Figure 3 shows a simplified model for a flexible rolling aircraft. The fuselage is modeled as...

 

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Figure 3 shows a simplified model for a flexible rolling aircraft. The fuselage is modeled as a pinned rigid body with a concentrated moment of inertia J and is free to rotate about the longitudinal axis passing through the black point at the center of the schematic. The mass distribution of each half wing is modeled as a concentrated mass mw that can translate vertically. The bending elasticity of the wing is modeled as massless flexible element connecting the mass to the fuselage. The flexible element is treated as an equivalent translational spring of constant 3E1/1³, where El is the element bending stiffness and I is its length. Assuming anti-symmetric behavior with respect to the aircraft centerline and small-amplitude motions, answer the following questions: 1. Identify and justify generalized coordinate choices that result in (a) Inertially decoupled equations (b) Elastically decoupled equations 2. For each generalized coordinate choice in Question 1, derive (a) The potential energy (b) The kinetic energy (c) The virtual work (d) The generalized equations of motion in matrix form using Lagrange's equations 3. Discuss the characteristics of the two sets of equations obtained in Question 2 4. Derive the transformation matrices between the two sets of coordinates in Question 1 5. Verify the transformation matrices from Question 3 by obtaining each set of equations from the other one To earn full points, clearly define the generalized coordinates and their positive directions in a drawing. F(t) m 3E1/1³ umim 3E1/1³ Figure 3: Schematic of a flexible rolling aircraft. F(t) 112 Figure 3 shows a simplified model for a flexible rolling aircraft. The fuselage is modeled as a pinned rigid body with a concentrated moment of inertia J and is free to rotate about the longitudinal axis passing through the black point at the center of the schematic. The mass distribution of each half wing is modeled as a concentrated mass mw that can translate vertically. The bending elasticity of the wing is modeled as massless flexible element connecting the mass to the fuselage. The flexible element is treated as an equivalent translational spring of constant 3E1/1³, where El is the element bending stiffness and I is its length. Assuming anti-symmetric behavior with respect to the aircraft centerline and small-amplitude motions, answer the following questions: 1. Identify and justify generalized coordinate choices that result in (a) Inertially decoupled equations (b) Elastically decoupled equations 2. For each generalized coordinate choice in Question 1, derive (a) The potential energy (b) The kinetic energy (c) The virtual work (d) The generalized equations of motion in matrix form using Lagrange's equations 3. Discuss the characteristics of the two sets of equations obtained in Question 2 4. Derive the transformation matrices between the two sets of coordinates in Question 1 5. Verify the transformation matrices from Question 3 by obtaining each set of equations from the other one To earn full points, clearly define the generalized coordinates and their positive directions in a drawing. F(t) m 3E1/1³ umim 3E1/1³ Figure 3: Schematic of a flexible rolling aircraft. F(t) 112

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1 Identify and justify generalized coordinate choices a Initially decoupled equations For initially decoupled equations we want to choose coordinates ... View the full answer