Possibility of using Simulink in the dynamics of lifting equipment

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Figure 2. Top view of the lift mechanism

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Figure 1 and Figure 2 shows the used labeling:

- MOT - motor;

- SP - clutch;

- BR – brake;

- OK1....OK4 – cogwheels 1....4;

- BUB – drum;

- L – rope;

- KL – pulley;

- KAB – car;

- PRZ – counterweight;

- H1....H7 – shafts 1....7.

2. Formulation of Dynamic Motion Equation Mechanical System

Mechanical system of a passenger lift represents a mechanism with one degree of freedom.

To compile the dynamic motion equation of this mechanism we use the method of weight and force variables reduction.

To ensure a dynamic equilibrium of original and reduced system, conditions have to be met

(1)

where: Ek, Ered – are kinetic energies in original and reduced system; A, Ared – are works of active forces in original and reduced system; P, Pred – are powers of active forces and moments in original and reduced system.

Motion of original system is reduced to motion of motor.

2.1. Calculation of Kinetic Energy of the System

Total kinetic energy of mechanical system is the sum of kinetic energies of its individual members, depending on the motion, which they perform. Figure 3 shows the diagram in terms of angular velocities about axes of rotation and members that perform translational motion

Figure 3. Labeling of angular velocities and velocities

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Kinetic energy of mechanical system is determined by:

(2)

If the kinetic energy is expressed in function of angular velocity of the motor (ω_MOT), after quantification of the masses of individual members of the system and their moments of inertia (members that perform rotational motion), then the kinetic energy of the original system is:

(3)

2.2. Calculation of Work of Active Forses and Moments

We determine the work of all active forces and moments that act on the individual members of the system (see Figure 3).

The work is expressed in a function of motion of selected system member, in this case the function of motion of the motor, thus its rotation φ_MO_T.

(4)

After substituting and expressing gear ratios the work of active forces and moments is:

(5)

2.3. Dynamic Motion Equation

Dynamic motion equation of reduced system is expressed:

(6)

after substitution from equations (1):

(7)

After substituting (3) and (5) into (6) the final dynamic motion equation of mechanical system is:

(8)

3. Results of the Solution Using Matlab-Simulink

Solution of differential equation (7) the mechanical system is performed using Simulink blocks. The scheme for solving equation (7) in Matlab/Simulink is in the Figure 4:

Figure 4. The block diagram in Simulink for solving and rendering selected variables

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Parameters of variables are saved to the Workspace Figure 5:

Figure 5. Workspace with variables data

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Figure 6. Angular acceleration

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Program in Matlab for graphic representation the results from Workspace is in the form:

Simulation results in Simulink from Scope block is in the Figure 7 and Figure 8.

The selected variables are angular velocity Figure 7 and angular displacement of motor Figure 8 of the mechanical system Figure 2.

Starting time is limited by time during which the lift car reaches traveling speed.

The shape of the curve of input torque of motor is in fact different from constant. Its course is part of the documentation to the supplying of motor and it is a commercial matt.

Figure 7. Angular velocity

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Figure 8. Angular displacement

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Figure 9. Setting parameters in block Scope

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