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Chapter 6: Problem 31

Consider an aircraft traveling at constant altitude and speed as it undergoes tight periodic rolling motion of the fuselage. Let the wings be modeled as equivalent rigid bodies with torsional springs of stiffness \(k_{T}\) at the fuselage wall. In addition, let each wing possess moment of inertia \(I_{c}\) about its respective connection point and let the fuselage of radius \(R\) have moment of inertia \(I_{o}\) about its axis. Derive the equations of rolling motion for the aircraft.

Short Answer

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Question: Describe the process of deriving the equations of rolling motion for an aircraft undergoing periodic rolling motion at constant altitude and speed, considering the wings as equivalent rigid bodies with torsional springs. Answer: To derive the equations of rolling motion for an aircraft, we need to apply the concepts of rigid body dynamics, particularly rotational motion. The process involves 4 main steps: (1) Identify the torques acting on each wing and the fuselage due to the torsional springs, (2) Apply Newton's second law for rotational motion for each wing, (3) Apply Newton's second law for rotational motion for the fuselage, and (4) Solve the obtained equations to find expressions for angular accelerations and displacements needed to describe the rolling motion.

Step by step solution

01

Identify the torques acting on each wing and the fuselage

In this problem, we need to consider the torques acting on each wing and the fuselage due to the torsional springs. For each wing, the torque is given by \(\tau_{T} = -k_T \theta_{i}\), where \(\theta_{i}\) is the angular displacement of the wing and \(k_{T}\) is torsional spring stiffness. Since the springs exert equal and opposite torques on the wings and fuselage, the torque on the fuselage is the sum of torques on the wings, which is given by \(\tau_{o} = -(\tau_{T_1} + \tau_{T_2})\).
02

Apply Newton's second law for rotational motion for each wing

Newton's second law for rotational motion states that the net torque acting on a rotational object is equal to its moment of inertia times its angular acceleration. Therefore, for each wing, \(I_{c} \alpha_{i} = \tau_{T_{i}} = -k_T \theta_i\), where \(\alpha_{i}\) is the angular acceleration of the wing, and \(I_{c}\) is the moment of inertia of the wing.
03

Apply Newton's second law for rotational motion for the fuselage

Likewise, we apply Newton's second law for rotational motion to the fuselage, \(I_{o} \alpha_{o} = \tau_{o} = -(\tau_{T_1} + \tau_{T_2})\). It is important to notice that the angular acceleration of the fuselage is equal to the sum of the angular accelerations of both wings, since the rolling motion is a result of the combined action of the wings. Therefore, we can write the equation for the fuselage as \(I_{o} (\alpha_1 + \alpha_2) = - (k_T \theta_1 + k_T \theta_2)\).
04

Solve the equations of motion

We have obtained three equations in step 2 and step 3 for the rotational motion of the aircraft. They are: 1. \(I_{c} \alpha_{1} = -k_T \theta_1\) 2. \(I_{c} \alpha_{2} = -k_T \theta_2\) 3. \(I_{o} (\alpha_1 + \alpha_2) = - (k_T \theta_1 + k_T \theta_2)\) We can solve these three equations to obtain expressions for \(\alpha_1\), \(\alpha_2\), and \(\theta_1\), and \(\theta_2\) needed to describe the rolling motion of the aircraft.

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