Lightweight vibrating structures (such as flapping wing micro air vehicle (FWMAV) designs) often require some form of control. To achieve controllability, local structural property changes (e.g., damping and stiffness changes) might be induced in an active manner. The stroke-averaged lift force production of a FWMAV wing can be modified by changing the structural properties of that wing at carefully selected places (e.g., changing the properties of the elastic hinge at the wing root as studied in this work). To actively change the structural properties, we investigate three different methods which are based on: (1) piezoelectric polymers, (2) electrorheological fluids, and (3) electrostatic softening. This work aims to gain simple yet insightful ways to determine the potential of these methods without focusing on the precise modeling. Analytical models of FWMAV wing designs that include control approaches based on these three methods are used to calculate the achievable lift force modifications after activating these methods. The lift force production as a result of a wing flapping motion is determined using a quasi-steady aerodynamic model. Both piezoelectric polymers and electrostatic softening are found to be promising in changing the structural properties and, hence, the lift force production of FWMAV wings. For the control of lightweight FWMAV designs, numerical simulations reveal a promising roll maneuverability due to the induced lift force difference between a pair of opposite wings. Although applied to a specific FWMAV design, this work is relevant for control of small, lightweight, possible compliant, vibrating structures in general.@en

Design sensitivities are used in various engineering methods to arrive at optimal or advanced designs. The design sensitivity of a system, also known as the derivative to a design variable, represents the change of the system response due to an adjustment of a design variable. Scientists and engineers often use modal design sensitivities to predict the change of a transient response. These sensitivities are the derivatives of the eigenvalues and eigenvectors of the system. Many researchers have successfully computed modal sensitivities using several methods. Nelson's method [1] is widely used even though computationally expensive. Fox's method [2] uses a modal superposition to represent the eigenvector sensitivities by the system eigenvectors, but is inaccurate when using only a few eigenvectors. Hence, several iterative methods were proposed to increase the accuracy of Fox's method (e.g., [3] or [4]). The primary focus of the previous research was on the computational algorithms rather than the interpretation of the eigenvector sensitivities. This interpretation depends highly on the chosen normalization. An approach for incorporating these normalizations was presented by Smith [5]. This paper shows the consequence and potential use for interpretation of the chosen normalization. This is illustrated on an example using a generic two-degrees-of-freedom system, as shown in FIGURE 2.@en

Design sensitivities are used in various engineering methods to arrive at optimal or advanced designs. The design sensitivity of a system, also known as the derivative to a design variable, represents the change of the system response due to an adjustment of a design variable. Scientists and engineers often use modal design sensitivities to predict the change of a transient response. These sensitivities are the derivatives of the eigenvalues and eigenvectors of the system. Many researchers have successfully computed modal sensitivities using several methods. Nelson's method [1] is widely used even though computationally expensive. Fox's method [2] uses a modal superposition to represent the eigenvector sensitivities by the system eigenvectors, but is inaccurate when using only a few eigenvectors. Hence, several iterative methods were proposed to increase the accuracy of Fox's method (e.g., [3] or [4]). The primary focus of the previous research was on the computational algorithms rather than the interpretation of the eigenvector sensitivities. This interpretation depends highly on the chosen normalization. An approach for incorporating these normalizations was presented by Smith [5]. This paper shows the consequence and potential use for interpretation of the chosen normalization. This is illustrated on an example using a generic two-degrees-of-freedom system, as shown in FIGURE 2.@en

Keywords: FWMAV control, eigensolution sensitivity, repeated eigenvalues, structural modifications, modal basis. Abstract: Several Flapping Wing Micro Air Vehicle (FWMAV) designs use structural resonating properties to increase energy efficiency. For practical use, control of the resonating structure, i.e., the eigensolutions, is required which is complicated by constraints on weight and power consumption. Hence, systematic ways to induce substantial changes in the resonance response are of high interest. Eigensolution sensitivity is used to approximate the change in the resonance vibration, i.e., the eigenmode, due to local structural modifications. Expressing these sensitivities in a modal basis, shows that (nearly) repeated eigenvalues are an indication for effective eigenmode control. The eigenmodes corresponding to repeated eigenvalues are not uniquely defined but any linear combination of these eigenmodes is also an eigenmode. This could lead to substantial changes in the eigenmode, or vibration response, if the eigenmodes become uniquely defined again if the corresponding eigenvalues become distinct due to local structural modifications. A smart structural design could use this idea to exhibit substantial changes in the eigenmodes as introduced by relatively limited control action. The response of a simple symmetric structure, with repeated eigenvalues, shows the response change if the structure becomes asymmetric. Additionally, the influence of structural damping on the effectiveness of the response change is shown.@en

Keywords: FWMAV control, eigensolution sensitivity, repeated eigenvalues, structural modifications, modal basis. Abstract: Several Flapping Wing Micro Air Vehicle (FWMAV) designs use structural resonating properties to increase energy efficiency. For practical use, control of the resonating structure, i.e., the eigensolutions, is required which is complicated by constraints on weight and power consumption. Hence, systematic ways to induce substantial changes in the resonance response are of high interest. Eigensolution sensitivity is used to approximate the change in the resonance vibration, i.e., the eigenmode, due to local structural modifications. Expressing these sensitivities in a modal basis, shows that (nearly) repeated eigenvalues are an indication for effective eigenmode control. The eigenmodes corresponding to repeated eigenvalues are not uniquely defined but any linear combination of these eigenmodes is also an eigenmode. This could lead to substantial changes in the eigenmode, or vibration response, if the eigenmodes become uniquely defined again if the corresponding eigenvalues become distinct due to local structural modifications. A smart structural design could use this idea to exhibit substantial changes in the eigenmodes as introduced by relatively limited control action. The response of a simple symmetric structure, with repeated eigenvalues, shows the response change if the structure becomes asymmetric. Additionally, the influence of structural damping on the effectiveness of the response change is shown.@en

Keywords: FWMAV control, eigensolution sensitivity, repeated eigenvalues, structural modifications, modal basis. Abstract: Several Flapping Wing Micro Air Vehicle (FWMAV) designs use structural resonating properties to increase energy efficiency. For practical use, control of the resonating structure, i.e., the eigensolutions, is required which is complicated by constraints on weight and power consumption. Hence, systematic ways to induce substantial changes in the resonance response are of high interest. Eigensolution sensitivity is used to approximate the change in the resonance vibration, i.e., the eigenmode, due to local structural modifications. Expressing these sensitivities in a modal basis, shows that (nearly) repeated eigenvalues are an indication for effective eigenmode control. The eigenmodes corresponding to repeated eigenvalues are not uniquely defined but any linear combination of these eigenmodes is also an eigenmode. This could lead to substantial changes in the eigenmode, or vibration response, if the eigenmodes become uniquely defined again if the corresponding eigenvalues become distinct due to local structural modifications. A smart structural design could use this idea to exhibit substantial changes in the eigenmodes as introduced by relatively limited control action. The response of a simple symmetric structure, with repeated eigenvalues, shows the response change if the structure becomes asymmetric. Additionally, the influence of structural damping on the effectiveness of the response change is shown.@en

Keywords: FWMAV control, eigensolution sensitivity, repeated eigenvalues, structural modifications, modal basis. Abstract: Several Flapping Wing Micro Air Vehicle (FWMAV) designs use structural resonating properties to increase energy efficiency. For practical use, control of the resonating structure, i.e., the eigensolutions, is required which is complicated by constraints on weight and power consumption. Hence, systematic ways to induce substantial changes in the resonance response are of high interest. Eigensolution sensitivity is used to approximate the change in the resonance vibration, i.e., the eigenmode, due to local structural modifications. Expressing these sensitivities in a modal basis, shows that (nearly) repeated eigenvalues are an indication for effective eigenmode control. The eigenmodes corresponding to repeated eigenvalues are not uniquely defined but any linear combination of these eigenmodes is also an eigenmode. This could lead to substantial changes in the eigenmode, or vibration response, if the eigenmodes become uniquely defined again if the corresponding eigenvalues become distinct due to local structural modifications. A smart structural design could use this idea to exhibit substantial changes in the eigenmodes as introduced by relatively limited control action. The response of a simple symmetric structure, with repeated eigenvalues, shows the response change if the structure becomes asymmetric. Additionally, the influence of structural damping on the effectiveness of the response change is shown.@en

The motion amplitude of a harmonically driven compliant structure is maximized if the driving frequency equals one of the structural resonance frequencies. For the effective use of resonating compliant structures, control of the eigensolutions, i.e., eigenfrequencies and eigenmodes, is often required. This work shows a design methodology in which eigenproblem sensitivity is used in a systematic way to design the most effective locations to apply local structural modifications for a required change in the eigensolutions. By applying control patches at these locations the power requirements are minimized. The modal basis of the structure is used as the preferred basis which leads to valuable insights in the possibilities and limitations of controlling specific eigensolutions. The influence on the eigensolutions due to the mechanical properties of the attached inactive control actuators is separated from the influence due to the active actuation of the control actuators. Whether or not a control actuator is actively actuated depends on the desired control state. Furthermore, it is argued that nearly repeated eigenfrequencies contribute to the effective control of eigenmodes. A simple compliant structure is used to demonstrate the potential of the presented design methodology. In this example, the uncontrolled eigensolutions are forced into different, independent control states using effectively distributed control actuators. Keywords: Resonance, Compliant Structures, Eigenproblem Sensitivity, Local Structural Modifications, Control States, Control patches@en

The motion amplitude of a harmonically driven compliant structure is maximized if the driving frequency equals one of the structural resonance frequencies. For the effective use of resonating compliant structures, control of the eigensolutions, i.e., eigenfrequencies and eigenmodes, is often required. This work shows a design methodology in which eigenproblem sensitivity is used in a systematic way to design the most effective locations to apply local structural modifications for a required change in the eigensolutions. By applying control patches at these locations the power requirements are minimized. The modal basis of the structure is used as the preferred basis which leads to valuable insights in the possibilities and limitations of controlling specific eigensolutions. The influence on the eigensolutions due to the mechanical properties of the attached inactive control actuators is separated from the influence due to the active actuation of the control actuators. Whether or not a control actuator is actively actuated depends on the desired control state. Furthermore, it is argued that nearly repeated eigenfrequencies contribute to the effective control of eigenmodes. A simple compliant structure is used to demonstrate the potential of the presented design methodology. In this example, the uncontrolled eigensolutions are forced into different, independent control states using effectively distributed control actuators. Keywords: Resonance, Compliant Structures, Eigenproblem Sensitivity, Local Structural Modifications, Control States, Control patches@en

The motion amplitude of a harmonically driven compliant structure is maximized if the driving frequency equals one of the structural resonance frequencies. For the effective use of resonating compliant structures, control of the eigensolutions, i.e., eigenfrequencies and eigenmodes, is often required. This work shows a design methodology in which eigenproblem sensitivity is used in a systematic way to design the most effective locations to apply local structural modifications for a required change in the eigensolutions. By applying control patches at these locations the power requirements are minimized. The modal basis of the structure is used as the preferred basis which leads to valuable insights in the possibilities and limitations of controlling specific eigensolutions. The influence on the eigensolutions due to the mechanical properties of the attached inactive control actuators is separated from the influence due to the active actuation of the control actuators. Whether or not a control actuator is actively actuated depends on the desired control state. Furthermore, it is argued that nearly repeated eigenfrequencies contribute to the effective control of eigenmodes. A simple compliant structure is used to demonstrate the potential of the presented design methodology. In this example, the uncontrolled eigensolutions are forced into different, independent control states using effectively distributed control actuators. Keywords: Resonance, Compliant Structures, Eigenproblem Sensitivity, Local Structural Modifications, Control States, Control patches@en

The motion amplitude of a harmonically driven compliant structure is maximized if the driving frequency equals one of the structural resonance frequencies. For the effective use of resonating compliant structures, control of the eigensolutions, i.e., eigenfrequencies and eigenmodes, is often required. This work shows a design methodology in which eigenproblem sensitivity is used in a systematic way to design the most effective locations to apply local structural modifications for a required change in the eigensolutions. By applying control patches at these locations the power requirements are minimized. The modal basis of the structure is used as the preferred basis which leads to valuable insights in the possibilities and limitations of controlling specific eigensolutions. The influence on the eigensolutions due to the mechanical properties of the attached inactive control actuators is separated from the influence due to the active actuation of the control actuators. Whether or not a control actuator is actively actuated depends on the desired control state. Furthermore, it is argued that nearly repeated eigenfrequencies contribute to the effective control of eigenmodes. A simple compliant structure is used to demonstrate the potential of the presented design methodology. In this example, the uncontrolled eigensolutions are forced into different, independent control states using effectively distributed control actuators. Keywords: Resonance, Compliant Structures, Eigenproblem Sensitivity, Local Structural Modifications, Control States, Control patches@en

In the quest for energy efficient flapping wing micro air vehicles (FWMAVs), the wing performance is of paramount importance. The wing performance is mainly determined by the wing planform and the wing-beat kinematics. Since the optimization of the wing planform and the wing-beat kinematics is complicated by the flapping wing aerodynamics, most FWMAV designs tend to use standard wing planform and kinematics inspired function of the wing planform and the kinematic pitching amplitude. For this purpose, a quasi-steady aerodynamic model is used to determine the aerodynamic loads. This model allows, opposed to the more computationally costly method of direct numerical simulation, its use in optimization. The average normalized lift force, the average normalized required power and the ratio between those two are visualized as a function of the design variables to define the required wing planform and pitching amplitude for optimal hovering performance. Using different optimization formulations, it was found that several different wing designs result in nearly equal performance. It is shown that there is a lot of design freedom with respect to the design variables. This freedom is also shown in nature by the presence of a huge variety of wing planforms.@en

Keywords: Compliant Structures, Eigensolutions, Systematic Control, Sensitivities, Local Structural Modifications Abstract. The motion amplitude of a harmonically driven compliant structure is maximized if the driving frequency equals one of the structural resonance frequencies. For practical use of resonating compliant structures control of the eigensolutions, i.e. eigenfrequencies and eigenmodes, is often required. In this work, a systematic way to control the eigensolutions of harmonically driven structures is presented while maintaining the advantages of a resonating structure. This control is realized using local structural modifications. Eigensolution sensitivities for these local structural modifications are used to indicate the locations for the most effective structural modifications, minimizing the required control power. This method uses the modal basis of the structure as the preferred basis. A simple harmonically driven compliant structure is used to show how local structural modifications are selected to obtain a desired change in the eigensolutions.The proposed method is found to be a convenient tool to determine effective local structural modifications to control the eigensolutions of resonating compliant structures. During the design phase it provides valuable insights in the possibilities and limitations of controlling specific eigensolutions.@en

Keywords: Compliant Structures, Eigensolutions, Systematic Control, Sensitivities, Local Structural Modifications Abstract. The motion amplitude of a harmonically driven compliant structure is maximized if the driving frequency equals one of the structural resonance frequencies. For practical use of resonating compliant structures control of the eigensolutions, i.e. eigenfrequencies and eigenmodes, is often required. In this work, a systematic way to control the eigensolutions of harmonically driven structures is presented while maintaining the advantages of a resonating structure. This control is realized using local structural modifications. Eigensolution sensitivities for these local structural modifications are used to indicate the locations for the most effective structural modifications, minimizing the required control power. This method uses the modal basis of the structure as the preferred basis. A simple harmonically driven compliant structure is used to show how local structural modifications are selected to obtain a desired change in the eigensolutions.The proposed method is found to be a convenient tool to determine effective local structural modifications to control the eigensolutions of resonating compliant structures. During the design phase it provides valuable insights in the possibilities and limitations of controlling specific eigensolutions.@en

Wing designs for Flapping Wing Micro Air Vehicles (FWMAVs) might use a properly tuned elastic hinge at the wing root to obtain the required passive pitching motion to achieve enough lift production to stay aloft. Practical use of this type of FWMAVs requires some form of control which can be achieved by actively adjusting the elastic hinge stiffness and, thus, the pitching motion and lift production of the wing. This paper studies an elastic hinge design consisting of stacked layers which can be sticked together using electrostatics. This sticking changes the bending stiffness of the hinge. The voltage-dependent behavior of this elastic hinge during the large pitching motion are described in detail. The passive pitching motion is governed by the equation of motion which is a function of the elastic hinge stiffness and the applied control voltage. The lift generated by the passive pitching wings is predicted by a quasi-steady aerodynamic model. Numerical simulations show significant changes of the passive pitching motion and, consequently, of the lift production, if slipping stacked layers stick together. Experiments are conducted to study the practical applicability of this method on FWMAVs. The experiments show similar trends as the numerical simulations in modifying the pitching motion although the effect is less significant which is mainly due to manufacturing difficulties. This approach is, in conclusion, promising to control FWMAV flight.@en

This thesis studies a controllability approach for general resonant compliant systems. These systems exploit resonance to obtain a specific dynamic response at relatively low actuation power. This type of systems is often lightweight, is scalable and minimizes frictional losses through the use of compliant hinges. Some insect-inspired Flapping Wing Micro Air Vehicle (FWMAV) designs are based on such a resonant compliant system. These designs are carefully tuned such that a particular resonance response of the system corresponds to the desired wing flapping motion. The system’s resonance response depends strongly on its structural properties (i.e., mass, damping, stiffness and their spatial distribution). Resonance response modifications can, thus, be controlled using carefully selected structural property changes. This thesis contributes to the controllability of resonant compliant systems in general and the compliant FWMAV design in particular. Although there is yet no physical, controllable FWMAV design with integrated active components, the current research strongly indicates the applicability of structural property changes to reach controllability of this type of systems. Effective control requires an integrated approach that considers simultaneously the structure, the wings, the kinematics, the methods to induce property changes and the desired controllability@en