D ynamical Modeling and Control of Motion System of the Gantry Crane to Minimize Swing Angle of the Payload

Kinetic analysis of the system
By applying Newton's second law to the free-body diagram of a cart shown in Fig. 3.

Fig. 3. Free-Body Diagram of the cart and the payload

(4)



(5)
Substituting R from (5) in (4) yield

(6)
For payload

(7)


(8)
From (7)
(9)
Substituting (9) in (6)




(10)

Equation (10) is the equation of the cart motion, which is a differential equation of the second degree and consists of four terms that represent four forces as follows:

- Inertia force created by cart motion.

- Coupling force created in the cart by action of the payload motion.

- Centrifugal force created by action of the payload swing on the cart.

- Damping force created by friction between the cart and the rail.

- The required external forces to overcome all the forces mentioned in order to move the cart.
For finding the equation of payload motion:
Substituting (9) in (8)

Multiplying it by the tan θ and distributing it then grouping similar terms yield

The third term is equal to zero because ( )

(11)
Neglecting friction between the cart and the rail, the equations of motion (10) and (11) can be set as follows:
(12)
Multiplying the second term in the equation (11) by






(13)
Equations (12) and (13) are the mathematical model of the motion system of the crane.
These two equations can be extracted using Lagrange's mechanics, which depend on both the kinetic energy and the potential energy of the system.
(14)
Where

- Total Kinetic Energy

- Total Potential Energy

- Dissipative Energy

- Generalized Coordinates ( )

- External Forces
Kinetic energy of both the cart and the payload
(15)

From the Equations (2)

(16)
Substituting (16) in (15)

(17)
- Kinetic energy, as if the cart is moving and the load is not.

- Kinetic energy as if the cart was fixed and the load swinging.

- Coupling kinetic energy as a result of the link between the two motions.
For the potential energy, we take an axis at which the potential energy is zero (Datum).





(18)
By neglecting the friction in the system motion (Dissipation) and substituting equations (17) and (18) in the Lagrange equation, and making the required differentiation relative to the variable x.