## Abstract

The gap between the windshield and hood allows windshield wipers to operate, but causes problems gathering leaves and snow. Active morphing approaches provide an opportunity to create a windshield cowling that addresses this issue by covering the gap normally and actively curling out of the way to allow wiper operation. Most existing morphing techniques lack simultaneous large force/stroke generation, cannot perform two-way actuation, or fail to rigidly hold their position against varying loads such as wind. This article studies a novel curling air surface based on hinged T-shaped tiles that improve upon existing technologies by adding straightening actuation to out-of-plane curling with large force and deflection, while also holding position rigidly. Through vacuuming an upper curling bladder enclosing the tiles and inflating lower straightening bladders spanning the hinge lines, the air surface uncovers and covers the gap against wind loads and holds its curled position rigidly using inter-tile hard stops. An analytical surface model aggregated from multiple instances of a first principle unit curling model predicts the air surface performance. This model includes additional kinematic effects, extending the range of applicability, and additional bladder effect phenomenological terms to improve accuracy. The model is validated across scales and enables design space visualization, which is applied to design a windshield cowling. The resulting design is validated and demonstrated in a full-scale prototype. This article provides the technology concept, supporting model, and design approach to broadly apply this useful air surface to other morphing applications.

## 1 Introduction

The exposed gap between the windshield and the hood of a car houses the windshield wipers but often presents issues with fallen leaves and winter snow sticking in this gap. This debris may interfere with the wiper function or clog the air intakes and drains located beneath. However, cleaning this gap is not easy due to the irregular shape of the gap and the wipers and requires lifting of the hood and manual removal. Additionally, exposure to the ultraviolet sunrays is harmful to the lifetime of the wipers. These issues suggest covering the gap. A fabric cover for the car or wipers only can be used when the car is parked and requires manual installation and removal. Alternatively, a wiper cowl [1–5], an exterior structure, is aesthetically appealing to avoid the gap breaking the car body curvature and may bring additional benefits for aerodynamics [3,6–8], and noise [9] and heat loss [10] reduction. Existing products in the market either do not fully cover the gap, leaving a small permanent gap to allow the wipers to function (Lund^{®} Ford F-150 2004–2008 Shadow^{™} Smoke Wiper Cowl), or fully cover the gap but leave the wipers exposed (Street Scene 950–70704 Wiper Cowl; KBD 37–3011 Wiper Cowl). A better solution to cover and protect the full gap and wipers is a windshield cowling that covers the full gap and wipers normally and can automatically open the gap to allow the wipers to operate. While there are various ways to cover and uncover a gap such as a movable plate with a linkage mechanism [5] or a similar configuration with in-plane sliding, curling out of the way can be an efficient way for a covering surface to open a gap to allow wiper operation in the restricted small gap area without the bulk and complexity of movable plate mechanisms.

There are many approaches to provide out-of-plane curling. Traditional motorized rigid mechanisms provide good force and control of their motion such as hinged flight control surfaces (ailerons and flaps) or robotic fingers [11,12], but their deformation is discrete and requires large actuators and transmission elements. Compliant structures feature reduced mechanical parts and smooth morphing profiles (with wing camber morphing examples to provide out-of-plane curling [13–16]), which is favorable for aerodynamics and aesthetics, while their actuation is still discrete, requiring large actuators and transmission elements and introducing the issue of concentrated local stresses.

Distributed actuation alleviates these issues when producing large out-of-plane curling under load. Smart materials provide distributed actuation without additional actuators or transmission elements, but they are limited in either deflection or speed depending on the materials used [17,18]. Piezoelectric materials [19] are fast with high stress and energy efficiency, but their potential for curling deformation is typically not large enough for the windshield cowling application. Shape memory materials (shape memory alloy wires [20–22] and shape memory polymers [23,24]) have high-energy density with shape memory effects, but the thermal actuation is slow in cooling, and in wind-exposed applications such as windshield cowling, there is heat loss. More recent thin-film or paper-like actuators [25–31] composed of organic or polymeric smart materials (including electro-active papers [25], liquid crystal polymer [30], etc.) are capable of large deformations, but they are low in stiffness and difficult to scale up to this external aero surface application.

Pneumatic structures are another type of actuation structure that provide distributed actuation and do not need extra specific actuators or transmission elements in environments already equipped with a pump. They are potentially capable of large deflection with significant force, and the distributed pneumatic actuation can be utilized in various ways. Most pneumatic curling (or bending) structures are soft, represented by the pneu-net bending actuators [32,33], which curl by asymmetrically inflated silicon rubber chambers and are commonly used in soft robotics for grippers, artificial muscles [34], and locomotion [35], but the soft structure cannot hold its shape rigidly against wind loads for the windshield cowling application. More research is investigating the integration of soft and rigid mechanisms with increased load support, stiffness, and ability to hold its position rigidly. Pressure-actuated cellular structures [36–38] are strong with rigid external surfaces and are capable of large curling deformations depending on their geometry, but the complex structure and manufacturing required to seal the many cells through rigid walls make it difficult to scale down to the scale of the windshield cowling application. Also, while stiffer than soft actuators, they still present significant compliance in their deformed position. Vacuum jamming makes use of negative pressure to rigidize granular, layer, or tile-based structures and thus hold their position rigidly [39,40], but their deformation is passive or requires additional actuators, often soft fluidic actuators [41,42]. Other structures driven by vacuum [43,44] use rigid walls for the air chambers and curl by vacuuming asymmetric air chambers, where negative pressure functions even more reliably for structures similar to pneu-nets, being less bulky and more robust, but they are either not stiff enough [43] or their compliant spine has tradeoff between ease of bending and the structural integrity [44]. Another origami-inspired structure [45] forms its curling surface with hinged tiles of triangular cross sections instead of a compliant plate, which are enclosed in an airtight bladder. The structure curls as air is vacuumed out of the bladder, pushing the triangular tiles close to each other and reaching a rigidly held position against the hard stop when the triangular tiles meet. However, the triangular tile shape limits its motion predetermined by the bladder length, where the actuation performance and the range of motion are coupled. Also, these structures only actuate in one direction without the ability to actively switch between two states and hold both positions rigidly.

This article, adapted from and extending a previous conference paper [46], explores a novel curling air surface architecture based on hinged T-shaped tiles, which improves on these issues by providing large motion out-of-plane curling in both curling and straightening directions and can switch between a flat and a rigidized curled position, enabling the design of a windshield cowling. An upper curling bladder encloses the hinged T-shaped tiles and pulls the T-protrusions close under vacuum, causing the air surface to curl. The T-shape provides sufficient space for the spanning bladder to deflect, and enables rigid tailorable inter-tile hard stops through chamfered tile edges. Separately controlled straightening bladders span the lower hinge lines to provide actuation in the opposite direction by pulling the tiles flat when inflated. Through vacuum and inflation of the two bladders, the air surface switches between a flat and a rigidly curled position to cover and uncover a gap, which applies well to the windshield cowling application. Because the air surface is constructed with tile units (single hinge systems with a membrane connecting the two protrusions), the air surface is modeled and designed based on its constituent tile units, where an air surface model is aggregated from multiple instances of a unit curling model that is derived from first principles with additional experimentally characterized phenomenological terms. This article extends the previous conference paper to include additional kinematic effects, extending the range of applicability, and additional bladder effect phenomenological terms to improve accuracy. Both the unit model and the surface model are experimentally validated using prototypes fabricated with easy-to-access 3D printing and heat-sealing techniques. By applying concentrated end loads at a variety of scales over extended ranges of curling angles, the model accuracy and the improved range of applicability are established. Model-based contour plots of the surface’s design space visualize an actuation effectiveness ratio for general curling applications against loads, which can be applied to the windshield cowling to meet actuation performance and packaging specifications. The selected design is tested and built into a device installed on a full-scale automobile. The hinged T-shaped tiled curling air surface is able to curl, straighten, and rigidly hold its deployed position against wind load, which is an enabler for the windshield cowling application as well as many other applications including a deployable air dam (demonstrated with an adapted full-scale prototype) and morphing wings that require active curling.

## 2 Hinged Tile-Based Curling Air Surface

The hinged tile-based air surface is a new curling surface architecture based on hinged T-shaped tiles that provide large out-of-plane curling and straightening and can rigidly hold a curled position under vacuum. The surface is constructed with a series of connected tile units (Fig. 1), such that the architecture, operation, and modeling of the tile unit set a foundation for the surface. The windshield cowling application provides an example of how the unit operation combines to produce the operation of the entire surface. This structure suggests a modeling approach, which also builds from a unit curling model to a model of the surface.

### 2.1 Hinged Curling Tile Unit.

A hinged curling tile unit features a curling vacuum bladder and a straightening inflation bladder built upon a pair of hinged T-shaped tiles. Vacuum and inflation of the two bladders enable the tile unit to switch between a flat and a rigidized curled form against an inter-tile hard stop in a four-state operation cycle.

#### 2.1.1 Unit Architecture.

The curling tile unit, Fig. 2, is composed of a pair of hinged T-shaped tiles with the upper protrusions connected by a membrane. A curling vacuum bladder (which may incorporate the membrane) encloses the tile structure, and a separate straightening inflation bladder spans the hinge line from below the surface (Fig. 2(b)). The T-shaped tiles are long, straight, and rigid tiles with uniform T-shaped cross sections along their length. The hinge connecting the tiles constrains the motion of the tile unit to allow only out-of-plane curling, where the tile edges are chamfered to provide a hard stop that restricts the maximum curling angle before the tile upper protrusions hit each other. The membrane, connecting the upper protrusions of the tiles, functions as a controllable medium to curl the tile unit by leveraging the vacuum pressure to pull the tiles. The T-shape of the tiles provides sufficient open internal space for this action to occur. The straightening inflation bladder spanning each hinge line across the bottom of the tiles pulls the tiles in the straightening direction under inflation, where the membrane stops the tile units from straightening beyond flat. A vacuum port is placed on the curling bladder, and an inflation port is placed on the straightening bladder to enable their operation.

#### 2.1.2 Unit Operation.

The tile unit operates in a four-state cycle with two static states and two transient states by vacuum and inflation of the curling and straightening bladders. This operation can occur in the presence of an external load, which tends to straighten the tile unit.

In the *fully straight state*, the curling bladder is open to the atmosphere such that the pressure inside the bladder *p*_{in} is equal to the pressure outside *p*_{out}, and the straightening bladder is inflated to actively hold the tile unit flat (Fig. 3(a)). Air pressure inside the inflation bladder results in bladder tension, which pulls the tiles on opposing sides of the bottom hinge line in the straightening direction, aided by the external load. Consequently, the membrane spanning the upper tile protrusions is in tension, restricting straightening beyond flat depending on the membrane extensibility. This pair of tensions keeps the tile unit fully straight.

To curl the tile unit against the external load (*curling state*), the straightening bladder is deflated, and the curling bladder is vacuumed (Fig. 3(b)). As the vacuum is applied, the pressure outside exceeds the pressure inside (*p*_{in} < *p*_{out}) and pushes the membrane into the space between the upper protrusions of the T-shaped tiles. Under this uniformly distributed pressure difference, the membrane deflects in a circular shape, pulling the tile protrusions toward each other. The pressure difference is also distributed along the perimeter of the tiles and pushes the tiles inward. The tile unit curls when the curling torques from the membrane tension and the pressure difference exceed any hinge resistance and the external load. As curling proceeds, the angle of this tension applied, i.e., tangent to the membrane, becomes steeper, reducing mechanical advantage. When the membrane becomes tangent to the upper protrusions, membrane wall contact occurs (not shown in Fig. 3), but the portion of the membrane not in contact remains active and circular. This membrane deflection shape determines the amount of curling, where any membrane extensibility reduces the amount of curling.

The tile unit reaches the *fully curled state* when the pressure difference becomes large (*p*_{in} ≪ *p*_{out}), and the hard stop surfaces contact each other (Fig. 3(c)). The chamfered tile edges stop the tile unit from curling beyond a predefined point and block against curling torques in excess of the external load. This allows the tile unit to hold its position rigidly and resist deflection or vibration under changing external loads. This rigidized curled form differentiates this new T-shaped tiled architecture from other curling architectures [32,43,44] and is useful in many applications requiring a tailorable rigid deflected shape.

The tile unit can be straightened again by relaxing the vacuum and inflating the straightening bladder (*straightening state*, Fig. 3(d)). As air moves back into the relaxed curling bladder, the membrane tension and the pressure difference are removed (*p*_{in} = *p*_{out}), eliminating the curling torques. When the inflated straightening bladder, aided by the external load, pulls the tile unit back to the fully straight position, the membrane is in tension again, and the operation cycle is complete.

### 2.2 Hinged Tile-Based Curling Air Surface.

A hinged tile-based curling air surface, Fig. 4 is constructed with a series of connected curling tile units. Multiple T-shaped tiles are hinged together with a single curling vacuum bladder, surrounding the entire assembly. A series of straightening bladders is interconnected through small air channels spanning the hinge lines creating one contiguous straightening bladder system (Fig. 4(b)). A single vacuum port and a single inflation port simultaneously activate all the tile units with distributed actuation and displacement.

The surface operates through the tile units. In the context of the windshield cowling, the surface covers the gap between the windshield and the hood and curls against the external wind load for the wipers to function. The cowling operates in a similar four-state cycle. In the *gap closed state*, the cowling is stowed, covering the wipers with one end fixed to the rear edge of the hood and the other resting on the windshield (Fig. 5(a)). The inflated straightening bladders actively pull the surface straight at every hinge, where the fully straightened surface extends slightly beyond flat due to the membrane extensibility. This overstraightening helps to secure the gap by applying a normal force against the windshield. When the wipers are needed, the gap opens, deploying the cowling, by deflating the straightening bladder and vacuuming the curling bladder (*gap opening state*, Fig. 5(b)). The curling angle of the tile units accumulates at each hinge to achieve large surface out-of-plane curling. The cowling opens a sufficient gap for the wipers to function when the surface is in the *gap opened state*, fully curled against its internal tile-tile hard stops (Fig. 5(c)). Excess curling torque from overpressurization enables the surface to hold the curled position rigidly against the wind load and prevents the structure from deflecting or fluttering in the changing environment during high-speed driving. In the *gap closing state*, the cowling stows and closes the gap by relaxing the vacuum and inflating the surface straightening bladder (Fig. 5(d)).

Identical repeated tile geometry within the surface provides uniform actuation across the surface. Although beyond the scope of this article, additional tailorability can be achieved by varying the tile height and width along the surface to produce differences in local actuation to create a particular motion and shape. For example, a surface can roll over itself with the tile units designed for incrementally stronger actuation. Such functionally graded surface design provides even more opportunity for enabling novel applications.

## 3 Modeling Approach

To design the air surface for different applications, an analytical model is used to quantitatively describe the surface performance including the amount of curling deflection generated and how much external load the structure can support as a function of the pressure, geometric variables, and material properties. Because the surface is built from tile units, a unit model that characterizes the torque–angle relationship from a single tile unit serves to develop the surface model combining the performance of its tile units. Internal torque from a tile unit comes from either the curling bladder or the straightening bladder. Since the external load is usually in the direction resisting curling and assisting straightening, accurate prediction of the torque from the curling bladder is important for design and is analytically modeled. The unit curling model predicts the torques from membrane tension and pressure difference based on first principles and includes additional phenomenological terms to capture additional bladder effects. Experimental characterization of the hinge stiffness, material properties, geometric end losses, and the phenomenological terms using unit prototypes validates the unit model across scales. By assembling the validated unit curling model using serial geometry and statics, the surface model analyzes the surface performance under different forms of external loads. The nomenclature for the unit curling model and the surface model is laid out in Table 1.

Varying parameters | |

Δp | Gauge pressure of the vacuum bladder |

θ | Curling angle, angle deflection from flat about a hinge |

φ_{i} | Orientation angle from flat of a tile in surface, $\phi i=\u2211j=1i\theta j(i=1,\u2026,n)$ |

h_{i} | Surface curled height above a hinge |

Fixed geometry parameters | |

H | Tile height |

W | Tile kinematic width excluding the tile wall thickness |

W_{t} | Tile width |

C | Hypotenuse length, $H2+W2$ |

α | Angle of the tile actuation right triangle, tan^{−1}(H/W) |

ϕ | Hard stop chamfer angle |

L | Length of tile (out of plane) |

W_{g} | Gap width that the surface covers |

n | Number of hinges or tile units, $n=\u230aWgWt\u230b$ |

Membrane parameters | |

$F^T$ | Membrane tension per unit length |

$\epsilon $ | Strain of the membrane, function of membrane tension $F^T$ |

l_{0} | Original membrane length, equal to 2W |

l | Stretched membrane length, equal to $l0(1+\epsilon (F^T))$ |

r | Membrane curvature |

β | Arc angle, half central angle for the membrane arc |

γ | Angle of membrane tension torque |

Δh | Membrane wall contact length at large curling angles |

Unit model terms and related parameters | |

T_{cur} | Torque from the curling vacuum bladder |

$T^mem$ | Torque from membrane tension (per unit length) |

$T^p$ | Torque from pressure difference along the tile outer perimeter (per unit length) |

$T^f$ | Phenomenological torque along the length presumably from the bladder friction (per unit length) |

T_{e} | Phenomenological torque at the ends presumably due to bladder end puckering |

T_{h} | Hinge resistant torque |

L_{e} | Geometric end loss |

L_{a} | Active length, L_{a} = L − L_{e} |

External load terms and related parameters | |

T_{ext} | Torque from external loads |

F_{d} | External force from a uniform distributed load |

T_{d} | External torque from a uniform distributed load, i.e., wind |

$Twe$ | External torque from a cantilevered concentrated end load |

$Twt$ | External torque from tile self-weight |

p_{ext} | Effective pressure due to (distributed) external load |

Varying parameters | |

Δp | Gauge pressure of the vacuum bladder |

θ | Curling angle, angle deflection from flat about a hinge |

φ_{i} | Orientation angle from flat of a tile in surface, $\phi i=\u2211j=1i\theta j(i=1,\u2026,n)$ |

h_{i} | Surface curled height above a hinge |

Fixed geometry parameters | |

H | Tile height |

W | Tile kinematic width excluding the tile wall thickness |

W_{t} | Tile width |

C | Hypotenuse length, $H2+W2$ |

α | Angle of the tile actuation right triangle, tan^{−1}(H/W) |

ϕ | Hard stop chamfer angle |

L | Length of tile (out of plane) |

W_{g} | Gap width that the surface covers |

n | Number of hinges or tile units, $n=\u230aWgWt\u230b$ |

Membrane parameters | |

$F^T$ | Membrane tension per unit length |

$\epsilon $ | Strain of the membrane, function of membrane tension $F^T$ |

l_{0} | Original membrane length, equal to 2W |

l | Stretched membrane length, equal to $l0(1+\epsilon (F^T))$ |

r | Membrane curvature |

β | Arc angle, half central angle for the membrane arc |

γ | Angle of membrane tension torque |

Δh | Membrane wall contact length at large curling angles |

Unit model terms and related parameters | |

T_{cur} | Torque from the curling vacuum bladder |

$T^mem$ | Torque from membrane tension (per unit length) |

$T^p$ | Torque from pressure difference along the tile outer perimeter (per unit length) |

$T^f$ | Phenomenological torque along the length presumably from the bladder friction (per unit length) |

T_{e} | Phenomenological torque at the ends presumably due to bladder end puckering |

T_{h} | Hinge resistant torque |

L_{e} | Geometric end loss |

L_{a} | Active length, L_{a} = L − L_{e} |

External load terms and related parameters | |

T_{ext} | Torque from external loads |

F_{d} | External force from a uniform distributed load |

T_{d} | External torque from a uniform distributed load, i.e., wind |

$Twe$ | External torque from a cantilevered concentrated end load |

$Twt$ | External torque from tile self-weight |

p_{ext} | Effective pressure due to (distributed) external load |

## 4 Unit Curling Modeling

*T*

_{cur}and are explicitly analyzed in the unit curling model:

*L*

_{a}(shown in Fig. 12), which is equal to the actual length

*L*minus a geometric end loss

*L*

_{e}due to an observable ineffective membrane portion resulting from bladder end puckering. $T^f$ is a phenomenological term distributed along the length presumably from bladder friction against the bottom of the tiles.

*T*

_{e}is a phenomenological term concentrated at the ends of the tiles presumably from puckering of the bladder. These phenomenological terms may differ between the curling and straightening directions. Finally,

*T*

_{h}is a resistive hinge stiffness torque, which is primarily dependent on the fabrication techniques for a particular prototype. Generally, all these torques may vary with angle depending on the model and the experimental identification of the phenomenological terms.

Unit prototypes are fabricated, and a custom experimental apparatus is used to both characterize the geometric end losses, phenomenological terms, and the hinge resistive torque as well as validate the complete model. While model parameter characterization is performed at a single scale (size), validation is performed across a range of scales. The unit curling model serves as a stepping stone to combine in series to predict the surface behavior.

### 4.1 Analytical First Principle Modeling.

Analytical first principle modeling is performed under a set of geometric and material assumptions. Geometry modeling produces a transcendental equation for the membrane arc angle *β* from two calculations of membrane curvature *r*, which depends on the membrane extensibility under tension. Force/torque equilibrium on the membrane and a curling tile produces an expression for the membrane tension, and eventually expressions for the curling torques from the membrane tension and the pressure difference. Since the tile wall may interfere with the membrane deflection at large curling angles, membrane wall contact also affects the geometry and static analysis.

#### 4.1.1 Assumptions.

The most important assumption of the unit model is that the membrane deflects in a circular shape because the pressure applies uniformly over the membrane, where the membrane is flexible with negligible bending stiffness. When membrane wall contact occurs after the membrane is tangent to the tile protrusions at large curling angles, the membrane lies flat along the tile protrusions for a contact length but remains circular in the middle. The tiles are assumed to be rigid such that they do not deform under the membrane tension, compression of the atmospheric pressure, or the contact at the inter-tile hard stop. The bladder is assumed airtight such that pressure within the vacuum bladder is uniform. The first principle modeling does not include the hinge stiffness and the bladder effects from friction and end effects, which are captured in the phenomenological terms characterized experimentally.

#### 4.1.2 Geometry Modeling.

Curling torques are analyzed using a cross-sectional view as torques per unit length shown in Fig. 6, since the tile unit is uniform along its length. The tile height *H* and kinematic width *W* are the two primary dimensions that determine the tile unit actuation. The height *H* is defined as the distance from the membrane connection point $M$ to the line of the tile base, and the kinematic width *W* is defined as the distance from the hinge point $O$ to the location of $M$ along the tile base. These two line segments form a right triangle, where the hypotenuse is *C* and the elevation angle of $M$ from $O$ is *α*. The hypotenuse *C* represents the geometric size of the tile, and therefore, the magnitude of the torque–angle relationship, while the angle *α* = atan(*H*/*W*), affects the shape of the torque–angle relationship. One tile is assumed fixed, allowing the other tile to curl, rotating around $O$ by an angle *θ* relative to flat up to a maximum curling angle *θ*_{max} limited by the tile chamfer angle *ϕ* of the hard stop.

Derivation of the torque from the membrane tension requires the angle *γ* between $M\u2032O\xaf$ and the membrane tangent line, which requires an expression for the angle *β* between $MM\u2032\xaf$ and the membrane tangent line. This membrane arc angle *β* is equal to half the central angle of the membrane arc. A transcendental equation for *β* is produced by equating two expressions for the membrane curvature *r*. The first of these expressions is obtained by equating two expressions for the membrane length *l*, and the other is obtained by equating two expressions for $PM\u2032\xaf$, the half distance between the tile upper protrusions.

*r*and central angle 2

*β*, while also elongating under tension, giving two expressions for the membrane length

*l*as follows:

*l*

_{0}is original length of the connecting membrane, which equals to 2W from the fully flat tile unit geometry, and $\epsilon (F^T)$ is the elongation strain of the membrane due to tension per unit length $F^T$, which is characterized experimentally. Equating these two expressions for the membrane length, the membrane curvature is obtained as follows:

*r*. $OP\xaf$ is perpendicular to $MM\u2032\xaf$ because $OP\xaf$ connects the midpoint of the base $MM\u2032\xaf$ in the isosceles triangle $\Delta MOM\u2032$. The center $Q$ of the membrane arc lies on the extended line of $OP\xaf$ because the tile unit is symmetric. This perpendicular relationship gives two right triangles that include $PM\u2032\xaf$: $\u22bfOPM\u2032$ and $\u22bfQPM\u2032$. In $\u22bfOPM\u2032$, the base $PM\u2032\xaf$ is calculated as follows:

*γ*is then obtained from $\u22bfOPM\u2032$ as follows:

#### 4.1.3 Force/Torque Equilibrium.

*p*=

*p*

_{out}−

*p*

_{in}, acts along the circular membrane, which is equivalent to the same pressure acting along the facing area $MM\u2032\xaf$. The membrane tension $F^T$ per unit length is tangent to the circular membrane pulling outward at both ends. In the direction normal to $MM\u2032\xaf$, the force equilibrium between the membrane tension and the effective pressure facing the area $MM\u2032\xaf$ gives

*r*from geometry (Eq. (7)), the membrane tension per unit length is expressed as follows:

*γ*is from geometry modeling (Eq. (9)); and the arc angle

*β*is solved from the transcendental geometry Eq. (8). When the angle of $F^T$ rotates past $OM\u2032\xaf(\gamma <0)$, the torque from the membrane tension becomes negative.

Each expression for each curling torque scales with pressure Δ*p* and with the square of the hypotenuse *C*^{2}. This torque–pressure scaling aligns with the expectation of increasing curling effects under stronger vacuum. Scaling with the geometric scale gives insight on the form of the phenomenological terms, which may also scale with *C*^{2}, allowing them to be condensed into dimensionless terms in the model along with the first principle torques. The phenomenological terms must be empirically characterized.

#### 4.1.4 Membrane Wall Contact.

*h*from the points $M$ to the points $N$ (Fig. 8). The remaining membrane length remains circular and tangent to the tile protrusions, resulting in a simple expression for the arc angle

*β*

*γ*using Eq. (9). However, this membrane wall contact also introduces the contact distance Δ

*h*that requires an additional transcendental equation to solve, which is derived in a similar process as for

*β*before the contact:

*h*:

*h*offsets the external pressure applied at this region in the free body diagrams, which shifts the effective pressure facing areas on the membrane and on the curling tile, resulting in new expressions for the two torques as follows:

The torque from the membrane tension in this case is always negative due to the angle of the tension *γ* (Eq. (9)), which switches its sign before membrane wall contact occurs. After membrane wall contact occurs, the torque–pressure scaling still applies, although the torques are close to but not strictly scaling with *C*^{2} as long as Δ*h* remains small.

In addition to the side walls, the membrane may also come into contact with the bottom of the tiles, requiring more complex geometry modeling similar to the approach used for membrane-side wall contact. For the T-shaped tiles, membrane–bottom wall contact only occurs when the aspect ratio of *H*/*W* gets very low (about 0.26 for *α* = 15 deg at *θ* = 85 deg), which is weak in actuation, difficult to fabricate, and thus not in a feasible region of design space. While the principal geometric design parameters *H* and *W* are defined by the relative position of the tile upper protrusions connecting the membrane $(M)$ with the hinge $(O)$, the specific tile cross-sectional design determines when and how the membrane contacts the tile. For a triangular tile cross section [45], torque from the membrane never switches its sign to negative as the curling angle increases, but the effective pressure area on the tile perimeter reduces drastically as the membrane contacts the triangular tile along its hypotenuse, which is why the T-shape is advantageous in delaying the occurrence of membrane–tile contact to maximize the torque from pressure difference and allow larger curling angles.

### 4.2 Unit Curling Model Experimental Methods.

To experimentally characterize and validate the unit curling model, unit prototypes implementing the curling vacuum bladder of the tile unit are fabricated across scales from the ** W** = 0.5 in. scale used in the windshield cowling surface to a larger 2 in. validation scale. A constant torque test measures the curling angle under different vacuum pressures using photogrammetry, assisted by a mechatronic balancing system to control the external applied moment arm. The torque–pressure scaling from the first principle modeling is leveraged such that the pressure-scaled torque–angle relationship of each prototype is captured by two curves over any pressure with hysteresis between the curling and straightening directions.

#### 4.2.1 Unit Prototype Fabrication.

Unit prototypes enable direct measurement of the curling torque from a tile unit with features adapted to apply external torques. Figure 9 illustrates the fabrication process. An L-channel, Fig. 9(a), is equivalent to half of a T-shaped tile and is easy to build and scale. No additional hard stop is built with the goal of characterizing the curling performance over its full range of motion. Lubricated door hinges minimize the resistive hinge torque. A woven cotton membrane with a 40S yarn count, and 120 GSM weight with low extensibility is glued using Loctite^{®} fabric adhesive onto the L-channel protrusions (Fig. 9(b)) with the membrane length carefully controlled because the membrane length affects the unit performance. Mounting shafts are threaded and screwed onto the side walls to mount the prototype and apply external loads. These must extend through holes in the bladder, requiring careful sealing with rubber washers compressed around the unthreaded middle portion of the mounting shafts (Fig. 9(c)). An air nozzle is installed through the bladder and screwed into and through the L-channels to ensure an open airway. Edges of the thermoplastic urethane (TPU)-coated nylon bladder are sealed with a heat press at 350 °F for 2 min (Fig. 9(d)).

To validate the unit curling model across scales, six-unit prototypes are built at three geometric sizes with two length variations, named with the L-channel lengths (≈*L*) and widths (≈*W*) as 0.5 ft–0.5 in. (windshield cowling surface scale), 1 ft–0.5 in., 0.5 ft–1 in., 1 ft–1 in., 1 ft–2 in., and 2 ft–2 in. The unit prototypes have roughly equal height *H* and kinematic width *W* from the symmetric geometry of the L-channel, but the exact dimensions of *H* and *W* are measured for each prototype according to their definition to include the dimensions of the mounted door hinge. The unit prototypes do not vary the ratio of *H* and *W*, but the unit model includes the effect of this ratio and is supported by the unit-based surface model validation where a surface prototype adopts a different ratio of *H* and *W*.

#### 4.2.2 Unit Curling Model Experimental Setup.

A constant torque test measures the curling angle of the unit prototypes under different vacuum pressures by applying a known external load. To apply the external torque, one L-channel is cantilevered with a weight hanging from a perpendicular mounting shaft at a fixed distance from the hinge. To maintain a constant moment arm and a perpendicular applied moment, the opposing L-channel is mounted to a movable fixture, which is part of a mechatronic balancing system (Fig. 10(a)). This system is controlled to maintain a horizontal orientation of the loaded and cantilevered L-channel as measured by an accelerometer and driven by a linear stepper through a linkage (Fig. 10(b)). A vacuum pump is used to actuate the unit prototypes, where the vacuum pressure is controlled manually by a regulator and measured with a pressure gauge. The side walls of both L-channels are marked with different colored dots to measure the curling angle, i.e., the angle difference between the two L-channels, through photogrammetry (Fig. 10(c)).

#### 4.2.3 Pressure Scaling of Torque–Angle Data.

Each unit prototype is run through loops of straightening and curling motions to obtain its curling torque–angle performance. To produce a loop, a high vacuum pressure is first applied to the bladder under no external load, fully curling it to a maximum curling angle *θ*_{max}. A weight is then hung from the mounting shafts on the loaded side of the prototype, producing a constant torque at which the unit remains fully curled at the initial high pressure. At this torque, the pressure is slowly decreased from its maximum allowing the unit to straighten until it is fully straight. During this motion, the mechatronic balancing system maintains a level orientation of the loaded side of the prototype to maintain a constant moment arm and therefore a constant torque, while accommodating the curling motion of the unloaded side of the prototype. The pressure is measured at relatively small intervals, where a photo is taken at each quasi-static interval to obtain the curling angle. Then the pressure is slowly increased continuing the incremental measurement procedure as the unit prototype curls until it is fully curled at high pressure. Typical repeatability errors of the captured data curves are around 4% with increased errors of around 8% at low angles for the small prototypes.

Figure 11(a) illustrates the tile unit curling performance, for example, 1 ft–2 in. prototype, which was measured twice: once with a 8.8 lb (4 kg) weight and once with a 13.2 lb (6 kg) weight, each with a fixed moment arm of 4 in. from the hinge. The resulting pressures and curling angles are plotted from these measurements with axes switched; the dependent variable (curling angle) is on the horizontal axis to display the performance from a force–deflection design perspective. Hysteresis exists between the straightening and curling directions, which is captured by the phenomenological terms in the model, which differ in the two directions.

Since it is expected from the first principle modeling that the torque scales with pressure, the data are replotted in Fig. 11(b) by replacing the pressure axis with the torque scaled by pressure. After scaling, the 8.8 lb and 13.2 lb constant torque data overlap within measurement noise, validating the torque/pressure scaling. The resulting two curves in the curling and straightening directions characterize the performance of each prototype independent of pressure such that model validation can be performed in this pressure-scaled form. In addition to these constant torque tests, constant pressure tests, measuring the curling angle under varying applied weights at a constant vacuum pressure, were performed and produced identical pressure-scaled torque–angle curves (not shown) also within measurement noise.

### 4.3 Unit Curling Model Characterization.

The various empirical and phenomenological terms in the unit curling model are experimentally characterized using the unit prototypes. The resistive hinge stiffness torque is measured directly and is found to be small. The tensile stiffness of the cotton membrane is measured separately from the prototype, relating tension per unit width to strain. The geometric end loss is measured from the observed parabolic shape of the inactive membrane region and scales with the geometric size *C*. The phenomenological terms $T^f$ and *T*_{e} are each captured by two constants (for curling and straightening), which are assumed to scale with *C*^{2} as does the first principle model, where $T^f$ in the curling direction is dominant, simplifying the proposed four phenomenological terms down to a single constant.

#### 4.3.1 Resistive Hinge Stiffness Torque.

The resistive torque of the unit prototype hinge is measured by a spring scale pulling it through its range of motion with the unit prototype held vertically to exclude gravity effects. The hinge resistive torque was found to increase linearly with the curling angle; for instance, the hinge resistance torque of a middle scale prototype (1 ft–1 in.) is *T*_{h} = 5.07 × 10^{−3}*θ* (lbf · ft), which is about 0.07 lbf · ft at the maximum curling angle, amounting to a maximum of 4% of the applied torque. This hinge resistance torque is considered small and not included in the unit curling model characterization.

#### 4.3.2 Membrane Tensile Stiffness.

A tensile test of the cotton membrane material used in the unit prototype is performed by loading a 4 in. wide and 1 ft long strip with a series of weights while measuring its length. The strip is wide enough with no apparent edge deformation, and the mounting of the fabric covers the full width to apply uniform and distributed stress. The cotton membrane strain relationship is nonlinear with the force per unit width and characterized as $\epsilon =6.56\xd710\u22123ln(F^T)\u22122.09\xd710\u22122$, where $F^T$ is presented in units of lbf/ft.

#### 4.3.3 Geometric End Loss.

*L*

_{e}is defined as the average inactive length along the width of the prototype for this parabolic shape. This average is taken as the centroid location of the parabolic region at each end. This centroid distance can be computed from the fabricated length

*L*and the measured length between the tips of the parabola end loss regions

*L*

_{tip}as follows:

*L*

_{e}against the geometric size of the prototype, represented by hypotenuse

*C*, produces a linear relationship (Fig. 12(b)):

*L*to produce the active length

*L*

_{a}in the unit curling model (

*L*

_{a}=

*L*−

*L*

_{e}).

#### 4.3.4 Phenomenological Correction Torques.

*T*

_{e}captures the bladder end effects likely due to bladder end puckering. These terms are proposed to scale with

*C*

^{2}similar to the first principle terms. The square relationship results from the force-producing effects scaling with size and the moment arm scaling with size. These effects are also assumed to scale with pressure, which impacts all bladder-related phenomena. By applying the scaling relationship with

*C*

^{2}and Δ

*p*, the unit curling model equation in Eq. (1) can be expressed as follows:

*k*

_{f}is dimensionless and

*k*

_{e}has units of length. Each of these terms may have different magnitudes in the curling and straightening directions. The two unknown terms

*k*

_{f}and

*k*

_{e}for each direction are obtained from two length variations for a given geometric size

*C*, by subtracting two instances of the scaled model (Eq. (21)) over the range of angles with the two lengths

*L*

_{1}and

*L*

_{2}:

*k*

_{f}for curling varies only by an average of 7.3% relative to its mean for the 2 in. scale. The small angle behavior below 20 deg is less predictable because of the large and varying membrane tension at near flat angles. Therefore, these terms are taken as constants, by averaging over angles larger than 20 deg and listed in Table 2.

k_{f} (or k_{f}���· 1 ft) | k_{e} (ft) | ||
---|---|---|---|

0.5 in. | Straightening | 1.12 × 10^{−2} | 9.71 × 10^{−3} |

Curling | −7.22 × 10^{−2} | 4.59 × 10^{−3} | |

1 in. | Straightening | −1.66 × 10^{−2} | 1.26 × 10^{−2} |

Curling | −4.19 × 10^{−2} | 7.54 × 10^{−3} | |

2 in. | Straightening | 8.07 × 10^{−4} | 6.99 × 10^{−3} |

Curling | −5.19 × 10^{−2} | 1.60 × 10^{−3} |

k_{f} (or k_{f}���· 1 ft) | k_{e} (ft) | ||
---|---|---|---|

0.5 in. | Straightening | 1.12 × 10^{−2} | 9.71 × 10^{−3} |

Curling | −7.22 × 10^{−2} | 4.59 × 10^{−3} | |

1 in. | Straightening | −1.66 × 10^{−2} | 1.26 × 10^{−2} |

Curling | −4.19 × 10^{−2} | 7.54 × 10^{−3} | |

2 in. | Straightening | 8.07 × 10^{−4} | 6.99 × 10^{−3} |

Curling | −5.19 × 10^{−2} | 1.60 × 10^{−3} |

Note: The distributed phenomenological constant *k*_{f} is dimensionless and multiplied with a length of 1 ft to compare with the end phenomenological constant *k*_{e}. *k*_{f} in the curling direction is found to be dominant.

*k*

_{e}and

*k*

_{f}in both directions, are compared with each other after converting to the same units by multiplying

*k*

_{f}by a typical characteristic length of 1 ft.

*k*

_{f}· 1 ft in the curling direction is consistently an order of magnitude larger than

*k*

_{e}, dominating the phenomenological bladder effects. The other three constants are relatively close to zero and are ignorable as indicated by their inconsistent signs across scales. In this way, the model can be reduced from four phenomenological terms to one as follows:

The single unknown constant *k*_{f} in the curling direction captures the torque hysteresis and can be computed from a single prototype (rather than requiring prototypes of two different lengths). The resulting *k*_{f} (curling) is also roughly constant at angles larger than 20 deg, and the averaged values are listed in Table 3. The constant is consistent in magnitude across prototypes of varying size and length, especially for the larger scales, confirming the scaling with *C*^{2} and *L*_{a}. Because of this, a single value of *k*_{f} can be measured from a single prototype and used to predict the torque for prototypes of any size or length, minimizing the required characterization experiments. Model prediction with four phenomenological constants and the single phenomenological constant is compared as part of validation (Fig. 13(c)).

0.5 ft–0.5 in. | 1 ft–0.5 in. | 0.5 ft–1 in. | 1 ft–1 in. | 1 ft–2 in. | 2 ft–2 in. | |
---|---|---|---|---|---|---|

k_{f} (Curling) | −6.22 × 10^{−2} | −6.75 × 10^{−2} | −5.93 × 10^{−2} | −5.00 × 10^{−2} | −5.00 × 10^{−2} | −5.10 × 10^{−2} |

0.5 ft–0.5 in. | 1 ft–0.5 in. | 0.5 ft–1 in. | 1 ft–1 in. | 1 ft–2 in. | 2 ft–2 in. | |
---|---|---|---|---|---|---|

k_{f} (Curling) | −6.22 × 10^{−2} | −6.75 × 10^{−2} | −5.93 × 10^{−2} | −5.00 × 10^{−2} | −5.00 × 10^{−2} | −5.10 × 10^{−2} |

### 4.4 Unit Curling Model Validation.

Given the combination of first principle modeling and experimental characterization, the unit curling model can predict the curling torque–angle curves under the action of a vacuum. The model is first validated against a single prototype. In this validation, the effects of the membrane elongation, the geometric end loss, and the phenomenological terms are introduced incrementally to the first principle model to investigate their impact on the curling performance. The model is also validated across scales by using the phenomenological constant obtained from one scale to predict the curling performance at the other scales. The ability of the model to predict low-angle performance is explored along with the possibility of friction as the source for the distributed phenomenological term.

#### 4.4.1 Single-Scale Validation.

Figure 13 shows an example model validation for the 1 ft–2 in. prototype. With only the first principle terms from the membrane tension and pressure difference on the tiles, the analytical unit curling model overpredicts the curling torque–angle relationship with errors of 14% and 48% in the straightening and curling directions if the membrane is assumed inextensible. The large error in the curling direction is due to the lack of hysteresis in the model. The cotton membrane elongation with strains on the order of 0.5% slightly reduces the curling torque with improved overprediction errors of 12% and 44%, and its impact decreases with the curling angle (Fig. 13(a)). The geometric end loss due to the inactive membrane at the ends has a larger impact, improving the prediction and no longer overpredicting the straightening curve (5.0% error), and reduces the curling direction error by almost half to 23% (Fig. 13(b)).

The phenomenological terms from bladder effects further correct the curves to characterize the performance in the curling and straightening directions. The full set of four phenomenological terms reduces the curling errors to 2.4% and the straightening errors further to 2.1%. Eliminating all but one phenomenological term (*k _{f}* in curling) achieves equally accurate prediction in the curling direction maintaining a 2.4% curling error, while without phenomenological terms in the straightening direction, the error only increases from 2.1% to 5.0% (Fig. 13(c)). This dominant distributed phenomenological term in the curling direction captures the hysteresis between the straightening and curling directions. Therefore, the single constant model in Eqs. (24) and (25) is used later as the unit curling model, predicting the experimental data with low single-digit errors. Similar results are achieved with single-scale validations at the other scales as the self-predictions in the scalability cross validation.

#### 4.4.2 Scalability Cross Validation.

The unit curling model is validated across scales using the six prototypes from the surface scale (0.5 ft–0.5 in.) to larger validation scales (2 ft–2 in.). The prediction errors are averaged from data at angles larger than 20 deg, where the phenomenological terms are computed and summarized in Fig. 14, while the low-angle performance is investigated separately. The first row of the bar graph shows the errors in the straightening direction, which involves no phenomenological scale-dependent characterization. Each of the other horizontal row bar graphs represents the model prediction errors obtained by characterizing the dominant phenomenological term from the shorter prototype at each geometric size (indicated in the titles) and using it to predict the curling performance of all the prototypes at all scales. In these bar graphs, the self-prediction errors are italicized.

The model predicts the straightening performance with consistently low errors of around 6% and slightly above the typical repeatability error of 4%. On the other hand, prediction in the curling direction includes the empirically characterized phenomenological constant *k*_{f}, which is measured at each scale and applied across scales. While the self-prediction errors (single-scale validations) are generally low, even across scales, the errors are mostly in the low single digits. The accuracy typically increases with the scale; bladder effects can have a larger impact on the smaller scales, where predictions for the two 0.5 in. L-channel width prototypes give errors around 10% consistently even for self-predictions, and a steeper shape is observed in their data than the model prediction curves at low angles. The low errors validate the scalability of the model from large scales to small scales and from small scales to large scales, proving its strength and credibility regardless of uncertainty in measurement and fabrication differences between prototypes.

#### 4.4.3 Low-Angle Validation.

The low-angle performance is useful to design for applications, but it is separated in the model validation because a different trend of the phenomenological terms is observed below 20 deg, which is not accurately captured by the phenomenological constant. A scalability cross validation plot similar to Fig. 14 is shown in Fig. 15 for errors below 20 deg. The straightening direction errors remain low within 10%. The large prototypes have low single-digit errors in both activation directions and over all angle ranges, showing that the model can capture the unit performance well when the scale is large. On the other hand, the low-angle curling performance for the small prototypes is predicted with up to 40% errors. The difference in the model errors shows where the scalability fails to work at low angles with potentially more physical effects such as bladder interactions with the tile unit. Even so, the 40% error is not extreme and can be captured in design with a safety factor or by allowing the nondimensional curling phenomenological term to vary as a function of angle.

#### 4.4.4 Friction as Source of Phenomenological Term.

*T*

_{f}in the model structure. The feasibility of this assumption is checked from both the magnitude and scalability perspectives. The friction force

*f*=

*μN*resulting from the normal force

*N*= Δ

*p*·

*L*·

*W*due to pressure pushing on the bottom of the L-channel wall produces a torque with a moment arm

*d*from the bottom surface to the hinge (Fig. 16):

*d*is measured to be around 0.4 in., and the width of the L-channel bottom is

*W*≈ 1 in. Since the phenomenological term produces a torque of 0.717 Nm (given an average Δ

*p*= 2 psi in the experiments), a friction coefficient

*μ*of 0.46 matches the magnitude exactly, which is plausible between the plastic L-channel surface and the TPU-coated inner side of the bladder [47,48]. The scalability of the torque due to friction and the phenomenological term also matches as they both scale linearly with the gauge pressure, and the geometric parameters (

*d*and

*W*), which scale in proportion to

*C*over the range of prototypes tested. Although the exact source of the phenomenological terms cannot be determined, this analysis of the magnitude and scalability shows the plausibility of this friction assumption.

## 5 Curling Air Surface Modeling

Based on the unit curling model, an air surface model predicts the curling performance of the entire surface under external loads by incorporating the geometry and statics of a series of hinged tile units. Surface prototypes are fabricated by 3D printing tiles directly onto heat-pressed bladders and are capable of curling rigidly against internal hard stops depending on the tile chamfered angles. To validate the air surface model, a surface prototype with internal hard stops is loaded with a concentrated end load, whose curling shape under different pressures is measured and predicted from flat to fully curled.

### 5.1 Analytical Surface Model.

Curling of the surface is a combined result of accumulated curling angles and torque transmissions across a number of hinged tile units. The surface geometry can be determined by the geometry and curling angles of its constituent tile units. A set of torque equilibria, one at each hinge, is used to form the surface model with the maximum curling angles limited by the hard stops. Equilibrium is described between the internal curling torque from the unit curling model for a hinge and the external torques applied to the remaining portion of the surface beyond that hinge. Numerically solving this set of equilibria, combined with the geometry, produces the curling angles at each hinge and thus the curling shape of the surface.

*n*hinged tile units of tile width

*W*

_{t}, covers a gap distance

*W*

_{g}and curls with

*n*curling angles

*θ*accumulated to a net orientation angle

*φ*over the surface (Fig. 17).

*W*

_{t}includes the thickness of the tile protrusion and twice the tile kinematic width, 2 W. The discrete number of tile units needed to cover a gap with

*n*+ 1 tiles (incorporating

*n*hinged tile units) can be determined from the quotient (rounded down) of the tile width and gap distance as follows:

*θ*at the local tile unit around its hinge, and an orientation angle

*φ*of this tile with respect to the first tile (usually fixed horizontally) is equal to the sum of the curling angle at previous hinges:

*φ*

_{n}may be used as a measure of total surface curling.

*i*th hinge (

*i*= 1, 2, …

*n*) on the free end of the remaining tiles in Fig. 18. Torques about the hinge balance between the internal torques from the unit curling model

*T*

_{cur,i}(Eqs. (24) and (25)) and other torques from external loads

*T*

_{ext,i}. Each torque equilibrium is used to form the surface model and is expressed at the

*i*th hinge as follows:

*θ*

_{max}=

*π*− 2

*ϕ*is limited by the hard stop chamfer angle

*ϕ*. At the fully curled position after the hard stop is reached, the curling torques in excess of the external torques are blocked by the chamfered tile edges.

*C*

_{d}= 2.05 is the drag coefficient for a uniform flow perpendicular to a long flat plate [49],

*ρ*is density of the air,

*U*is the flow speed, and

*A*=

*Lh*

_{i}is the area facing the flow, where

*h*

_{i}is the sum of the tile vertical height above the

*i*th hinge:

*w*

_{e}on the last tile of the surface provides an extreme yet controllable case of these externally applied loads and is used in the surface model validation, where

*w*

_{t}also resists curling (Fig. 18) with a total torque at the

*i*th hinge of

Given the form of load, the surface model is solved numerically to satisfy equilibrium and find the curling angles at each hinge. The surface model enables the prediction of the surface curling shape under an external load as a function of the geometric parameters (*H*, *W*, *W*_{t}, and *W*_{g}) and the actuation vacuum pressure Δ*p*.

### 5.2 Surface Prototype Fabrication.

Surface prototypes are fabricated with both curling and straightening bladders to validate the curling and straightening functionalities and the prediction accuracy of the surface model. The interconnected straightening inflation bladders are first made by a masked heat-sealing technique [50,51], where the desired shape of the bladders is masked by nonheat sealable Kapton^{®} tape, which is heat pressed between two TPU-coated nylon fabric sheets (Fig. 19(a)). In the absence of masking material, the TPU-coated nylon fuses together, leaving inflatable air bladders where the mask was placed. The T-shaped tiles are directly 3D printed onto the sheet containing the straightening bladders with the hinge lines aligned with the centerlines of the straightening bladders (Fig. 19(b)). A rigid TPU-based filament (NinjaTek^{®} Armadillo^{®}) is used such that it adheres to the TPU-coated nylon bladder. This lower sheet of straightening bladders forms the lower layer of the curling vacuum bladder. The top layer of the curling vacuum bladder is heat pressed onto the tops of the tile protrusions, but care must be taken not to overheat or overcompress and thereby permanently deform the tiles. Alternatively, they can also be glued together using Loctite^{®} fabric adhesive. Finally, the air nozzles are installed, and the bladders are sealed at all four edges with a linear heat sealer (Fig. 19(c)).

A surface prototype is fabricated with five hinged T-shaped tiles of 0.5 in. height, 0.4 in. kinematic width, 9 in. full width, and 6 in. length, whose tile chamfered edges provide a hard stop at $\theta max=26.5deg$. Figure 20 demonstrates the surface prototype’s curling and straightening functionalities through its four-state operation cycle, where vacuum and inflation of the two bladders are produced by two separate pumps, but could alternatively be produced by a single pump with appropriate venting valves. The surface prototype reaches its fully curled position as fast as in 0.3 s.

Due to the different fabrication processes, the surface prototype features different membrane extensibility and increased hinge stiffness relative to the similarly scaled unit prototype and is experimentally characterized to determine the empirical model terms. The measured surface hinge resistive torque is of comparable magnitude to the applied external torque in the surface validation and therefore cannot be ignored as in the unit model validation. While all efforts were made to minimize or characterize these uncertainties, it is expected that they will cause some additional validation error.

### 5.3 Surface Loaded Shape Validation.

To validate the surface model, the surface prototype is loaded with a concentrated end load and curls under increasing vacuum pressure. The surface curling shape is measured through photogrammetry and compared with the surface model prediction using the applied torque and actuation pressure.

To apply a concentrated end load, the surface prototype is clamped horizontally at one end with a weight hung on the last tile at the cantilevered end (Fig. 21). As the vacuum pressure increases, the surface curls from almost flat to fully curled against the internal tile-tile hard stop. The surface curling shape is measured via photogrammetry, where the curling angles at each hinge are obtained from the positions of the colored markers placed on each upper tile protrusion. The surface internal tile shapes are plotted at four different values of vacuum pressures for a given 0.22 lb (0.1 kg) weight based on the curling angles and the tile geometry. The measured locations (dotted contours) are close to the locations predicted from the surface model (solid contours) over the entire range of motion. The model error accumulates along the hinges, causing the surface shape to appear to deviate more near the loaded end, but the curling angle errors remain relatively small (approximately 1–3 deg) at each hinge. Since the curling torque scales with pressure, the results in Fig. 21 also indicate that the 6 in.-long surface prototype can lift up to 7 lb cantilevered end load to 35 deg of surface curling, and 2.5 lb to over 90 deg curling (corresponding to approximately 14 lb and 5 lb distributed load). This is larger than the weight of typical debris. For example, a 6 in. depth of packed snow would weigh 4.7 lb distributed over the 6 in. prototype, and a 2 in. depth of wet leaves would weigh only 0.64 lb [52].

The air surface prototype is run through the full range of curling motion from flat to fully curled under three different weights of 0.22 lb, 0.44 lb, and 0.66 lb. The total curling angle, or equivalently the orientation angle of the last and loaded tile, is plotted against the vacuum gauge pressure along with the model prediction curves in Fig. 22, which has an average error of 13.3% over the full 106 deg of motion. This error distributed among the four tile units is similar to that seen in the unit prototypes; therefore, it can be concluded that additional uncertainties in the surface prototype fabrication are not significant.

The model prediction curves show several nonsmooth points and horizontal and vertical discontinuities, which correspond to different physical behaviors in the measured data. A discrete increase in the slope occurs each time one of the tile units reaches its hard stop; the last tile unit (at the loaded end) reaches its hard stop first, and then each tile unit incrementally reaches its hard stop until the first tile (at the fixed end) reaches its hard stop, and the surface prototype is fully curled at a total curling angle of 106 deg. An instability jump appears in the model curves as a horizontal discontinuity as the last tile becomes close to vertical, and the rate at which the horizontal moment arm decreases with the curling angle is faster than the rate at which the torque produced by the last hinge decreases, causing the angle to jump to 90 deg, after which the weight is effectively cantilevered from the second to last tile. This instability jump does not occur in the data because the hard stops of the fabricated surface are not perfectly rigid such that the surface curling data curves are smooth regardless of the discrete hinges. Except for the instability jump, the shape of the model curve is well reflected in the measured data including discrete changes in slope, demonstrating the predictive ability of the model.

## 6 Curling Surface Model-Based Design

The geometric design parameters of the hinged tile-based air surface can be selected to meet actuation performance and packaging requirements. A visualization of the design space helps to understand the surface actuation capability and to make design selections over a wide range of design parameters. A dimensionless design space contour is developed using the surface model in an example design context for a distributed external load. A windshield cowling design case study converts this dimensionless design contour into a dimensioned plot to drive design selection. The selected design is validated by building and testing a morphing air surface segment demonstrating the wind retention ability and a full-scale prototype windshield cowling operating on a full-size sedan and is also demonstrated on a similar deployable air dam application on a sedan.

### 6.1 Design Space Visualization.

The model and the experimental validation have shown that the surface performance scales with the actuation pressure and its geometric size, while the actuation performance is also affected by the ratio betweeen the height and kinematic width of the tiles (*H*/*W*), which is necessary to design the surface to meet both the actuation performance and packaging (or even fabrication) requirements. A dimensionless design plot excludes the impacts of the scaling variables to investigate the effects of the tile dimensions, which visualizes an actuation effectiveness ratio in the design space of a dimensionless tile height and width scaled with the overall size of the surface (the gap distance *W _{g}*). This dimensionless plot can be easily dimensioned to include the scaling variables and specifications for a specifc design scenario.

An example dimensionless design contour is developed in the design context of a distributed external load such as air drag, which applies to the windshield cowling application. As a measure of surface actuation performace, the actuation effectivess ratio is defined as the ratio of an equivalent pressure from the external load *p*_{ext} and the actuation vacuum pressure Δ*p*, where *p*_{ext} in this design context can be obtained using expressions for distributed loads in Eqs. (31) and (33) as load over area. For distributed loads, the lowest hinge holds the most load when the surface is curled up to its maximum height *h* above the lowest hinge. At this worst-case condition, the actuation effectiveness ratio is solved from torque equilibrium using the surface model, where the culing torque includes the first principle and phenomenological terms from the unit curling model. The hinge stiffness and the geometric end loss are not included since they are not scalable with pressure and geometric size like other terms. Furthermore, their impact can be relatively small for applications such as the windshield cowling; when the surface is long, the end loss is not significant, and when the actuation vacuum pressure is large, the hinge stiffness’s impact is less significant compared to the curling torques. A safety factor may be applied to the torque equilibrium to account for these two effects and other sources of errors including the model errors, and once a design is selected, either from the dimensionless or the dimensioned plots, and the impact of these effects can be evaluated.

Assuming a fully curled position when the last tile is vertical (for a total curling angle of $\phi =90deg$), a dimensionless design contour is generated in Fig. 23. In this visualization, contours of actuation effectiveness ratio are drawn over the design space of tile width and height each normalized with the gap width. The tile protrusions’ width is assumed to be 0.1 *W _{t}* to ensure its strength such that the ratio between the kinematic width and the tile width is

*W*/

*W*

_{t}= 0.45. The zigzag shape of the contour curves results from the discrete number of tiles and occurs when the number of tiles (equal to reciprocal of

*W*

_{t}/

*W*

_{g}) changes. At each zigzag, the dimensionless tile height required to achieve the same actuation effectiveness (or to support the same distributed load under the same actuation vacuum pressure) increases with the relative tile width. As tiles get wider for a given number of tiles, the over surface is wider than the gap until it is wide enough to cover the gap with one fewer tile. The resulting height drops significantly when this last tile is removed, reducing the external torque, producing vertical discontinuities where the next zigzag begins. Overall, the actuation effectiveness ratio increases with the dimensionless tile height such that the surface can support effective external pressure of 5–40% of the actuation vacuum pressure. An infeasible region appears when the height is close to half the total surface gap distance, which is a result of surface self-interference when the upper tile protrusions interfere with each other before the desired total curling of 90 deg is achieved. The dimensionless design contour elucidates the design space and serves as a good reference to evaluate the surface feasibility for particular types of applications.

### 6.2 Windshield Cowling Design Case Study.

*U*

_{vehicle}to represent the surface actuation performance, where a vehicle speed of 80 MPH should satisfy the actuation requirement for most driving conditions. When the cowling is curled, it is essentially perpendicular to the streamlined airflow over the car and therefore acts as an object experiencing drag. At the gap area between the windshield and the hood, the velocity of the flow

*U*near the gap area is lower than the vehicle speed, where $U=66%Uvehicle$ according to computational fluid dynamics computation results [53]. The flow speed

*U*is related to the effective external pressure using the air drag formula for a plate in Eq. (31):

*ρ*= 0.075 lbm/ft

^{3}is used since moist air density is less than that of dry air [54], and the drag coefficient

*C*

_{d}for long flat plate perpendicular to the air flow is 2.05 [49]. The external pressure

*p*

_{ext}is equal to the actuation effectiveness ratio times the actuation vacuum pressure Δ

*p*, chosen at 7 psi, which is equivalent to half the atmospheric pressure and is easily achievable for most pumps. The gap distance between the windshield and the hood is measured from a typical vehicle to be 4.5 in. and applied to the dimensionless tile width and height.

Applying the specifications for the windshield cowling, a dimensioned plot is converted from the dimensionless design plot (Fig. 23) in Fig. 24, where each contour line shows the maximum driving speed at which the surface can curl up to 90 deg against the air drag under 7 psi vacuum to cover a gap distance of 4.5 in. with a safety factor of 3. The contour line for 80 MPH (bolded) shows where the surface meets the actuation requirement. With the zigzag shape inherited from the dimensionless plot due to the discrete number of tiles, the design requirement can be met with lower height at the valleys of the zigzags, where an integer number of tiles exactly spans the gap. The final design is chosen at the valley for a five-tile (4-hinge) design such that the morphing shape is relatively smooth with five tiles, and a surface with 0.90 in. tile width and 0.54 in. height is within the feasible range for fabrication with the processes and materials used and meets the packaging requirements. In addition, the surface design providing 90 deg total curling is checked to provide sufficient gap distance (Fig. 5(c)) to allow wiper operation, and the impacts of the hinge stiffness and geometric end loss are determined from the model to be well within the selected safety factor, thus ensuring proper operation.

### 6.3 Prototype Validation.

The windshield cowling design is validated by building and testing a short-length prototype to test the actuation performance against a wind load, and a full-scale automotive device to demonstrate its functionalities.

To test the surface’s ability to curl and hold its curled position rigidly against air drag, a short length (0.5 ft) prototype of the selected dimensions was fabricated and placed out of the window of a driving vehicle by mounting it fixed to a bar and held out of the passenger window (Fig. 25). Cameras were fixed to the same bar and the front passenger seating to record the prototype performance from two angles during the tests. The cowling’s ability to deploy and stow was tested at a vehicle speed of 25 MPH, where the surface actuation was smooth and complete. Its ability to hold its curled position at a higher speed (up to 50 MPH limited by test feasibility) is also tested, where the deployed prototype did not deflect or vibrate under 7 psi of vacuum at all tested speeds as measured by video (frame captured in Fig. 25(b)). By holding out of the window, the tested driving speed is roughly equal to the flow speed at the prototype; however, since $U=66%Uvehicle$ at the wiper area, 50 MPH in this test is equivalent to 75 MPH. Up to this tested speed, the surface has demonstrated a strong actuation performance without any deflection or vibration, and with the safety factor, it should hold the designed performance at 80 MPH.

To demonstrate this surface architecture as a solution for covering the gap between the windshield and the hood, a full-scale automotive device is manufactured and installed on a vehicle (Figs. 26(a) and 26(b)). To accommodate the hood curvature, the full-scale cowling is built with eight 7.5 in. long segments to cover the entire 5 ft wide windshield. Since the windshield is not flat, extra bladder material and angles are designed between the segments to accommodate the required curvature change to create the 3D motion. The full-scale cowling can perform the full four-state operation cycle to deploy and stow itself through vacuum and inflation, and opens a sufficient gap for the wipers to function. The actuation of the automotive device is monolithic and aesthetically pleasing due to the distributed actuation. While the full-scale prototype is relatively slow to operate (deploys in 1 s) given the faster 0.3 s speed of the small prototype, optimized pneumatic routing can improve this dramatically. Once installed to a pre-existing pump inside the vehicle, the cowling is potentially an effective and low-cost add-on device to solve the problem with the gap between the windshield and the hood.

In addition to the windshield cowling, the surface can be useful for many other external aero surface applications, including a straightforward modification to a deployable air dam [55,56]. A deployable air dam is useful to break the tradeoff between improving aerodynamic drag and maintaining high ground clearance. Figures 26(c) and 26(d) show the full-scale deployable air dam adapted from the cowling prototype by installing it upside down under the car front. The specifications are similar for the two applications; the air dam also needs to hold its shape under air drag at high-speed driving. Using different design constraints and specifications, an air dam design can be obtained following the same steps to generate a dimensioned design plot of a total deployed height, for example, from the dimensionless plot to meet its actuation performance and packaging requirements.

## 7 Conclusion

This article introduced and explored a curling air surface architecture based on hinged T-shaped tiles, which provide both curling and straightening functions with large force and deflection, and can hold its position rigidly against external loads. With the ability to cover the gap normally and actively curl out of the way to allow wiper operation, the tile-based air surface structure is a viable solution for the gap problem between the windshield and hood, protecting the gap from debris when the wipers are not in use. The air surface architecture combines rigid structures with inflatables by sealing hinged T-shaped tiles in an airtight vacuum bladder to provide out-of-plane curling and adding inflation bladders spanning the bottom hinge lines to provide straightening in the opposite direction. The two-bladder surface structure can be produced with simple fabrication processes using 3D printing and a masked heat-sealing technique to produce multiple functionalities in a unified structure.

The repeated unit structure enables the accumulation of curling over the air surface, which can be operated, modeled, and designed based on a hinged tile unit, defined as a single hinge system with a membrane connecting the two protrusions. The tile unit curls as the vacuumed bladder pulls the T-protrusions under external air pressure. The rigid T-shaped tiles provide sufficient space for the vacuumed inextensible membrane to sink, generating large curling torques and angles from the membrane tension, while maintaining a significant curling torque from the pressure differential on the tile walls. The circular membrane deflection shape under uniform pressure keeps the geometry of the tile unit simple and enables analytical modeling of these two main sources of curling torques based on first principles, forming a unit curling model with additional phenomenological terms. Membrane wall contact is also analytically modeled and discussed to capture the performance at large curling angles, which, by delaying the contact, highlights the advantage of the T-shaped tile cross section. As indicated by the first principle terms, scaling of the curling torques with pressure differential and square of the geometric size is supported experimentally and gives insights for the form of the phenomenological terms. The phenomenological terms capturing bladder effects are studied in all possible types: both along the length and at the ends for both directions, where the distributed term in the curling direction dominates over the other three terms and matches in magnitude with frictional effects from the bladder. Representable by a single dimensionless constant, this phenomenological term captures the hysteresis and can be obtained from one scale (size) to predict performance in other scales. With experimentally characterized material properties, geometric end loss due to bladder end effects, and the single phenomenological term, the unit curling model is validated with average errors of 6.2% above 20 deg curling angle across scales. The low-angle performance below 20 deg is equally good with average errors of 4.2% in the straightening direction and for the large prototypes, and bounded with 40% error for the small prototypes in the curling direction. The air surface model is aggregated from multiple instances of the unit curling model and predicts the deflected shape of air surface prototypes under load with average errors of 13%.

The validated model enables a scalable dimensionless design space visualization of the tile geometries for general curling applications against loads. With a dimensioned vehicle speed limit contour converted from the dimensionless design plot, a windshield cowling design is selected to meet the packaging and actuation requirements. This design is validated by building and testing: (1) a morphing air surface segment demonstrating 80 MPH wind retention force and (2) a full-scale prototype windshield cowling operating on a full-size sedan. The same full-scale prototype is directly applicable to a deployable air dam with the ability to support high wind load when deployed in the rigidized curled form and stow in-plane beneath the car when not in use, e.g., ensuring ground clearance when parked. This article provides the technology concept and the supporting model and design approach to more broadly apply this useful air surface architecture to applications in automotive (air dam, adaptive seating), aerospace (morphing wing), architecture (self-assembly shelters), and other domains.

## Acknowledgment

The authors would like to thank the General Motors/University of Michigan Multifunctional Vehicle Systems Collaborative Research Lab on Smart Materials and Structures for support of this work.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.