Theoretical background

Theoretical background 4.1

4

Introduction

In the early days of bird-proof structure design and in the absence of today’s powerful low-cost computers, only theoretical (analytical) and experimental approaches were used for the mechanical study of bird impact and its characteristics, including the loads and the pressure applied by the bird on the target, the deformations of the bird and the impacted structure, and the resulting damage to the component concerned (Hedayati & Ziaei-Rad, 2011a; Hedayati & Ziaei-Rad, 2012c). Experimental tests are costly and require time-consuming procedures (Hedayati & Ziaei-Rad, 2011b). Especially, a bird-strike test requires costly testing equipment, very accurate measurement devices, and an intact aircraft component (which is very expensive and is useless after being damaged in the tests). Therefore, many researchers attempted to approach the birdstrike problem theoretically to mitigate the high costs of practical testing. Analytical approach to bird-strike problem has its own limitations. An impacting bird severely deforms after the initial instant of contact. The deformations of the bird and of the target and their interactions act simultaneously to create an “impact scenario.” The problem becomes much more complicated if on the inner side of the target, the structure geometry is complex or if the material behavior of the target is non-linear. Permanent damage of the target, which is a result of material plasticity, makes the problem even more difficult to engage with. The architecture of different aircraft structures can be very different in terms of their size, surface curvature, geometry of their inner connected parts, etc. As a result, for each aircraft type, for each component, and for each material (e.g. composites or metals), the lengthy exhausting procedure of derivation of the analytical formulas has to be repeated. Therefore, due to its intrinsic complexity, the derivation of analytical formulations for realistic bird-strike events has remained somewhat infrequent (Hedayati, Ziaei-Rad, Eyvazian, & Hamouda, 2014). Since the loads generated in a bird-strike problem depend on the deformation of both the target and the bird, some researchers have suggested decoupling the loads from the target response (Wilbeck, 1978; Barber, Taylor, & Wilbeck, 1975) in order to simplify the theory of a bird-strike problem. This was accomplished by studying bird strikes against rigid surfaces in 1970s. The results of such analysis could then be generalized to deformable targets by defining extra geometrical and material parameters. However, not much on the analytical studies of non-rigid targets was published after 1970s. The theoretical investigation of “bird-rigid target” impact is very helpful to understand the involved parameters in a bird-strike problem and finding the most effective parameters. Even after the development of powerful computers since the 1990s, the theoretical results obtained in 1970s have retained their importance. Analytical results are useful for benchmarking the numerical codes and for understanding the principal physical controls of a bird-strike problem (Senthilkumar, 2014). Bird Strike. http://dx.doi.org/10.1016/B978-0-08-100093-9.00004-2 Copyright © 2016 Elsevier Ltd. All rights reserved.

50

Bird Strike

During the impact of a bird onto a target, the bird is highly deformable and tends to flow over the target. An impactor flows over the target if the generated stress during the impact greatly exceed the yield stress of the impactor. When a stiff material, such as a steel or aluminum projectile, impacts a metal plate, the generated stresses are small enough to maintain the overall shape of the impactor, but large enough to create local holes or damage in the target. However, if the strength of the solid impactor is much smaller than that of the target plate, the impactor heavily deforms. These types of materials are referred to as “soft body” materials. Examples of soft body impacts include insects hitting an automobile windshield, birds striking an aeroengine, tire fragments or ice particles striking aircraft fuselage, and snowballs falling on windows (Wilbeck, 1978). Due to the fluid-like behavior of birds in bird strike, the bird impact problem can be solved analytically using hydrodynamic theory. The initial works in the field of hydrodynamic theory were mostly on the investigation of the impacts of liquid drops on solid surfaces (Heymann, 1969; Lesser, 1981), the impact of open sea waves against rigid vertical walls (Bagnold, 1939; Weggel & Maxwell, 1970), the impact of a column of water on a water hammer (Cook 1928), the impact of spheres onto rigid targets (Bowden & Field, 1964), and water-jet cutting (Johnson & Vickers, 1973). Not many theoretical investigations have been carried out on the bird-strike problem. Despite the scarcity of analytical studies in the field of bird strike, the existing analytical relationships are referred to in most of the numerical researches for validation purposes. This fact demonstrates the importance of reviewing the bird-strike theories. As one of the initial attempts, MacCauley (1965) and Mitchell (1966) in Canada studied birdimpact phenomenon theoretically. MacCauley assumed the impacted bird’s body behaves as a pure fluid, while Mitchell considered the bird as a semi-rigid projectile (Wilbeck, 1978). In both the works, several approximations and simplifying assumptions were considered, and neither of the two researchers validated their analytical results through experimental tests. As a result, their works did not attract much attention. Only two theoretical works by Wilbeck (1978) and Barber, Taylor, and Wilbeck (1978) have shown good accordance with the experimental data. Notably, the theoretical conclusions and results presented by Wilbeck (1978) have kept their importance till now in such a way that several numerical case studies such as Hedayati, Sadighi, and Mohammadi-Aghdam (2014), Hedayati and Ziaei-Rad (2012a), Hedayati and ZiaeiRad (2012b), Ivancˇevic´ and Smojver (2011), Heimbs (2011), Mao, Meguid, & Ng (2009), Jenq, Hsiao, Lin, Zimcik, & Ensan (2007), Airoldi and Cacchione (2006), Johnson and Holzapfel (2003), Guida, Marulo, Meo, Grimaldi, and Olivares (2011), Hedayati, Ziaei-Rad, et al. (2014) have used the same shape and characteristics for birds as was suggested in Wilbeck’s original report (Hedayati & Ziaei-Rad, 2013). The mechanisms dominating the mechanical response of a material in an impact mainly depend on its velocity. Based on the speed range and the acting mechanism, Hopkins and Kolsky (1960) categorized all the impacts into five main groups, namely elastic, plastic, hydrodynamic, sonic, and explosive impacts. In elastic impacts, the generated stresses are well below the strength of the material, and both the conservation of momentum and conservation of mechanical (kinetic + potential) energy are observed (no energy is dissipated). In these impacts, the generated stresses solely depend on the elastic moduli, the material densities, and the wave speeds of the

Theoretical background

51

materials, as well as the difference in the initial speed of the impacting bodies. By increasing the impact velocity, the generated stresses exceed yield stress causing permanent plastic deformation. The elastic properties of the material are still the dominating factors. By further increasing the impact speed, the generated stresses highly exceed the yield stress, and the material behaves like a flow. The fluidic behavior of the materials (for example, that of a bird in a bird-strike impact) suggests the use of a hydrodynamic approach. In these types of impacts, instead of material strength and elastic modulus, the material density determines the response of the impactor (Wilbeck, 1978). In this chapter, first the 2D hydrodynamic theory of formulating bird strike against a rigid plate is introduced, and explicit relationships for Hugoniot and the steady pressures are given. Modifications to the results of this theory for yawed and inclined impacts are also presented. Impact on non-rigid targets is very important due to the fact that aircraft components are usually made from ductile metals or composite materials. A projectile’s porosity is also very effective on the values related to Hugoniot and steady pressures. These two subjects will also be discussed. Since the distribution of pressure in a normal or oblique cylindrical impact is in fact three-dimensional, a 3D fluid dynamic approach will be introduced. The dynamic forces generated during a bird strike on an engine fan blade are highly non-linear in nature. An analytical solution to bird impact on a set of rotating fan blades will be presented in the final section of this chapter.

4.2

2D hydrodynamic theory

The generated stresses in a bird impacting a very stiff target severely exceed its strength, and as stated above, the problem can be solved using hydrodynamic theory. In hydrodynamic theory, the strength and viscosity of the bird’s material is neglected, and the stress of the bird at any time can be obtained using a relationship relating energy, velocity, pressure, and density (Wilbeck, 1978). When a bird impacts a relatively rigid surface, its frontal particles which go in contact with the impacted surface are brought to rest, and therefore, create a “shock wave” at that location. This shock wave starts moving backwards (into the succeeding layers of the bird) attempting to decrease their speeds. The velocity of this reversing shock wave is so high that the bird’s particles far enough away from the rigid surface do not have time to be affected by the free boundary conditions in their periphery. This means that their behavior is similar to the behavior of a semi-infinite medium which is deformed in a plane-strain condition. Therefore, afterwards, the initial shock is assumed to influence the bird material in a plane-strain process (Wilbeck, 1978). Due to the high initial velocity of the bird, the pressure of the formed compression wave is great and is constant throughout each layer at the initial time of impact. As the shock wave propagates into the forthcoming projectile particles, the material located in the periphery of each layer are subjected to a very high pressure gradient; their pressure at the inner side is that of the compression shock wave, while the pressure level is that of the atmosphere at the outer side. This large pressure gradient spreads the periphery particles radially, thus relieving the radial pressure gradient.

52

Bird Strike

Due to the effects of the compression wave and the radial pressure release, a very complex state of stress is formed. Lateral movement of the particles at the external surface of the bird causes both shear—due to the relative lateral displacement of each layer with respect to their following material layer—and tensile—due to expansion of the material—stresses. At any region, the bird flows when its stress greatly exceeds its strength. Since this phenomenon is established throughout the impact process, the material flowing continues until the final instant of the impact. Since the strength of the bird is negligible, the bird can be considered as a fluid. For these materials, to a first approximation, the material strength can be neglected so that they can be considered to behave as fluids (Wilbeck & Barber, 1978; Wilbeck, 1978). High-speed photography employed by Deping and Qinghong (1990) to record the evolution of the bird torso and the large deformations of the targets such as aeroengine fan blades further strengthened the fluidic property hypothesis of the bird tissue under high-speed impact scenarios. More recent studies such as Lavoie, Gakwaya, Ensan, & Zimcik (2007) and Salehi, Ziaei-Rad, and Vaziri-Zanjani (2010) have also demonstrated the fluidic behavior of the impacted bird in experiments as well. When the compression wave reaches the free end of the bird, it reflects back until it again reaches the other end of the bird which is in contact with rigid surface. After several fast reflections, the shock wave gradually loses its strength and finally disappears. After that, a steady state condition of the bird flowing is formed. The velocity and pressure field remain constant in the space and the bird material flows along fixed paths into space, which are called “streamlines” (Wilbeck, 1978). Finally, when all the bird material has passes through the streamlines, the impact process is ended. Several simplifications have been used in the analytical derivations presented in the following. Distribution of body mass density is different in the bodies of different bird species. However, water is the main composition of birds’ bodies (Ellis & Jehl, 1991), and the strength and viscosity of the birds’ materials can be neglected. Therefore, in order to simplify the analysis, the bird material is considered homogenous. For the bird geometry, a straight-ended cylinder is considered. By assuming the target to be rigid, the effect of target deformation on the generated forces during impact is eliminated. Another simplification in the presented theory is ignorance of shear frictional forces between the bird and the target surface (Wilbeck, 1978). According to what was stated above, the bird impact can be divided into four main phases (Fig. 4.1): (a) Shock regime, when the first compression wave is formed and propagates back into the bird material. (b) Release regime, when the bird’s particles located in its periphery tend to be released radially. (c) Steady flow regime, when bird particles flow in fixed streamlines in space. (d) Impact termination, when all the bird particles have reached the target surface and the pressure descends to zero.

4.2.1

Shock regime

For the normal impact of a cylinder on a rigid plate, the flow across the generated shock can be considered one-dimensional (1D), adiabatic, and irreversible. Figure 4.2a illustrates a shock wave propagating into the fluid at rest, where us is

Theoretical background

53

Figure 4.1 Illustration of shock and release waves in soft body impactor (Heimbs, 2011) according to Wilbeck, (1978). Reprinted by the permission of the publisher (Elsevier).

Shock wave

Shock wave

u0

Release wave

(a)

(b)

(c)

(d)

us u 2 = up

u1 = 0 (2) (1)

(a) us –up

u2 = 0

u1 =up (2) (1)

(b) u2 =us –up

u1 =us

(2) (1)

(c) Figure 4.2 One-dimensional shock flow: (a) shock propagating into a fluid at rest; (b) flow brought to a rest across the shock; (c) standing shock (according to Wilbeck, (1978))

54

Bird Strike

defined as the velocity of the shock propagating into the fluid at rest, and up is the velocity of the particles behind the shock in the global reference system. From this figure, it can be seen that the particle velocity is actually the change in velocity across the shock. Figure 4.2b illustrates the case for which the velocities are all measured relative to the fluid in the shocked state. This case is synonymous with the impact of a cylinder on a rigid plate. The projectile’s initial velocity is seen to be u0 and it is brought to rest behind the shock (Wilbeck, 1978). In this case, the equations of conservation of mass (continuity) and momentum may be written as (Wilbeck, 1978):
 ρ 1 us ¼ ρ2 us  u p

(4.1)


2 P1 + ρ1 u2s ¼ P2 + ρ2 us  up

(4.2)

Combining these two equations, the pressure behind the shock is found to be: P 2  P 1 ¼ ρ 1 us up

(4.3)

The pressure difference in the shocked region, given by Eq. 4.3, is often referred to as the Hugoniot pressure. Throughout the remainder of this chapter, this pressure will be represented by the notation PH. For the impact of a cylinder on a rigid plate, it can be seen that up ¼ u0 . Thus, in this case, Eq. 4.3 becomes (Wilbeck, 1978): P H ¼ ρ1 us u0

(4.4)

For very low impact velocities, the shock velocity us can be approximated by the isentropic wave speed in the material c0. Thus, for low impact velocities, Eq. 4.4 may be approximated by the relationship: P H ¼ ρ1 c 0 u0

(4.5)

Although this relationship may be adequate for very low-velocity impacts, it deviates markedly from Eq. 4.4 in higher velocities. Figure 4.3 demonstrates the differences in pressures found using these two relationships for water (Wilbeck, 1978).

4.2.2

Release regime

Although very high pressure values are generated at the initial moments of a bird impact onto a rigid target, it lasts only for several hundred microseconds (Wilbeck & Barber, 1978). The zero partial pressure in the outer side and the very high pressure value in the inner side of the external surface of the soft body creates a high-pressure gradient which is prone to accelerate the material radially outward. This radial

Theoretical background

55

Figure 4.3 Effect of compressibility on the Hugoniot pressure for water (Wilbeck, 1978)

7000

Hugoniot pressure, PH (MN/m2)

6000

5000

4000

P = p usuo

3000

2000 P = p couo 1000

0

0

300

600

900

1200

1500

Impact velocity, u0 (m/s)

acceleration forms a radial release wave. Unlike the first phase which was considered one-dimensional, the bird material deformation in this phase is two-dimensional and axisymmetric (compare Fig. 4.1a and Fig. 4.1b) (Barber et al., 1978). Over time, the radial released region advances to the central axis of the bird projectile. Propagation of the release wave decreases the pressure of the bird material greatly. The explained procedure can be better seen in Fig. 4.4 which demonstrates the formation of release waves for a cylinder with length to diameter ratio of DL ¼ 2. Formation of the almost 2D shock wave in the projectile just after the impact is shown in Fig. 4.4b. The theoretical value for pressure in this region is given in Eq. 4.4. The generated release waves which are negligible in the beginning (Fig. 4.4b) proceed inward until they converge at the central point on the target B (Fig. 4.4c). With further propagation of the release waves, they resolve the fully shocked region in them. When the intersection of the radial shock waves reaches point C, the shocked region is severely weakened with decreased pressure and velocity values (Fig. 4.4d) (Wilbeck, 1978). After a while, when the radial pressure gradient is decreased sufficiently, the release waves themselves disappear and a steady flow regime is established. The existence of the steady phase is dependent on the length of the projectile. If the length of the projectile is short, the impact process ends before the propagation of release waves forms a steady phase. Although there are not analytical solutions to the complex behavior of release regime, Barber et al. (1978) obtained and formulated a few effective parameters on this phase.

56

Bird Strike

u0

C B

(a)

A

(b)

B

A

C

C

(c)

B

A

(d)

B

A

Figure 4.4 Shock and release waves in fluid impact (Wilbeck, 1978)

The duration of the initial peak pressure can be estimated by measuring the time it takes by the initial release waves to travel the distance from A to B in Fig. 4.4c. After that, as stated above, the initial shock wave is disappeared, and the pressure decreases very quickly which marks the end of the peak pressure phase. The velocity of the release waves is equal to the speed of sound in the shocked region, which is given by (Wilbeck, 1978) c2r

  dP ¼ dρ PH

(4.6)

the speed of sound in the shocked region cr is the slope of the isentropic pressuredensity curve at the Hugoniot state. Using Eq. 4.6, the required time for the release waves to reach the center of impact is: tB ¼

a cr

(4.7)

where a is the radius of the bird before impact. Figure 4.5 demonstrates the wave speeds in the shocked and uncompressed regions. Increase of density in the shocked region slightly increases the wave speed, especially in larger impact velocities. Using Fig. 4.5 and Eq. 4.7, the duration of the peak pressure phase for different projectile radii can be obtained (as shown in Fig. 4.6). The duration of release wave regime is equal to the time it takes for release waves to completely capture the shock wave (after they have been converged). Since the velocity of the (release) waves in the shocked regime is higher than the velocity of the shock waves themselves, the release waves finally capture the shock waves

Theoretical background

57

2500

cr Wave speed (m/s)

2000

us

Figure 4.5 Comparison of the shock velocity us and the sound speed in the shocked region cr (Wilbeck, 1978)

1500

1000

500

0

0

100

200

300

Impact velocity, u0 (m/s)

Figure 4.6 Duration of impact vs. impact velocity for different cylinder radii (Wilbeck, 1978)

25

Duration, tB (ms)

20

15

a = 0.03 m

10

a = 0.02 m a = 0.01 m

5

0 0

50

100 150 200 Impact velocity, u0 (m/s)

250

300

(Wilbeck, 1978). By considering the geometrical dimensions and the wave speeds, obtaining the time of shock decay is feasible. The release wave initially formed at point A must first travel the radial distance AB and then the axial distance BC to reach point C (Fig. 4.4). At the time the intersection occurs, the shock has propagated the distance (Wilbeck, 1978): xs ¼ ðus  u0 Þtc

(4.8)

where us  u0 is the relative velocity of the shock wave with respect to the target plate (Fig. 4.2b). Using Eq. 4.8, the total distance between the initial location of the release wave (point A in Fig. 4.4b.) and its location at the instance the steady regime starts (point C in Fig. 4.4d) is: xr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffi x2s + a2

(4.9)

58

Bird Strike

This distance is traveled by the release wave in the time (Wilbeck, 1978): tC ¼

xr cr

(4.10)

Replacing all the unknowns in Eq. 4.10 gives: a tC ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2r  ðus  u0 Þ2

(4.11)

If the length of the projectile is very short, the impact process is terminated before a steady regime is initiated. The minimum length the projectile requires for having a steady regime is called critical projectile length LC. In a projectile with length LC, the time it takes for the release wave to reach point C is equal to the time required for the shock wave to reach the end of the projectile. Therefore, from geometry we have (Wilbeck, 1978): L C ¼ us t C

(4.12)

Replacing tC from Eq. 4.11 into Eq. 4.12 gives: us ðL=DÞC ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c2r  ðus  u0 Þ2

(4.13)

For projectiles with L=D < ðL=DÞC , the shock wave reflects back from the rear end of the bird before they are completely captured by the release waves. When the compressive shock reaches the free end of the bird, it subsequently reflects back in the form of a tensile wave. The tensile wave decreases the velocity of the incoming materials and disturbs the radial release process (Wilbeck, 1978). For projectiles with L=D > ðL=DÞC , the shock wave is captured by the release waves and disappears before it reaches the end of the projectile creating a steady state flow fluid afterwards. The values of both us and cr for water in different impact velocities are given in Fig. 4.5. Replacing the two values from Fig. 4.5 into Eq. 4.13, the critical length is readily obtained. Variation of critical length in different impact velocities is shown in Fig. 4.7. In impact velocities higher than 100 m/s, the critical length of the bird is smaller than its diameter.

4.2.3

Steady flow regime

In the release regime, radial release waves decrease the pressure of the bird material greatly by conducting the bird material radially outward. Convergence of the waves at the central axis of the bird and their interaction with shock waves is another source of sharp drop in the high initial shock pressures. After being sufficiently weakened, the shock wave disappears and a set of streamlines are established throughout the bird

Theoretical background

59

Figure 4.7 Variation of critical length with impact velocity for water (Wilbeck, 1978)

2.0

Critical length (L/D)c

1.5

1.0

0.5

0

0

50

100

150

200

250

300

350

Impact velocity, u0 (m/s)

material. If the effects of shear forces (viscosity) and body forces (inertia) are neglected and the variation of fluid properties in the material is continuous, the well-known Bernoulli’s equation can be applied to the streamlines of the bird (Wilbeck, 1978): ð ð dP + u du ¼ K (4.14) ρ where K is the constant of Bernoulli’s equation which is unvaried along each streamline but can be different between different streamlines. In a cylinder impacting a rigid target, in a distance sufficiently far from the target plate, the flow field is uniform and not yet affected by the shock waves, which implies that the value of K is equal throughout the entire flow (Wilbeck, 1978). Therefore, the velocity of the material at any point u can be related to the pressure at that point P by the following relationship: ðP

dP + P0 ρ

ðu

u du ¼ 0

(4.15)

u0

where P0 and u0 are the pressure and velocity of any particular point of the streamline whose properties are known. In locations far from the impacted surface, the velocity of the material is identical to the initial velocity of the bird, and since the material is not yet affected by the shock waves, the pressure is equal to the atmospheric pressure. In order to obtain the material pressure at the target surface using Eq. 4.15, two other unknowns are still required: the velocity at that point, u, and the density at that

60

Bird Strike

point, ρ. The density of the material at any point is given through the equation of state, ρ ¼ ρðPÞ (to be discussed in detail in Chapter 6). During the impact of a symmetric projectile onto a target, the material velocity in the centerline is parallel to the projectile initial pathway. Moreover, at the center of impact on the target plate, the material is allowed to move neither radially (due to axial symmetry) nor axially (due to the existence of the target). Therefore, at that point, the velocity is zero and the pressure at that point is called the stagnation pressure PS. Inserting the velocity and pressure of the center of impact in Eq. 4.15 yields: ð PS + P0 dP u20 (4.16) ¼ 2 ρ P0 Assuming the fluid being incompressible, integrating the above equation gives: 1 Ps ¼ ρu20 2

(4.17)

The high pressures created during the initial phase of impact increase the density of the material, which subsequently increases the pressure of the material at steady regime. Therefore, due to the compressibility of the fluid, it can be concluded that: 1 Ps  ρu20 2

(4.18)

The other state of material (other than the state before impact and far from the target plate) that can be considered for obtaining the K coefficient in a streamline in the steady regime is the state of material after impact and at a far distance radially from the impact point. At those points, the pressure is zero and the material possesses (low) kinetic energy. The impulse applied by the target plate to the system (i.e. the bird) must equal the change in the momentum of the system (Wilbeck, 1978). In other words: ðu ð tD F dt ¼ M du (4.19) u0

0

where F and M are the applied force and the impactor mass, respectively. Using Eq. 4.19, it is possible to obtain the force applied by the target plate to the bird material during the steady flow regime. As shown by prior experimentation (see Chapter 5), the applied force during steady regime is almost constant and variations in its value can be neglected. Integrating Eq. 4.19 yields: FtD ¼ Mðu  u0 Þ

(4.20)

where tD is the duration of impact and can be estimated by the time required for the bird with initial velocity u0 to travel a distance equal to its length, L, i.e.: tD ¼

L u0

(4.21)

Theoretical background

61

According to Wilbeck (1978), the rebound velocity after impact, u, is so small that it can be ignored. For a bird with cross-sectional area of A and density of ρ, inserting u ¼ 0 and geometrical dimensions in Eq. 4.20 yields: F ¼ ρAu20

(4.22)

As obvious, the total force applied by the bird into the target is the spatial integral of pressure over the contact area, i.e. (Wilbeck, 1978): ð1 2π 0

Pr dr ¼ ρAu20

(4.23)

Any assumed expression for P must satisfy the above equation. Moreover, the expression must yield PS at r ¼ 0 (i.e. the stagnation point) and zero at r ¼ 1. The pressure must always be decreasing from r ¼ 0 to r ¼ 1. Two relationships have been proposed by Banks and Chandrasekhara (1963) and Leach and Walker (1966) for radial distribution of pressure imposed by a water jet on the target. In the relationship presented by Banks and Chandrasekhara (1963), the pressure at any radius r of the target is: 1 P ¼ ρu20 eζ1 2


r 2 a

(4.24)

where a is the initial radius of the jet. The constant ζ 1 is used to make Eq. 4.24 conform to Eq. 4.23. In the case of a bird strike, ζ 1 ¼ 0:5 (Wilbeck, 1978). The more complex formula presented by Leach and Walker (1966) is: (  2  3 ) 1 2 r r + P ¼ ρu0 1  3 2 ζ2 a ζ2 a

(4.25)

where the constant ζ 2 is used to make Eq. 4.25 conform to Eq. 4.23 and is ζ 2 ¼ 2:58 for a bird strike.

4.2.4

Termination of impact

As stated previously, in the steady state condition, the particles of material move along streamlines. Along each streamline, the gage pressure is first zero but its value gradually increases (due to a gradual decrease in velocity) and it reaches its maximum value near the surface of the target. The streamlines are turned in the locations near the target surface radially. In the final stages of steady state regime, when the rear part of bird reaches the field of increasing local pressure, it disrupts the field due to the very low pressure behind it (Wilbeck, 1978).

62

4.3

Bird Strike

Inclined impacts

In section 4.2, a comprehensive theory was presented for the normal impact of cylindrical projectiles onto rigid targets. However, the bird impacts are not usually perpendicular, and in fact in most cases, there is an angle between the centerline of the bird and the normal of the target surface (see for example Fig. 6.21 that shows a real birdstrike situation). Therefore, knowing the response of a bird’s form in inclined impacts is critical. To understand the inclined impacts, first, the impact of a yawed projectile is studied and then its results are used for studying oblique impact.

4.3.1

Projectile yaw

Figure 4.8a shows the impact of a yawed projectile with an initial velocity parallel to the normal of the rigid target surface. The centerline of the projectile has an angle ϕ with respect to its initial velocity vector. The amplitude of shock pressure is close to that in a normal impact with the same initial velocity. However, the duration of this pressure at different points of the target can be different if the angle ϕ is larger than a critical value ϕcr (Wilbeck, 1978). Figure 4.8b shows a yawed projectile in the initial phase of impact when a shock wave has just been formed and traveled back into the projectile. Consider an arbitrary point B on the frontal surface of the cylinder. If the shock wave reaches point B prior to point B reaching the target surface, due to the establishment of very high pressure gradient at that point, a release wave is created. Similarly, at each point of the frontal surface of the cylinder, similar release waves are formed. The release waves interact with the shock waves and weaken it considerably. Therefore, the only region that experiences the initial Hugoniot pressure is a very small area around the initial point of impact. Even at the initial point of impact, the duration of peak pressure is shortened with respect to that in a normal impact because of the rapid effects of the release waves generated in the neighbor particles (Wilbeck, 1978).

f

u0

u0 B CL

(a)

(b)

C

f A

Figure 4.8 Normal impact of a yawed projectile into a rigid target (Wilbeck, 1978)

Theoretical background

63

On the other hand, if point B reaches the target surface before the initial shock wave reaches it, the total frontal surface of the projectile experiences peak pressures with similar duration and amplitudes to those in a normal impact (Wilbeck, 1978). At ϕ ¼ ϕcr , the time it takes for the shock wave travel (with speed us) the distance AB is equal to the time it takes for point B travel (with the speed u0) the distance BC. This gives (Wilbeck, 1978): For ϕ ¼ ϕcr , Δt ¼

BC u0 ¼ AB us

(4.26)

The trigonometric relations in the triangle ABC (Fig. 4.8b) gives: sin ðϕcr Þ ¼

BC AB

(4.27)

from which the critical angle can be obtained: ϕcr ¼ sin 1

4.3.2

  u0 us

(4.28)

Oblique impact

The oblique impact of a yawed projectile onto a rigid target is shown in Fig. 4.9a. In this impact, the initial velocity vector is parallel to the axis of the cylinder. If the frictional forces between the bird and the target are neglected, this impact is equivalent to the normal impact of a yawed cylinder onto the rigid target with the initial velocity of u0 sin(α). Correspondingly, the peak pressure is identical to that of a normal impact with an initial velocity of u0 sin(α). Figure 4.10 demonstrates the effect of the impact angle on the variation of Hugoniot pressure in different velocities. The angle that the projectile makes with the target surface is complementary to the angle it makes with the target surface normal, i.e. ϕ ¼ 90°  α. Like the impact of the yawed projectile, release waves are generated at the frontal face of a cylinder

u0

u0sina

α

(a)

α

(b)

u0cosa

Figure 4.9 Oblique impact of a yawed projectile onto a rigid plate (Wilbeck, 1978)

64

5000 Hugoniot pressure, PH (MN / m2)

Figure 4.10 Effect of impact angle on the variation of Hugoniot pressure with respect to velocity for cylindrical water (Wilbeck, 1978)

Bird Strike

a = 90⬚ (Normal)

4000

a = 45⬚

3000

2000

a = 30⬚ 1000

a = 15⬚ 0

0

300

600

900

1200

1500

Impact velocity, u0 (m/s)

u0

a

CL

S

Figure 4.11 Steady flow phase of an oblique impact (Wilbeck, 1978)

impacting a rigid surface with an oblique angle, if ϕ > ϕcr . In that case, the shock pressure is only sensed at the initial impact point. For ϕ < ϕcr , the distribution and duration of Hugoniot pressure on the target surface is similar to that in a normal impact (Wilbeck, 1978). The steady flow regime in an oblique impact is shown in Fig. 4.11. To conserve the momentum of the projectile, the majority of fluid flows downstream. This time, the stagnation pressure is not along the axis of the projectile and is shifted upstream (Fig. 4.11) (Wilbeck, 1978). The velocity and pressure of the particles in the stagnation point is identical to those in the stagnation point of a normal impact. Therefore, the stagnation pressure can be obtained using Eq. 4.17 again. No analytical solution has been presented for the pressure distribution in the steady flow phase of an inclined impact. However, Taylor (1966) developed the pressure distribution for a twodimensional jet of water impinging the target surface with a 30 ° angle. As can be seen in Fig. 4.12, the stagnation point is located in the intersection point of the target plate edge and the lower edge of the water jet.

Theoretical background

65

30⬚ C L

S

Figure 4.12 Steady flow pressure distribution for a 30° impact of a plane jet of water (Wilbeck, 1978)

Considering the momentum conservation principle, Eq. 4.22 is also useful for obtaining the impact force that the target plate imposes on an inclined impactor, providing that u0 is replaced by u0 sin(α): F ¼ ρAu20 sin ðαÞ

4.4

(4.29)

Flexible targets

In the previous chapters, the target plate was considered rigid which greatly simplified the analysis of the impact process. However, in reality, none of the aircraft components can be considered rigid. All the components show some degrees of flexibility against the bird impact. If the generated stresses in the impacted structure do not exceed yield stress, the component deforms elastically. At the very initial moments of impact, only the material located directly under the projectile-target interface is affected by the bird impact (Wilbeck, 1978). The formulas obtained for a rigid target can also be used for flexible targets with some modifications. After the target was deformed to some extent, its complex behavior, which is accompanied by the effect of its interaction with the impinging bird, makes the analysis of the impact very complex. In Fig. 4.13, regions 2 and 3 are the shocked domains in the projectile and the target, respectively. In the bird impact with a flexible target, the particle velocity up is not identical to the initial velocity u0. Therefore, Eq. 4.4 must be modified to: PH ¼ ρ1 us up

(4.30)

Moreover, the formed Hugoniot pressures in the target and the soft body are not equal. Implementing Eq. 4.30 for regions 2 and 3 gives (Wilbeck, 1978): P2 ¼ ρp usp upp

(4.31)

P3 ¼ ρt ust upt

(4.32)

66

Bird Strike

Projectile u s – u0 p

1

u0

2 3 4 Target

usT

Figure 4.13 Impact on an elastic target during the early shock regime (Wilbeck, 1978)

where usp and upp are the shock wave and material velocities in the projectile, respectively. Similarly, ust and upt are the shock wave and the material velocities in region 3 of the target plate, respectively. Continuity of material and Newton’s second law at the contact interface imply that: P2 ¼ P3

(4.33)

u 2 ¼ u3

(4.34)

From the definition of particle velocity, we have (Wilbeck, 1978): up p ¼ u 0  u 2

(4.35)

u p t ¼ u3  u4 ¼ u3

(4.36)

Inserting Eq. 4.36 into Eq. 4.32 and then equating Eq. 4.31 and Eq. 4.32 yields:
 ρp usp upp ¼ ρt ust u0  upt

(4.37)

from which we have: ( up p ¼ u 0

ρt ust ρp us p + ρt us t

) (4.38)

By inserting Eq. 4.38 into Eq. 4.31, the Hugoniot pressure of the impact of a cylindrical projectile onto a flexible target is found as (Wilbeck, 1978): ( PH ¼ ρp usp u0

ρt ust ρp usp + ρt ust

) (4.39)

Comparison of the Hugoniot pressure given in Eq. 4.39 with the Hugoniot pressure obtained for a rigid target (i.e. Eq. 4.4) can be of interest. In thick flexible targets, it was seen that the obtained Hugoniot pressure for bird impact with an initial velocity

Theoretical background

67

of 200 m/s onto flexible targets made of steel, titanium, aluminum, and polycarbonate are respectively 4%, 8%, 11%, and 35% lower than that in rigid targets (Wilbeck, 1978). In flexible targets, the thickness of the target is very effective on the Hugoniot pressure. In thin targets, the compressive wave is reflected back from the rear surface of the target in the form of a tensile wave. Interaction of the reflected tensile waves and the incoming compressive waves severely decreases the Hugoniot pressure. Moreover, contact forces between the thin target plate and the bird material moves the target plate at a high speed along the initial velocity of the bird, which in turn decreases the relative velocity of the projectile with respect to the impacted surface. This is another source of decrease in the Hugoniot pressure in flexible structures (Wilbeck, 1978).

4.4.1

Effect of porosity

The experimental results have demonstrated that porosity of a projectile has a significant effect on the impact characteristics. For example, 10% porosity can decrease the Hugoniot pressure of a gelatine projectile by 50% (see Chapter 5, Fig. 5.9). As suggested by Wilbeck (1978), the pressure–density relationships developed by Torvik (1970) for shock compression can be useful for bird-strike studies. In the relationships obtained by Torvik (1970), the material is assumed to be macroscopically homogenous and isotropic. A porous material is simply defined as the material that contains some pores or tiny holes. Pores inside a liquid can exist only if they are filled by a gas such as air, otherwise they are simply eliminated by the very high gage pressure around them. Porosity of a material is defined by volume fraction of pores in it and is denoted as z here. If the mass density of the matrix material (material with no porosity) is represented by ρf, the macroscopic average density of the mixture of material and pores is given by: ρz ¼ zρair + ð1  zÞρf

(4.40)

According to Wilbeck (1978), the relationship between the medium densities before and after impact is:   1 B ρ1 P2 +1 ¼ ð1  z Þ + z ð1  qÞ ρ2 A

(4.41)

where: ρ1 c20 4k  1 B ¼ 4k  1 ρ2 1 ¼ ρ1 1  q A¼

(4.42)

68

Bird Strike

where k is an experimental constant. The parameter q is defined as (Wilbeck, 1978): q¼1

ρ1 ¼ q1  q2 ρ2

(4.43)

with: ρ1 c20 P0 q1 ¼
2Pk rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   ρ1 c20 2 2 2

2Pk + P0  4P k q2 ¼
2 2Pk
+ 2Pk

(4.44)

P 2 ρ c2 q P
¼ ¼ 1 0 P1 P0 ð1  kqÞ2 If we use Eq. 4.41 along with Eq. 4.1 and Eq. 4.2, the Hugoniot pressure can be predicted more accurately. The shock velocity for a porous material is found by isolating ρ1 in Eq. 4.1 and Eq. 4.41 and equating them, whilst utilizing Eq. 4.2 for the pressure ρ2 P2. Using the obtained shock pressure and Eq. 4.5, the Hugoniot pressure can be calculated (Lavoie et al., 2007). Variations of shock velocity and shock pressure in different impact velocities are depicted in Fig. 4.14 for non-porous and 10% porous water. Figure 4.14 shows that the lower the impact velocity is, the higher effect the porosity has on the shock pressure. At the impact velocity of 120 m/s, which is typical of a bird strike, the Hugoniot pressure of porous water is about half of that for non-porous water.

Shock velocity (m/s)

2000

Vsh, z = 10%

Vsh, z = 0%

Psh, z = 10%

Psh, z = 0%

500 400

1500

300 1000

200

500

100

0 0

50

100

150

200

250

Shock pressure (MPa)

600

2500

0 300

Impact velocity (m/s)

Figure 4.14 Variations of shock velocity and shock pressure in different impact velocities (Lavoie et al., 2007) Copyright © 2007, Praise Worthy Prize S.r.l. Reprinted, with permission of Praise Worthy Prize S.r.l.. from the International Review of Mechanical Engineering, IREME Vol. 1 no. 4.

Theoretical background

69

The question is that what porosity should be used for a bird model having a mass m? The mass density of bird models, with feathers removed, has been enumerated by the databases of the International Birdstrike Research Group (Seamans, Hamershock, & Bernhardt, 1995) as: ρ0 ¼ 1148  63 log 10 ð1000mÞ

(4.45)

In another study conducted by Guida et al. (2011), the relationship between the bird’s mass, and its equivalent diameter was found to be: log 10 d ¼ 1:095 + 0:335log 10 m

(4.46)

where d is the diameter in meters and m is the mass in kg.

4.5

3D hydrodynamic theory

The relationships and formulas introduced in this subsection are taken from the work published by Barber et al. (1978). The distribution of pressure in an oblique cylindrical impact is difficult to be analysed as it is a three-dimensional (3D) fluid dynamic problem. Barber et al. (1978) used 3D potential flow theory to develop a model for predicting the pressure distribution produced by the steady flow of a cylindrical jet impacting on a flat plate. The model was based on superposition of two elementary solutions to the Laplace equation (Barber et al., 1978): Δ2 φ ¼

@2φ @2φ @2φ + + @x2 @y2 @z2

(4.47)

which is the governing equation for steady, incompressible, irrotational flow. The two elementary solutions used were: first, the uniform flow of a fluid in a round duct, and second, the uniform distribution of planar sources over an elliptical area. The coordinate system used to model the flow is shown in Fig. 4.15. Let (0, η, ξ) represent the coordinates of the location of a point source in the yz plane. The velocity components induced by this source are given by (Karamcheti, 1966): q x i3 4π h 2 x2 + ðy  ηÞ2 + ðz  ξÞ2

u¼

v¼

q yη h i3 4π 2 x2 + ðy  η Þ2 + ðz  ξÞ2

w¼

q 4π h

zξ x2 + ðy  ηÞ2 + ðz  ξÞ2

i3 2

(4.48)

70

Bird Strike

v

y

u

w

Uniform distribution of sources

q

x a

z

Ua (Uniform flow) 0 r

A

z2 (y sinq)2 + =1 a2 a2

Ua0 cosq –Ua0 sinq

Section A–A

A

Figure 4.15 Oblique impact potential flow model (Barber et al., 1978)

where q is the strength of the source. The velocity field induced by a uniform surface distribution of sources in the yz plane of strength q 00 per unit area is given by (Barber et al., 1978): q00 uðx, y, zÞ ¼ 4π q00 vðx, y, zÞ ¼ 4π

wðx, y, zÞ ¼

q00 4π

ξð2 ηð2

ξ1 η1

dηdξ h

x2 + ðy  ηÞ2 + ðz  ξÞ2

ξð2 ηð2

ξ1 η 1

h

ðy  ηÞdηdξ x2

2

i3 2

i3 2 2

(4.49)

+ ðy  η Þ + ðz  ξ Þ

ξð2 ηð2

ðz  ξÞdηdξ h i3 2 2 2 ξ1 η 1 x 2 + ð y  η Þ + ð z  ξ Þ

There is no closed-formed solution for the integration of the three equations given in Eq. 4.49 over the elliptical area bounded by:   ξ2 y sin θ 2 + ¼1 a2 a

(4.50)

Theoretical background

71

which is the projection of the cylindrical jet on the plane. However, it is possible to discretize the area using square elements for definite integrations. The velocity field induced by the uniform distribution of sources over a rectangular element whose corners are located at (η1, ξ1), (η1, ξ2), (η2, ξ1), and (η2, ξ2) in the yz plane is given by the following expressions (Kellogg, 1929): q00 ðz  ξ2 Þðy  η2 Þ ðz  ξ1 Þðy  η1 Þ tan 1 + tan 1 xr3 xr1 4π

ðz  ξ 1 Þðy  η 1 Þ ðz  ξ 2 Þðy  η 1 Þ  tan 1 tan 1 xr2 xr4

00 q ½r3 + ðξ2  zÞ½r1 + ðξ1  zÞ vðx, y, zÞ ¼ ln 4π ½r4 + ðξ2  zÞ½r2 + ðξ1  zÞ

00 q ½r3 + ðη2  yÞ½r1 + ðη1  yÞ wðx, y, zÞ ¼ ln 4π ½r4 + ðη2  yÞ½r2 + ðη1  yÞ uðx, y, zÞ ¼

(4.51)

where: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + ðy  η1 Þ2 + ðz  ξ1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 ¼ x2 + ðy  η2 Þ2 + ðz  ξ1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r3 ¼ x2 + ðy  η2 Þ2 + ðz  ξ2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r4 ¼ x2 + ðy  η1 Þ2 + ðz  ξ2 Þ2

r1 ¼

(4.52)

In order that the yz plane represents a surface across which no mass flows, that is, a flat plate, the round jet flow and the flow due to the sources on all the square elements (whose sum approximates the elliptical area) must be superimposed such that u is zero on yz plane. This condition is satisfied by setting the strength of the surface distribution q 00 over each square element equal to (Barber et al., 1978): q00 ¼ 2U1 sin θ

(4.53)

With the surface source strength per unit area so chosen, the U-component of velocity is identically zero over the entire yz plane at x ¼ 0. The V-component of velocity of the superimposed flow in the y-z plane at x ¼ 0 over the elliptical area is given by (Barber et al., 1978): V ð0, y, zÞ ¼ U1 cos θ +

V1 sin θ X Vk ð0, y, zÞ 2π k

(4.54)

where the summation is taken over each of the square areas comprising the elliptical area. The W-component of velocity of the superimposed flow in the yz plane at x ¼ 0 is given by (Barber et al., 1978):

72

Bird Strike

W ð0, y, zÞ ¼

U1 sin θ X Wk ð0, y, zÞ 2π k

(4.55)

The pressure on the plate over the elliptical area is then given by Bernoulli’s equation (Barber et al., 1978): o 1 n pð0, y, zÞ ¼ p + ξ1 ½V ð0, y, zÞ2 + ½W ð0, y, zÞ2 2

(4.56)

Since p is the atmospheric pressure, Bernoulli’s equation can be written in terms of a pressure coefficient (equivalent to the non-dimensionalized pressure) cp (Barber et al., 1978): cp ¼

p  p1 1  ¼ 2 V2 + W2 1 2 U1 ξU 2 1

(4.57)

Barber et al. (1978) wrote a computer program to calculate the pressure coefficient cp. Figure 4.16 shows the variation of the pressure coefficient calculated along the major axis of the elliptical impact area and plotted as a function of r, the projection of y in the yz plane at x ¼ 0 onto a plane perpendicular to the axis of the jet (i.e. r ¼ ysin θ). The pressure coefficient at any point on the surface can be readily calculated. Since the model does not contain the vorticity which undoubtedly occurs, it does not reliably predict coefficients near the boundary of the jet (y ¼ a=sin θ). However, over the central portion of the jet, the predictions should be reasonably accurate (Barber et al. 1978).

1.00 85⬚ 0.75 75⬚ Pressure coefficient, cp

Figure 4.16 Pressure coefficient (2P/ρv2) vs. nondimensional radius along the major axis of the impact for oblique impacts (Barber et al., 1978)

60⬚

0.50

45⬚ 0.25 25⬚ 0 –0.25 –0.50 –0.75 –0.8

–0.4

0

0.4

Nondimensional radius

0.8

Theoretical background

4.6

73

Turbofan bladed-rotor

The relationships and formulas introduced in this subsection are taken from the work published by Sinha, Turner, & Jain (2011). Based upon the bird size and the inlet area of the engine, a bird can impact multiple blades in one sector of the bladed-rotor, which would create unacceptable levels of rotor imbalance. Thus, a bird strike may not only result in sudden thrust decrease, it could also apply significant torque and imbalance loads on the fan shaft, which need to be considered during the design phase of these components. When the bird impacts a turbofan, it is sliced into several parts the number of which is dependent on several parameters including the number of blades on the fan rotor, rotational speed, aircraft speed, etc. For analytical formulations, the impact of a cylindrical bird projectile with mass M at a span height “s” on a rotating bladedrotor with Nb number of blades on it is considered (Fig. 4.17). For a bird with an initial axial velocity of Va, the mass of each of its slices after impacting the blades is (Sinha et al., 2011): Bs ¼

60 Va ðRPMÞNb

(4.58)

where RPM is the rotational speed of the fan rotor and is equal to 60Ω/2π. From the geometrical profile of the blade, the “stagger angle φ(s)” at the impact location due to pretwist φ0 in the airfoil is expressed as (Sinha et al., 2011): φ ð s Þ ¼ φ 0 + φ0 s

(4.59)

eˆr º eˆx

Direction of local eˆ r and eˆ x

–jL Bird dia. =BD

Va BL s ˆ i

L

eˆt

Angular Velocity = W

Wt kˆ

n

jˆ

tatio eˆa of ro Axis Twist at root = –j0

(R-L)

R

Angular velocity

=W

Bird-strike location

=s

Bird mass density

= rB

Blade mass density

=r

Blade length

= L (x-axis)

Blade width

= C (y-axis)

Blade thickness

= h (z-axis)

Blade stagger angle j

= jL–j0

Blade flexural rigidity

=D

Blade z-Deflection

= w(x, y, t)

Number of blades

= Nb

Figure 4.17 Bird cylinder coming in contact with rotating turbofan blades and local coordinate system (Sinha et al., 2011). Copyright © 2007, Praise Worthy Prize S.r.l. Reprinted with permission of Praise Worthy Prize S.r.l.

74

Bird Strike

In order to obtain the force imposed by the bird on the turbofan blades, it is first necessary to know the relative acceleration vector of the bird with respect to the blades in contact with it. The components of relative acceleration vector are: An ¼ Ay sin θ + Az cos θ

(4.60)

Aθ ¼ Ay cos θ  Az sin θ

(4.61)

A r ¼ Ax

(4.62)

with: Ax ¼ s€ Ω2 ðR  L + sÞ  Ωθ_ ðr0 + r 0 sÞsin ðφ0 + φ0 s + 2θÞ
 + 2Ω φ0 s_ + θ_ ðr0 + r 0 sÞsin ðφ0 + φ0 s + θÞ
  2Ω φ0 s_ + θ_ ðr0 + r 0 sÞsin ðφ0 + φ0 s + 2θÞ

(4.63)

 2Ωr0 s_ cos ðφ0 + φ0 s + 2θÞ + 2Ωr0 s_ cos ðφ0 + φ0 s + θÞ Ay ¼ s€½r 0 sin θ + ðr0 + r 0 sÞðcos θ  1Þφ0  + θ€½2 cos θ  1ðr0 + r 0 sÞ + 2φ0 r 0 ðs_Þ2 ðcos θ  1Þ  ðφ0 s_Þ ðr0 + r 0 sÞsin θ + 4r 0 s_θ_ cos θ
2  4 θ_ ðr0 + r 0 sÞsin θ  4φ0 s_θ_ ðr0 + r 0 sÞsin θ 2

(4.64)

+ 2Ωs_ sin ðφ0 + φ0 s + θÞ  2r 0 s_θ_  Ω2 ðr0 + r 0 sÞsin ðφ0 + φ0 s + θÞ  ½cos ðφ0 + φ0 s + θÞ  cos ðφ0 + φ0 s + 2θÞ Az ¼ s€½r 0 ðcos θ  1Þ  φ0 ðr0 + r 0 sÞsin θ  2ðr0 + r 0 sÞsin θθ€  2φ0 r 0 ðs_Þ2 sin θ
2 _ θ  4ðr0 + r 0 sÞcos θ θ_  4r0 s_θsin 0  2Ωs_ cos ðφ0 + φ s + θÞ  4φ0 s_θ_ ðr0 + r 0 sÞcos θ
2 + φ0 s + θ_ ðr0 + r 0 sÞ  ðr0 + r 0 sÞðφ0 sÞ cos θ 2

 Ω2 ðr0 + r 0 sÞcos ðφ0 + φ0 s + θÞ  ½cos ðφ0 + φ0 s + 2θÞ  cos ðφ0 + φ0 s + θÞ

(4.65)

Theoretical background

75

where R and L are two dimensions of the problem geometry and are shown in Fig. 4.17. Knowing the value of An is necessary for obtaining the contact force. Due to some geometrical considerations explained in Sinha et al. (2011), the first two acceleration components Ar and Aθ must be set to zero. By doing this, a set of coupled second-order non-linear differential equations with two unknowns (s, θ) as a function of time “t” are obtained, which determine the time-history of the bird-slice mass trajectory. Once the coupled equations have been solved for the time-dependent parameters (s, θ), they are used to determine the acceleration “An” of the bird slice mass M in the direction normal to the concave pressure surface of the blade, which yields the time history of Coriolis forces (Sinha et al., 2011) as follows: Ftravel ðtÞ ¼ MAn

(4.66)

Sinha et al. (2011) solved the coupled set of non-linear differential equations of motion with Ar ¼ 0 and Aθ ¼ 0 numerically by a sixth-order Runge–Kutta method (Fehlberg, 1964). The initial conditions for the numerical solution of the contactimpact forces were described as (Sinha et al., 2011): s ð 0Þ ¼ s i θ ð 0Þ ¼ 

(4.67) ϑ 2

Vθ ðsi  ϑ=2Þ θ_ ð0Þ ¼ r

 Vn  Va sin ðφ + θÞ ΩðR  L + si Þcos ðφ + 2θÞ  r sin θθ_ s_ð0Þ ¼ ½φ0 rsin θ  r 0 ð1  cos θÞ

(4.68) (4.69)

(4.70)

References Airoldi, A., & Cacchione, B. (2006). Modelling of impact forces and pressures in Lagrangian bird strike analyses. International Journal of Impact Engineering, 32, 1651–1677. Bagnold, R. A. (1939). Interim report on wave-pressure research. Journal of the Institute of Civil Engineers, 12, 201–226. Banks, R. B., & Chandrasekhara, D. V. (1963). Experimental investigation of the penetration of a high-velocity gas jet through a liquid surface. Journal of Fluid Mechanics, 15(1), 13–34. Barber, J. P., Taylor, H. R., & Wilbeck, J. S. (1978). Bird impact forces and pressures on rigid and compliant targets (No. UDRI-TR-77-17). Dayton Univ OH Research Inst. Barber, J. P., Taylor, H. R., & Wilbeck, J. S. (1975). Characterization of bird impacts on a rigid plate: Part 1. Air Force Flight Dynamics Laboratory. Bowden, F. P., & Field, J. E. (1964). The brittle fracture of solids by liquid impact, by solid impact, and by shock. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 331–352.

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Theoretical background

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