Question

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

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

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 ${\tau }_o=-({\tau }_{T_1}+{\tau }_{T_2})$.

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.

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 $I_o({\alpha }_1+{\alpha }_2)=-(k_T{\theta }_1+k_T{\theta }_2)$.

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.

Key concepts

Torsional Springs

Torsional springs play a crucial role in the dynamics of systems involving rotation, such as the rolling motion of aircraft. These springs exert a restoring torque proportional to the angular displacement from their equilibrium position, which in this case, is the wings' rest position relative to the fuselage. The spring's stiffness is denoted by the parameter $k_T$, which indicates how much torque is generated for a given angular displacement $\theta$. This relationship is expressed by the torque formula:

• ${\tau }_T=-k_T{\theta }_i$

In this equation, the negative sign signifies that the torque works to counteract the displacement, thus attempting to restore the wings to their original position. Understanding torsional springs is essential for modeling how the aircraft responds during maneuvering by assessing the generated torques and how they influence motion.

Moment of Inertia

The moment of inertia is a fundamental concept in rotational dynamics that reflects an object's resistance to changes in its rotational motion. It plays a role similar to mass in linear motion but applies to rotating bodies. For our airplane model, each wing possesses its moment of inertia, denoted as $I_c$, about its attachment to the fuselage, while the fuselage itself has a moment of inertia $I_o$ along its axis.

• Moment of inertia of a wing: $I_c$
• Moment of inertia of the fuselage: $I_o$
• Influences angular acceleration: $\alpha$

This concept is critical as it determines how much torque is needed to achieve a certain angular acceleration for both the wings and the fuselage. The greater the moment of inertia, the greater the torque required to effect a given change in rotation rate.

Angular Displacement

Angular displacement refers to the angle through which a body rotates around a specific point or axis. For the wings of the aircraft, it is characterized by angles ${\theta }_1$ and ${\theta }_2$ for each, respectively. This displacement points to how much the wing has rotated compared to its resting state.Understanding angular displacement is vital for analyzing how torsional springs influence the aircraft's rolling motion. As the wings move, the angular displacement changes and affects the restoring torque applied by the springs, according to:

• Restoring torque: ${\tau }_T=-k_T{\theta }_i$

Thus, knowing the angular displacements helps calculate the forces in action during aircraft roll, critical for designing stable and responsive flight systems.

Newton's Second Law for Rotational Motion

Newton's second law for rotational motion establishes a foundation for understanding how torques influence an object's rotational acceleration. It states that the net torque $\tau$ acting on a body is equal to the product of the body's moment of inertia $I$ and its angular acceleration $\alpha$:

• $\tau =I\alpha$

For our aircraft scenario, this law applies to both the wings and the fuselage:

• For each wing: $I_c{\alpha }_i=-k_T{\theta }_i$
• For the fuselage: $I_o({\alpha }_1+{\alpha }_2)=-(k_T{\theta }_1+k_T{\theta }_2)$

Through this law, we can derive the motion equations that describe how the aircraft behaves as a result of the torques from the torsional springs acting on it. This understanding enables engineers to analyze and predict the aircraft's rolling motion dynamics under different conditions.