. Study of magnet fall through conducting pipes using a data logger. SN Appl Sci. 2019;1:1–8.
DOI  Google Scholar
12

Itahashi K, Kishigi K. Practice of flipped classroom and development of teaching materials for fall motion of magnet in metal pipe. Bull Sojo Univ. 2020;45:29–34.
Google Scholar
13

Kishimoto K, Honda K. Falling motion of a neodymium magnet in an aluminum pipe. Res Rep Kanagawa Inst Tech. 2004;B28:95–7, Japanese.
Google Scholar
14

Amrani D. Determination of magnetic dipole moment of permanent disc magnet with two different methods. Phys Educ. 2015;31(1):9.1–6.
Google Scholar
15
Most read articles by the same author(s)

Barenten Suciu,
Computation of the Negative Damping Associated to the Hunting Motion of the Railway Wheelset, without Using Geometrical or Tribological Restrictions into the Model , European Journal of Engineering and Technology Research: Vol. 5 No. 2: FEBRUARY 2020 
Barenten Suciu,
Influence of the Drop Volume and Applied Magnetic Field on the Wetting Features of Waterbased Ferrofluids , European Journal of Engineering and Technology Research: Vol. 5 No. 9: SEPTEMBER 2020 
Barenten Suciu,
Sota Karimine,
Visualization of Chladni Patterns at LowFrequency Resonant and NonResonant Flexural Modes of Vibration , European Journal of Engineering and Technology Research: Vol. 9 No. 3 (2024)