Evaluation of the Damping Produced by the Motion of a Permanent Magnet inside of a Conductive Pipe

In this study, a method to evaluate the electro-magnetic 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 magnet-pipe 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.

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 electro-magnetic 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 electro-magnetic 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 force-displacement 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 thin-walled 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 north-pole and the south-pole surfaces of the magnet. In fact, flux density depends on the measurement point selected on the surface. Thus, high-class equipment, using a three-dimensional probe attached to a Gauss-meter, 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 dash-double-dot lines.

Fig. 1. Schematic view of the magnet falling inside of a conductive pipe, showing the geometrical parameters and the forces acting on the magnet.

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) $$\begin{array}{r}m\frac{dv}{dt}=mg-F_m=mg-cv,\end{array}$$

where m is the magnet mass, g = 9.8 m/s² 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) $$\begin{array}{ll}z(t=0)=0;& v(t=0)=0\end{array}$$

variation against time of the axial coordinate z (t), velocity v (t), and acceleration a (t) can be expressed as:

(3) $$\begin{array}{r}\{\begin{array}{l}z\left(t\right)={\displaystyle \frac{g}{{\overline{c}}^2}}[\overline{c}t-1+\exp (-\overline{c}t)]\\ v\left(t\right)={\displaystyle \frac{dz}{dt}}\left(t\right)={\displaystyle \frac{g}{\overline{c}}}[1-\exp (-\overline{c}t)],\\ a\left(t\right)={\displaystyle \frac{d^{2}z}{d{t}^2}}\left(t\right)=g\cdot \exp (-\overline{c}t)\end{array}\end{array}$$

where $\overline{c}=c/m$ 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) $$\begin{array}{r}\{\begin{array}{c}z\left(T\right)=L-h={\displaystyle \frac{g}{{\overline{c}}^2}}[\overline{c}T-1+\exp (-\overline{c}T)]\\ v\left(T\right)={\displaystyle \frac{g}{\overline{c}}}[1-\exp (-\overline{c}T)]\\ a\left(T\right)=g\cdot \exp (-\overline{c}T)\end{array}\end{array}$$

Note that the 1^st expression of (4) leads to the following transcendent equation in $\overline{c}$:

(5) $$\begin{array}{r}(L-h){\overline{c}}^2=g[\overline{c}T-1+\exp (-\overline{c}T)],\end{array}$$

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 $\exp (-\overline{c}T)\cong 0$ can be neglected, as will be later proved in Section IV, (5) can be fairly approximated to the following quadratic equation:

(6) $$\begin{array}{r}\frac{L-h}{g}{\overline{c}}^2-T\cdot \overline{c}+1=0.\end{array}$$

Solving (6) and selecting the appropriate larger solution, one obtains the specific damping coefficient as follows:

(7) $$\begin{array}{r}\overline{c}=\frac{g}{2(L-h)}[T+\sqrt{T^2-4\frac{L-h}{g}}].\end{array}$$

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.

Table I. Thickness of the Pipes Used in the Falling Tests

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.

Table II. Properties of the Magnets Used in the Falling Tests

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. 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. 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.

Fig. 2. Schematic view of the test rig used in the falling experiments of the permanent magnet inside of the conductive pipe.

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 shock-absorbing 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 shock-absorbing 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 $c=\overline{c}\cdot m$ 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 $\overline{c}$ and damping coefficient c, are illustrated for all the copper pipes P1–P6 in Tables III–VIII, respectively.

Table III. Results Obtained for Magnets M1–M12 Versus Pipe P1

Magnettype	Mean fallingtime, T (s)	Specific damping coefficient, $\overline{c}$ (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

Table IV. Results Obtained for Magnets M1–M12 Versus Pipe P2

Magnet type	Mean falling time, T (s)	Specific damping coefficient, $\overline{c}$ (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

Table V. Results Obtained for Magnets M1–M12 Versus Pipe P3

Magnet type	Mean falling time, T (s)	Specific damping coefficient, $\overline{c}$ (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

Table VI. Results Obtained for Magnets M1–M12 Versus Pipe P4

Magnet type	Mean falling time, T (s)	Specific damping coefficient, $\overline{c}$ (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

Table VII. Results Obtained for Magnets M1–M12 Versus Pipe P5

Magnet type	Mean falling time, T (s)	Specific damping coefficient, $\overline{c}$ (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

Table VIII. Results Obtained for Magnets M1–M12 Versus Pipe P6

Magnet type	Mean falling time, T (s)	Specific damping coefficient, $\overline{c}$ (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 $\exp (-\overline{c}T)$ of (5).

Except for the combinations of M7 with pipes P1 and P2, in which case $\exp (-\overline{c}T)\cong 2\times {10}^{-180}$ and $3\times {10}^{-285},$ for all other combinations of magnets and tubes, $\exp (-\overline{c}T)=0.$ Thus, evaluation method of the specific damping coefficient, based on (6) and (7), appears to be appropriate.

Moreover, by substituting $\exp (-\overline{c}T)=0$ 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) $$\begin{array}{lll}v\left(T\right)={\displaystyle \frac{g}{\overline{c}}}={\displaystyle \frac{mg}{c}}& ;& a\left(T\right)=0\end{array}$$

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.

Fig. 3. Variation of the magnet mass versus the magnet diameter, for a fixed magnet height of 10 mm, and versus the magnet height, for a fixed magnet diameter of 11.5 mm.

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 $\pm$4.55 mT, i.e., with a fluctuation of only $\pm$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. 4. Variation of the magnetic residual flux density versus the magnet diameter, for a fixed magnet height of 10 mm, and versus the magnet height, for a fixed magnet diameter of 11.5 mm.

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. 5. Variation of the damping coefficient versus the magnet diameter, found for a fixed magnet height and various values of the pipe thickness.

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. 6. Variation of the damping coefficient versus the magnet height, found for a fixed magnet diameter and various values of the pipe thickness.

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 saturation-like dependence of the damping coefficient on the pipe thickness.

Fig. 7. Variation of the damping coefficient versus the pipe thickness, found for a fixed magnet height and various values of the magnet diameter.

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. Variation of the damping coefficient versus the pipe thickness, found for a fixed magnet diameter and various values of the magnet height.

Fig. 8 shows also a nonlinear, saturation-like 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 electro-magnetic 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.

Fig. 9. Variation of the damping coefficient versus the clearance between the pipe and magnet, determined for a fixed magnet height and various values of the pipe thickness.

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. 1) Nonlinear, concave-like augmentation pattern against the magnet diameter,
2. 2) Nonlinear, saturation-like augmentation pattern versus the pipe thickness,
3. 3) Nonlinear, concave-like 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:

1. 4) Linear augmentation versus the magnet diameter,
2. 5) Nonlinear, saturation-like augmentation pattern versus the pipe thickness.