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Your question is an essence a question about aerodynamic damping in the roll axis. As the aircraft rolls, the rolling motion increases the angle-of-attack of the descending wing and increases the angle-of-attack of the rising wing, eventually causing the lift created by each wing to be equal despite the deflected ailerons. At this point roll torque is zero and the roll rate can no longer increase. If the stick is brought back to center, the roll torque from aerodynamic damping will soon cause the rate will soon drop to zero, or to near zero. (Yes, it is possible for an airplane to tend to roll toward a steeper or shallower bank angle with the stick centered, but the roll rate is much lower than we see with the stick highly deflected to one side.)

Read more about in this section of John S. Denker's excellent "See How It Flies" website.

For the same wing area, roll damping will be greater with a large wingspan than with a small one. This is why airplanes with smaller wingspans generally achieve higher roll rates than aircraft with larger wingspans.

For the same external physical shape, a high moment of inertia in the roll axis (due for example to having the wings full of fuel or loaded with bombs) will cause the roll damping effect to take more time to bring the roll rate to zero (or near zero) after centering the stick.

To get actual formulae, you are probably going to have take a deep dive into some textbooks on flight dynamics or aircraft design.

Rolling also creates an adverse yaw torque, even with ailerons centered, which creates sideslip which interacts with any dihedral or sweep that is present to slow the roll rate. See this section of the "See How It Flies" website. But that's probably a finer-scale effect than you are looking to take into account. (The complex subject of the roll torque generated by the interaction between sideslip and the aircraft's 3-dimensional geometry has been touched on in several different questions on the ASE site.)

You'll find plenty of other content in the "See How It Flies" website that should help you with your project as well.

Your question is an essence a question about aerodynamic damping in the roll axis. As the aircraft rolls, the rolling motion increases the angle-of-attack of the descending wing and decreases the angle-of-attack of the rising wing, eventually causing the lift created by each wing to be equal despite the deflected ailerons. At this point roll torque is zero and the roll rate can no longer increase. If the stick is brought back to center, the roll torque from aerodynamic damping will soon cause the rate will soon drop to zero, or to near zero. (Yes, it is possible for an airplane to tend to roll toward a steeper or shallower bank angle with the stick centered, but the roll rate is much lower than we see with the stick highly deflected to one side.)

Read more about in this section of John S. Denker's excellent "See How It Flies" website.

For the same wing area, roll damping will be greater with a large wingspan than with a small one. This is why airplanes with smaller wingspans generally achieve higher roll rates than aircraft with larger wingspans.

For the same external physical shape, a high moment of inertia in the roll axis (due for example to having the wings full of fuel or loaded with bombs) will cause the roll damping effect to take more time to bring the roll rate to zero (or near zero) after centering the stick.

To get actual formulae, you are probably going to have take a deep dive into some textbooks on flight dynamics or aircraft design.

Rolling also creates an adverse yaw torque, even with ailerons centered, which creates sideslip which interacts with any dihedral or sweep that is present to slow the roll rate. See this section of the "See How It Flies" website. But that's probably a finer-scale effect than you are looking to take into account. (The complex subject of the roll torque generated by the interaction between sideslip and the aircraft's 3-dimensional geometry has been touched on in several different questions on the ASE site.)

You'll find plenty of other content in the "See How It Flies" website that should help you with your project as well.

Your question is an essence a question about aerodynamic damping in the roll axis. As the aircraft rolls, the rolling motion increases the angle-of-attack of the descending wing and increases the angle-of-attack of the rising wing, eventually causing the lift created by each wing to be equal despite the deflected ailerons. At this point roll torque is zero and the roll rate can no longer increase. If the stick is brought back to center, the roll torque from aerodynamic damping will soon cause the rate will soon drop to zero, or to near zero. (Yes, it is possible for an airplane to tend to roll toward a steeper or shallower bank angle with the stick centered, but the roll rate is much lower than we see with the stick highly deflected to one side.)

Read more about in this section of John S. Denker's excellent "See How It Flies" website.

For the same wing area, roll damping will be greater with a large wingspan than with a small one. This is why airplanes with smaller wingspans generally achieve higher roll rates than aircraft with larger wingspans.

For the same external physical shape, a high moment of inertia in the roll axis (due for example to having the wings full of fuel or loaded with bombs) will cause the roll damping effect to take more time to bring the roll rate to zero (or near zero) after centering the stick.

To get actual formulae, you are probably going to have take a deep dive into some textbooks on flight dynamics or aircraft design.

Rolling also creates an adverse yaw torque, even with ailerons centered, that creates sideslip which interacts with any dihedral or sweep that is present to slow the roll rate. See this section of the "See How It Flies" website. But that's probably a finer-scale effect than you are looking to take into account. (The complex subject of the roll torque generated by the interaction between sideslip and the aircraft's 3-dimensional geometry has been touched on in several different questions on the ASE site.)

You'll find plenty of other content in the "See How It Flies" website that should help you with your project as well.

Your question is an essence a question about aerodynamic damping in the roll axis. As the aircraft rolls, the rolling motion increases the angle-of-attack of the descending wing and increases the angle-of-attack of the rising wing, eventually causing the lift created by each wing to be equal despite the deflected ailerons. At this point roll torque is zero and the roll rate can no longer increase. If the stick is brought back to center, the roll torque from aerodynamic damping will soon cause the rate will soon drop to zero, or to near zero. (Yes, it is possible for an airplane to tend to roll toward a steeper or shallower bank angle with the stick centered, but the roll rate is much lower than we see with the stick highly deflected to one side.)

Read more about in this section of John S. Denker's excellent "See How It Flies" website.

For the same wing area, roll damping will be greater with a large wingspan than with a small one. This is why airplanes with smaller wingspans generally achieve higher roll rates than aircraft with larger wingspans.

For the same external physical shape, a high moment of inertia in the roll axis (due for example to having the wings full of fuel or loaded with bombs) will cause the roll damping effect to take more time to bring the roll rate to zero (or near zero) after centering the stick.

To get actual formulae, you are probably going to have take a deep dive into some textbooks on flight dynamics or aircraft design.

Rolling also creates an adverse yaw torque, even with ailerons centered, which creates sideslip which interacts with any dihedral or sweep that is present to slow the roll rate. See this section of the "See How It Flies" website. But that's probably a finer-scale effect than you are looking to take into account. (The complex subject of the roll torque generated by the interaction between sideslip and the aircraft's 3-dimensional geometry has been touched on in several different questions on the ASE site.)

You'll find plenty of other content in the "See How It Flies" website that should help you with your project as well.

Your question is an essence a question about aerodynamic damping in the roll axis. As the aircraft rolls, the rolling motion increases the angle-of-attack of the descending wing and increases the angle-of-attack of the rising wing, eventually causing the lift created by each wing to be equal despite the deflected ailerons. At this point roll torque is zero and the roll rate can no longer increase. If the stick is brought back to center, the roll torque from aerodynamic damping will soon cause the rate will soon drop to zero, or to near zero. (Yes, it is possible for an airplane to tend to roll toward a steeper or shallower bank angle with the stick centered, but the roll rate is much lower than we see with the stick highly deflected to one side.)

Read more about in this section of John S. Denker's excellent "See How It Flies" website.

For the same wing area, roll damping will be greater with a large wingspan than with a small one. This is why airplanes with smaller wingspans generally achieve higher roll rates than aircraft with larger wingspans.

For the same external physical shape, a high moment of inertia in the roll axis (due for example to having the wings full of fuel or loaded with bombs) will cause the roll damping effect to take more time to bring the roll rate to zero (or near zero) after centering the stick.

To get actual formulae, you are probably going to have take a deep dive into some textbooks on flight dynamics or aircraft design.

Rolling also creates an adverse yaw torque, even with ailerons centered, that creates sideslip which interacts with any dihedral that is present to slow the roll rate. See this section of the "See How It Flies" website. But that's probably a finer-scale effect than you are looking to take into account.

You'll find plenty of other content in the "See How It Flies" website that should help you with your project as well.

Your question is an essence a question about aerodynamic damping in the roll axis. As the aircraft rolls, the rolling motion increases the angle-of-attack of the descending wing and increases the angle-of-attack of the rising wing, eventually causing the lift created by each wing to be equal despite the deflected ailerons. At this point roll torque is zero and the roll rate can no longer increase. If the stick is brought back to center, the roll torque from aerodynamic damping will soon cause the rate will soon drop to zero, or to near zero. (Yes, it is possible for an airplane to tend to roll toward a steeper or shallower bank angle with the stick centered, but the roll rate is much lower than we see with the stick highly deflected to one side.)

Read more about in this section of John S. Denker's excellent "See How It Flies" website.

For the same wing area, roll damping will be greater with a large wingspan than with a small one. This is why airplanes with smaller wingspans generally achieve higher roll rates than aircraft with larger wingspans.

For the same external physical shape, a high moment of inertia in the roll axis (due for example to having the wings full of fuel or loaded with bombs) will cause the roll damping effect to take more time to bring the roll rate to zero (or near zero) after centering the stick.

To get actual formulae, you are probably going to have take a deep dive into some textbooks on flight dynamics or aircraft design.

Rolling also creates an adverse yaw torque, even with ailerons centered, that creates sideslip which interacts with any dihedral or sweep that is present to slow the roll rate. See this section of the "See How It Flies" website. But that's probably a finer-scale effect than you are looking to take into account. (The complex subject of the roll torque generated by the interaction between sideslip and the aircraft's 3-dimensional geometry has been touched on in several different questions on the ASE site.)

You'll find plenty of other content in the "See How It Flies" website that should help you with your project as well.