Preliminary design of a mechanism for flapping flight - durability analysis and vibration modes Francesca Negrello1, Paolo Silvestri2, Aleramo Lucifredi2, Joel E Guerrero1, Alessandro Bottaro1 1Department of Civil, Chemical and Environmental Engineering, University of Genova, Italy E-mail: Alessandro.Bottaro@unige.it 2Department of Mechanical Engineering, University of Genova, Italy E-mail: lucifredi@unige.it Abstract The object of the paper is the kinematic and structural design of a flapping wing UAV, in order to develop an Unmanned Aerial Vehicle, a drone capable of executing reconnaissance and video-surveillance missions. To define the characteristic dimensions of the vehicle a biological study was initially carried out, analyzing, for example, the weight-wingspan ratio for the correct kinematics of the flight. On the other hand, several mechanisms apt to reproduce flapping flight were analyzed, searching for the best solution in terms of wings’ articulation.

Then, an optimization of the length of the different parts of the mechanism was needed to reproduce the kinematic law, provided by CFD (computational fluid dynamics) simulations.

The results of the optimization were the starting point for the design of the mechanism parts and for the diagnostic aspects. The stress resistance of the mechanical parts and fatigue life were verified in a FEM environment developing several simulations of the working conditions of the wing mechanism. 1 - Introduction The UAV is an aircraft with no pilot on board; it can be a remotely controlled aircraft or can fly autonomously based on pre-programmed flight plans or more complex dynamic automation systems. There is a wide variety of drone shapes, sizes, configurations and characteristics. UAVs are currently used for a number of missions, including reconnaissance and attack roles. They are predominantly deployed for military applications, but may be used for numerous indoor and outdoor civil applications, to monitor buildings, forests, cities, or to make possible an intervention in dangerous environments; moreover, for military applications, where discretion is important, the biomimetic aspect provides added value (spying and investigating).

The aim of the research is the development of a UAV, the main specification for the design was a biomimetic appearance and consequently the flying “style” chosen for it was the flapping flight.

Since this kind of propulsion is not yet well defined in its physics it was not possible to follow any established rule or design method, so the design was carried out trying to consider all the aspects of this multi-physics problem. This paper is mainly devoted to the study of the structural aspects.

The structure of the UAV was inspired to the birds’ anatomy, which have developed a skeletal system suitable for it. During the design phase, fluid dynamics and structural aspects related to the wings were considered: these should be long and slender to obtain a proper aspect ratio, even if the cantilever presents a large bending. For this reason a ribbed structure with a high moment of area is preferable. The structure of the mechanism is birds-based because they have optimal performances in terms of resistance/weight ratio [1].

From a kinematic point of view, birds to fly should produce lift and a negative resistance for having propulsion. Both of these force, lift and resistance, are generated from the movement of the wings; depending on the type of bird (dimension and weight) the first 50-60% of the wing semi-span generates the lift while the remaining 40-50% generates the thrust.

From a kinematic point of view the wing has four degrees of freedom:

• flapping: angular motion around the aircraft axis • lagging: angular motion around a vertical axis permitting a back and forth wing motion • feathering: angular motion permitting a variation of the wing attack angle • spanning: expansion and dilatation motion of the wing; it implies the articulation of the

wing into parts.

Often for propulsion and lift the model airplanes use only flapping, but also a passive feathering motion due to the fluid-structure interaction may be individuated. Utilizing only the flapping e feathering degrees of freedom, three parameters must be taken into account during the wing motion: the beat frequency, the beat amplitude and the wing attack angle versus its position. When properly coordinated, the three parameters may generate lift not only in the down stroke but also in the upstroke. 2 - Kinematic and structural analysis For the definition of the mechanism it was necessary to consider the number of degrees of freedom to be given to the wings and the dimension of the latter, as functions of the required forces for lift and thrust; this part is examined in detail in another related paper [2].

As anticipated the task of the wings is to generate thrust and lift in the same time; the fluid-dynamic study provided a parameterization of the motion of the wing and the corresponding optimal values. The parameters of interest for the beat are:

• α1 angular stroke of the internal semi-wing, • α2 relative angular stroke between the two semi-wings • S time extent of the up-stroke during the period.

Considering the angular ranges required and the beat frequency, the inertia of the wings should be low, to ensure a sufficient operation of the battery; therefore a solution was searched such to permit to concentrate the mass related to the actuation, i.e. the electric motor, inside the aircraft body. On the other side the wings must be enough robust to carry the loads and to limit the deformations arising as a consequence of the beat. Each wing has been designed into two parts, having a relative motion, that will be indicated as “internal semi-wing” (the part nearer to the fuselage) and as “external semi-wing” (the portion including the final tip; its aerodynamic optimization has been the object of a separated study). These preliminary considerations were followed by the synthesis of the mechanism.

The goal of the kinematic synthesis is to individuate a single degree of freedom mechanism providing the required motion of each of the two semi-wings; the internal semi-wing oscillates with respect to the frame, while the external semi-wing makes a rotation with respect to the internal semi-wing.

To satisfy the required characteristics, the simplest system between the possible kinematic chains was adopted, i.e. four binary members. The four bar linkage provides form-coupling, good behavior at high speed and doesn’t suffer of wear problems due to elevated contact forces. The articulated system is better also under the aspect of the weight; a cam in fact would require a spring having an high stiffness and would make more complicated the structure and more heavy the structural strength problems, due to additional load caused by the spring.

A four bar linkage was therefore adopted for each semi-wing. The configuration chosen for the four bar linkage driving the internal semi-wing is of the crank-rocker type, having at input the continuous rotational motion of the engine and at the output an oscillating arm; the mechanism was designed according to the Grashov rule; for the external semi-wing, on the contrary, a rocker-rocker configuration is adopted (non Grashov condition).

In summary, the choice was a kinematic chain consisting of two four bar linkages in series, where the conrod of the first four bar linkage is the input of the second. The internal semi-wing is the rocker of the first four bar linkage, while the external semi-wing coincides with the rocker of the second four bar linkage.

For the preliminary sizing, the lengths of the first four bar mechanism was determined, to obtain the variables of interest:

The length assumed as the scale parameter of the mechanism. An

optimization was performed to get mechanism lengths such to reproduce with the minimum error the law of motion obtained from the fluid-dynamic simulations; this study is described in another related paper [3].

Structural analysis has been performed through the support of the flexible motion design application of LMS Virtual.lab®. So, after defining the geometry of the parts, the mesh was created and the modeshapes were computed through Patran® e Nastran® software. On the basis of the previously elaborated data, the flexible parts were introduced into LMS Virtual.lab®, performing a flexible multibody analysis. Results were obtained in terms of stresses and displacements. The used solutor was the Nastran Craig-Bampton solution feature implemented in the flexible body design application of Virtual.Lab®; it’s based on the Craig-Bampton algorithm for the modeshapes superposition [4].

Each part was created trying to conjugate resistance and lightness; the tendency to take into a higher consideration the latter brought to the choice of configurations often to the limits of feasibility, as confirmed also by the first results. As a consequence, a redesign was performed aimed to reduce the criticalities. In particular, considerations related to the beat frequency (3 Hz) and to the stresses of the members of the mechanism suggested to perform more advanced studies, in particular about fatigue and to individuate the various vibration modes and the frequencies of the structure. 3 - Durability analysis The FEM model of the mechanism has been used, in a subsequent phase of the structural activity, for a durability analysis.

Durability analysis allows detecting the degradation of a material due to repeated cyclic loading. For most machinery, the length of time to the beginning of the cracks is typically much longer than the length of time that it takes for the cracks to propagate and to fracture the component. This means that the useful life of the component can be nearly the same as the length of time it takes to initiate cracks. LMS Virtual.Lab allows estimating the crack initiation time of structures through an analysis of the stresses or of the strains in the structure and of the material properties characteristics.

Two basic approaches are available in the software: the stress-life and the strain-life approach. The stress-life approach was adopted in the present analysis as the stresses are below the elastic limit of the material (this can be stated considering Von Mises value on nodes). For durability analysis the software requires fatigue data in the form of curves of stress vs. cycles to failure N obtained from specific tests on standardized smooth specimens. To obtain this information, a database available in LMS Virtual.Lab was used.

The following figures show some fatigue results for all the parts of the mechanism. Preliminarily to fatigue analysis for each part of the mechanism, the colormap corresponding to tension maxima (computed through the uni-axial equivalent stresses, e.g. according to Von Mises) has been computed, which are reached in the mesh nodes during the overall simulation or during a full beat of the wing. In this case we considered a mechanism operation condition characterized by a 3 Hz frequency (the maximum one) and the presence of aerodynamic forces, considered as concentrated loads applied to proper points both of the internal and of the external wing. The maximum stress value is an indicator of the degree of stress of the mechanism, and provides an information on the most critical zones and therefore on the zones which may be subjected to fatigue

To evaluate the fatigue behavior of the virtual prototype, single analyses on each part constituting the mechanism have been performed, using the loads evaluated through the multibody simulation. It was also necessary to introduce into the model the parameters related to the fatigue behavior of the material: the stress/strain curves and the Wöhler curves. The stresses computation method chosen in this specific case was the one known as Critical plane, Goodman correction.

Here we report and analyze first a result of the durability computation referred to the rocker, since it is one of the most stressed parts, therefore prone to be critical under the structural point of view.

The “fatigue life” is computed, an index of the life of the component in the operation conditions considered before the onset of a fatigue crack. This is expressed as a multiple of the simulated cycle of the multibody analysis, which in the considered case was corresponding to two full beats of the wing.

Fig.1 - Colormap of the maximum stress of the rocker

By analyzing the fatigue behavior of the rocker, one can note the existence of most critical zones, in general corresponding to the most stressed ones; the software forecast indicates fatigue life values higher than 3 104 cycles.

Fig.2 - Colormap of the fatigue life (time to failure) of the rocker

For the rocker a hot spots analysis was made. Hot spots are areas of locally bad performing result values. One value is assigned to each element by searching the highest value of the nodes of the element. Each hot spot is characterized by one “center element”, which has the highest value in a certain neighborhood. If such an element has been found, then the neighborhood of this element is

analyzed. If the elements in this neighborhood have a value which is close enough to the damage of the center element, then they are added to the hot spot.

The following figure shows the rocker divided into 3 colored zones (red, black, and brown) as a result of the hot spots analysis. In each of these zones the nodes (hot spots) corresponding to the highest values of stress and representing the full zone are put into evidence (e.g. the black zone presents its hot spot in a node corresponding to the Y junction of the rocker – node 112696). The zones identified may be defined as the areas having an equal level of criticality for the rocker.

Fig.3 - The hot spots visualization: the figure shows the colored zones characterized by a similar fatigue behavior (up -> zone 1 (red), down -> zone 2 and 3 (black and brown)) and the hot spots, representing the individuated zones, put into evidence through yellow markers (nodes 675805, 112696 and 79988). For the most critical points in the structure, a refined stress history analysis has been performed (identification of local stress state, stress in critical planes, etc.). Locally the influence of the applied loads at the hot spots has been analyzed to find out which loads have a high influence on fatigue.

Next figure shows the local stress tensor history and the stress in critical plane for the hot spot related to black zone (node 112696).

Fig.4 - (L) Local stress tensor 3D history, (R) Stress Critical Plane (node 112696)

LMS Virtual.Lab allows inserting virtual strain gages into the developed FE model. For calculation of the results, first the stress tensors are calculated. Then the strain tensor is calculated in each time step using Hooke’s law. The Young’s and the Poisson modulus are read from the finite element definition. It is possible to define the strain gage orientation by creating a vector and an axis-system where this vector is defined. The figure shows the virtual sensor placed on the Hot spot 2: the orientation is defined by z axis.

Fig.5 - Strain gage orientation definition through an axis system (Specific case of Hot Spot 2)

The results of the virtual strain gages are displayed in the following figure. These results can be visualized and exported and in the next step of the activity compared to real measurements when the real prototype will be available in order to validate the numerical analysis.

Fig.6 - (L) Strain Gage 0-45-90 - 1x3 Time Series; (R) Strain Gage 0-60-120 - 1x3 Time Series - Hot spot 2 Next figure shows the cumulative range pair diagram where the cumulated stress amplitude is plotted as a curve. The lower amplitudes are cumulated one after the other to higher stress amplitudes. The diagram is plotted for Von Mises stress and is related to the previous 3 hot spots. The other following figure reports the modal participation factors related to the first 12 modes.

Fig.7 - (L) Cumulative Range Pair (Von Mises); (R) modal participation factor (modes 1-12) Through the “load contribution to damage” display it is possible to identify the loads causing the largest damages. This display calculates the relative decrease of the damage at a selected position if

a load is omitted. The following figure (related to node 112696) indicates that load number 2 (corresponding to mode participation factor of mode 2) gives the highest contribution to the local damage. In this case when the load is included in the calculation the damage gets 2.2 times larger than in the case where it is excluded from the calculation. Load number 4 has a significant influence and load number 10 even has a small beneficial influence (its presence slight improves the durability behavior). The load contribution to the stress over time display shows at each point of the simulation time what is the von Mises stress created by each load: here it is possible to see that the load number 2 provides the highest contribution

Fig.8 - (L) Load contribution to damage for node 112696 (Hot spot 2); (R) Load contribution to stress over time for node 112696 (Hot spot 2) By analyzing the crank, criticalities are apparent for the fatigue life in the zone between the crank pin and the frame pin. Comparing this behavior with the one of the maximum stress, in this case a strong correlation doesn’t always exists, since the maximum stress values (localized in the crank pin and in the frame pin) don’t coincide with those of the maximum fatigue damage.

Fig.9 - Colormap of the maximum stress and of the fatigue life of the crank The conrod exhibits well-defined zones where the fatigue damage is more evident; comparing the behavior with the one of the other parts of the mechanism, the evidence of higher values of fatigue durability could be justified by lower values of maximum stress values.

Fig.10 - Colormap of the maximum stress and of the fatigue life of the conrod The following figure reports the fatigue analysis performed on the lower bar: the component doesn’t exhibit particular criticalities, except a very limited zone at the beginning of the circular stiffening element. Stress analysis shows a peak of stress in this area.

Fig.11 - Colormap of the maximum stress and of the fatigue life of the lower bar (detail) The external conrod looks to be fatigue critical in the sections near to the hinges. As in the case of the previous part, we are in presence of points exhibiting high stress values and fatigue damage in the zone at the beginning of the cylindrical stiffening element.

Fig.12 - Colormap of the (L) maximum stress and of the (R) fatigue life of the external conrod As for the case of the rocker, hot spots analysis has been performed on the external conrod, identifying 3 more critical zones under the fatigue behavior aspect. The load contribution to stress over time shows that the load numbers 5, 7 and 12 contribute most.

Fig.13 - (L) Visualization of the hot spots for the external conrod; (R) Load contribution to stress over time display for node 953202 (Hot Spot 2) The results of the virtual strain gages positioned on the Hot Spot 2 are displayed in the following figure.

Fig.14 - Strain Gage (L) 0-45-90 Time Series, (R) 0-60-120 Time Series - Hot Spot 2 The following figure reports damage time histories and their derivatives, which allow detecting the time instance the damage occurs.

Fig.15 - Damage time histories and their derivatives for (L) Hot Spot 1 and (R) Hot Spot 2 Globally evaluating the mechanism from the point of view of the fatigue behavior, the frame pivot and crank pin condition is individuated as the most critical; the limited values of the fatigue life term reached in these zones (lower than 2000 cycles i.e. 600 s of operation at the beat frequency 3 Hz) seem to require a redesign of these parts to make acceptable the mechanism life. Some zones of the rocker and other near the hinge points of the external conrod, present some criticalities. It has to be pointed out that the simulation has been done at the maximum beat frequency and therefore in the heaviest operation condition. An operation at lower frequencies provides a relevant reduction of the inertial loads and therefore a better fatigue behavior of all the system.

Fig.16 - Colormap of the fatigue life of the linkage

4 - Vibration modes in the different configurations of the linkage during the wing beat The results of a sequence of modal analyses of the mechanism in the various configurations assumed during the wings beat are reported hereafter. To get detailed information on the modal behavior during the beat, 20 different positions of the crank were considered, spaced of 18 degrees.

The methodology adopted for this analysis was developed in the LMS Samcef simulation environment and is based on the possibility of “chaining” two numeric analyses: in the particular case the “Implicit Non Linear” evaluating the kinematic of the linkage for large crank rotations and a subsequent linear modal analysis performed at the final instant of the first analysis. Doing so, in a fully automatic way, it is possible to locate the mechanism into a particular configuration and to perform the modal analysis in this configuration. To perform the numeric modal analysis it was necessary to modify some restraints of the system. In particular the “Hinge” present in the linkage has been substituted by “Bushing” since the previous elements introduce a non-linear behavior incompatible with the “linearization analysis” used for the computation of the vibration modes.

θ = 0 θ = 90°

θ = 180° θ = 270° Fig.17 – Linkage configurations corresponding to 4 values of the crank angle θ (0°, 90°, 180°, and 270°) The adopted methodology made possible to numerically evaluate how the frequency of the vibration modes vary in the different configurations assumed by the mechanism during the wing beat from one extreme position to the other. The following plots report the modeshapes of the first two modes (13.58 Hz and 20.77 Hz) in the configuration corresponding to the crank angle 0deg: in both cases the modeshapes have bending characteristics.

Fig.18 – Modeshapes of the first and second modes in the crank position θ = 0° of the linkage: both show bending characteristics (in different planes): in the first mode the modeshape belongs to the mechanism plane, in the second mode to a plane orthogonal to the previous one.

Analyzing the frequency of the first mode versus the crank rotation angle shows that it assume slow values in the configurations where the linkage results extended (θ = 0÷150°). This may be justified by the elevated value, in these conditions, of the moment of inertia of the system with respect to a direction normal to the mechanism plane. The increment of the inertial term implies a reduction of the frequency.

Analogous considerations hold to justify the opposite behavior, i.e. the increment of the frequencies for the contracted wing configurations (reduction of the moment of inertia). The same considerations may be extended to the second mode, which has too a bending modeshape. The modes 3 and 4 present opposite behaviors with respect to the previous ones. This may be justified by the torsional characteristics of the modeshape; the extension of the wing causes a reduction of the moment of inertia with respect to the longitudinal direction of the linkage, justifying the increase of the frequencies of the modes.

Modes 5 and 6 have modeshapes having local bending characteristics in the external part of the wing: in this case the low sensitivity of the inertial terms to the position of the linkage in the vibration mode implies a low sensitivity of the frequency to the variations of configuration of the linkage during the wing beat.

0 50 100 150 200 250 300 35010

15

20

25

30

35

40

45

50

teta [deg]

f [Hz

]

mode 1mode 2mode 3

0 50 100 150 200 250 300 35060

70

80

90

100

110

120

130

140

150

teta [deg]

f [Hz

]

mode 4mode 5mode 6

Fig.19 – Frequency variation of the vibration modes during the motion of the mechanism Considering the frequencies of the first mode of vibration and the frequency 3 Hz of the wing beat with its multiple frequencies, it is possible to evaluate if resonance conditions of the system may take place.

Analyzing the following plot reporting the fundamental frequency in the different configurations of the linkage, this frequency is always higher than the order 4X of the wing beat frequency; it seems possible to deduce that the system dynamics is acceptable, since all the modes are little excited. In detail, the first mode (which having the lower frequency is the mostly problematic one) is excited only by superior orders of the wing beat frequency which are at high frequencies and generally are characterized by low energies (the plot indicated only the 5X as a possible order having the possibility of exciting the first vibration mode (13.5÷14.8 Hz) when the beat frequency is slightly lower than the maximum one, which is 3 Hz (5 ⋅ 3= 15 Hz). 5 - Conclusion The numerical models developed for the mechanism are an advanced tool of dynamic analysis and made possible the study of its structural characteristics also in terms of fatigue life and vibration modes.

Using the LMS Durability module it was possible to forecast the fatigue behavior of the virtual prototype, using the loads on the various parts evaluated through a multibody simulation of an operation condition representing the real operation. The results have been reported in terms of “fatigue life” referred to the length of the simulated cycle, i.e. 2 full beats of the winds. The analyses, performed through LMS SAMCEF®, offered useful indications which made possible to

deepen the dynamic behavior. Studying not the single parts separately, but the mechanism as a whole in the different positions assumed during the wing beat, it was possible to evaluate the values of the frequencies of the vibration modes and to correlate them to the characteristics of the forcing function.

A possible future development is the dynamic analysis of the mechanism when created using carbon fibers, to be performed using LMS SAMCEF®, since this tool of analysis, through internal specific modules, makes possible to model this type of material and its particular structure.

0 50 100 150 200 250 300 3500

5

10

15

teta [deg]

f [Hz

]

mode 11X2X3X4X5X

Fig.20 – Variation of the fundamental frequency in the different configurations of the mechanism and values of the orders 1X, 2X, 3X, 4X e 5X (f = 3 Hz); modeshape of the first mode in the minimum (θ = 36°) and maximum (θ = 216°) condition of the frequency (the figure reports also the undeformed shape for θ = 0) Bibliography [1] Pennycuick, C.J., “Modelling the Flying Bird”, Elsevier (2008). [2] Guerrero, J.E., Negrello, F., Lucifredi, A., Silvestri, P., Bottaro, A., “Preliminary design of a

small-size flapping UAV. II. Aerodynamic Performance and Flight Stability”, Convegno AIMETA, 2013 - Torino - Italy.

[3] Guerrero, J.E., Pacioselli, C., Pralits, J. O., Negrello, F., Silvestri, P., Bottaro, A., “Preliminary design of a small-size flapping UAV. I Kinematic and structural aspects”, Convegno AIMETA, 2013 - Torino - Italy.

[4] Roy Craig and Mervyn Bampton, Coupling of Substructures for Dynamics Analysis, AIAA Journal, Volume 6, No. 7, 1968, Pages 1313-1319.

[5] D'Ippolito, R., Hack, M., Donders, S., Hermans, L., A reliability analysis approach to improve the fatigue life of a vehicle knuckle, 9th International Conference on Recent Advances in Structural Dynamics, Southampton, UK, 18-19 July 2006

[6] Society of Automotive Engineers. Fatigue Design Handbook, 2nd Edition, Rice, R. C., B. N. Leis, and D. V. Nelson Eds., SAE, Warrendale, PA, 1988.

[7] American Society for Testing and Materials, Ed. Standard Definitions of Terms Relating to Fatigue. Annual Book of ASTM Standards. Vol. E 1150-87, ASTM, Philadelphia, 1987.

[8] LMS Virtual Lab R12 On-Line Help, Leuven, Belgium - 2013 [9] LMS SAMCEF Field 8.4 On-Line Help, Liège, Belgium.- 2013