Moment Response in Dynamics

Questions and Answers

What happens to the roll acceleration when the moment of inertia about the roll axis Ix increases?

It increases substantially.

It becomes negligible.

It decreases. (correct)

It remains constant.

Gyroscopic precession is primarily caused by which combination?

Linear momentum and friction.

Angular momentum and friction.

Linear momentum and applied torque.

Angular momentum and applied torque. (correct)

In the context of a rolling pull-up maneuver, what is the role of the negative yaw rate r?

It stabilizes the aircraft.

It contributes to adverse yaw. (correct)

It neutralizes rolling moments.

It increases roll acceleration.

What do coupling terms in rotational dynamics refer to?

Relationships between disturbances in different axes. Signup and view all the answers

Which equation relates rolling moment to angular acceleration in roll?

ṗIx = L Signup and view all the answers

How does an increase in Iz affect the gyroscopic precession phenomenon?

It increases the rate of precession. Signup and view all the answers

What does the term 'inertial products' denote in the context of rotational dynamics?

The effective inertia between axes. Signup and view all the answers

In the moment response equation, what does the variable L represent?

The net torque applied. Signup and view all the answers

Which equation relates the body rotation rates to the Euler rates?

$Ω_f = p i + q j + r k = Ψ + Θ + Φ$ Signup and view all the answers

What is the transformation required for rotating the angular position Ψ?

A positive rotation through Θ followed by Φ Signup and view all the answers

If an aircraft has Euler angles of Ψ = 0 deg, Θ = 0 deg, and Φ = 90 deg, what is the significance of this configuration in motion?

It establishes a neutral orientation in the flight path. Signup and view all the answers

According to the kinematic equations provided, which of the following represents the body rate q?

$q = ext{sin}(Φ) ext{cos}(Θ)Ψ̇ + ext{cos}(Φ)Θ̇$ Signup and view all the answers

What can be said about the relationship between the rotational moments and the Euler angles?

They directly correspond as per the kinematic equations. Signup and view all the answers

What is the mathematical representation of the relationship between p, q, and r?

$p^2 + q^2 + r^2 = Ψ̇^2 + Θ̇^2 + Φ̇^2$ Signup and view all the answers

How do kinematic equations derive from the conditions of motion?

They arise from the need to balance unknowns in rotational dynamics. Signup and view all the answers

What factor is considered essential to close the equations of motion in rotational kinematics?

The number of Euler angles involved Signup and view all the answers

What effect does a roll in either direction have on an aircraft with a negative Ixz?

Results in a pitch up Signup and view all the answers

Which moment equation corresponds to the pitching motion in an aircraft?

q̇Iy + pr(Iz − Ix) + (p² − r²)Ixz = MA + MF Signup and view all the answers

In the longitudinal equations of motion, what contributes to the x-force equation?

Weight component and aerodynamic forces Signup and view all the answers

What is primarily described by the lateral equations of motion?

Dynamics related to lateral forces and moments Signup and view all the answers

Which term is NOT included in the moment equations of motion for aircraft?

Weight vector contribution Signup and view all the answers

What does the term coupling in roll coupling refer to in aircraft dynamics?

Interrelationship of roll and pitch motion Signup and view all the answers

What control forces are depicted in the lateral moment equations?

Combination of aerodynamic and thrust forces Signup and view all the answers

Which of the following components is required for wings-level flight according to the longitudinal equations?

All forces must be in equilibrium Signup and view all the answers

Study Notes

Moment Response - Angular Acceleration

Rolling moment equation: ṗIx + qr (Iz - Iy) - (r˙ + pq)Ixz = L
When gyro and coupling terms are small, simplify to ṗIx = L.
Larger moment of inertia about the roll axis (Ix) results in smaller roll acceleration.

Moment Response - Gyroscopic Precession

Gyroscopic precession involves rotation at an angle relative to the axis.
Occurs from angular momentum combined with an applied moment or torque.

Moment Response - Gyroscopic Precession Example

In a rolling pull-up maneuver, with negligible angular acceleration and coupling terms, the equation reduces to −Iy qr = L.
Positive rolling moment (e.g., from ailerons) leads to negative yaw rate (r), causing adverse yaw.

Moment Response - Coupling Terms

Coupling terms define inertial relationships, illustrating how disturbances in one axis can affect others.
Incorporate inertial products like Ixy to represent asymmetries in inertial resistance.

Moment Response - Coupling Terms Example

Pitching moment equation simplifies to p² Ixz = -Iy q̇ when ignoring gyro precession and yaw rate.
Negative Ixz means off-axis mass distribution is more rearward, predicting a roll will induce pitch up (roll coupling).

Applied Moments

Applied moments consist of rolling, pitching, and yawing moments from aerodynamic forces (LA, MA, NA) and thrust (LF, MF, NF).
Weight vector does not contribute as it aligns with the center of mass.
The moment equations total six:
- ṗIx + qr (Iz - Iy) - (r˙ + pq)Ixz = LA + LF
- q̇Iy + pr (Iz - Ix) + (p² - r²)Ixz = MA + MF
- r˙Iz + pq(Iy - Ix) + (qr - ṗ)Ixz = NA + NF

Longitudinal Equations of Motion

Three primary equations govern longitudinal (x-z) plane motion, including forces and moments.
Key equations involve:
- m (u̇ + qw - rv) = -mg sin(Θ) + [-W cos(α) + A sin(α)] + F cos(iF)
- q̇Iy + pr (Iz - Ix) + (p² - r²)Ixz = MA + MF
- m (ẇ + pv - qu) = mg cos(Φ) cos(Θ) + [-W sin(α) - A sin(α)] - F sin(iF)

Lateral Equations of Motion

Remaining equations address lateral forces and moments:
- ṗIx + qr (Iz - Iy) - (r˙ + pq)Ixz = LA + LF
- m (v̇ + ru - pv) = mg sin(Φ) cos(Θ) + RAY + RFY
- r˙Iz + pq(Iy - Ix) + (qr - ṗ)Ixz = NA + NF

Kinematic Equations

Three additional kinematic equations close the motion equations due to extra unknowns from Euler angles.
Establish relationship between rotational moments and Euler angles.

Kinematic Conditions

Magnitudes of body rotation rates should equal magnitudes of the Euler rates: Ω f = p î + q ĵ + r k̂ = Ψ + Θ + Φ.
Consistent relationship: p² + q² + r² = Ψ̇² + Θ̇² + Φ̇².

Coordinate Transforms

Use coordinate transformation matrices to switch angular rate vectors between Earth and body axes.
Transformations defined for each Euler angle, ensuring correct rotational sequence.

Required Kinematic Equations

Essential kinematic equations to determine body rates:
- p = -sin(Θ)Ψ̇ + Φ̇
- q = sin(Φ) cos(Θ)Ψ̇ + cos(Φ)Θ̇
- r = cos(Φ) cos(Θ)Ψ̇ - sin(Φ)Θ̇

Kinematic Equations Example

Given:
- Ψ = 0 degrees, Ψ̇ = 10 degrees/s
- Θ = 0 degrees, Θ̇ = 0 degrees/s
- Φ = 90 degrees, Φ̇ = 0 degrees/s
Calculate body rates based on provided Euler angles and rates.

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Description

This quiz explores the concepts of moment response, angular acceleration, and gyroscopic precession in dynamic systems. It highlights the relationships between applied rolling moments and the resulting angular accelerations. Understand the implications of gyroscopic effects in mechanical motion.