∎

11institutetext: Elio J. Challita 22institutetext: Chemical and Biomolecular
Engineering, Georgia Institute of Technology, Atlanta, GA 30311, USA
Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30311,
USA 33institutetext: Symone Alexander 44institutetext: Chemical and
Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30311,
USA 55institutetext: Sarah Han 66institutetext: Department of Biology,
Integrated Bioscience Program, The University of Akron, Akron, OH 44325 , USA
77institutetext: Todd A. Blackledge 88institutetext: Department of Biology,
Integrated Bioscience Program, The University of Akron, Akron, OH 44325, USA
99institutetext: Jonathan A. Coddington 1010institutetext: Smithsonian
Institution, National Museum of Natural History, 10th and Constitution, NW
Washington, DC 20560, USA 1111institutetext: Sunghwan Jung 1212institutetext:
Department of Biological and Environmental Engineering, Cornell University,
Ithaca, NY 14853, USA 1313institutetext: M. Saad Bhamla 1414institutetext:
Chemical and Biomolecular Engineering, Georgia Institute of Technology,
Atlanta, GA 30311, USA
1414email<EMAIL_ADDRESS>
# Slingshot spiders build tensed underdamped webs for ultrafast launches and
speedy halts

Elio J. Challita  Symone Alexander *These authors contributed equally Sarah
Han  Todd A. Blackledge  Jonathan A. Coddington  Sunghwan Jung  M. Saad Bhamla

(Received: date / Accepted: date)

###### Abstract

We develop a mathematical model to capture the web dynamics of slingshot
spiders (Araneae: Theridiosomatidae), which utilize a tension line to deform
their orb webs into conical springs to hunt flying insects. Slingshot spiders
are characterized by their ultrafast launch speeds and accelerations
(exceeding 1300 $m/s^{2}$), however a theoretical approach to characterize the
underlying spatiotemporal web dynamics remains missing. To address this
knowledge gap, we develop a 2D-coupled damped oscillator model of the web. Our
model reveals three key insights into the dynamics of slingshot motion. First,
the tension line plays a dual role: enabling the spider to load elastic energy
into the web for a quick launch (in milliseconds) to displacements of 10-15
body lengths, but also enabling the spider to halt quickly, attenuating
inertial oscillations. Second, the dominant energy dissipation mechanism is
viscous drag by the silk lines - acting as a low Reynolds number parachute.
Third, the web exhibits underdamped oscillatory dynamics through a finely-
tuned balance between the radial line forces, the tension line force and
viscous drag dissipation. Together, our work suggests that the conical
geometry and tension-line enables the slingshot web to act as both an elastic
spring and a shock absorber, for the multi-functional roles of risky predation
and self-preservation.

###### Keywords:

Ray orbweavers Theridiosomatidae Underdamped oscillator Spider biomechanics
Arachnid locomotion

## Introduction

The sit and wait strategy of many orb-weaving spiders is well known for the
ability to effectively snare prey using sticky silk and the rapid dissipation
of the captured prey’s kinetic energy (Kelly et al., 2011; Sensenig et al.,
2012; Yu et al., 2015; Das et al., 2017). Slingshot spiders, orb-weavers known
for extending and releasing their webs like slingshots to capture prey,
evolved an orb web modified with a tension line attached at its central hub
(Alexander and Bhamla, 2020; Coddington, 1986; Hingston, 1932; Wienskoski,
2010; Alves et al., 2007; Eberhard, 1981, 1990, 1986; Coddington, 2005). The
tension line enables the spider to deform its web into a 3-D conical
structure, loading elastic energy into the radial lines that ultimately
facilitates rapid accelerations ($>1300\;m/s^{2}$) for capturing flying prey
(SI Movie 1), or possibly avoiding predation (Alves et al., 2007). Although in
previous work, we described how the slingshot spider loads the web as a spring
to achieve ultrafast motion (Alexander and Bhamla, 2020), it remains unclear
how the spider leverages its unique web and tension line to decelerate quickly
and come to a halt, either after successful capture or missing of prey
(Eberhard, 1990), and to reset its web to potentially fire again. What are the
relevant physical forces governing energy storage and energy dissipation in
the web? How does the conical geometry affect the slingshot spatial and
temporal dynamics? The role of the tension line in enabling the spider to load
energy into the web as a spring is known, but what role does the tension play
in stopping the spider’s motion? To address these open questions, we develop a
mathematical model of the slingshot spider web dynamics in this paper.

The slingshot spider grips the center of its web with four rear legs while
incrementally pulling and twisting the tension line and coiling the silk with
claws on its anterior legs and pedipalps (non-locomotor anterior appendages)
to store elastic energy in the radial lines of the web (Fig. 1a) (Alexander
and Bhamla, 2020; Coddington, 1986). Upon sensing external stimuli (for
example a finger snap), the spider releases the tension line, catapulting both
the web and spider backwards (Fig. 1b). Multiple trajectories ($n=5$) are
provided in Fig. 1c, with the full displacement taking place in around 30 ms.
The dominant movement is mostly in the y-direction (10 - 15 mm or 10-15
$\times$ body lengths) with smaller movements ($\pm 1.1$ mm, less than one
body length) in the x-direction. Spiders achieve vertical speed of up to 4.2
m/s ($v_{max}=4.16\pm 0.07\;m/s$) and accelerations exceeding 1300$\;m/s^{2}$
($a_{max}=1163\pm 144\;m/s^{2}$) (Alexander and Bhamla, 2020). Despite this
ultrafast millisecond motion, we observe that the resultant vibrational
response of the web attenuates within milliseconds as well - suggesting that
the web design facilitates both energy storage and speed as well as energy
dissipation, which could potentially improve prey capture, reduce the
probability of damage to spider or web, and/or facilitate rapid reset and
reloading of the web.

Figure 1: Slingshot spider web dynamics. a The spider loads elastic energy by
pulling on the tension line, thereby stretching the radial lines and deforming
the web and forming a conical web structure. The spider remains in this
position awaiting its prey. We note from field observations that the spider
web does not seem to have a specific preference for orientation with respect
to gravity. b Upon sensing a stimulus the slingshot spider releases the web
moving explosively, almost 9 times its body length in less than 30 ms. The web
is oriented horizontally with the tension line at the bottom of the frame, so
that the primary motion upon release in the positive $y$ direction facing
upwards in this case. c Quantification of the multiple trajectories of the
same single slingshot spider for repeated launches (triggered by snapping
fingers). The red line denotes the average trajectory of the multiples
displacement curves. We adjust trajectories so that they all start at
$(x=0,y=0)$ at $t=0$. d Movement in the $x$ direction is negligible before the
first oscillation.

Motivated by the rapid attenuation of the slingshot spiders motion as well as
its non-planar configuration, we set to theoretically model the dynamics of
the slingshot spider web. We treat the slingshot motion like the dynamic
response of a step input typically used in process control design (Kuo, 1987).
Specifically, we focus on spatiotemporal parameters such as the rise time,
overshoot and settling time. Modeling the forces in the radial lines and the
tension line allows us to explore how they enable both rapid movement and
quick return back to the equilibrium position. The mathematical model
described herein offers insights about the balance of elastic forces in
slingshot spider webs and the opportunity to understand their design in ways
that are difficult to achieve experimentally.

## Methods

### Field Videography

Field work was conducted in Puerto Maldonado, Peru at the Tambopata Research
Center ($-13.134^{\circ}$, $-069.609^{\circ}$). Research permit no.
654-2018-GOREMAD-GRRNYGMA-DRFFS was obtained by the Gerencia Regional Forestal
y de Fauna Silvestre. The spiders were located by scanning dead branches and
leafy plants for their conical webs and then snapping fingers near to the web
to confirm slingshot motion. A Chronos 1.4 high speed camera (Krontech) was
utilized for high speed video recording (up to 38,500 fps) in conjunction with
a field portable Zaila high intensity light with portable battery packs. Field
Videos were captured at 1057 fps.

### Field specimen and silk collection

The spider specimen was identified to the best of our ability as an
undescribed species in the genus Epeirotypus sp. (Araneae: Theridiosomatidae)
(see SI text for images of organism including epigynum). Videography was
performed utilizing spiders that had built webs in their natural habitat.
After observation and videography, spider specimens were collected and stored
in 200 proof ethanol for species identification and further analysis. Silk
samples were collected using a notched microscope slide.

### High speed video analysis

Matlab was employed to analyze the highspeed video obtained in the field. The
code was written to identify the spider in each frame using an intensity
threshold and record the location of its centroid. Accurate tracking was
verified utilizing a binary output video highlighting the spider as white and
setting the background as black. The code converted the units of the centroid
measurements from pixels to meters and calculated elapsed time from the frame
rate and number of frames to calculate displacement, velocity, and
acceleration.

## Mathematical Model

We mathematically model the spatiotemporal dynamics of the motion of the
slingshot spider as a 2D mass-spring model in $x$-$y$ direction (Fig. 2a). The
model consists of three springs: two symmetric springs extended in the
radial/horizontal $x$ direction ($F_{r1,2}$) and one in the vertical $y$
direction that represents the tension line ($F_{t}$). We model the spider as a
point mass ($m_{s}$) located at the intersection of the three springs (Fig.
2). In the following subsections, we describe different components of the
model in detail along with relevant assumptions and limitations.

Figure 2:  Schematic of the mathematical model. a We model the conical web
structure as two symmetric horizontal radial springs and a vertical tension
line located in a 2D plane. We assume that the slingshot spider is a point
mass having a characteristic length $D_{s}$ on the order of $\sim 1.3\;mm$
located at the intersection of radial lines and the tension line. We define
$L_{r,eq}$ and $L_{t,eq}$ as the equilibrium length of the the radial lines
and the tension line respectively.The spider starts at $(x_{o},y_{o})$ at
$t=0$. Each force has a projected component in the $x$ and $y$ directions. Our
model ignores the effects of gravity on the slingshot dynamics and the
positive $y$ direction is set facing the upward direction. b We consider the
case of a pure 1D motion where at $t<0$, the radial lines forces are balanced
by the tension line force. At $t=0$, the spider releases the tension resulting
in a net force in the positive (upward in this particular case) y-direction
caused by the radial lines and opposed by the air drag. When the spider
crosses the equilibrium point, the tension line starts stretching, resulting
in a restoring downward force. The spider continues oscillating, while slowly
approaching equilibrium.

### 2D web geometry

We define $\phi_{r1}$ and $\phi_{r2}$ as the angles formed between the radial
lines and the $x$-axis and $\phi_{t}$ as the angle between the tension line
and the $y$-axis (Fig. 2b). We prescribe two equilibrium points to the model
(Fig. 2b). The first equilibrium point is before the spider launches ($t<0$,
$\phi_{r1,2}=\phi_{r0}$, Fig. 2c), when the spider has deformed the web into a
cone and waits for a prey to come within striking distance. At this point the
forces in the radial springs are balanced by the force in the tension line and
the system is in a static equilibrium. The second equilibrium point is defined
when the all the web lines (radial and tension line) are at their equilibrium
lengths ($L_{r}$=$L_{r,eq}$ , $L_{t}$=$L_{t,eq}$ , $\phi_{r1,2}$ = $\phi_{t}$
= 0, Fig. 2c). This point occurs when the spider’s motion has ceased and all
the web forces are equal to zero (Fig. 2c). We note that in the actual
slingshot spider web these equilibrium points may be shifted due to the
asymmetric architecture of the web. However for the purposes of our model,
this assumption is valid as we are assuming that the springs are linear and
that the variations in the forces are more important to the dynamics compared
to the absolute base forces themselves. In a sense, this is similar to the
treatment of a vertical mass-spring system where the effects of gravity are
ignored since the equilibrium point is defined when the weight is balanced by
the initial extension of the spring.

### Elastic silk springs

We make two assumptions to model the silk web lines (radial and tension lines)
as elastic force-generating springs. First, the silk lines are approximated as
linear Hookean springs ($F=K\cdot\Delta L$) with no internal viscous damping.
We recognize that this assumption is a significant simplification since actual
silk fibers behave as complex and nonlinear viscoelastic materials (Tietsch et
al., 2016; Yu et al., 2015; Gosline et al., 1986, 1984; Kelly et al., 2011).
The spring constant $K$ is defined by $K=EA/L_{eq}$ , where $A$ is the cross-
sectional area of the silk measured using SEM images (See SI text Fig. 4), and
$L_{eq}$ is the equilibrium length and $E$ is the silk’s young modulus. The
silk’s young modulus ranges between $\sim 1-10$ GPa and is obtained from the
nonlinear stress-strain curves of the major ampullate (MA) silk of various
spider species (Gosline et al., 1999; Ko et al., 2011) .

Second, unlike conventional springs that can exert a push or pull force
depending on compression or extension, our modeled silk springs react only to
extension to exert a pulling force - they cannot push (Jung et al., 2014).
Mathematically, we incorporate this through the Heaviside function
$\Theta(\Delta L)$, which is defined as $\Theta(\Delta L)=1$ when the silk
spring is extended ($\Delta L>0$) and $\Theta(\Delta L)=0$ for compression
$\Delta L<0$.

### Energy dissipation to the environment through aerial drag

The energy of the system is dissipated by viscous drag to the environment by
the fast moving spider and web. We ignore any friction that might arise
between the spider pedipalps. Viscous damping in the silk lines itself is also
ignored as discussed earlier. The aerodynamic drag of an object traveling
through a fluid (air) depends on several factors such as its geometry,
dimensions and flow conditions. The Reynolds number, which determines the
effect of inertial forces with respect to viscous forces, is typically
calculated to identify the type of damping involved in the system. The
Reynolds number is estimated as $Re=\rho$ $V_{max}$D/$\mu$, where $\rho$=
1.225 kg/m3 is the density of air, $V_{max}$ is the spider speed, D is the
characteristic length and $\mu$ =$1.81\times 10^{-5}$ kg/(m s) is the
viscosity of air. The Reynolds number is calculated for both the spider and
the web below.

The characteristic length of the spider is estimated to be
$D_{s}\sim\;1.75\;mm$ (fig. 2a) yielding a $Re_{s}$ $\sim\;$ 280\. The spider
drag is approximated by flow around a sphere at finite Reynolds number as
$F_{d,s}=C_{s}V_{s}^{2}$, where $C_{s}=\frac{1}{2}C_{d}A_{s}\rho_{air}$,
$C_{d}$ $\sim$ 1.25 is the drag coefficient at $Re\sim\;$280 and
$A_{s}=\frac{\pi}{4}D_{s}^{2}$ is the characteristic cross-sectional area, and
$V_{s}$ is the spider instantaneous speed (Vogel, 2009).

Since the silk fibers are orders of magnitude smaller than the spider itself,
we expect the Reynolds number for the silk to be much lower. Assuming each
silk line as a cylinder that pivots around a stationary substrate so that the
web moves at half the speed as the spider ($V_{w}=V_{s}/2$), we obtain a
Reynolds number $Re_{w}\sim$ 0.1. For these low Reynolds numbers, we can
estimate the drag on the silk lines using slender body theory in Stoke flows
(Gary Leal, 2007) as: $F_{dw}=\tfrac{4\pi L_{w}\mu
V_{w}}{ln{\frac{L_{w}}{D_{w}}}}$, where $L_{w}$ and $D_{w}$ are the web length
and diameter, respectively (see Table 1). Due to the principle of linearity in
low Reynolds flow, we can add up the drag contributions from all the radial
and capture lines by measuring them and summing up the drag forces. A summary
of the values used in the model are provided in Table 1.

|  | Simulation 1 | Simulation 2 |
---|---|---|---|---
| Parameter | Value | Value | Source
Spider | Mass, $m_{s}$ (Kg) | 1.6$\times 10^{-3}$ | 1.6$\times 10^{-3}$ | Measured
| Characteristic Length, $D_{s}$ (m) | 1.3$\times 10^{-3}$ | 1.3$\times 10^{-3}$ | Measured
| Body Length, BL (m) | 1.75$\times 10^{-3}$ | 1.75$\times 10^{-3}$ | Measured
| Damping coefficient, $C_{s}$ (Kg/s) | 8.3$\times 10^{-7}$ | 8.3$\times 10^{-7}$ | Calculated
Radial lines | Number, $N_{r}$ | 8 | 8 | Field Data
| Young’s Modulus, $E_{r}$ (Pa) | 0.35$\times 10^{10}$ | 0.45$\times 10^{10}$ | (Gosline et al., 1999)
| Diameter, $D_{r}$ (m) | 1$\times 10^{-6}$ | 1$\times 10^{-6}$ | SEM
| Length, $L_{r}$ (m) | 4.5$\times 10^{-2}$ | 4.5$\times 10^{-2}$ | Measured
| Equilibrium Length, $L_{r,eq}$ (m) | 2.34$\times 10^{-2}$ | 2.34$\times 10^{-2}$ | Measured
Tension line | Young’s Modulus, $E_{t}$ (Pa) | 0.35$\times 10^{10}$ | 0.45$\times 10^{10}$ | (Gosline et al., 1999)
| Diameter, $D_{t}$ (m) | 1$\times 10^{-6}$ | 1$\times 10^{-6}$ | SEM
| Length, $L_{t}$ (m) | 4$\times 10^{-2}$ | 4$\times 10^{-2}$ | Measured
| Equilibrium Length, Lt,eq (m) | 6$\times 10^{-2}$ | 6$\times 10^{-2}$ | Measured
Capture lines | Number per radial line | 13 | 13 | Measured
| Length, $L_{c}$ (m) | 6.5$\times 10^{-3}$ | 6.5$\times 10^{-3}$ | Calculated
Total web lines | Total Length, $L_{w}$ (m) | 1.03 | 1.03 | Calculated
| Damping coefficient, $C_{w}$ (Kg/s) | 2.41$\times$10-4 | 2.41$\times$10-4 | Calculated
Initial Conditions |  $x_{o}$ (m) | -4.5$\times 10^{-4}$ | -18.5$\times 10^{-4}$ |
|  $y_{o}$ (m) | -12$\times 10^{-3}$ | -12$\times 10^{-3}$ |
Table 1: Simulation parameters and their values. Summary of the geometrical
parameters and physical properties used in the simulation. The source of these
parameters is either from literature or measure from field data. The results
of simulations 1 and 2 are shown in Fig. 4a and Fig. 5a, respectively.

### 2-D Equations of Motion

Considering all the forces (inertia, web elasticity, and drag) along $x$ and
$y$ directions, we write the 2D equations of motion that describe the
trajectory of the slingshot spider as a function of time as follows:

$\begin{split}\vspace{-2em}\textbf{x-axis}&:m_{s}\ddot{x}=-F_{r1x}+F_{r2x}-F_{tx}-F_{dsx}-F_{dwx}\\\
\textbf{y-axis}&:m_{s}\ddot{y}=-F_{r1y}-F_{r2y}-F_{ty}-F_{dsy}-F_{dwy}\\\
\end{split}$ (1) where: $\displaystyle F_{r1,2x}=\Delta
L_{r1,2}\;K_{r}\;\cos\phi_{r1,2}\;,\;F_{r1,2y}=\Delta
L_{r1,2}\;K_{r}\;\sin\phi_{r1,2}$ $\displaystyle\;F_{tx}=\Delta
L_{t}\;K_{t}\;\sin\phi_{t}\;,\;F_{ty}=\Delta L_{t}\;K_{t}\;\cos\phi_{t}$
$\displaystyle
F_{dwx}=C_{w}\;\frac{\dot{x_{s}}}{2}\;,\;F_{dwy}=C_{w}\;\frac{\dot{y_{s}}}{2}$
$\displaystyle F_{dsx}=C_{s}\;\
(\dot{x_{s}}^{2}+\dot{y_{s}}^{2})\cos(\arctan{\frac{\dot{y_{s}}}{\dot{x_{s}}})},\;F_{dsy}=C_{s}\;(\dot{x_{s}}^{2}+\dot{y_{s}}^{2})\sin(\arctan{\frac{\dot{y_{s}}}{\dot{x_{s}}})}$
$\displaystyle\Delta
L_{r1,2}=L_{r1,2}-L_{r,eq}\;,\;L_{r1,2}=\sqrt{L_{r,eq1,2}^{2}\pm
x^{2}}-L_{r,eq1,2}$
$\displaystyle\sin\phi_{r1,2}=\frac{y}{L_{r1,2}}\;,\;\cos\phi_{r1,2}=\frac{L_{r,eq1,2}\pm
x}{L_{r1,2}}\;,\;\sin\phi_{t}=\frac{x}{L_{t}}$ $\displaystyle
K_{r1,2}=\frac{E_{rA_{r}}}{L_{r,eq1,2}}\;,\;K_{t1,2}=\frac{E_{tA_{t}}}{L_{t,eq1,2}}$

At $t<0$, the spider is at static equilibrium at a position ($x_{o}$,$y_{o}$)
where the radial spring forces ($F_{r1,2}$) are opposed by the force ($F_{t}$)
exerted by the tension line (Fig. 2c1). At $t=0$, the tension force is set to
zero resulting in a net upward motion. This is akin to the spider launching
itself when triggered by an external stimulus. The $x$-$y$ trajectories are
obtained by numerically solving the equations (1) and (2) using the 4th order
Runge-Kutta approach in Matlab. The system starts from static equilibrium at
($x_{o}$,$y_{o}$) at $t=0$. The simulation is stopped after $200\;ms$ has
elapsed.

## Results and Discussion

### Web dynamics in 1D motion

Figure 3: Defining key parameters of slingshot spider trajectories. We focus
on several spatial and temporal parameter to assess the fit between the
mathematical model and the field data. The equilibrium point is set as the
final displacement ($y_{eq}$) when the spider comes to a halt. We define the
rise time ($t_{s}$) as the duration from $y=0$ to the first time the spider
cross the equilibrium point ($y_{eq}$) and the peak time ($t_{p}$) is the time
taken to reach the maximum displacement ($y_{max}$). We measure the overshoot
as the ratio between the max displacement and the equilibrium point
($\frac{y_{max}-y_{eq}}{y_{eq}}$). These variables are highlighted here on the
average normalized displacements ($n=5$) in the vertical direction of a single
slingshot spider. Finally, we define the settling time as the time it takes
the spider to finish one oscillation with an amplitude that falls between
$\pm$ one body length of the spider.

To understand the dynamics of the system, we first consider the limiting 1D
case, where the motion is purely in the y-direction ($\phi_{t}=0$ ,
$x=0\;,\;\forall t$ ) (Fig. 2c). This assumption may be valid during the
initial stages of the motion, where the displacement is almost one-dimensional
in the y-direction and relatively negligible in other dimensions (Fig. 1c-d).
However, this assumption breaks down during the later stages of the
displacement as motion gets more complex. Nevertheless, this approach helps to
shed light on several key features of the slingshot motion as well as
providing guidance to the iterative exercises of fitting the model to the
data.

We assess the goodness of the fit by using four different spatiotemporal
parameters: peak time ($t_{p},ms$), settle time ($t_{s},ms$), overshoot
($\%$), and rise time ($t_{r},ms$), which are graphically defined in Fig. 3.
The peak time is duration that the spider takes to go from initial position to
the maximum displacement $y_{max}$. The overshoot is the percentage offset the
spider travels with respect the equilibrium point
$\frac{y_{max}-y_{eq}}{y_{eq}}$. The settling time ($t_{s}$) is the duration
that the spider take to go from the initial peak displacement to the first
oscillatory peak that falls between $\pm$ on the body length (BL) above the
equilibrium point. We assume that at this point, the slingshot motion has
practically come to a halt and the spider is able to reset the elastic
loading.

The rise time $t_{s}$ corresponds to the time the spider takes to go from
$y=y_{o}$ to $y=y_{eq}$ during the first oscillation (Fig. 3). During this
time $t<t_{s}$, the spatiotemporal dynamics are mainly governed by the radial
line forces $F_{r}$ along with drag $F_{d}$, while the tension line force
$F_{t}$ is zero ($\Delta L_{t}<0$). The initial rise time is
$t_{s}=2\pi$/$\omega_{d,n}$ where $\omega_{d,n}$ is the damped natural
frequency. Once $\Delta L_{t}\geq 0$, the tension line becomes engaged and the
restoring force $F_{t}$ starts influencing the dynamics of the system.
Specifically, the damped natural frequency becomes proportional to the
combined effect of both the radial lines and the tension line (
$\omega_{n}\propto(K_{r}+K_{t})^{\frac{1}{2}}$ when $y>y_{eq}$ ).

The mass subsequently oscillates around the equilibrium point with diminishing
amplitudes before eventually stopping as the kinetic and potential energies
get dissipated by drag. An interesting geometrical feature of the model is
that as the mass approaches the equilibrium point ($y\rightarrow 0$,
$\phi\rightarrow 0$), the projected radial forces in the y direction drop
sharply.

### Slingshot Dynamics in a 2D motion

Next, we solve the equations of motion in 2D while changing the physical and
geometrical parameters to better fit the displacement of the spider. To avoid
major variations in geometry and initial conditions, we validate the
simulation outputs with multiple firing events ($n=5$) of a single spider
(Fig. 1b-d). By matching the initial conditions of the model $(x_{o},y_{o})$
to the experimental data, we show the output of the model in Fig. 4a, referred
to as Simulation 1. The model captures the rise time, peak time and maximum
peak with less than $5\%$ error as quantified in Table 2. However, the model
does not accurately capture the secondary oscillations ($t>30$ ms), and also
predicts a $\sim\;20\;ms$ longer setting time compared to experimental
observations. In the x-direction, the model captures both the amplitudes and
frequency of the oscillations with a phase shift, as seen qualitatively with
the $x$-$y$ map (Fig. 4d).

The temporal evolution of the underlying forces is also computed and
highlighted in conjunction with the y-displacement. The radial forces
projected into the y-direction start with a combined maximum value of around
$\sim 7.5\times$10-4 N or $75$ dynes. Interestingly, this falls within the
force ranges previously measured in the radial silk lines of slingshot spider
webs using a custom-built tension apparatus by Coddington (Coddington, 1986).
The net radial forces in the $x$-direction are relatively smaller and become
more significant at $t>30\;ms$ as the spider approaches equilibrium. The
tension line force $F_{t}$ is only enabled when $\Delta L_{t}>L_{t,eq}$, and
is observed as a sharp increase in the negative direction (retarding spider
motion) around $t=\;14\;ms$ (Fig. 4b). The drag forces in the $y$-direction
start from zero and rapidly increase in magnitude as the spider approaches
$V_{s,max}$ before decreasing as the mass comes to a halt. We note that the
radial web forces and drag forces are always acting in opposite directions -
the web forces driving motion and drag damping it.

Figure 4:  Simulation 1: Slingshot spider dynamics in 2-D. a We obtain the
$x$-$y$ displacement of the spider by solving the equations of motion (Eq. 1).
The simulation output matches the rise time, maximum peak and maximum output
$y_{max}$, as well the minimal oscillations in the $x$ direction. However,
beyond the first fluctuation, the model does not predict any subsequent
oscillations in the $y$-direction. b The temporal evolution of the forces
reveals that the system starts with forces in the radial lines in the $x$-$y$
directions. As the system picks up speed, drag from the web resists the motion
- drag from spider is negligible. The restoring force in the tension line
which has been zero so far, steps in as the tension line becomes stretched
($t\sim 15\;ms$). c,d  The simulation also predicts the apparent 2D
displacement of the spider starting from from ($x_{o},y_{o}$) at $t=0$. The
simulation show a qualitatively similar trajectory in the $x$-$y$ plane, as
the spider initially catapults in the $y$-direction before undergoing low
amplitude oscillations in the $x$-direction as it slowly approaches
equilibrium.

Two underlying assumptions made in this model are that the trajectory of the
slingshot spider is confined in a 2D plane and that the spider behaves as a
point mass object. In reality, the slingshot motion is a more complex 3D
motion, but we only record a projection of the motion in 2D with a single
high-speed camera in the field. Moreover, the mass of the spider is not
equally distributed (asymmetric body) with most of its mass not facing the
tension line. This unequal distribution of mass causes the spider to behave
like a 3D inverted pendulum that can rotate around the point of intersection
between the spider and the silk (as seen in SI Movie 1) (Han et al., 2019).
The model also ignores any displacement biases that might arise due to
asymmetries in the web structure, since the spider constructs the web in small
plant branches. To highlight the consequences of displacement biases in an
orthogonal direction ($z$ direction), we consider an exaggerated case in the
initial value of $x$ where we run the simulation after multiplying $x_{0}$ by
4. The results shown in Fig. 4 show better agreement between the model and
field data, and the secondary oscillations in the experimental data are
matched well by the model, including matching of the settling time (Table 2).

Figure 5:  Simulation 2 with larger contributions from $x$-motion. a To capture the dynamics of the subsequent vibrations in the $y$-direction, we run a second simulation with an exaggerated initial displacement in the x-direction (4 $\times$ the one used in simulation 1). This highlights the additional layer of complexity imposed by an amplified orthogonal motion. The model matches the field data better in terms of the temporal parameters as well as the subsequent low amplitude fluctuations after the first overshoot. In reality, the spider motion is 3D, which may include twisting and rotation due to asymmetries in mass and geometry (See SI Movie 1). These extraneous influences become more significant at low amplitudes and frequencies. b The evolution of the forces in the $x-y$ plane show similar dynamics as seen earlier in Fig. 4b. Interestingly, the tension force in the $x$-direction becomes more pronounced here as the influence in the $x$-direction is amplified due to the initial conditions. Parameter | Field data | Simulation 1 | Rel. error % | Simulation 2 | Rel. error %
---|---|---|---|---|---
Rise time, $t_{r}$ | 9.48 ms | 9.6 ms | 1.25% | 9.6 ms | 1.25%
Peak, $y_{max}$ | 14.58 mm | 15.07 mm | 3.36% | 14.81 mm | 1.57%
Peak time, $t_{p}$ | 20.81 ms | 20.25 ms | 2.76% | 20.5 ms | 1.53%
Overshoot % | 17.93 % | 30.5 % | 12.57% | 23.89 % | 5.96%
Settling time, $t_{s}$ | 37.84 ms | 60 ms | 58.5% | 40 ms | 5.71%
Table 2: Comparison of simulation to single representative field data point
Summary of the spatiotemporal parameter outputs of simulations 1 and 2.

In summary, despite the major dimensional reductions assumed in developing
this simplified mathematical model, our model still captures salient
spatiotemporal dynamic features that arise due to the complex design of the
slingshot spider web. The values used in the model fall within the biological
boundaries in terms of physical properties and geometrical constraints (Tables
1,2). Next we explore the rich parameter space provided by this model, by
evaluating the web dynamics under the influence of changing web stiffness
$K_{t}$, tension line stiffness $K_{r}$, and drag force coefficient $C_{w}$.

### Slingshot spider dynamics as a function of web parameters
($K_{t},K_{r},C_{d}$)

Here we explore the effect of three parameters - radial stiffness $K_{r}$,
tension line stiffness $K_{t}$ and web drag $C_{w}$ \- on the dynamics of the
slingshot motion. These parameters capture most of the mechanical and
geometrical characteristics of the web and dictate the significant dynamics of
the system. Simulations are performed by changing one of these parameters
while keeping the others constant. For a convenient physical interpretation,
we normalize the output displacements by the spider body length (BL) while
using the results and parameters ($K_{t,ss}$, $K_{r,ss}$ and $C_{w,ss}$) from
Simulation 2 as a reference. We showcase the resulting normalized displacement
curves with respect to the reference plot in Fig. 6 and compare the
sensitivity of the model outputs in terms of the aforementioned spatiotemporal
parameters in Fig. 7.

Figure 6: Exploring web dynamics as a function of model parameters. We explore
the impact of the different forces on the slingshot dynamics obtained from
Simulation 2. The output displacements are normalized by the spider body
length (BL) and parameters ($K_{r,ss}$, $K_{r,ss}$ and $C_{w,ss}$) from
Simulation 2 (Fig. 5a) as a reference. a Increasing the radial lines stiffness
$K_{r}$ leads to both a faster increase in the initial displacement as well as
higher overshoots ($\sim$4 BL above equilibrium) with subsequent increasingly
large amplitude vibrations around the equilibrium point. An order of magnitude
decrease in the radial lines stiffness causes the system to slowly approach
the equilibrium point without any observable overshoot. b Changing the tension
line stiffness $K_{t}$ only affects the dynamics past the equilibrium line.
Without any tension force the system achieves a higher overshoot and more
subsequent vibrations. As the tension line stiffness increases, the vibrations
peaks become sharper above the equilibrium with larger and slower undershoot.
c. The model reveals high sensitivity to web drag $C_{w}$. Halving the damping
coefficient causes the system to significantly vibrate over two body lengths
around the equilibrium line. Doubling the damping coefficient leads to a small
overshoot as the system quickly approaches equilibrium without any noticeable
oscillations, similar to critically damped oscillator.

Fig. 6a shows the normalized displacement curves when the radial line’s
stiffness $K_{r}$ is changed while everything else remains the same. We
observe that, for an order of magnitude decrease in the value of $K_{r}$, the
web loses it’s ability to oscillate and slowly asymptotes to the equilibrium
point. This is similar to an overdamped system where drag dissipation
dominates. Increasing $K_{r,ss}$ causes a faster rise in the displacement as
well as a larger overshoot. These results are quantified in Fig. 7 which shows
that as the radial stiffness increases, the spider travels a longer distance
in a shorter time duration and takes a shorter time to settle. We (and others
(Eberhard, 1986; Coddington, 1986)) have observed that slingshot spiders fire
their webs in response to external vibratory/sound clues (finger snapping). It
is therefore possible that spiders could release their webs in response to the
nearby frequency of an insect’s wing beat. Furthermore, it is key that the
spider resets its web rapidly, as to not miss out on potential prey. Our model
reveals that the slingshot $K_{r}$ appears to balance finite overshoot at
smaller peak and settle times, which could facilitate reaching a flying prey
at a distance quickly, with minimizing oscillations to reset, and repeat
motion if unsuccessful.

From the biomechanics context of the slingshot spiders motion, what does it
mean to change parameters such as the stiffness of the radial lines? Stiffness
depends on intrinsic molecular and geometric properties and is defined as
$K_{r}=\frac{N_{r}E_{r}A_{r}}{L_{r,eq}}$. These properties vary among
different species of spiders, and have never been measured specifically for
Epeirotypus sp.. However, the young’s modulus ranges between 1-10$\times
10^{9}$ Pa in the literature, depending upon external mechanical conditions
such as such as strain, strain rate and environmental conditions such as
temperature and humidity (Su and Buehler, 2016; Yazawa et al., 2020; Agnarsson
et al., 2009). Another strategy to modify the radial stiffness would be
changing by other geometrical parameters such as the equilibrium length
$L_{r,eq}$ or the area $A_{r}$. For example Eberhard reports direct
observations of Epeirotypus sp. using its legs to adjust the tension in the
radial and sticky lines (by effectively changing $L_{r,eq}$) during web
construction (Eberhard, 1981).

Next, we examine the role of the tension line highlighted in Fig. 6b. Since
the tension line is not originally stretched ($\Delta L_{t}<L_{t,eq}$),
varying the tension line stiffness does not affect the rise time $t_{s}\approx
10$ ms. Beyond this point, the tension line is engaged and applies a pulling
force on the spider. In the limit of no tension line ($K_{t}\;=0$), the system
overshoots to almost two spider BL above the equilibrium point before slowly
oscillating back to equilibrium. As the stiffness of tension increases, the
overshoot decreases while the undershoot increases. For instance, at
$10\;K_{t,ss}$, the maximum displacement decreases to less than one body
length above the equilibrium before bouncing into almost two body lengths
below the equilibrium point. This is due to the fact that at slightly above
equilibrium, the time scale is mostly determined by the properties of the
tension line, namely
$t=\frac{2\pi}{\omega_{nd}}\sim\sqrt{\frac{m_{s}}{K_{t}}}$. Fig. 7b summarizes
these results showing a monotonic decrease from 30$\%$ to almost 8$\%$ across
four order of magnitudes in $K_{t}$. Meanwhile, the peak time is not as
strongly influenced, decreasing by only $\sim 10\;ms$ over four order span in
$K_{t}$. Interestingly, we observe a minimum of around $40\;ms$ near the
reference value in the settling time due to the increase in the undershoot as
$K_{t}$ increases.

From a biomechanics context, our model suggests that, besides allowing the
spider to load elastic energy and keep the web in static equilibrium, the
tension line also plays an integral role during the slingshot motion. Without
a tension line, the slingshot spider is completely at the mercy of air drag
near the equilibrium point as radial forces become negligible due to geometry
(see SI text Fig. 2). The tension line stiffness assists in preventing too
much overshoot while decreasing the settling time. Controlling the overshoot
and settling the web faster may allow the spider to reach the insect in its
web faster. It may also reduce the time to reset and repeat its hunting motion
in case of a missed prey capture. The tension line also allows the spider to
control its vertical displacement at intermediate tension line lengths
actively as observed in field experiments (See SI Movie 2).

Finally, we look at the effect of the web drag on the slingshot dynamics.
Compared to the other two parameters, we observe that the model is highly
sensitive to the drag coefficient.At a high damping coefficient (2 to
10$\times C_{w,ss}$), the system goes rapidly to an overdamped regime whereby
the system gradually approaches equilibrium without any oscillations. At lower
damping coefficient (0.1 to 0.5$\times C_{w,ss}$), the system becomes highly
underdamped and vibrates with several body lengths around the equilibrium
point. This is further highlighted in Fig. 7c, which shows that overshoot may
reach up to 60$\%$ at 0.01 $C_{w,ss}$. At values larger than 10 $C_{w,ss}$,
the overshoot is minimal or even non existent. The high sensitivity of the
model to damping may be further exemplified in the settling time which shows a
minimum near the reference value $C_{w,ss}$. Above the minimum value, the
system is overdamped and takes a long time to reach the $\pm$ 1BL envelope
around equilibrium. Below that minimum, the system becomes highly underdamped
and fluctuates with high amplitudes around equilibrium.

Figure 7: Summary of web dynamics as a function of model parameters. We
quantify the changes of the various simulations by focusing on the percentage
overshoot, peak time ($t_{p}$) and settling time ($t_{s}$). We highlight the
reference values in red a. The overshoot increases monotonically from 0 (no
peak) to 80 $\%$ with the increase in $K_{r,ss}$. At low $K_{r,ss}$, the
system does not vibrate around the equilibrium point causing $t_{p}$ and
$t_{s}$ to go beyond the max simulation time of 200 ms. This behavior is
tantamount to an overdamped dynamical system as the damping forces overcome
the spring forces causing the .  b. The overshoot decreases by almost 35$\%$
as $K_{t}$ increases from $0.01$ to $100\times K_{t}$. Similarly, the peak
time $t_{p}$ slightly decreases from $24\;ms$ to $15\;ms$ as $K_{t}$
increases. Interestingly, the settling time ($t_{s}$) decreases to a minimum
of $40\;ms$ around the reference value of $K_{t}$ before sharply increasing.
This is mainly due to the larger undershoot causes by a higher $K_{t}$ c. The
simulation is highly sensitive to the damping coefficient. The overshoot
decreases from around 60$\%$ to almost no overshoot as damping increases. The
settling time shows a minimum around the reference point as low drag causes
the system to oscillate violently around the equilibrium point while a high
drag causes it to slowly approach the equilibrium. Both effects yield an
increase in the settling time.

### Limitations of the model and future work

Our work aims to provide a reduced-order approach to modeling the complex
dynamics in slingshot spider webs. However, we make certain assumptions in our
model that can be further improved in future work.

#### Dissipation due to aerodynamic drag and molecular friction

Our model shows that aerodynamic drag at an intermediate to low Reynolds
number adequately describes the damping experienced by the spider and its web.
The model ignores the possible contribution of viscous dissipation within the
viscoelastic spider threads. The relative effect of these two sources of
dissipation in oscillating spider webs is underexplored and still an active
area of research. Past work highlighted the importance of aerodynamic drag in
the context of “ballooning spiders” (Sheldon et al., 2017; Suter, 1992) and
the stopping of flying insects by orb weavers webs (Sensenig et al., 2012;
Kelly et al., 2011). Other work however completely dismissed aerodynamic drag
and attributed dissipation primarily to internal viscous damping (Ko, 2004;
Aoyanagi and Okumura, 2010). A summary of studies exploring the role of
dissipation, internal and external, is provided in SI Table 1.

Owing to the fineness of spider silk having a diameter $<4\;\mu m$ (Ko, 2004),
the characterization of the mechanical properties of the spider silk had been
mainly experimentally limited to tensiometry or impulsive loading at lower
strain rates. Replicating the high strain rates faced by the webs of the
slingshot spiders studied in this work ($>60\;s^{-1}$) is experimentally
challenging. In addition, the mechanical properties and structural geometries
of spiders’ web silk are highly dependent on environmental conditions such as
temperature, humidity, wind conditions, water content (supercontractility) and
surrounding conditions such as the location and flexibility of the substrate
(Gosline et al., 1999). This poses further difficulties in replicating the
native conditions of spiders to experimentally examine the mechanical
properties of the spider silk. Overall, our model reveals that viscous damping
(in silk) is not necessary to capture the underdamped behaviour in the
slingshot spider, but it could still be important and will be focus of future
work (See SI text fig. 3).

#### Geometrical constraints

Our simulations also assume no coupling between the parameters. For instance,
orb webs vary tremendously across species in the numbers of radii and rows of
capture spirals (Craig, 1987; Sensenig et al., 2010)), but slingshot spider
webs are relatively similar to one another in having relatively few radii and
around ten rows of capture silk in their webs (Eberhard, 1986). This might
suggest geometric constraints on slingshot spider web topology and dimensions.
Thus, this model may be extended to examine other web topologies and silk
dimensions to test the hypothesis that the evolution of theridiosomatid web
architecture is limited in part by optimizing the amount of aerial damping
during the slingshot motion

## Conclusion

We develop a 2D mathematical model to simulate the dynamics of the slingshot
spider powered by its conical web geometry and tension line. We validate the
model with experimental results and explore the sensitivity of various
physical parameters governing web forces. We find that web parameters are
finely-tuned to yield an underdamped oscillating web, that enables the spider
to displace finite distances quickly, while minimizing residual oscillations
in the web. These design parameters may enable the spider to exploit a risky
hunting strategy of catching flying insects in mid-air while minimizing
oscillations in its web due to its rapid movements. At the same time, the
spider should be able to sense the vibrations induced by the flying preys and
discern them from possibly faulty ones induced by the surrounding ambient air
(Craig et al., 1985; Eberhard, 1981). Though we have presented a first
approach for understanding this fascinating slingshot spider dynamics, open
questions remain about the molecular structure of the radial silk and tension
line, as well as their 3D-web topology that enables their agile power packed
performance. These research questions open up rich avenues for
multidisciplinary research, while furthering our knowledge and appreciation of
arachnids and their ingenious engineering strategies for locomotion and
survival.

## Declarations

### Funding

S.J. acknowledges funding support from the NSF under grant no. CBET-2002714.
M.S.B acknowledges funding support through NSF award number 1817334 and CAREER
1941933 and National Geographic Foundation (NGS-57996R-19). T.A.B.
acknowledges funding support from the NSF (IOS-1656645)

### Conflicts of interest

The authors declare no competing interests.

### Availability of data and material

All Matlab codes and data for this article are accessible here:
https://github.com/bhamla-lab/slingshotspider2021

### Ethics approval

All applicable international, national, and institutional guidelines for the
care and use of animals were followed.

### Author’s contribution

SA, MSB collected field data. EJC, SA, SJ developed mathematical model. EJC,
SA and SH analyzed data. EJC conducted simulations. All authors contributed to
editing, interpreting and writing the manuscript. TAB, SJ and MSB managed
funding and resources.

###### Acknowledgements.

We thank Jaime Navarro for his excellent field guide services in the Peruvian
Amazon Rainforest.

## References

  * Agnarsson et al. (2009) Agnarsson I, Dhinojwala A, Sahni V, Blackledge TA (2009) Spider silk as a novel high performance biomimetic muscle driven by humidity. The Journal of experimental biology 212(Pt 13):1990–1994
  * Alexander and Bhamla (2020) Alexander SLM, Bhamla MS (2020) Ultrafast launch of slingshot spiders using conical silk webs. Current biology: CB 30(16):R928–R929
  * Alves et al. (2007) Alves DdA, Pioker FC, Ré-Jorge L, do Nascimento SM (2007) Funcoes do compartamento disparo da teia de naatlo sp.(aranea), theridiosomatidae. Ecologia da mata Atlantica
  * Aoyanagi and Okumura (2010) Aoyanagi Y, Okumura K (2010) Simple model for the mechanics of spider webs. Phys Rev Lett 104:038102
  * Coddington (1986) Coddington JA (1986) The genera of the spider family theridiosomatidae. Smithson Contrib Zool
  * Coddington (2005) Coddington JA (2005) Theridiosomatidae. Spiders of North America: an identification manual
  * Craig (1987) Craig CL (1987) The ecological and evolutionary interdependence between web architecture and web silk spun by orb web weaving spiders. Biological Journal of the Linnean Society 30(2):135–162
  * Craig et al. (1985) Craig CL, Okubo A, Andreasen V (1985) Effect of spider orb-web and insect oscillations on prey interception. Journal of theoretical biology 115(2):201–211
  * Das et al. (2017) Das R, Kumar A, Patel A, Vijay S, Saurabh S, Kumar N (2017) Biomechanical characterization of spider webs
  * Eberhard (1981) Eberhard WG (1981) Construction behaviour and the distribution of tensions in orb webs. Bulletin of the British Arachnological Society 5(5):189–204
  * Eberhard (1986) Eberhard WG (1986) Ontogenetic changes in the web of epeirotypus sp. (araneae, theridiosomatidae). The Journal of arachnology 14(1):125–128
  * Eberhard (1990) Eberhard WG (1990) Function and phylogeny of spider webs. Annual review of ecology and systematics 21(1):341–372
  * Gary Leal (2007) Gary Leal L (2007) Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press
  * Gosline et al. (1984) Gosline JM, Denny MW, DeMont ME (1984) Spider silk as rubber. Nature 309(5968):551–552
  * Gosline et al. (1986) Gosline JM, DeMont ME, Denny MW (1986) The structure and properties of spider silk. Endeavour 10(1):37–43
  * Gosline et al. (1999) Gosline JM, Guerette PA, Ortlepp CS, Savage KN (1999) The mechanical design of spider silks: from fibroin sequence to mechanical function. The Journal of experimental biology 202(Pt 23):3295–3303
  * Han et al. (2019) Han SI, Astley HC, Maksuta DD, Blackledge TA (2019) External power amplification drives prey capture in a spider web. Proceedings of the National Academy of Sciences of the United States of America 116(24):12060–12065
  * Hingston (1932) Hingston RWG (1932) A Naturalist in the Guiana Forest. Longmans, Green
  * Jung et al. (2014) Jung S, Clanet C, Bush JW (2014) Capillary instability on an elastic helix. Soft matter 10(18):3225–3228
  * Kelly et al. (2011) Kelly SP, Sensenig A, Lorentz KA, Blackledge TA (2011) Damping capacity is evolutionarily conserved in the radial silk of orb-weaving spiders. Zoology 114(4):233–238
  * Ko (2004) Ko FK (2004) Engineering Properties of Spider Silk Fibers, Springer US, Boston, MA, pp 27–49
  * Ko et al. (2011) Ko FK, Kawabata S, Inoue M, Niwa M, Fossey S, Song JW (2011) Engineering properties of spider silk. MRS Online Proceedings Library 10.1557/PROC-702-U1.4.1
  * Kuo (1987) Kuo BC (1987) Automatic Control Systems, 5th edn. Prentice Hall PTR, USA
  * Sensenig et al. (2010) Sensenig A, Agnarsson I, Blackledge TA (2010) Behavioural and biomaterial coevolution in spider orb webs. Journal of Evolutionary Biology 23(9):1839–1856
  * Sensenig et al. (2012) Sensenig AT, Lorentz KA, Kelly SP, Blackledge TA (2012) Spider orb webs rely on radial threads to absorb prey kinetic energy. J R Soc Interface 9(73):1880–1891
  * Sheldon et al. (2017) Sheldon KS, Zhao L, Chuang A, Panayotova IN, Miller LA, Bourouiba L (2017) Revisiting the Physics of Spider Ballooning. Composites Part A: Applied Science and Manufacturing Women in Mathematical Biology pp 163–178
  * Su and Buehler (2016) Su I, Buehler MJ (2016) Spider silk: Dynamic mechanics. Nature materials 15(10):1054–1055
  * Suter (1992) Suter RB (1992) Ballooning: Data from spiders in freefall indicate the importance of posture. The Journal of Arachnology 20(2):107–113
  * Tietsch et al. (2016) Tietsch V, Alencastre J, Witte H, Torres FG (2016) Exploring the shock response of spider webs. J Mech Behav Biomed Mater 56:1–5
  * Vogel (2009) Vogel S (2009) Glimpses of Creatures in Their Physical Worlds. Princeton University Press
  * Wienskoski (2010) Wienskoski E (2010) The genus naatlo (araneae: Theridiosomatidae): distribution and taxonomic history. Revista Brasileira de Biociências 8(2)
  * Yazawa et al. (2020) Yazawa K, Malay AD, Masunaga H, Norma-Rashid Y, Numata K (2020) Simultaneous effect of strain rate and humidity on the structure and mechanical behavior of spider silk. Communications Materials 1(1):10
  * Yu et al. (2015) Yu H, Yang J, Sun Y (2015) Energy absorption of spider orb webs during prey capture: A mechanical analysis. J Bionic Eng 12(3):453–463