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(solution) This is a system modeling problem for my advanced dynamics class


This is a system modeling problem for my advanced dynamics class that I need help with.  Thank you!!!!


ME 450

 

OUT: 11/06/15 Computer Project 2

 

Gantry-Crane System Fall 2015

 

DUE: 12/04/15 (2 students work as a team) Fig. 1 Gantry-Crane System

 

(INDCO, Inc) Fig. 2 Simplified Mechanical Model of

 

Gantry-Crane System Gantry-Crane systems of various forms (e.g., Fig. 1) are widely used in industry. Their dynamics

 

can be simply represented by the above mechanical system consisting of a cart moving on a track

 

with a rod attached to the cart body through a rotary joint with very little rotational damping (Fig.

 

2). mc represents the mass of the cart and mr represents the mass of the rod. xc represents the

 

position of the cart, Fm represents a control force that will be applied to the cart by a motor

 

installed on the cart, represents the rotational angle of the rod with respect to the vertical axis,

 

and xr and xp are the positions of the Center of Mass (CoM) and the tip of the rod, respectively.

 

There are viscous friction (bc) between the cart and the track (neglect Coulomb friction) and

 

viscous friction (br) between the cart and the rod at the rotary joint. bc and br are linear and

 

rotational damping coefficients, respectively. System parameters of a scaled-down gantry-crane

 

model are given in the Appendix. Model this system by the following procedure: Model development

 

(1) Identify the reference point and the positive direction, and then draw the FBD for each element

 

shown in Fig. 2.

 

(2) Write down all the elemental equations and interconnecting equations. Check to make sure

 

that the number of equations is the same as the number of unknown variables. Note: You may find that the original system dynamics are nonlinear. A linearized model can be

 

obtained by assuming that the rotation angle of the rod is small so that sin and cos1

 

and all higher order terms can be neglected.

 

(3) Combine the elemental equations and derive the EoMs (make sure the number of EoMs

 

matches the DoF). Now if we are interested in the tip movement of the rod xp and set it as the output of the system,

 

and use motor control force Fm as the input. Instead of directly obtaining the input-output model

 

and the corresponding transfer function G(s) relating the output xp to the input Fm , here we can

 

use the following method of block diagram to construct the input-output model:

 

(4) Consider the total system consisting of two subsystems: 1) Cart dynamics, with horizontal

 

forces (summation of the motor force and horizontal contact force between the cart and the rod) as

 

input and cart position xc as output, find the corresponding transfer function G1(s). 2) Rod

 

dynamics, with cart position xc as input and rod angular position as output, find the

 

corresponding transfer function G2(s). (5) It is easy to see that these two subsystems are coupled, and it can be shown that, for example,

 

the system can be represented by the following block diagram. Fig. 3 Block Diagram of the Gantry Crane System

 

Either develop your own block diagram for the system or use the one shown in Fig. 3 and find

 

the corresponding G1, G2, G3, G4 and G5. Simulation

 

(6) Construct the block diagram derived above in Simulink.

 

(7) If the system starts with the cart in the center of the track and the rod held vertically, try

 

different inputs of Fm (for example, impulse, step or pulse (rectangular) functions) and show you

 

results. Comment on how are the responses of the Gantry-Crane system different from that of a

 

standard second order system introduced in the class. (8) Find the profile of the input force that moves the tip of the rod a distance of approximate10cm

 

(and then stops). You may notice that finding this profile is tedious and requires trial and error,

 

and that it is difficult for us to specify any performance specification (e.g., %OS and settling time).

 

A better solution is to design a feedback controller that uses the measured position of the rod tip

 

and generates the profile of input force according to some control algorithm, which also guarantees

 

certain performance specifications. The theory and design of such a controller will be introduced

 

in ME455 and some introduction will be provided in ME450. Model analysis

 

(9) Change the length of the rod (assume other parameters remain the same) and comment on its

 

effect on the response of the system.

 

(10) Change the damping coefficients br (assume other parameters remain the same) and comment

 

on its effect on the response of the system.

 

Finally, find at least one application of the Gantry-Crane system in industry, and provide a brief

 

description of this application. Appendix

 

Parameters of Gantry-Crane Systems

 

CART mc

 

bc Mass of Cart + Additional Mass

 

0.911

 

kg

 

Damping Coefficient of Cart due to the viscous friction between the cart and the track

 

1.9

 

kg/sec

 

ROD mr

 

L

 

g

 

br Mass of Rod

 

0.231

 

kg

 

Length of rod

 

0.64

 

m

 

Gravitational acceleration

 

9.81

 

ms-2

 

Rotational damping coefficient at the rotary joint due to viscous friction between the rod and the cart

 

0.00014

 

Nms/rad

 


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