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

First Equation




(19)

It's the same equation (12)

second Equation

Substituting equations (17) and (18) in the Lagrange equation, and making the required differentiation relative to the variable θ.

Dividing by

(20)

It's the same equation (13)

SIMPLIFICATION OF THE MODEL

We will simplify the mathematical model by converting it from nonlinear equations into linear equations so that we can perform linear control by avoiding nonlinear variables in the system like the friction between the cart and the rail, and consider the payload swing angle is very small, so we can take sinθ → θ, cosθ → 1, tanθ → θ.

Substituting this in the equations (19) and (20), we obtain the final form of the kinetic equations that represent the mathematical model of the gantry crane motion.

(21)
(22)
It can be configured in a Compact Matrix Form
(23)
This reminds of the well-known equation of the Mass-Spring System,

Therefore

(24)
Where

M – Mass Matrix

K – Stiffness Matrix

q – Generalized Coordinate Vector

F – Generalized Force Vector
Equation (24) is a linear equation of the second degree that represents the crane system from practical reality to theoretical. Thus, the kinematic and dynamic response of the gantry crane can be studied and analyzed as a result of any external forces that can be applied to it.