A vibration absorber with variable eigenfrequency for turboprop aircraft

Aerospace Science and Technology 13 (2009) 165–171

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A vibration absorber with variable eigenfrequency for turboprop aircraft S. Keye a,∗ , R. Keimer b , S. Homann c a b c

German Aerospace Center (DLR), Institute for Aerodynamics and Flow Technology, Lilienthalplatz 7, D-38108 Braunschweig, Germany German Aerospace Center (DLR), Institute for Composite Structures and Adaptive Systems, Lilienthalplatz 7, D-38108 Braunschweig, Germany Hauni Maschinenbau AG, Kurt-A.-Körber-Chaussee 8-32, D-21033 Hamburg, Germany

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 13 October 2006 Received in revised form 16 August 2007 Accepted 13 October 2008 Available online 25 October 2008

Typical turboprop noise spectra exhibit a series of characteristic peaks which are directly related to the product of propeller rpm and number of propeller blades. These blade passage frequencies contribute significantly to the overall sound pressure level both outside and inside the aircraft. Their contribution to cabin noise is usually reduced by appropriately adjusted mass dampers. However, since the engine rpm varies for different flight stages, any fixed eigenfrequency absorber will merely be a sub-optimal compromise. The Tunable Vibration Absorber (TVA) introduced in this article has a variable resonant frequency which enables an adaptation to different flight phases providing largely improved performance. Frequency tuning is achieved through a piezo-electric stack actuator, which applies a pressure force to a pair of leaf springs thus reducing their effective bending stiffness. Among the main advantages of this particular approach are a static control signal and low power consumption. To enable a light-weight construction the components which generate the pressure loading are incorporated into the oscillating mass. The TVA allows to cover a wide frequency range using only a single device. Additionally, it features damping control capability and optional active multi-mode operation. Structural-acoustic simulations have indicated a noise reduction potential of approximately 10 dB. This article gives a short overview of different tuneable vibration absorber concepts, lines out the theoretical background of the proposed approach, discusses the general components layout and describes the experimental verification of a prototype TVA for the Airbus A400M. © 2008 Elsevier Masson SAS. All rights reserved.

Keywords: Mass damper Vibration absorber Tuneability

1. Introduction Most leading text books on mechanical vibrations discuss the basic equations of dynamic vibration absorbers to some extend, e.g. [13,14,21,31,36]. Among the pioneering publications providing an in-depth theoretical treatment are those by Ormondroyd and Den Hartog [23] and Den Hartog [9]. For linear dynamic vibration absorbers a closed theory is available, but due to the large number of system parameters and varying technical applications with different requirements no unique solution exists. Generally, a significant influence of damping on the vibration reduction performance is observed. The problem of attaching a vibration absorber to a discrete multi-degree-of-freedom or continuous structure has been outlined in many papers and monographs by Bishop and Welbourn [3], Warburton [37], Hunt [11], Gerasch and Natke [22], Snowdon [32], Frik [6], Korenev and Rabinovic [18] and Aida et al. [1] to name but a few. Applications to absorbers subjected

*

Corresponding author. E-mail address: [email protected] (S. Keye).

1270-9638/$ – see front matter doi:10.1016/j.ast.2008.10.001

© 2008

Elsevier Masson SAS. All rights reserved.

to random loads have been described by Wirsching and Campbell [38], Hunt and Nissen [12] or Gupta and Chandraskearan [8]. Nonlinear vibration absorbers have been investigated by Kolovsky [17], Kauderer [15], Pipes [25], Roberson [27] and Korenev and Rabinovic [19]. Additionally, some work on auto-parametric vibration absorbers has been published by Sevin [30] or Haxton and Barr [10]. Passive vibration absorbers have been successful used for many years in civil engineering for protecting tall buildings, mainly in Japan and the United States, from wind and earthquake loads [16,28]. Vibrations of power lines are reduced by the well-known Stockbridge-Damper [33,34]. In mechanical engineering much work during the 1930s to 1950s was dedicated to reducing torsional vibrations of engine crankshafts [4,20,24,35]. These systems can by considered to be semi-active, because their construction caused the absorber frequency to change with the shaft rotation frequency. In recent years various active vibration absorber systems have been developed for both fixed-wing aircraft and helicopter applications. Gardonio [7] provides an extensive overview of noise excitation mechanisms including both active and passive solution

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approaches and control algorithms and shortly illustrates some applications for tunable vibration absorbers. Notable contributions were also made by von Flotow [5] who describes various concepts of mostly motor-driven tunable absorbers generally featuring a large number of mechanical components which require precise and therefore costly manufacturing and by design are subject to wear. The disadvantage of most current systems is the large amount of energy needed to control the absorber resonant frequency. Additionally, application-related problems like manufacturing tolerances, backlash, mechanical wear or the influence of operational temperature variations are generally not covered in the literature.

and the maximum bending stiffness is obtained for P → 0. In this case the system acts like a conventional beam which is clamped at both sides with bending stiffness given by c ( P → 0) = 12 ·

EI l3

.

(6)

2.2. Eigenfrequency To find a relation between bending stiffness and eigenfrequency an elastic one-degree-of-freedom system consisting of a weightless spring with stiffness k and an attached mass m, Fig. 2, is considered. This system has a natural frequency [13]



k

2. Theoretical background

ω0 =

The TVA is basically regarded as a one-degree-of-freedom system consisting of a mass attached to the tip of a cantilever beam. Resonant frequency variation is achieved by changing the beams effective bending stiffness by means of an axial compressive force.

Assuming the TVAs leaf spring to be weightless and introducing a point mass m in x = l one obtains a mechanically equivalent elastic system where the eigenfrequency is derived from Eq. (7) by replacing the spring stiffness k with the beam bending stiffness c ( P )

2.1. Bending stiffness

ω0 ( P ) =

In order to derive a relation between bending stiffness and compressive force the system shown in Fig. 1 is considered. An elastic beam of length L which is fixed at the left end and axially guided at the right end, cf. Euler’s fourth buckling mode (boundary condition: fixed/fixed), is subjected to an axial compressive force P and a transverse force Q at x = L /2. The deflection perpendicular to the beam axis [2,26] is given by Q

y (x) =

kP

      kl − cos(kx) − 1 · tan + sin(kx) − kx 2

(1)

with



k=

P

m

(7)

.



c( P ) m

(8)

.

Using Eq. (6) the highest eigenfrequency becomes



ω0,max = 12

EI ml3

.

The principal relation of eigenfrequency and compressive force is plotted in Fig. 3. With regard to the later technical implementation as a tuneable vibration absorber the system shall further on be supported in x = l and the mass shall be equally distributed to the free ends, Fig. 4. This allows for fast and easy system integration as only a single attachment point is needed on the aircraft like it is the case with conventional tuned mass dampers. Additionally, the components needed to apply the compression force become part of the

(2)

EI

where E is the modulus of elasticity or Young’s modulus and I is the second moment of inertia of the beam cross-section. The largest deflection at x = l is u=



Q

 

2 tan

kP

kl 2



− kl

(3) Fig. 2. One-degree-of-freedom system.

and the resulting bending stiffness becomes Q

c( P ) =

u

=

kP 2 tan( kl2 ) − kl

.

(4)

Eq. (4) describes a decrease in bending stiffness as the compressive load is increased. The first root is at

 P=

π l

2 EI

(5)

Fig. 1. Clamped beam under compressive load.

(9)

Fig. 3. Relation of eigenfrequency and compressive force.

S. Keye et al. / Aerospace Science and Technology 13 (2009) 165–171

167

Fig. 4. Basic design of tuneable vibration absorber (TVA).

oscillating mass thus reducing the dead weight. The new system is mechanically equivalent to the one sketched in Fig. 1. As a result Eqs. (4) to (9) remain valid. 3. Components layout The TVAs structural dynamic behavior is specified through its resonant frequency range, Eq. (8), and the relation between base point force F and excitation amplitude u 0 given by the product of mass and quality factor m·q=m

u u0

=

F u 0 ω02

.

(10)

Fig. 5. Axial tip deflection.

In order to derive the primary system parameters of the governing one-degree-of-freedom system, i.e. mass and stiffness, the excitation amplitude must be known and a consistent estimate on the amount of damping in the ultimate TVA structure is needed. Now the oscillating mass m=

F u ω02

(11)

and the beams bending stiffness at the upper frequency limit c = mω02,max

(12)

are obtained. Additionally, the compression force required to tune the system down to the lower frequency limit is needed. Since Eq. (4) cannot be solved for P the value where c ( P ) = mω02,min

(13)

must be computed. 3.1. Leaf spring The leaf spring element comprises the TVAs resonant frequency tuning feature and therefore is regarded as the systems core component. The geometric dimensions and material have to be chosen such that both the maximum bending stiffness defined by Eq. (9) and the required effective stiffness variation, Eqs. (4) and (13), are achieved. To reduce the number of variables a rectangular outline with constant cross-section will be considered further on. The resulting unknown design parameters are length, width, thickness and Young’s modulus. To derive some preliminary information on how to choose these parameters their relation to other system components will be studied in this section. It will be shown that in some cases contradictory requirements are found for an individual system parameter and acquiring a working compromise is the key issue with designing a TVA complying to the manufacturerspecified system parameters. 3.1.1. Leaf spring length First, the relation between the leaf spring length and the compressive force incorporated in Eq. (4) is investigated. Computing P while varying l it can be shown that for any given frequency range ω0 ( P ) = const. P ∝ l.

(14)

Fig. 6. Axial tip deflection as a function of leaf spring length.

This implies that the leaf spring length should be chosen as short as possible in order to minimize the compression force required for frequency tuning. Next, in a working system the leaf spring will be subjected to finite transverse deflection amplitudes as opposed to the nondeformed state assumed in theory. The tip deflection results in an ‘S’-shaped deformation which is associated with an axial tip deflection lgeo. , Fig. 5. For a specific value of u this is a function of leaf spring length, Fig. 6. Given the typical free stroke of piezo-electric stack actuators being in the order of some tens of micrometers it becomes obvious that from these geometric considerations the leaf spring should be made as long as possible. Finally, the compressive load P leads to an additional, elastic axial tip deflection component

lel. = εax · l =

σ E

·l =

P AE

·l

(15)

where A denotes the leaf springs cross-section. Because of Eq. (14)

lel. ∝

l2 AE

,

(16)

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S. Keye et al. / Aerospace Science and Technology 13 (2009) 165–171

Fig. 7. Tuneable vibration absorber with tension loop.

i.e. to reduce losses due to elastic deformation a short leaf spring will be beneficial. 3.1.2. Leaf spring width and height Eq. (15) reveals that the elastic tip deflection component also is a function of the leaf springs cross-section area A. To examine this in more detail Eq. (6) is used and solved for b and h, respectively. Introducing in Eq. (15) and keeping in mind that P ∝ l yields

lel. ∝ h2

(17)

and

lel. ∝ b−2/3 .

(18)

This suggests that for a given leaf spring length the corresponding second moment of inertia should be realized through a wide rather than a high cross-section. If needed, e.g. to withstand the high compression forces usually associated with covering a wide frequency range, multiple leaf springs can be arranged in a stack of two or more. This increases the effective width without excessively enlarging integration space. Additionally, the stacked arrangement helps to force the leaf spring into the desired ‘S’-shape. 3.2. Tension loop Up to now no details have been provided how the compressive forces are introduced into the leaf springs. It was mentioned earlier that in order to enable a light-weight construction the loading mechanism will be part of the oscillating mass. Therefore, it is not possible to simply ground the actuator to the aircraft base structure. Instead, a tension loop is used to sustain the outward facing reaction forces, Fig. 7. The design is much less involved than in case of the leaf spring. Two main issues, both arising from the actuation devices small stroke, need to be taken into account: (1) The axial stiffness must be sufficiently high. For given lengthwise dimensions of leaf spring and actuator stack axial stiffness is determined by Young’s modulus, i.e. choice of material, and cross-section area. (2) The thermal expansions of tension loop and inner components must be matched. This subject will be addressed in Section 3.4. 3.3. Actuation Within the scope of the general components layout the piezoelectric actuation device shall for now be characterized by its maximum free stroke and blocking force. Since the number, type and configuration of actuators largely depend on the desired operating conditions, i.e. resonant frequency, frequency tuning range, oscillating mass, vibration amplitude and leaf spring geometry, only the basic procedure is illustrated here. The operating conditions determine a required compressive force P req. and the total elongation ltot. = lgeo. + lel. as described previously within this section. These define a working

Fig. 8. Actuator selection.

point, Fig. 8, and an appropriate actuation device must be selected such that the specified working point falls within the actuators range of operation. In cases where no solitary actuator type provides the necessary performance a parallel configuration of actuators enables to increase the blocking force while a serial arrangement provides larger the free stroke. 3.4. Thermal expansion considerations With the dimensions and tuning forces usually associated with representative aircraft applications both the leaf spring and tension loop will most likely have to be manufactured from carbon fiber to meet the high demands regarding axial stiffness and low weight. In a realistic design the inner main components will be supplemented by additional elements needed to hold actuator and leaf springs in place, transmit compression forces from the brittle actuator stacks to leaf springs and tension loop and to adjust the axial preload. The choice of materials is primarily governed by functionality and manufacturing issues and carbon fiber is less likely to be used for these components. As a result a difference in coefficients of thermal expansion (CTE) between the tension loop and the elements in the inner load string occurs. For realistic TVA dimensions and commonly required temperature ranges these differences in thermal expansion easily exceed the actuators free stroke. Therefore, it will be necessary to balance the inner and outer CTEs to ensure full tuning capability throughout the entire operation temperature range. From the CTE values listed in Table 1 it becomes clear that even with low-CTE materials like Invar® it is not possible to match the collective CTE of the inner components to the negative tension loop CTE. Consequently, the tension loops CTE must be increased. Unfortunately, all alternative fiber materials with positive CTE have a much lower modulus of elasticity and higher density than carbon fiber causing problems relate to both axial stiffness and mass. A better approach is to combine different fibers such that their

S. Keye et al. / Aerospace Science and Technology 13 (2009) 165–171

Table 1 Coefficients of thermal expansion for various construction materials. Material

169

Table 2 Design-related manufacturer requirements.

Density [g/cm3 ]

Young’s modulus [GPa]

CTE [10−6 /K]

Aluminium Dispal® Steel Invar® PZT

2.70 2.70 7.83–7.85 8.00 ∼7.0

70.6 100 204–208 140–150 ∼40

+23.8 +12.0 +11.1 +1.7–+2.0 −5–+10

Carbon (HM) Glass Aramid (HM) Basalt

1.80–1.96 2.50–2.60 1.45 2.75

300–500 73–87 127 89

−1.2–−1.5 +4.0–+5.3 −4.1 +8.0

Property

Value

Resonant frequency range Base point force Oscillating mass Quality factor

87.3–97.4 Hz 150 N 0.90 kg 150

Overall dimensions Total massa Electrical power consumption Operation temperature range Service life

280 × 70 × 100 mm3 1.08 kg <1 W −40 ◦ C–+70 ◦ C 30000 flight hours

a

Including support, housing and cables.

Along with the appropriate choice of materials for the internal components using a carbon/basalt compound enables to almost completely compensate the difference in CTE between inner components and tension loop while maintaining a high stiffness and low density. This ensures full frequency tuning capability over the entire operation temperature range. 4. Demonstrator

Fig. 9. Computation of combined coefficient of thermal expansion (CTE).

desired properties – high modulus of elasticity and CTE – are preserved and the unwanted eliminated. With this in mind the noticeably high CTE of 8.0 · 10−6 /K suggests basalt fibers as a promising partner for a compound system with carbon. To optimize the combined coefficient of thermal expansion, Young’s modulus, density, etc. parallel unidirectional strands of carbon and basalt are mixed in varying volume fractions. For strands of equal length the joint modulus of elasticity is given by E eff. = η · E b + (1 − η) · E c

(19)

where

η = A b / A tot.

(20)

is the ratio of basalt cross-section to total cross-section. Computation of the combined coefficient of thermal expansion requires to take into account the different fiber stiffnesses, Fig. 9. This leads to

αeff. =

ηαb E b + (1 − η)αc E c E eff.

.

(21)

Eqs. (19) and (21) are plotted in Fig. 10. Symbols denote those cross-section ratios which can be manufactured from integer numbers of strands. Density is determined in analogy to Eq. (19).

Following the layout procedure described in Section 3 a prototype TVA was designed and built in order to experimentally validate the analytical and finite-element models and develop a system controller. The structural layout is based on realistic operating conditions in the Airbus A400M military aircraft and takes into account specific technical requirements regarding system integration, operating conditions and safety issues as defined in EUROCAE ED14/RTCA DO160 as well as all aviation standards relevant to certification for aircraft application. Some requirements relating to the TVA’s structural design are listed in Table 2. The demonstrator was developed in close cooperation with OHB System AG, Bremen, Germany. More details on engineering and manufacturing issues are provided in [29]. Fig. 11 shows the final prototype. The leaf springs are made from single layers of unidirectional CN-60 carbon fiber with top and bottom layers of T300 fabric oriented at ±45◦ . This ensures both high axial stiffness and strength. The package is held in position by the central mounting bracket and two clevises which also transfer the actuators axial compression force to the leaf spring tips. Additional glass fiber layers are used to further reinforce the areas of clamping. Stack actuator and preload mechanism are protected from transverse loads through carbon fiber bushings with ±60◦ fiber orientation featuring a low axial stiffness and high

Fig. 10. CTE and Young’s modulus for carbon/basalt compound system.

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S. Keye et al. / Aerospace Science and Technology 13 (2009) 165–171

Fig. 11. TVA demonstrator for validation and controller development. Table 3 Frequency deviations caused by manufacturing tolerances and material parameter uncertainties. Parameter

Tolerance

Frequency deviation

Leaf spring length Leaf spring width Leaf spring height Young’s modulus Density

0.10 mm 0.10 mm 0.10 mm 30.0 GPa 0.03 kg/m3

0.19 Hz 0.08 Hz 10.5 Hz 1.60 Hz 0.32 Hz

shear stiffness. To facilitate a high axial stiffness the tension loop is also built from unidirectional high-modulus CN-60 carbon fiber. Two pressure heads hold the belt and transmit the actuator force. 4.1. Compensation of manufacturing tolerances Given that the TVA’s highest eigenfrequency is obtained for P → 0, Fig. 3, frequency tuning using an axial compression force, either from the preloading mechanism or the piezoelectric actuation device, will always result in lowering the eigenfrequency. Therefore, in order to enable the elimination of influences from manufacturing uncertainties the demonstrator must be designed for a higher than specified maximum frequency which after assembly is tuned down to the desired value using the preloading mechanism. Based on an eigenfrequency sensitivity analysis with respect to the relevant design parameters, Eq. (9), and an assessment of tolerances related to manufacturing and material parameters, Table 3, a safety margin of approximately 10.6 Hz was determined. As a result the TVA demonstrator has been designed for a frequency range of 98 Hz to 108 Hz. 5. Experimental validation The two main purposes of the experimental investigations were to measure the TVAs eigenfrequency as a function of actuator voltage and to determine the amount of damping. Tests were performed with the demonstrator mounted on a single-axis vibration table. Acceleration sensors were located at the central mounting bracket and two laser triangulators measured vibration amplitudes at the tension loop near the actuator and preloading mechanism locations, respectively. A sine-sweep base excitation with constant deflection amplitude was used. Amplitude was set such that when excited at the resonant frequency the tension loop deflection was 1 mm. Fig. 12 shows the measured frequency response for four different actuator voltages. The largest value corresponds to 70% of the

maximum allowable actuator driving voltage and was limited by the controller electronics which had been designed for a different actuator type. The highest eigenfrequency was found to be 106.1 Hz which, due to a minimum of axial preloading that was necessary to prevent an unintentional disassembly during operation, is approximately 1.8% lower than the design eigenfrequency computed for zero axial load. The lowest tunable frequency was 99.4 Hz and good frequency linearity with respect to actuator voltage was observed. An extrapolation to the maximum actuator voltage of 160 V leads to a minimum resonant frequency of 96.5 Hz resulting in a tuning range of 9.6 Hz which is in good agreement to the design frequency range of 10.0 Hz. The smaller peaks around 90 Hz correspond to a rotation mode around the suspension bracket. It is excited due to a minor asymmetrical distribution of oscillating mass and does not interfere with the tuning process. Quality factors range between 39.3 dB (92.6) and 41.1 dB (112.9) falling somewhat short of the required value of 150. Tests at different excitation amplitudes revealed that the level of damping increased with vibration amplitude. For very small amplitudes quality factors up to 300 were measured. The effect was thought to be due to microscopic slip movements in the joints which had not been fastened permanently on the test specimen to enable subsequent disassembly and inspection. Increasing the axial preload led to improved quality factors supporting the assumption of friction losses. The consistency of theoretical and experimental results shows that both the analytical and numerical models are physically correct and yield accurate predictions with respect to both resonant frequency and tuning range. 6. Conclusions A vibration absorber featuring a variable eigenfrequency for application in turboprop aircraft has been introduced. The device basically consists of an oscillating mass attached to a pair of leaf springs in a symmetrical arrangement. Frequency tuning is realized through subjecting the leaf springs to an adjustable axial pressure force which alters the leaf springs effective bending stiffness. The pressure force is generated using a piezo-electric stack actuator which in order to minimize supplementary weight is part of the oscillating mass. The theoretical background of a beam under compressive axial load has been outlined and layout of the leaf springs, tension

S. Keye et al. / Aerospace Science and Technology 13 (2009) 165–171

171

Fig. 12. Measured frequency response.

loop and actuation system has been discussed. To overcome thermal expansion problems due to wide operation temperature range and the piezo-electric stack actuators small stroke a compound of carbon and basalt fibers is used for the tension loop. A prototype compliant to Airbus A400M operating conditions, safety issues and certification standards has been built and tested. A good correlation of experimental and analytical results with respect to both resonant frequency and tuning range has been observed confirming the reliability and accuracy of the analytical and numerical models. References [1] T. Aida, T. Aso, K. Nakamoto, K. Kawazoe, Vibration control of shallow shell structures using shell-type dynamic vibration absorber, J. Sound Vibration 218 (1998) 245–267. [2] Akademischer Verein Hütte: Hütte, Des Ingenieurs Taschenbuch, 28 Auflage, Verlag Wilhelm Erst & Sohn, Berlin, 1955. [3] R.E.D. Bishop, D.B. Welbourn, The problem of the dynamic vibration absorber, Engineering 174 (1952) 796. [4] K. Desoyer, A. Slibar, Zur Berechnung von Pendel-Schwingungstilgern, Ingenieur-Archiv 21 (1953) 208–212. [5] A.H. von Flotow, A. Beard, D. Bailey, Adaptive tuned vibration absorbers: Tuning laws, tracking agility, sizing, and physical implementations, in: Noise-Con 94, Ft. Lauderdale, Florida, 1994, pp. 437–454. [6] M. Frik, Zur Abstimmung von Mehrmassentilgern, in: Vortrag b. d. Tagung: Dynamische Systeme, Oberwolfach, Oct. 7th–13th 1979. [7] P. Gardonio, Review of active techniques for aerospace vibro-acoustic control, J. Aircraft 39 (2) (2002) 206–214. [8] Y.P. Gupta, A.R. Chandraskearan, Absorber systems for earthquake excitations, in: Proceedings of the Fourth World Conference on Earthquake Engineering, Santiago, Part B. 3, 1969, pp. 139–148. [9] D.B. Den Hartog, Mechanical Vibrations, fourth ed., McGraw-Hill, New York, 1956. [10] R.S. Haxton, A.D.S. Barr, The autoparametric vibration absorber, J. Engng. Industry (February 1972) 119–125. [11] J.B. Hunt, Dynamic Vibration Absorbers, Mechanical Engineering Publications, London, 1979. [12] J.B. Hunt, J.C. Nissen, The broadband dynamic vibration absorber, J. Sound Vibration 83 (1982) 573–578. [13] D.J. Inman, Engineering Vibration, Prentice Hall, Englewood Cliffs, 1996. [14] H. Irreteir, Grundlagen der Schwingungstechnik 2, Vieweg Verlag, Braunschweig, 2001. [15] H. Kauderer, Nichtlineare Mechanik, Springer Verlag, Berlin, 1958.

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