Fundamental considerations on introducing damping to passively magnetically stabilized rotor systems

Abstract

When designing passively magnetically stabilized rotor systems, introducing sufficient damping is key. We investigated two different dynamic systems—with one mass and two masses—to determine their theoretical optimal behavior and how they can be realized considering real-world specifications and limitations. Based on dimensionless formulations of reduced dynamic systems, we present fundamental correlations between, and the optimal choice of, physical parameters. Furthermore, we compare the two systems in terms of feasibility and efficiency of different damping methods. Our investigations used an eddy current damper and viscoelastic damping elements as exemplary damping methods for the one-mass and two-mass systems, respectively. Passive stabilization was realized by means of permanent magnetic bearings. We found that the two-mass system is preferable due to the broader range of damping possibilities.

Keywords

Passive magnetic bearing system design permanent magnetic bearing passive damping

Introduction

Passive stabilization

Contact-free support of magnetically levitated rotor systems has several advantages. The lubrication- and friction-free concept of magnetic levitation is optimal for applications in which high rotational speeds, clean-room conditions, or hermetic encapsulation of the rotor are desired. Additionally, passive magnetic bearing systems combine outstanding reliability with relatively low complexity of the bearing element (especially in comparison with active magnetic bearings). Passive stabilization can be achieved using superconducting materials,^1–3 electrodynamic effects,^4–8 or permanent magnetic bearings,^9–13 but the common problem is a lack of damping. Electrodynamic and superconducting bearings even introduce instability to the supercritical regime.⁷ Thus, several approaches have been developed to damp passive magnetic bearing systems.

Damping

Passive damping concepts keep the constructive simplicity of passively stabilized systems low, and eddy current dampers are the first choice for contact-free damping of the rotor.^14–17 The damping values of eddy current dampers are relatively low given the installation space they require, and so, in Filatov and Maslen,¹⁸ an operational amplifier circuit was introduced to improve the damping value at the cost of a lowered cut-off frequency. Another approach to damping was shown in MacHattie,¹⁹ where a needle surrounded by oil is magnetically coupled with the rotor movements. In Beams,²⁰ oil-damping was used to reduce the oscillations of a magnetically supported system. In the approach described in Fremerey et al.,²¹ the stator magnets of the permanent magnetic bearings are web-mounted so they can move radially. This free movement is damped by a special friction device. Another passive vibration control approach in rotor dynamics uses viscoelastic materials.²² In previous studies,^23–25 concepts were presented in which the stator rings of permanent magnetic bearings are directly mounted on viscoelastic elements. The authors of earlier works^26–28 introduced whole systems with permanent magnetic radial and tilt stabilization and viscoelastic damping elements.

Each damping method has drawbacks as well as advantages. However, the damping concept is not the only factor to be considered when designing a well-damped system. Several other aspects must be taken into account.

Design of passive systems

Active systems generally have an area of operation within which adjustments are possible by means of proper controller design. This also means that there is a defined range within which the system can be altered after construction. Such an adjustment is not easily possible with passively stabilized dynamic systems. Thus, to avoid having to identify the best design by trial and error, a very detailed model of the whole system (including passive bearings, rotor dynamics, damping mechanism, etc.) and a thorough understanding of the dynamic behavior are essential to perform optimizations and find the most robust design. Depending on the damping mechanism, different dynamic systems must be considered: a one-mass resonator system if eddy current dampers are used, and a two-mass system if a magnetically coupled mass is used for damping or if the whole stator is mounted on a damping device.

In the referred literature, different damping methods or innovative concepts of passively stabilized rotor systems are presented. They investigate a certain damping system or present new constructive approaches. What is missing to a great extent are investigations on a system level that help the engineer to understand and identify the important factors and relations. In this article, we present fundamental findings on the optimal choice of physical parameters (mass, stiffness, and damping ratio). These are based on dimensionless formulations of simplified dynamic systems and a comparison of examples of one-mass and two-mass resonators that considers real-world specifications and limitations. For passive stabilization, we chose permanent magnetic ring bearings. Nevertheless, most of our findings have general validity.

Thus, the main objective of this article is not to introduce new concepts or methods but to illustrate aspects which are especially important for the design of passively levitated systems. Therefore, we compare the potential of different dynamic concepts (one- and two mass) and consider their feasibility.

Dynamic systems

When two permanent magnetic ring bearings with an axial displacement (see Figure 1) are used, all radial and tilt degrees of freedom $(r, φ)$ can be stabilized passively. The axial degree of freedom (which can, in such a case, not be stabilized by permanent magnets^29,30) can be stabilized, for example, by an active magnetic bearing.³¹ For a system, as pictured in Figure 1, the radial and tilt degrees of freedom are decoupled if the center of mass of the rotor is in the middle of the two identical permanent magnetic radial bearings. Furthermore, assuming rotational symmetry and omitting gyroscopic effects, the radial behavior can be investigated by a reduced dynamic system with an overall radial stiffness and a single radial degree of freedom. The tilt behavior can be examined analogously.

Figure 1.

Basic system with two radially stable permanent magnetic bearings. The stabilization of the axial degree of freedom is not shown.

One-mass system

This simplified system, used to investigate the radial degree of freedom of a permanent magnetic stabilized rotor, is shown in Figure 2. As a contact-free damping method, an eddy current damper is assumed. As shown in Tonoli and Amati,¹⁷ if the cut-off frequency of the eddy current damper $ω_{co} = R / L$ (R—resistance, L—inductivity) is much higher than the highest frequency to be damped ( $Ω_{\max}$ for the case of unbalance excitation), the damping value d can be expected to be constant.

Figure 2.

One-mass resonator with contact-free damping.

Assuming rotational symmetry for all components, the rotor will always follow a circular orbit, and thus, only one direction of movement must be investigated. The magnitude of the harmonic movement of this simplified system corresponds to the orbit of the three-dimensional system. In such a model, a static mass unbalance can be described as a force

F (t) = e_{u} m_{r} Ω^{2} \cos (Ω t)

(1)

where $e_{u}$ is the eccentricity of the principal axis of inertia, $m_{r}$ is the mass, and $Ω$ is the angular speed of the rotor. The solution of the dynamic equation

m_{r} \overset{\cdot\cdot}{x} + d \overset{\cdot}{x} + kx = F (t)

(2)

with the force given in equation (1) can be found in standard literature. Its maximum amplitude related to the eccentricity of the principal axis due to mass unbalance is

V (η) = \frac{\hat{x}}{e_{u}} = \frac{η^{2}}{\sqrt{{(1 - η^{2})}^{2} + {(2 D η)}^{2}}}

(3)

where

ω_{r} = \sqrt{\frac{k}{m_{r}}}, η = \frac{Ω}{ω_{r}}, and D = \frac{d}{2 m_{d} ω_{r}} with m_{d} = m_{r}

(4)

With the angular speed at resonance for the undamped system $ω_{r}$ , the parameter $η$ is a normalized rotational speed, and D is a dimensionless measure of the damping. The mass of the damped body $m_{d}$ is the rotor’s mass. $\hat{x} (t)$ is the magnitude of the general solution of the linear system equation (2) with harmonic excitation

x (t) = \hat{x} \cos (Ω t + γ)

(5)

The transfer function of the one-mass resonator (equation (3)) is shown in Figure 3 for different values of the damping ratio D. Another kind of excitation to be considered when designing a passively levitated system is an external vibration

ξ (t) = \hat{ξ} \cos (Ω t)

(6)

as illustrated in Figure 4. In Krämer,³² it was shown that

V_{ξ} (η) = \frac{\hat{x}}{\hat{ξ}} = \frac{\hat{x}}{e_{u}} = equation (3)

(7)

Figure 3.

Dimensionless gain function $V (η)$ of the one-mass resonator for different values of the damping ratio D.

Figure 4.

External excitation of a one-mass resonator.

From this theoretical point of view, it becomes clear that robustness against internal and external disturbances can be achieved with a sufficiently high value of D. Thus, the limiting factor of this concept is the damping ratio achievable with the contact-free eddy current–based dampers. An example from the literature is given in the “Design examples” section.

Two-mass system

A simplified two-mass system where the stator is mounted on a damping device is shown in Figure 5.

Figure 5.

Two-mass system where the stator is mounted on a damping device.

From the equations of motion for $ξ (t) = 0$

\begin{matrix} m_{r} ({\overset{\cdot\cdot}{x}}_{s} + \overset{\cdot\cdot}{x}) + kx = F (t) \\ m_{s} {\overset{\cdot\cdot}{x}}_{s} + d {\overset{\cdot}{x}}_{s} + k_{d} x_{s} - kx = 0 \end{matrix}

(8)

and the unbalance excitation as defined in equation (1), the transfer function

V (η) = \frac{\hat{x}}{e_{u}} = \sqrt{η^{4} \frac{{(r^{2} - η^{2})}^{2} + {(2 D η)}^{2}}{{(η^{2} (1 + μ - η^{2}) + r^{2} (η^{2} - 1))}^{2} + {(2 D η (η^{2} - 1))}^{2}}}

(9)

can be deduced. In addition to D and $η$ , the two dimensionless parameters $μ$ and r are introduced. $μ$ puts the rotor mass in relation to the stator mass, and r is defined as the ratio of the two critical angular speeds

μ = \frac{m_{r}}{m_{s}}, r = \frac{ω_{s}}{ω_{r}} = \sqrt{μ \frac{k_{d}}{k}} with ω_{s} = \sqrt{\frac{k_{d}}{m_{s}}}

(10)

Since the damped mass is now the stator mass, the damping ratio D (cf. equation (4)) is now defined using the stator mass, $m_{d} = m_{s}$ . Using the Routh–Hurwitz criterion, it can be shown that the System (equation (8)) is asymptotic stable for $d > 0$ . The influence of each dimensionless parameter is shown in Figure 6. For $m_{r} = m_{s}$ and $k = k_{d}$ ( $μ = 1$ and $r = 1$ ), the transfer function is shown in Figure 6(a). The main difference from the one-mass behavior is that basically two resonance peaks are present, one below and one above $η = 1$ . With rising values of the damping ratio D, the peak values decrease, and at $D \approx 0.85$ , the peaks vanish. If D is further increased, a single peak around $η \approx 1$ emerges because at high damping values, the stator is not able to oscillate and the dynamic behavior of the two-mass oscillator tends toward the undamped one-mass system. Thus, a non-trivial optimal value for the damping ratio can be found. Figure 6(b) plots the transmission behavior when the damping ratio is set to $D = 0.1$ and parameter r is varied. It can be seen that r weights the peak values of the two resonances. If the maximum peak over the whole frequency range is to be minimized, an optimal value can also be found for r, namely, where both peaks have the same magnitude. For the case shown, this is at $r \approx 1.2$ . How the optimal values of r and D correlate with the mass ratio $μ$ is given by the empirical equations (11) and (12) (the relations were identified by means of the HeuristicLab software, available at http://dev.heuristiclab.com)

r = 4.33 - \frac{1}{0.3 + 0.02 μ}

(11)

D = 0.05 + 0.31 \sqrt{μ} + 0.394 μ

(12)

Figure 6.

Transmission behavior $V = \hat{x} / e_{u}$ , equation (9), as a function of the dimensionless parameters $η, μ, r, and D$ : (a) $μ = 1, r = 1$ ; (b) $μ = 1, D = 0.1$ ; (c) $r = equation (11), D = equation (12)$ ; and (d) $r = 1, D = 0.1$ .

Note that these equations minimize $V (η)$ only for mass unbalance excitations. The optimal behavior as a function of the mass ratio $μ$ is shown in Figure 6(c). It can be seen that with rising $μ$ , a better design can be achieved. This means that the stator should be lightweight compared to the magnetically levitated rotor. For a vanishing stator mass and a very high stiffness k of the permanent magnetic bearing, that is, $μ$ is large and $r / \sqrt{μ}$ is small, and the transfer function equation (9) becomes equation (3)—the transfer function of the one-mass resonator. As illustrated in Figure 6(d), the mass ratio $μ$ for fixed values of r and D influences mainly the distance between the two resonance peaks. That a small value of $μ$ performs better in the case considered is due to the choice of $r = 1$ and $D = 0.1$ . Calculations based on equations (11) and (12) show that the selected parameters of r and D are close to the optimal values for small mass ratios but far from optimal for large values of $μ$ .

External excitation of the two-mass resonator leads to a different behavior than unbalance excitation. From the corresponding equations of motion

\begin{matrix} m_{r} (\overset{\cdot\cdot}{ξ} + {\overset{\cdot\cdot}{x}}_{s} + \overset{\cdot\cdot}{x}) + kx = 0 and \\ m_{s} (\overset{\cdot\cdot}{ξ} + {\overset{\cdot\cdot}{x}}_{s}) + d {\overset{\cdot}{x}}_{s} + k_{d} x_{s} - kx = 0 \end{matrix}

(13)

the transfer function

V_{ξ} (η) = \frac{\hat{x}}{\hat{ξ}} = \sqrt{η^{4} \frac{r^{4} + 4 D^{2} η^{2}}{{(η^{2} (1 + μ - η^{2}) + r^{2} (η^{2} - 1))}^{2} + {(2 D η (η^{2} - 1))}^{2}}}

(14)

can be calculated. Based on equations (11) and (12), the transmission behavior due to external excitation is illustrated in Figure 7 together with the unbalance response from Figure 6(c). Figure 7 shows that although the relations used for r and D are not optimized for external excitation, adjusted relations can be expected to provide only marginal improvements because the resonance peaks are already almost equal in height. In fact, when optimizing r and D for $μ = 0.1$ for base-point excitation, $V_{ξ}$ always remains above $V_{ξ} > 11$ . From Figure 7, it also becomes apparent that especially for small values of $μ$ , external excitation leads to poorer system behavior than mass unbalance of the rotor. Again, increasing values of $μ$ lead to significant improvements in the theoretically reachable optimum.

Figure 7.

Response function for unbalance (thin) and external excitation (thick) for different values of $μ$ and parameters r and D calculated using equations (11) and (12).

In the considerations above, only the relative displacement between stator and rotor was taken into account. Depending on the specifications, also the movement of the stator $x_{s}$ or the absolute displacement of the rotor in the inertial frame, $x_{I} = x_{s} + x$ , could be optimized. Nevertheless, it turned out that if the relative displacement x behaves well, also the stator movements remain within reasonable limits.

Design examples

The previous section showed that, generally, a proper choice of design parameters leads to robust dynamic behavior of the two systems investigated. The key parameter of the two-mass system is—aside from the damping ratio—the mass ratio $μ$ . Considered purely theoretically, optimum performance should be achieved by a one-mass system. The question now arises to what extent specifications and practical feasibility limit the ability to reach the theoretical optima. To illustrate this problem, design examples with an example specification are presented in this section.

A maximum mass unbalance of grade G6.3 corresponding to DIN ISO 1940-1 is defined as excitation. This means that the product of eccentricity $e_{u}$ and angular speed $Ω_{N}$ is $e_{u} Ω_{N} \leq 6.3 \times 10^{- 3} m / s$ . We assume a nominal speed of $n_{N} = 10, 000 r / \min$ . Thus, the maximum eccentricity is $e_{u, \max} = 6.016 μ m$ . From the construction, a mass of the rotor of $m_{r} = 0.5 kg$ and a mechanical air gap between stator and rotor of $h = 0.5 mm$ are given. A further requirement is that if the system is arranged horizontally, the static deflection must be less than a fifth of the air gap. This means that

\frac{g m_{r}}{k} \leq \frac{h}{5} \Rightarrow k \geq 5 g \frac{m_{r}}{h} = 49, 050 N / m

(15)

with the gravitational acceleration $g = 9.81 m / s^{2}$ . First, we design a one-mass system where the rotor does not touch the stator at resonance $(\hat{x} |_{η = 1} < h)$ , with maximum unbalance and a minimum absolute damping value d. All specifications are summarized in Table 1.

Table 1.

Summarized specifications of the design example.

$n_{N} = 10, 000 r / \min$	$e_{u} \leq 6.016 μ m$
$m_{r} = 0.5 kg$	$k \geq 49, 050 N / m$
$h = 0.5 mm$	$d \dots$ minimum

One-mass system

For better visualization, a new transfer function $V_{h}$ is defined, where the relative displacement of the rotor $\hat{x}$ is related to the air gap dimension h instead of the eccentricity $e_{u}$

V_{h} (η) = \frac{\hat{x}}{h} = \underset{equation (3)}{\underset{︸}{V (η)}} \frac{e_{u, \max}}{h}

(16)

The problem is solved assuming that the damped resonance lies exactly at $ω_{r}$ . The validity of this assumption, especially at low grades of damping, can be seen from Figure 2. The objective is then

V_{h} (η = 1) = \frac{e_{u, \max}}{2 Dh} < 1 \Rightarrow D > \frac{e_{u, \max}}{2 h} = 0.00602 = D_{\min}

(17)

For $D > 0.00602$ , the rotor does not touch the stator, even when running at resonance frequency. To find a realization, k and d must be chosen such that the desired grade of damping is reached. From equation (4), the relation $d = 2 D \sqrt{m_{r} k}$ can be obtained. If d is to be minimized, then k also has to be minimized. From the specification for the static sag of a horizontal arrangement, the condition $k > 49, 050 N m$ was calculated. Thus

\begin{array}{l} d > 2 D_{m i n} \sqrt{m_{r} k} = 2 \cdot 0.00602 \sqrt{0.5 \cdot 49, 050} \\ = 1.8855 N s / m \end{array}

Note the strong correlation of the result with the, in essence, arbitrarily defined condition of the static sag. Let us assume that no requirements are defined for the static sag and k could be chosen without any restrictions. Thus, for a very small value of k, also small values of d would be sufficient to realize the same damping ratio D. If the system is then mounted horizontally, causing the radial direction to align with the gravitational field of the earth, the rotor is likely to rest in the touchdown bearings. If, however, the static sag is specified to be very small, an accordingly high value of d is necessary. For eddy current dampers, this means more mass and installation space. Specifications generally have significant influence on the achievable designs, and thus, additional conditions must be defined in order not to end up with trivial solutions that mostly do not work in practice. Specifying overly conservative requirements commonly leads to realizations which are far from an optimal solution for practical use.

Implementation example

For a system as pictured in Figure 1, one permanent magnetic bearing unit as shown in Figure 8 could be realized. To prevent damage of the permanent magnetic rings, the air gap of the bearing is chosen to be slightly larger than the desired mechanical gap. Thus, for the bearing, an air gap of $h_{b} = 0.7 mm$ is chosen. Using two such bearings, a radial stiffness of $k = 2 \cdot 24, 813 N / m = 49, 626 N / m$ is achieved. The stiffness was calculated by the method presented in Marth et al.³³ In Bender and Post,³⁴ an eddy current damper for use in a flywheel energy storage system was presented. With overall dimensions of about $80 mm$ in diameter and $25 mm$ in the axial direction, an almost constant damping ratio of $1.5 N s / m$ can be established. Note that this design has a relatively large air gap between the exciting magnets and the copper disk where the eddy currents are induced. Nevertheless, this example should provide an idea of the capabilities of eddy current dampers. The dashed box in Figure 8 indicates the construction space of this eddy current damper.

Figure 8.

Proportional drawing of a bearing unit which could be used in our example; dimensions: $a = 2 mm, b = 2.5 mm, h_{b} = 0.7 mm, r_{h} = 13 mm$ ; magnet material: $B_{r} = 1.0 T$ ; and resulting stiffness of the bearing unit: $k_{r} = 24, 813 N / m$ .

Two-mass system

Our next aim is to find an optimal design for the two-mass resonator with damped stator using the same specifications as for the one-mass system. Furthermore, the absolute damping value from the optimization of the one-mass system, $d = 1.8855 Ns / m$ , is assumed. Again, the objective is to minimize the resonance peaks, where the design parameters are $k_{d}$ and $m_{s}$ . As for the one-mass system, we use the modified transfer function

V_{h} (η) = \frac{\hat{x}}{h} = \underset{equation (9)}{\underset{︸}{V (η)}} \frac{e_{u, \max}}{h}

(18)

For a structured approach, the findings from the “Dynamic systems” section are used. If the resonance peaks are to stay below a given limit over the whole frequency range, both peaks should be weighted equally (cf. Figure 6(b)). This is the case if equation (11) is fulfilled, which can be written in the following form

\frac{k_{d}}{k} = \frac{{(0.0866 m_{r} + 0.299 m_{s})}^{2}}{{(0.02 m_{r} + 0.3 m_{s})}^{2}} \frac{m_{s}}{m_{r}} \propto \frac{1}{μ}

(19)

Here, $k_{d}$ and $m_{s}$ can still be chosen freely. The bearing stiffness k is limited by the specification of the static sag. From the general investigations (Figure 6(c)), it follows that $μ$ should be large. This means that the mass of the stator $m_{s}$ must be small. From the definition of the damping ratio of the two-mass system, equation (4) with $m_{d} = m_{s}$

D = \frac{d}{2 m_{s} \sqrt{\frac{k}{m_{r}}}}

it follows that D reaches a maximum if k is minimized for given values of d, $m_{r}$ , and $m_{s}$ . Thus, for any value of $m_{s}$ , the bearing stiffness is chosen to be $k = 49, 050 N / m$ . Finally, with equation (12), the optimally tuned support stiffness $k_{d}$ can be calculated as a function of the stator mass $m_{s}$ . The corresponding unbalance response for different values of $m_{s}$ is shown in Figure 9. It can be seen that for a mass ratio of $μ \geq 1.67$ , the maximum peak is lower than that of the one-mass system. In particular, the second resonance peak moves significantly toward a higher frequency value—which is dangerously close to the nominal speed region in the example shown. The adjusted values for $k_{d}$ used in Figure 9 are given in Table 2. The minimum stator mass commonly emerges from the construction of the system. The restriction on the stiffness of the mounting element, $k_{d}$ , is similar to that on the permanent magnetic ring (i.e. maximum stator displacement). The major difference from the one-mass system is that the damping device need not work in a contactless manner. Thus, a broad range of damping concepts with substantially higher damping capabilities can be used.

Figure 9.

Response of the two-mass resonator for different values of $m_{s}$ and adjusted values of $k_{d}$ using equation (19): $k = 49, 050 N / m$ , $m_{r} = 0.5 kg$ , and $d = 1.8855 N s / m$ . For comparison, the response of the one-mass system is also shown.

Table 2.

Adjusted values for $k_{d}$ as used for Figure 9.

$m_{s}$ (kg)	$μ$	$\frac{k_{d}}{k}$	$k_{d}$ $(N / m)$
0.1	5	0.6698	32,853
0.3	1.667	1.0613	52,059
0.5	1	1.4520	71,222
0.7	0.714	1.8457	90,529

Implementation example

In this example, the stator is mounted on an annular element with rectangular cross section made of viscoelastic material, as shown in Figure 10. The radial stiffness of such an element can easily be calculated assuming pure shear leading to

k_{d} (f) = \frac{E (f)}{2 (1 + ν)} d_{m} π \frac{b_{d}}{a_{d}}

(20)

with the frequency-dependent Young’s modulus $E (f)$ , Poisson’s ratio $ν$ , and the geometric parameters, as indicated in Figure 10. With the frequency-dependent loss factor of the material $\tan (δ (f))$ , the damping value of the element can be calculated by Bormann³⁵

d (f) = \frac{\tan (δ (f)) k_{d} (f)}{2 π f}

(21)

Figure 10.

Proportional drawing of the system mounted on a viscoelastic damping element; dimensions of the damping element: $a_{d} = 4 mm$ , $b_{d} = 6 mm$ , and $d_{m} = 48 mm$ .

Modeling the frequency-dependent behavior in the dynamic system is possible, for example, using a Prony series representation of the damping element.³⁶ However, obtaining the necessary master curves of the viscoelastic material is difficult. For this investigation, we keep the simple spring–damper model and calculate the parameters for three different excitation frequencies as given in Table 3. For the Sylomer^® SR55³⁷ material used, Poisson’s ratio of $ν = 0.5$ is assumed. From Table 3, we can see that much higher damping values can be achieved than with eddy current dampers. The dynamic behavior of our system using this viscoelastic element is shown in Figure 11, using the material parameters for 1, 50, and $100 Hz$ . To facilitate choosing the significant curve, the top abscissa of Figure 11 shows the absolute rotor frequency. Thus, the real behavior will be a combination of the 50-Hz curve and the 100-Hz curve, essentially at the first and second resonance peaks, respectively.

Table 3.

Frequency-dependent parameters of the Sylomer^® SR55 material $(E, \tan δ)$ and the final damping element $(k_{d}, d)$ .

f (Hz)	E $(N / m m^{2})$	$\tan δ$	$k_{d}$ $(N / m)$	d $(N s / m)$
1	0.47	0.11	35,437	1254.00
50	0.89	0.21	67,104	47.88
100	0.95	0.25	71,628	28.50

Figure 11.

Transfer behavior with viscoelastic material damping (material: Sylomer^® SR55) and $μ = 1$ .

An optimization for a system similar to that in Figure 2 but considering all passively stabilized degrees of freedom, including gyroscopic effects, and using viscoelastic damping was presented in Marth et al.³⁸

Conclusion

The design process of passively stabilized systems should be supported by as much knowledge as possible to obtain a robust and efficient system. A fundamental understanding of the relevant design parameters and their effects on system behavior is therefore key. As demonstrated in this article, in addition to the theoretical relations, also the practical feasibility of the chosen concept must be taken into account from the beginning of the design process. The design examples have shown that the two-mass system is preferable to the one-mass system, essentially due to the broader range of possible damping. Furthermore, the effect of proper system specifications on the final design was emphasized. If all aspects presented are borne in mind when starting the design of a passively stabilized rotor system, a detailed optimization is likely to be successful.

Footnotes

Academic Editor: Anand Thite

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was in part supported by the Austrian COMET-K2 program of the Linz Center of Mechatronics (LCM) and was funded by the Austrian federal government and the federal state of Upper Austria.

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