Answer to Question #153900 in Mechanical Engineering for Suleman butt

Question

Question: - Derive 6DOF Equations of motion for the Vehicle.

The equation of motion for the vehicle are mathematical models, which express the motion law of 

the vehicle. Based on the models, one may analyze and simulate the motion of a vehicle. In 

addition, based on small disturbance theory, one may derive linear longitudinal small disturbance 

motion equation and lateral small disturbance motion equations from the dynamic equation. 

Motion of the vehicle follows Newton’s Laws. Newton’s law formulates the relations between the 

summation of external forces, the acceleration, and the relations between the summation of 

external moments and the angular acceleration.

Your work should follow the following assumptions

1. The earth is considered as an inertial reference, i.e. it is stationary.

2. Earth’s curvature is neglected, and earth-surface is assumed to be flat.

3. The vehicle is assumed to be rigid body. Any two points on or within the airframe retain 

fixed with respect to each other. Ignore the aero-elastic effects of the vehicle.

4. The mass of the vehicle is assumed to retain constant.

5. The vehicle is considered as symmetry about Oxbyb plane. The product of inertia Ixy and Izy

vanish.

Assume that the moving coordinate frame with an angular velocity ω as shown in the figure 1. The 

vector ω is resolved into three component p, q, r in this coordinate frame as follows. Where i, j, k

are unit vectors respectively along xb, yb and zb axes

𝝎 = 𝑝𝒊 + 𝑞𝒋 + 𝑟k

You have to derive relations for

1. Force Equations

2. Moment Equations

3. Kinematic Equations

a. Equation for Center of Mass

b. Angular Motion Equations

Guess

Consider changeable vector a(t). The a(t) is resolved into three component ax, ay, az in the 

coordinate frame thus

𝒂 = 𝑎𝑥𝒊 + 𝑎𝑦𝒋 + 𝑎𝑧𝒌 (2)

Taking derivative of a(t) with respect to time t yield 

𝑑𝒂

𝑑𝑡

=

𝑑𝑎𝑥

𝑑𝑡

𝒊 +

𝑑𝑎𝑦

𝑑𝑡

𝒋 +

𝑑𝑎𝑧

𝑑𝑡

𝒌 + 𝑎𝑥

𝑑𝒊

𝑑𝑡

+ 𝑎𝑦

𝑑𝒋

𝑑𝑡

+ 𝑎𝑧

𝑑𝒌

𝑑𝑡

(3)

Theoretical mechanics presents that if a rigid body rotates at an angular velocity 𝝎 about fixed 

point, the velocity of arbitrary point P in rigid body is given by

𝑑𝒓

𝑑𝑡

= 𝝎 × 𝒓 (4)

Where r is a vector radius from the origin point O to the point P

For vector radius

𝑑𝒊

𝑑𝑡

= 𝝎 × 𝒊 (5)

𝑑𝒋

𝑑𝑡

= 𝝎 × 𝒋 (6)

𝑑𝒌

𝑑𝑡

= 𝝎 × 𝒌 (7)

Similarly

𝑑𝒂

𝑑𝑡

=

𝑑𝑎𝑥

𝑑𝑡

𝒊 +

𝑑𝑎𝑦

𝑑𝑡

𝒋 +

𝑑𝑎𝑧

𝑑𝑡

𝒌 + 𝝎 × (𝑎𝑥

𝑑𝒊

𝑑𝑡

+ 𝑎𝑦

𝑑𝒋

𝑑𝑡

+ 𝑎𝑧

𝑑𝒌

𝑑𝑡) (8)

i.e.

𝑑𝒂

𝑑𝑡

=

𝛿𝒂

𝛿𝑡 + 𝝎 × 𝒂 (9)

Where

𝛿𝒂

𝛿𝑡 =

𝑑𝑎𝑥

𝑑𝑡

𝒊 +

𝑑𝑎𝑦

𝑑𝑡

𝒋 +

𝑑𝑎𝑧

𝑑𝑡

𝒌

𝛿𝒂

𝛿𝑡 is called “relative derivative”

𝑑𝒂

𝑑𝑡

is called “absolute derivative

Answer 1

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