# Viscoelastic Characterization of Long-Eared Owl Flight Feather Shaft and the Damping Ability Analysis

Authors: Jia-li Gao; Jin-kui Chu; Le Guan; Hai-xin Shang; Zhen-kun Lei
Journal: Shock and Vibration
 (2014)

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## Abstract

Flight feather shaft of long-eared owl is characterized by a three-parameter model for linear viscoelastic solids to reveal its damping ability. Uniaxial tensile tests of the long-eared owl, pigeon, and golden eagle flight feather shaft specimens were carried out based on Instron 3345 single column material testing system, respectively, and viscoelastic response of their stress and strain was described by the standard linear solid model. Parameter fitting result obtained from the tensile tests shows that there is no significant difference in instantaneous elastic modulus for the three birds’ feather shafts, but the owl shaft has the highest viscosity, implying more obvious viscoelastic performance. Dynamic mechanical property was characterized based on the tensile testing results. Loss factor (tanδ) of the owl flight feather shaft was calculated to be 1.609 ± 0.238, far greater than those of the pigeon (0.896 ± 0.082) and golden eagle (1.087 ± 0.074). It is concluded that the long-eared owl flight feather has more outstanding damping ability compared to the pigeon and golden eagle flight feather shaft. Consequently, the long-eared owl flight feathers can dissipate the vibration energy more effectively during the flying process based on the principle of damping mechanism, for the purpose of vibration attenuation and structure radiated noise reduction.


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## Body

## 1. Introduction

Noise pollution is a major problem in the main urban traffic area recently, especially the airport neighborhoods, adversely affecting the lives of millions of people and animals. Evolving a silent flight, most genera of owls are not audible to man and, more importantly, to their prey. How flight noise can be efficiently reduced may be learned from owls.Since more than a century ago, the research regarding the mechanisms that enable the nearly silent flight of owls has remained an interesting field for theoretical and experimental research. Mascha and Graham discovered and innovatively concluded three mechanisms of the owl feathers that supposedly enabled the silent flight: (1) comb-like structure at the leading edge of the wings, (2) long and soft fringes at the trailing edge, and (3) soft downy upper surface of the feathers [1, 2]. These three mechanisms have been gradually identified [3–6] and accepted by the bionic researchers [7–13]. The reported biological studies of the noiselessly flying owls show that owl wing feather has significantly long soft feather material and unique morphological structure, which are different from other bird feathers.However, material research of the owl feathers is still not fully carried out and is away from material biomimetic synthesis so that bionic studies about the owl feather are almost all based on the outline edges. Though Bachmann et al. experimentally measured the flexural stiffness of the owl primary feather shaft cortex [14], a more comprehensive mechanical characterization of owl feather is necessary. In addition, its comparison with other bird feathers is still insufficient.Stiff, strong, but lightweight shaft is the key force bearing structure in an owl feather for its gliding flight. Interacting with the flow field during flight, feather shafts are subjected to bend mainly under aerodynamic loads [15, 16]. Because relative movement caused by the feather vibration and friction during flight would lead to the radiation noise [17], amplitude stability of the feather shaft directly affects the vibration of the whole feather structure and noise radiation. Damping factor as an important physical parameter can be used to characterize the vibration attenuation capability of the owl feather shaft and compared with other bird’s shafts. Consequently, on account of the assumption that the investigation of biomimetic materials inspired by the quiet flying owls might lighten a new way to obtain more excellent muffling performance in addition to the morphological bionic design, we present our study on the viscoelastic characterization of owl flight feather materials and the analysis of their vibration damping ability, so as to provide further insight into the owl's silent flight mechanism and bionic design in our further research in view of material acoustics [18, 19].The research presented in this paper focuses on the viscoelastic characterization of flight feather shaft of long-eared owl (Asio otus, Strigiformes), pigeon (Columba livia, Columbiformes), and golden eagle (Aquila chrysaetos, Falconiformes) to reveal the damping ability of their dynamic feathers on wings. Here, long-eared owl is the main research object with noise reduction function [20]. Its flight generally consists of an initial flapping phase followed by a gliding phase and a short flapping during touchdown [21]. Pigeon is a typical flapping flight bird which differs from the long-eared owl and has a smaller size than the owl. Golden eagle involves the same flight style as the long-eared owl and its size is larger than the long-eared owl. Both of the pigeon and golden eagle do not have the silent flight ability. The aim is to compare the viscoelastic performance of owl feather with other noisy flying birds without the impact of flying style and bird size on noise generation.Feather shaft can be regarded as a slender beam, so the shear effect on its bending deformation can be ignored. Hence, normal stress accounts for the bending deformation of feather shaft. Therefore, uniaxial tensile tests of the three types of the prepared flight feather shaft specimens were conducted on Instron 3345 single column materials testing system to investigate the relationship of strain and normal stress of the shafts at first. Then, viscoelasticity of feather shaft material was studied employing the standard linear viscoelastic solid model. And then, based on the viscoelastic material parameters obtained from the tensile testing results, dynamic mechanical property at a constant temperature was analyzed. Andtanδ of the long-eared owl, pigeon, and golden eagle flight feather shaft was calculated for a damping ability comparison between the three types of flight feathers. Relationship between damping ability and vibration attenuation for owl feather shaft was briefly discussed at last.


## 2. Materials and Methods

### 2.1. Feather Material and Anatomy of Feather Shafts

Flight feathers of long-eared owl, pigeon, and golden eagle were obtained from Dalian Forest Zoo (Liaoning, China). It is noted that feathers are collected with no harm to the birds’ life, complying with the Animal Protection Act. Barbs on both sides of the feather shaft were removed by the scalpel blade. Trimmed feather shafts were rinsed twice with deionized water for 30 minutes and naturally dried on clean filter paper at room temperature. All the specimens were cleaned and prepared in the cleanroom of key laboratory for precision and nontraditional machining of Ministry of Education, Dalian University of Technology.The tested specimens were selected from the feather shafts according to the following rules:(a)
the same relative position that the testing samples are selected in the same proportion position on flight feather shafts of the long-eared owl, pigeon, and golden eagle,(b)
the same aspect ratio that the ratio of the shaft length to maximum cross-sectional diameter is unified for all the testing parts: one end of the testing section is marked according to the selection method (a) at first, and its maximum diameter size is measured by a spiral micrometer subsequently. The other end is located away from the marked position with a gauge length of fifteen times the maximum diameter.After marking the testing portion in accordance with the above methods, specimen including the testing section and sufficient extension parts on its two sides for clamping is cut out, as is shown in Figure1. Before the test is started, reinforcement of the clamping parts is necessary in order to avoid damage of the specimen caused by gripping. Two extension parts were inserted in flat boxes which are full of liquid epoxy structural adhesive (WD3003) with high bonding strength and excellent performance to resist the compression. Clamping structures were obtained when the liquid adhesive was solidified into a block after 24 hours. Meanwhile, thin aluminium sheets with strong ductility were pasted on gripping interface of the adhesive to prevent contamination of the experimental equipment.Schematic drawings of flight feather and the sampling of tensile tested specimen. (a) Complete feather. (b) Feather shaft after removing the vane with scissors. (c) Tested specimen with extension parts on its two sides for clamping.
(a)
(b)
(c)


### 2.2. Experimental Instrument and Procedures

Instron 3345 single column materials testing system (Instron Corporation, the United States) is used for uniaxial tensile test, as is shown in Figure2. The whole system is equipped with a 5 kN load cell, a displacement sensor, a speed-controlled guide screw driving device (range: 0.005 mm/min to 1000 mm/min, accuracy: ±5% of the point value), and wedge clamps with the gripping force of 50 kN that one clamp is connected to the load cell while the other is secured to the machine. Both the load accuracy and the strain accuracy are ±5% of indicated value for the materials testing system.Figure 2
Schematic of the tensile test. The test specimen (feather shaft) was center-mounted onto the wedge clamps. Tensile force (F) was measured while the specimen was pulled at a constant rate of 0.5 mm/min. Tensile process was recorded by a high speed camera and the strain of the specimen was calculated from DIC analysis of the CCD images. DIC analysis was applied with the purpose of excluding the influence caused by the sample debonding, clamping, and the measurement error of sample’s initial length.During the experiments, the material testing system recorded the stretching force every 0.1 seconds and the one-to-one sampling time until the testing specimen was fractured. Digital image correlation (DIC) was applied to analyze the deformation mechanisms [22]. Firstly, two high contrast markers were spotted on the lateral wall of the specimen. Secondly, high-speed camera was used to obtain charge coupled device (CCD) images before and after the tensile deformation in the experimental process. The corresponding elongated deformation of the specimen was computed from the extremum of correlation coefficient by the correlation calculation of the CCD images finally. CCD images of each loading in the whole test were recorded and the real strain at each observing time was calculated. All samples were tested at a constant room temperature of 22°C and humidity of 55%.


## 2.1. Feather Material and Anatomy of Feather Shafts

Flight feathers of long-eared owl, pigeon, and golden eagle were obtained from Dalian Forest Zoo (Liaoning, China). It is noted that feathers are collected with no harm to the birds’ life, complying with the Animal Protection Act. Barbs on both sides of the feather shaft were removed by the scalpel blade. Trimmed feather shafts were rinsed twice with deionized water for 30 minutes and naturally dried on clean filter paper at room temperature. All the specimens were cleaned and prepared in the cleanroom of key laboratory for precision and nontraditional machining of Ministry of Education, Dalian University of Technology.The tested specimens were selected from the feather shafts according to the following rules:(a)
the same relative position that the testing samples are selected in the same proportion position on flight feather shafts of the long-eared owl, pigeon, and golden eagle,(b)
the same aspect ratio that the ratio of the shaft length to maximum cross-sectional diameter is unified for all the testing parts: one end of the testing section is marked according to the selection method (a) at first, and its maximum diameter size is measured by a spiral micrometer subsequently. The other end is located away from the marked position with a gauge length of fifteen times the maximum diameter.After marking the testing portion in accordance with the above methods, specimen including the testing section and sufficient extension parts on its two sides for clamping is cut out, as is shown in Figure1. Before the test is started, reinforcement of the clamping parts is necessary in order to avoid damage of the specimen caused by gripping. Two extension parts were inserted in flat boxes which are full of liquid epoxy structural adhesive (WD3003) with high bonding strength and excellent performance to resist the compression. Clamping structures were obtained when the liquid adhesive was solidified into a block after 24 hours. Meanwhile, thin aluminium sheets with strong ductility were pasted on gripping interface of the adhesive to prevent contamination of the experimental equipment.Schematic drawings of flight feather and the sampling of tensile tested specimen. (a) Complete feather. (b) Feather shaft after removing the vane with scissors. (c) Tested specimen with extension parts on its two sides for clamping.
(a)
(b)
(c)


## 2.2. Experimental Instrument and Procedures

Instron 3345 single column materials testing system (Instron Corporation, the United States) is used for uniaxial tensile test, as is shown in Figure2. The whole system is equipped with a 5 kN load cell, a displacement sensor, a speed-controlled guide screw driving device (range: 0.005 mm/min to 1000 mm/min, accuracy: ±5% of the point value), and wedge clamps with the gripping force of 50 kN that one clamp is connected to the load cell while the other is secured to the machine. Both the load accuracy and the strain accuracy are ±5% of indicated value for the materials testing system.Figure 2
Schematic of the tensile test. The test specimen (feather shaft) was center-mounted onto the wedge clamps. Tensile force (F) was measured while the specimen was pulled at a constant rate of 0.5 mm/min. Tensile process was recorded by a high speed camera and the strain of the specimen was calculated from DIC analysis of the CCD images. DIC analysis was applied with the purpose of excluding the influence caused by the sample debonding, clamping, and the measurement error of sample’s initial length.During the experiments, the material testing system recorded the stretching force every 0.1 seconds and the one-to-one sampling time until the testing specimen was fractured. Digital image correlation (DIC) was applied to analyze the deformation mechanisms [22]. Firstly, two high contrast markers were spotted on the lateral wall of the specimen. Secondly, high-speed camera was used to obtain charge coupled device (CCD) images before and after the tensile deformation in the experimental process. The corresponding elongated deformation of the specimen was computed from the extremum of correlation coefficient by the correlation calculation of the CCD images finally. CCD images of each loading in the whole test were recorded and the real strain at each observing time was calculated. All samples were tested at a constant room temperature of 22°C and humidity of 55%.


## 3. Results

### 3.1. Morphological Characterization of the Feather Shaft Cross Section

Intersecting surface of the long-eared owl, pigeon, and golden eagle flight feather shaft was morphologically characterized, as shown in Figure3. The general morphology of the rachis is very similar in the three species. All shafts have an approximate shape of a box girder; each shaft consists of a solid compact shell (cortex) and is filled by the medulla, which contains air-filled keratinized epithelial cells with an average diameter of 10–20 μm [23]. Noteworthy is the asymmetry of the cross section of a considerable part of the flight feather shaft.Flight feather shaft cross-section and scanning cross-section of the cortex. (a) Cross-section of long-eared owl feather. (b) Cross-section of pigeon feather. (c) Cross-section of golden eagle feather. C: Cortex; M: Medulla; C-s: Cross-section of the cortex.
(a)
(b)
(c)According to the reported literatures, the feather shaft gives rise to X-ray diffraction patterns of great complexity that they have been described as the most complex known for the naturally occurring fibrous substances. And further study suggests thatβ-keratin structure is proposed to account for the principal features of the X-ray pattern and for some physical properties of the feather shaft. Cortex is composed of two different β-keratin fiber layers where the inner layer fiber direction is along the shaft and the outer keratin fiber direction is circumferential. Due to its high hardness and great flexibility, cortex bears most bending stresses in the flight process. The compact keratin was found to be approximately 100 times stiffer than the medullary foam [24], consistent with an earlier finding of Purslow and Vincent that the medullary foam and transverse septa contribute only 16% to the overall bending stiffness of the feather shaft [25]. Because the medullary foam appears to play a minor structural role, calculation of the medulla structure is negligible in this study and our research is focused on the cortex for all flight feather shafts.Cross-sectional area of the fractured position of the feather shaft cortex was calculated using image extraction method. Cross-sectional area of the tested long-eared owl specimen is1.360±0.103 mm2, cross-sectional area of pigeon specimen is 0.308±0.024 mm2, and cross-sectional area of golden eagle specimen is 1.511±0.273 mm2.


### 3.2. Mechanical Behavior of the Flight Feather Shafts

Figure4 gives a description of the flight feather shaft of long-eared owl, pigeon, and golden eagle specimens before and after experiments, respectively. Fractured positions are mostly located near to the thin end of the gauge length for shaft specimens. The fracture mode of the shafts is ductile fracture. Influence caused by the sample fixation, clamping, and adhesive debonding can be avoided since tensile strain is calculated by DIC analysis.Experimental specimens in the clamps, scale bar: 1 mm. (a1) Fixed long-eared owl specimen before test. (a2) Breaking long-eared owl specimen after tensile process. (a3) Breaking point close-up of the owl specimen. (b1) Fixed pigeon specimen before test. (b2) Breaking pigeon specimen after tensile process. (b3) Breaking point close-up of the pigeon specimen. (c1) Fixed golden eagle specimen before test. (c2) Breaking golden eagle specimen after tensile process. (c3) Breaking point close-up of the golden eagle specimen.
(a)
(b)
(c)Figure5 presents typical stress-strain curves of long-eared owl’s and pigeon’s flight feather shaft. Stress (σ) of the shaft relative to the cross-sectional area (A) is given by the formula σ=F/A where F is the tensile force. In addition, the formula for the strain (ε) is also given as ε=L′/L where L is the feather shaft’s initial gauge length and L′ is the drawn shaft length, respectively. Nonlinear stress-strain relationship is a common feature for all shafts in Figure 5. And it can also be found in Figure 5 that stress versus strain curve curvature of the owl shaft is larger than the other two curves.Figure 5
Relationships of the stress and strain for long-eared owl and pigeon shafts.


## 3.1. Morphological Characterization of the Feather Shaft Cross Section

Intersecting surface of the long-eared owl, pigeon, and golden eagle flight feather shaft was morphologically characterized, as shown in Figure3. The general morphology of the rachis is very similar in the three species. All shafts have an approximate shape of a box girder; each shaft consists of a solid compact shell (cortex) and is filled by the medulla, which contains air-filled keratinized epithelial cells with an average diameter of 10–20 μm [23]. Noteworthy is the asymmetry of the cross section of a considerable part of the flight feather shaft.Flight feather shaft cross-section and scanning cross-section of the cortex. (a) Cross-section of long-eared owl feather. (b) Cross-section of pigeon feather. (c) Cross-section of golden eagle feather. C: Cortex; M: Medulla; C-s: Cross-section of the cortex.
(a)
(b)
(c)According to the reported literatures, the feather shaft gives rise to X-ray diffraction patterns of great complexity that they have been described as the most complex known for the naturally occurring fibrous substances. And further study suggests thatβ-keratin structure is proposed to account for the principal features of the X-ray pattern and for some physical properties of the feather shaft. Cortex is composed of two different β-keratin fiber layers where the inner layer fiber direction is along the shaft and the outer keratin fiber direction is circumferential. Due to its high hardness and great flexibility, cortex bears most bending stresses in the flight process. The compact keratin was found to be approximately 100 times stiffer than the medullary foam [24], consistent with an earlier finding of Purslow and Vincent that the medullary foam and transverse septa contribute only 16% to the overall bending stiffness of the feather shaft [25]. Because the medullary foam appears to play a minor structural role, calculation of the medulla structure is negligible in this study and our research is focused on the cortex for all flight feather shafts.Cross-sectional area of the fractured position of the feather shaft cortex was calculated using image extraction method. Cross-sectional area of the tested long-eared owl specimen is1.360±0.103 mm2, cross-sectional area of pigeon specimen is 0.308±0.024 mm2, and cross-sectional area of golden eagle specimen is 1.511±0.273 mm2.


## 3.2. Mechanical Behavior of the Flight Feather Shafts

Figure4 gives a description of the flight feather shaft of long-eared owl, pigeon, and golden eagle specimens before and after experiments, respectively. Fractured positions are mostly located near to the thin end of the gauge length for shaft specimens. The fracture mode of the shafts is ductile fracture. Influence caused by the sample fixation, clamping, and adhesive debonding can be avoided since tensile strain is calculated by DIC analysis.Experimental specimens in the clamps, scale bar: 1 mm. (a1) Fixed long-eared owl specimen before test. (a2) Breaking long-eared owl specimen after tensile process. (a3) Breaking point close-up of the owl specimen. (b1) Fixed pigeon specimen before test. (b2) Breaking pigeon specimen after tensile process. (b3) Breaking point close-up of the pigeon specimen. (c1) Fixed golden eagle specimen before test. (c2) Breaking golden eagle specimen after tensile process. (c3) Breaking point close-up of the golden eagle specimen.
(a)
(b)
(c)Figure5 presents typical stress-strain curves of long-eared owl’s and pigeon’s flight feather shaft. Stress (σ) of the shaft relative to the cross-sectional area (A) is given by the formula σ=F/A where F is the tensile force. In addition, the formula for the strain (ε) is also given as ε=L′/L where L is the feather shaft’s initial gauge length and L′ is the drawn shaft length, respectively. Nonlinear stress-strain relationship is a common feature for all shafts in Figure 5. And it can also be found in Figure 5 that stress versus strain curve curvature of the owl shaft is larger than the other two curves.Figure 5
Relationships of the stress and strain for long-eared owl and pigeon shafts.


## 4. Discussions

### 4.1. Viscoelastic Behavior and Model Analysis

The above uniaxial tensile test results reveal a nonlinear material behavior. Based on the reported achievements in relation to natural biological composite materials, asymmetrical geometry, viscoelastic property, and inhomogeneous density are inherent characteristics for feather shaft, leading to the nonlinear response between stress and strain during the loading process. In this study, the selected testing samples from long-eared owl, pigeon, and golden eagle feather shaft were straight as far as possible in length direction and the cortex sustained most of the tensile stress. Therefore, asymmetrical geometry and inhomogeneous density of the material are not the determinate factors for nonlinear mechanical response. As a result, it is suggested that viscoelastic property might be the major factor causing the nonlinear stress-strain relationship of the feather shaft.In consideration of the nature of the tested material and the applied testing method, we employed a three-parameter standard linear solid [26] to describe the nonlinear stress-strain relation of feather shaft. This nonlinear model is subject to uniaxial tensile test. The model consists of a single elastic element (spring) (elastic modulus E1) and a Kelvin element with an elastic element (elastic modulus E2) and a viscous element (dashpot) (coefficient of viscosity η) in parallel. The constitutive law in differential equation form for the standard linear solid model is
(1)σ+ηE1+E2σ˙=E1E2E1+E2ε+E1ηE1+E2ε˙.
Delay time (τc=η/E2) and relaxation time (τR=η/(E1+E2)) are two important internal time parameters for viscoelastic material. And for the standard linear solid model, τc and τR only relate to material constants (E1,E2,η).In our test, shaft specimens were tested at a constant rate. Therefore, mechanical behavior was studied under the condition of constant strain ratio(T). Stress response was investigated under the whole strain history. Strain expression is as follows:
(2)ε(t)=Tt.
Employing Boltzmann’s superposition method, constitutive equation of the stress response with a loading history of constant strain ratio is obtained:
(3)σ(t)=∫τ=∞τ=∞E(t-τ)dε(τ)=E(t)*dε(t)=E1TE1+E2[E2t+E1ηE1+E2(1-e-((E1+E2)/η)t)].
Dynamic behavior of viscoelastic materials is analyzed when alternating stress (4) with angular frequency (ω) is applied. Thus, strain response of the viscoelastic solid can be expressed in (5):
(4)σ(t)=σ0eiωt,(5)ε(t)=G(iω)σ(t)=G(iω)σ0eiωt.G(iω) is defined as complex compliance, an elastic constant which is equal to the ratio of strain and stress. It changes only with frequency and is a characterization of dynamic mechanical properties of material in stable vibration. Complex compliance is used and is to be presented as follows:
(6)G(iω)=G1(ω)+iG2(ω),
where G1 and G2 are identified as storage compliance and loss compliance, respectively. Equations (4) and (5) are put into (1); thus, complex compliance G(iω) of the standard linear solid model can be obtained in:
(7)G(iω)=E1+E2+ηiω(E1E2)2+(E1ηω)2×[(E1+E2)E1E2+E1(ηω)2-E12ηiω].
Viscoelastic parameters can be derived from the storage compliance and loss compliance. One parameter is the loss factor (tanδ), where δ is an indication of the phase lag between the stress and strain. tanδ represents the damping ability of the viscoelastic material and can be quantified by the ratio of the loss modulus to storage modulus:
(8)tanδ=-G2(ω)G1(ω)=((η/E2)-(η/(E1+E2)))ω1+(η/E2)·(η/(E1+E2))ω2=(τc-τR)τcτR·ωτcτR1+τcτRω2.
From (8), it can be easily found that the maximum of tanδ is obtained when τcτR=1, and tanδ is close to zero when ω approaches either zero or extremum (∞).


### 4.2. Determination of the Material Constants(E1,E2,η)

Least square method was employed to formulation regression for material constants prediction. The method of least squares is a standard approach to the approximate solution of overdetermined systems, mostly used in data fitting. The best fit in the least squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns.In our test, the measured strain performs a linear enlargement with the increase of time. Relationship between the measured stress and testing time exhibits a nonlinear characteristic due to stress relaxation for the viscoelastic materials. SPSS statistical software was used following the principle of least square method for the linear fit of strain and loading time according to (2) firstly. Strain ratio (T) is determined by the slope of the fitted straight line. And then non-linear fitting of the stress and loading time is conducted on the base of (3) for viscoelastic parameters E1,E2,η.


### 4.3. Fitted Material Parameters of Long-Eared Owl and Pigeon Flight Feather Shafts

Uniaxial tensile tests of the long-eared owl, pigeon, and golden eagle flight feather shaft specimens were performed and their fitted material parameters based on the three-parameter standard linear solid model were listed in Table1. Five specimens were tested for each shaft sample group.Table 1
Fitting parameters based on the viscoelastic model from tensile tests for the long-eared owl, pigeon, and golden eagle flight feather shafts.
Sample
Cross-sectional areaA (mm2)
Gauge lengthL (mm)
Elastic modulusE1 (MPa)
Elastic modulusE2 (MPa)
Coefficient of viscosityη (MPa·s)
owl1
1.432
38.94
8210
625
1620290
owl2
1.474
34.28
8989
554
1684931
owl3
1.279
29.54
8318
896
1966866
owl4
1.384
38.40
8760
860
1747912
owl5
1.230
34.16
9930
673
1510543
pigeon1
0.280
19.84
10087
2411
597359
pigeon2
0.324
22.98
9590
2314
981513
pigeon3
0.330
21.32
11103
3851
836727
pigeon4
0.282
21.08
10201
3373
680685
pigeon5
0.322
23.46
10404
2296
896814
eagle1
1.197
39.48
7023
1333
1047637
eagle2
1.784
42.14
6674
1073
1107921
eagle3
1.710
43.48
7255
1197
1265107
eagle4
1.624
37.86
8396
1455
938756
eagle5
1.242
38.97
7016
1515
1141217From Table1, it can be found that the golden eagle feather shaft has the lowest instantaneous elastic modulus (E1) ((7273±661) MPa) and that of the pigeon feather shaft ((8841±687) MPa) is the highest. E1 of the owl feather shaft falls in between the pigeon and golden eagle shafts. The conclusion is in accordance with Worcester’s investigation that, among species, larger birds have more flexible primaries than smaller birds [27]. However, material parameters of Kelvin element (elastic modulus (E2) of the spring and viscosity coefficient (η) of the dashpot) have greater dissimilarities. E2 of long-eared owl flight feather shaft was (722±149) MPa, lower than the one of pigeon feathers (2849±718) Mpa and golden eagle feathers (1315±182) Mpa. η of long-eared owl feathers feather shaft was (17.06±1.70) * 105 Mpa·s, higher than that of pigeon feathers (7.99±1.57) * 105 Mpa·s and golden eagle feathers (11.00±1.20) * 105 Mpa·s. This indicates that the long-eared owl flight feather shaft is more like a typical viscoelastic material, but the pigeon and golden eagle flight feather shafts are more like elastic materials with a lower viscosity. Thus, more internal energy dissipation would occur by long-eared owl flight feather due to its molecular friction during the deformation process on the basis of viscoelastic theory.


### 4.4.tanδ of Long-Eared Owl and Pigeon Flight Feather Shafts

tan
δ of the tested long-eared owl, pigeon, and golden eagle flight feather shafts is calculated for damping ability characterization from the fitted material parameters of uniaxial tensile test results according to (8), when alternating stress is applied to the specimens at constant testing temperature.Figure6 illustrates the loss tangent (tanδ) of the long-eared owl, pigeon, and golden eagle flight feather shafts. And Table 2 lists their peak value of tanδ. It is found that tanδmax⁡ of the long-eared owl flight feather shafts (1.609±0.238) is higher than the pigeon feather shafts (0.896±0.082) and golden eagle shafts (1.087±0.074). In other words, damping ability of the long-eared owl flight feather is better than the pigeon and golden eagle feather. Therefore, long-eared owl flight feather would dissipate more energy from the wing vibration in the flying process. The owl feather can restrain the resonance response more effectively due to its large damping factor. Thus, high damping ability is concluded to be an important factor to reduce or eliminate the mechanical vibration and free vibration of the owl feathers as soon as possible. As a result, noise caused by wing vibration and feather friction from the relative motion would effectively be suppressed by the long-eared owl.Table 2
Peak value of the loss tangent (tanδ) of the feather shaft specimens.
Owl specimen
tan
⁡
δ
max
⁡
Pigeon specimen
tan
δ
max
⁡
Eagle specimen
tan
δ
max
⁡
1
1.746
1
0.919
1
1.052
2
1.954
2
0.914
2
1.157
3
1.447
3
0.932
3
1.140
4
1.522
4
0.754
4
1.109
5
1.376
5
0.963
5
0.976Figure 6
Internal friction spectrum of feather shaft specimens.


## 4.1. Viscoelastic Behavior and Model Analysis

The above uniaxial tensile test results reveal a nonlinear material behavior. Based on the reported achievements in relation to natural biological composite materials, asymmetrical geometry, viscoelastic property, and inhomogeneous density are inherent characteristics for feather shaft, leading to the nonlinear response between stress and strain during the loading process. In this study, the selected testing samples from long-eared owl, pigeon, and golden eagle feather shaft were straight as far as possible in length direction and the cortex sustained most of the tensile stress. Therefore, asymmetrical geometry and inhomogeneous density of the material are not the determinate factors for nonlinear mechanical response. As a result, it is suggested that viscoelastic property might be the major factor causing the nonlinear stress-strain relationship of the feather shaft.In consideration of the nature of the tested material and the applied testing method, we employed a three-parameter standard linear solid [26] to describe the nonlinear stress-strain relation of feather shaft. This nonlinear model is subject to uniaxial tensile test. The model consists of a single elastic element (spring) (elastic modulus E1) and a Kelvin element with an elastic element (elastic modulus E2) and a viscous element (dashpot) (coefficient of viscosity η) in parallel. The constitutive law in differential equation form for the standard linear solid model is
(1)σ+ηE1+E2σ˙=E1E2E1+E2ε+E1ηE1+E2ε˙.
Delay time (τc=η/E2) and relaxation time (τR=η/(E1+E2)) are two important internal time parameters for viscoelastic material. And for the standard linear solid model, τc and τR only relate to material constants (E1,E2,η).In our test, shaft specimens were tested at a constant rate. Therefore, mechanical behavior was studied under the condition of constant strain ratio(T). Stress response was investigated under the whole strain history. Strain expression is as follows:
(2)ε(t)=Tt.
Employing Boltzmann’s superposition method, constitutive equation of the stress response with a loading history of constant strain ratio is obtained:
(3)σ(t)=∫τ=∞τ=∞E(t-τ)dε(τ)=E(t)*dε(t)=E1TE1+E2[E2t+E1ηE1+E2(1-e-((E1+E2)/η)t)].
Dynamic behavior of viscoelastic materials is analyzed when alternating stress (4) with angular frequency (ω) is applied. Thus, strain response of the viscoelastic solid can be expressed in (5):
(4)σ(t)=σ0eiωt,(5)ε(t)=G(iω)σ(t)=G(iω)σ0eiωt.G(iω) is defined as complex compliance, an elastic constant which is equal to the ratio of strain and stress. It changes only with frequency and is a characterization of dynamic mechanical properties of material in stable vibration. Complex compliance is used and is to be presented as follows:
(6)G(iω)=G1(ω)+iG2(ω),
where G1 and G2 are identified as storage compliance and loss compliance, respectively. Equations (4) and (5) are put into (1); thus, complex compliance G(iω) of the standard linear solid model can be obtained in:
(7)G(iω)=E1+E2+ηiω(E1E2)2+(E1ηω)2×[(E1+E2)E1E2+E1(ηω)2-E12ηiω].
Viscoelastic parameters can be derived from the storage compliance and loss compliance. One parameter is the loss factor (tanδ), where δ is an indication of the phase lag between the stress and strain. tanδ represents the damping ability of the viscoelastic material and can be quantified by the ratio of the loss modulus to storage modulus:
(8)tanδ=-G2(ω)G1(ω)=((η/E2)-(η/(E1+E2)))ω1+(η/E2)·(η/(E1+E2))ω2=(τc-τR)τcτR·ωτcτR1+τcτRω2.
From (8), it can be easily found that the maximum of tanδ is obtained when τcτR=1, and tanδ is close to zero when ω approaches either zero or extremum (∞).


## 4.2. Determination of the Material Constants(E1,E2,η)

Least square method was employed to formulation regression for material constants prediction. The method of least squares is a standard approach to the approximate solution of overdetermined systems, mostly used in data fitting. The best fit in the least squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns.In our test, the measured strain performs a linear enlargement with the increase of time. Relationship between the measured stress and testing time exhibits a nonlinear characteristic due to stress relaxation for the viscoelastic materials. SPSS statistical software was used following the principle of least square method for the linear fit of strain and loading time according to (2) firstly. Strain ratio (T) is determined by the slope of the fitted straight line. And then non-linear fitting of the stress and loading time is conducted on the base of (3) for viscoelastic parameters E1,E2,η.


## 4.3. Fitted Material Parameters of Long-Eared Owl and Pigeon Flight Feather Shafts

Uniaxial tensile tests of the long-eared owl, pigeon, and golden eagle flight feather shaft specimens were performed and their fitted material parameters based on the three-parameter standard linear solid model were listed in Table1. Five specimens were tested for each shaft sample group.Table 1
Fitting parameters based on the viscoelastic model from tensile tests for the long-eared owl, pigeon, and golden eagle flight feather shafts.
Sample
Cross-sectional areaA (mm2)
Gauge lengthL (mm)
Elastic modulusE1 (MPa)
Elastic modulusE2 (MPa)
Coefficient of viscosityη (MPa·s)
owl1
1.432
38.94
8210
625
1620290
owl2
1.474
34.28
8989
554
1684931
owl3
1.279
29.54
8318
896
1966866
owl4
1.384
38.40
8760
860
1747912
owl5
1.230
34.16
9930
673
1510543
pigeon1
0.280
19.84
10087
2411
597359
pigeon2
0.324
22.98
9590
2314
981513
pigeon3
0.330
21.32
11103
3851
836727
pigeon4
0.282
21.08
10201
3373
680685
pigeon5
0.322
23.46
10404
2296
896814
eagle1
1.197
39.48
7023
1333
1047637
eagle2
1.784
42.14
6674
1073
1107921
eagle3
1.710
43.48
7255
1197
1265107
eagle4
1.624
37.86
8396
1455
938756
eagle5
1.242
38.97
7016
1515
1141217From Table1, it can be found that the golden eagle feather shaft has the lowest instantaneous elastic modulus (E1) ((7273±661) MPa) and that of the pigeon feather shaft ((8841±687) MPa) is the highest. E1 of the owl feather shaft falls in between the pigeon and golden eagle shafts. The conclusion is in accordance with Worcester’s investigation that, among species, larger birds have more flexible primaries than smaller birds [27]. However, material parameters of Kelvin element (elastic modulus (E2) of the spring and viscosity coefficient (η) of the dashpot) have greater dissimilarities. E2 of long-eared owl flight feather shaft was (722±149) MPa, lower than the one of pigeon feathers (2849±718) Mpa and golden eagle feathers (1315±182) Mpa. η of long-eared owl feathers feather shaft was (17.06±1.70) * 105 Mpa·s, higher than that of pigeon feathers (7.99±1.57) * 105 Mpa·s and golden eagle feathers (11.00±1.20) * 105 Mpa·s. This indicates that the long-eared owl flight feather shaft is more like a typical viscoelastic material, but the pigeon and golden eagle flight feather shafts are more like elastic materials with a lower viscosity. Thus, more internal energy dissipation would occur by long-eared owl flight feather due to its molecular friction during the deformation process on the basis of viscoelastic theory.


## 4.4.tanδ of Long-Eared Owl and Pigeon Flight Feather Shafts

tan
δ of the tested long-eared owl, pigeon, and golden eagle flight feather shafts is calculated for damping ability characterization from the fitted material parameters of uniaxial tensile test results according to (8), when alternating stress is applied to the specimens at constant testing temperature.Figure6 illustrates the loss tangent (tanδ) of the long-eared owl, pigeon, and golden eagle flight feather shafts. And Table 2 lists their peak value of tanδ. It is found that tanδmax⁡ of the long-eared owl flight feather shafts (1.609±0.238) is higher than the pigeon feather shafts (0.896±0.082) and golden eagle shafts (1.087±0.074). In other words, damping ability of the long-eared owl flight feather is better than the pigeon and golden eagle feather. Therefore, long-eared owl flight feather would dissipate more energy from the wing vibration in the flying process. The owl feather can restrain the resonance response more effectively due to its large damping factor. Thus, high damping ability is concluded to be an important factor to reduce or eliminate the mechanical vibration and free vibration of the owl feathers as soon as possible. As a result, noise caused by wing vibration and feather friction from the relative motion would effectively be suppressed by the long-eared owl.Table 2
Peak value of the loss tangent (tanδ) of the feather shaft specimens.
Owl specimen
tan
⁡
δ
max
⁡
Pigeon specimen
tan
δ
max
⁡
Eagle specimen
tan
δ
max
⁡
1
1.746
1
0.919
1
1.052
2
1.954
2
0.914
2
1.157
3
1.447
3
0.932
3
1.140
4
1.522
4
0.754
4
1.109
5
1.376
5
0.963
5
0.976Figure 6
Internal friction spectrum of feather shaft specimens.


## 5. Conclusion

Flight feather shafts of long-eared owl, pigeon, and golden eagle were mechanically characterized based on Instron 3345 single column material testing system at a constant loading rate. Nonlinear response of their stress and strain obtained from the uniaxial tensile tests was described by the standard linear solid model to reveal the viscoelastic characteristic of dynamic feathers on owls, pigeons, and golden eagle. The parameter fitting result of the three-parameter model indicates that the long-eared owl flight feather shaft is more like a typical viscoelastic material, but the pigeon and golden eagle flight feather shafts are more like elastic materials with a lower viscosity. Thus, more internal energy dissipation would occur by long-eared owl flight feather due to its molecular friction during the deformation process on the basis of viscoelastic theory.In order to compare the damping ability of the three birds’ feather shaft more intuitively, dynamic mechanical property at a constant temperature was characterized based on the parameter fitting result of the three-parameter model. Loss factor(tanδ) of the long-eared owl flight feather shaft was calculated to be 1.609±0.238, far greater than those of the pigeon (0.896±0.082) and golden eagle (1.087±0.074). It is concluded that the long-eared owl flight feather has more outstanding damping ability compared to pigeon and golden eagle flight feather shaft. Consequently, the flight feathers of the long-eared owl can dissipate the vibration energy more effectively in the form of heat during the flying process based on the principle of damping mechanism. Because relative movement caused by the feather vibration and friction would lead to the radiation noise, large damping factor which is efficient in rapid vibration attenuation could be assumed to reduce the flight noise greatly for the flight feather shaft.In this paper, noise reduction mechanism for long-eared owl was investigated from the viscoelastic viewpoint of flight feather materials. We hope the conclusion would offer a new perspective for the research in this area.



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*Source: 709367-2014-08-28.xml*