Joseph Calogero

Analytical model and stability analysis of the leading edge spar of a passively morphing ornithopter wing

This paper presents the stability analysis of the leading edge spar of a flapping wing unmanned air vehicle with a compliant spine inserted in it. The compliant spine is a mechanism that was designed to be flexible during the upstroke and stiff during the downstroke. Inserting a variable stiffness mechanism into the leading edge spar affects its structural stability. The model for the spar-spine system was formulated in terms of the well-known Mathieus equation, in which the compliant spine was modeled as a torsional spring with a sinusoidal stiffness function. Experimental data was used to validate the model and results show agreement within 11%. The structural stability of the leading edge spar-spine system was determined analytically and graphically using a phase plane plot and Strutt diagrams. Lastly, a torsional viscous damper was added to the leading edge spar-spine model to investigate the effect of damping on stability. Results show that for the un-damped case, the leading edge spar-spine response was stable and bounded; however, there were areas of instability that appear for a range of spine upstroke and downstroke stiffnesses. Results also show that there exist a damping ratio between 0.2 and 0.5, for which the leading edge spar-spine system was stable for all values of spine upstroke and downstroke stiffnesses.

A dynamic spar numerical model for passive shape change

A three-dimensional constraint-driven dynamic rigid-link numerical model of a flapping wing structure with compliant joints (CJs) called the dynamic spar numerical model is introduced and implemented. CJs are modeled as spherical joints with distributed mass and spring-dampers with coupled nonlinear spring and damping coefficients, which models compliant mechanisms spatially distributed in the structure while greatly reducing computation time compared to a finite element model. The constraints are established, followed by the formulation of a state model used in conjunction with a forward time integrator, an experiment to verify a rigid-link assumption and determine a flapping angle function, and finally several example runs. Modeling the CJs as coupled bi-linear springs shows the wing is able to flex more during upstroke than downstroke. Coupling the spring stiffnesses allows an angular deformation about one axis to induce an angular deformation about another axis, where the magnitude is proportional to the coupling term. Modeling both the leading edge and diagonal spars shows that the diagonal spar changes the kinematics of the leading edge spar verses only considering the leading edge spar, causing much larger axial rotations in the leading edge spar. The kinematics are very sensitive to CJ location, where moving the CJ toward the wing root causes a stronger response, and adding multiple CJs on the leading edge spar with a CJ on the diagonal spar allows the wing to deform with larger magnitude in all directions. This model lays a framework for a tool which can be used to understand flapping wing flight.

Understanding the Relationship Between Pitch Agility and Propulsive Aerodynamic Forces in Bio-Inspired Flapping Wing Vehicles

Ornithopters, or flapping wing mechanical birds, represent a unique category of aerial vehicles that fill a need for small-scale, agile, long range, and payload-capable flight vehicles. This study focuses on understanding the relationship between the propulsive aerodynamic forces and pitch agility in these flapping wing vehicles. Using analytical methods, the aerodynamic moment acting upon a wing undergoing elastic flapping was calculated. A method to determine the pitch stiffness of the vehicle was then derived using a preexisting stability analysis. This method was used to demonstrate that pitch agility in flapping wing birds is intricately tied to the flapping cycle with different parts of the cycle creating stabilizing and destabilizing effects. The results indicated that pitch agility, and propulsive force generation, have a dependency on the shape of the wing, and that deformations such as bend and sweep are capable of making the vehicle more agile. Contact-aided compliant mechanisms with nonlinear stiffness were designed and inserted into the wing of an ornithopter to induce controlled morphing. These elements have varying stiffness during the upstroke and downstroke parts of the cycle which introduces an asymmetry between the two halves of the flapping cycle. The resulting flapping motion exhibited a two fold increase in horizontal propulsive force over the baseline case. A motion tracking system was used to capture the free flight response of the ornithopter in steady level flight. This information was then used to calculate the pitch stiffness of the ornithopter with a rigid spar, and, one with a nonlinear compliant element inserted into the spar to induce a desired shape change. The results revealed that an upstroke in which the aerodynamic forces are similar in magnitude to that of the downstroke, may be necessary to make the vehicle more agile, and, that there is a compromise between vehicle agility and flight propulsive forces.Copyright

A Dynamic Spar Numerical Model for Passive Shape Change

A three-dimensional constraint-driven numerical dynamic model of a flapping wing structure called the Dynamic Spar Numerical Model (DSNM) is introduced and implemented. The model currently includes a leading edge spar and a diagonal spar, attached to a body by revolute and spherical joints, respectively. The spars consist of a user-specified number of rigid links connected by compliant joints (CJs): spherical joints with distributed masses and three axis nonlinear torsional spring-dampers. The goal of this model is to quickly simulate mechanisms in a test platform to see how their CJ design properties and spatial distribution affect passive shape change and physical performance metrics. The results of this model can be used as a starting point for further refinement in compliant joint design for passive shape change. Previous research leading to and assumptions made for modeling CJ are presented. The constraints are established, followed by the formulation of a state model used in conjunction with a forward time integrator, and finally several example runs. Modeling the CJs as linear springs produces a nearly symmetric rotation angles through the flapping cycle, while bi-linear springs show the wing is able to flex more during upstroke than downstroke. Increasing damping ratio reduces high frequency oscillations during the flapping cycle and the number of cycles required to reach steady state. Coupling the spring stiffnesses allows an angle about one axis to induce an angle about another axis, where the magnitude is proportional to the coupling term. Modeling both the leading edge and diagonal spars show that the diagonal spar changes the kinematics of the leading edge spar verses only considering the leading edge spar, causing much larger axial rotations in the leading edge spar. The kinematics are very sensitive to CJ location, where moving the CJ toward the wing root causes a stronger response, and adding multiple CJs on the leading edge spar with a CJ on the diagonal spar allow the wing to deform with larger magnitude in all directions. Future work includes implementing a performance metric, experimental verification, applying loads to represent ambient and flight conditions, and using the model as an optimization tool for parameter and spatial optimization.Copyright

Stability Analysis of the Wing Leading Edge Spar of a Passively Morphing Ornithopter

This paper presents a stability model for the wing leading edge spar of a test ornithopter. The long-term goal of this research effort is to passively improve the performance of ornithopters during steady level flight by implementing a set of wing kinematics found in natural flyers. The desired kinematics is achieved by inserting a compliant mechanism called a compliant spine into the wing leading edge spar to mimic the function of an avian wrist. The stiffness of the compliant spine is time varying and given the nature of flapping flight, it is periodic. Introducing a variable stiffness compliant mechanism into the leading edge spar of the ornithopter affects its structural stability. Therefore, a stability analysis is required. In order to start the stability analysis, an analytical model of the ornithopter wing leading edge spar with a compliant spine inserted in is necessary. In the model, the compliant spine is modeled as a torsional spring with a sinusoidal stiffness function. Moreover, the equations of motion of the wing leading edge spar-spine system can be written in the form of non-homogeneous Mathieu’s equations, which has well-known stability criteria. The analytical system response is then validated using experimental data taken at NASA Langley Research Center. Results show that the analytical spine angular deflection agrees with the experimental angular deflection data within 11%. Stability was then demonstrated using both analytical and graphical proving that the response of leading edge spar with a compliant spine design inserted at 37% of the wing half span is bounded.Copyright

Optimization of a bend-twist-and-sweep compliant mechanism

The overall goal of this research is to develop design optimization methodologies for compliant mechanisms that will provide passive shape change. Our previous work has focused on designing two separate contact-aided compliant elements (CCE): one for bend-and-sweep deflections, called the bend-and-sweep compliant element (BSCE), and another for twist deflection, called the twist compliant element (TCE). In the current paper, all three degrees of freedom, namely bending, twist, and sweep, are achieved simultaneously using a single passive contact-aided compliant mechanism. A new objective function for a contact-aided compliant mechanism is introduced and the results of the optimization procedure are presented. A bend-twist-and-sweep compliant element (BTSCE) can be inserted into the leading edge spar of an ornithopter, which is an avian-scale flapping wing un-manned air vehicle. The multiple objective functions of the optimization problem presented in this paper are: for upstroke, maximize tip bending and sweep deflections, maximize twist angle, and minimize the mass and peak von Mises stress in the BTSCE, and for downstroke, minimize tip bending and sweep deflections, minimize twist angle, and minimize the mass and peak von Mises stress in the BTSCE. This allows a designer to select a CCE from a set of optimal designs to accomplish all three displacement goals. The BTSCE was modeled using a commercial finite element program and optimized using NSGA-II, a genetic algorithm. The results for a single angled compliant joint (ACJ) for quasi-static upstroke loading conditions are presented. Two optimal designs are discussed and compared, one with a moderate peak stress and moderate deflections, the other with a high peak stress and large deflections. The optimization results are then compared to the previous results for the two independent CCEs. A design study showed that the angle of the ACJ needs to be obtuse to achieve a positive twist angle during upstroke, and an acute contact angle reduces peak stress. The deflection objective functions were relatively insensitive to eccentricity for upstroke and downstroke compared to the other parameters, and a high stress penalty was paid for any gains in deflection. The downstroke objective functions were relatively insensitive to all parameters compared to the upstroke objective functions, and were much smaller in magnitude. The optimization showed that under simplified upstroke loading conditions, the BTSCE with a single ACJ allowed bending deflection near 30% of the length of the BTSCE, twist angle near 0.14 radians, and sweep deflection near 5% of the length of the BTSCE.Copyright