King fahd university of petroleum & minerals college of computer sciences & engineering

Experiment #5: Function Generation Using the Analog Computer

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Experiment 6: Analog Simulation of a System of Coupled Masses

Experiment #5: Function Generation Using the Analog Computer

OBJECTIVES
The objective of this experiment is to synthesize a number of time functions (signals) using the analog computer.
INTRODUCTION
The analog computer can be utilized to generate the various functions or waveforms e.g. sinusoidal signal, saw-tooth, square waveform etc. which are used in various applications. These signals can be synthesized as outputs of linear differential equations whose solutions are the required function.
Therefore, the first step in synthesizing such time functions, is to recover the differential equation governing its behavior. There are many ways to get this differential equation. The simplest way is to write the solution of the differential equation in terms of the required time function and to walk back to the differential equation by differentiation. It is also important here to keep track of the initial condition. Another approach is to represent the input and output of the differential equation corresponding to the required time function in the s-domain (using Laplace transforms) and to recover the differential equation by taking inverse Laplace transform. Yet one can also recover the differential equation from experience.
EQUIPMENT:
GP-6 Analog Computer

X-Y Plotter

BACKGROUND
The time functions we are interested in synthesizing are the following:

(a)

(b)

(c)

(d)

where A, and are given constants.

PROCEDURE
(i) First write the obtain the differential equations whose solutions are the corresponding time functions.

For example, for the first waveform (a), this can be obtained as follows. Let

x(t) =

, which implies that x(0) = 0.

Then
and

And

Therefore, the required differential equation whose solution is the given time function is

(ii) Now draw the analog block diagram to simulate the above system and generate the required time function.

(iii) Simulate the differential equation and plot its output using the X-Y plotter as explained in the previous experiments. The result should correspond with the desired time function.

(iv) Repeat the above procedure (i)-(iii) for the time functions (b) and (c). The following Hint will also help you for the case of the time function (iv).
Hint: Functions (a) and (b) require identical unforced second-order differential equations with initial conditions. Function (c) requires a first order unforced differential equation, and function (d) requires a first order differential equation having the forcing function given by (c). Alternatively, you can generate (d) from a second order linear differential equation with constant coefficients whose auxiliary equation has equal roots. Please refer to the introductory part of this manual on linear differential equations.
REPORT
Your report should include the following:

(i) The analog simulation diagrams with labels.

(ii) A plot of the response of the simulations for all the time functions.

(iii) Compare these plots with the actual time functions and make comments.

(iv) Give comments and conclusions on your observations.
Exercise
(i) For the time function (a), if you increase A, and separately, what happens to the output of the analog computer?

(ii) Similarly, for the time function (c), if you increase , what happens to the output of the analog computer?

Experiment #6: Analog Simulation of a System of Coupled Masses

OBJECTIVES
The objective of this experiment is to study the behavior of a system of coupled mass-spring systems using the analog computer.
INTRODUCTION
Many engineering systems have more than one input and more than one output. For example, an aircraft has as its outputs its horizontal speed, its vertical speed and its altitude; a 6 degree of freedom robot manipulator has as its output three Euler Angles and three spatial coordinates of its end effector (Px, Py, Pz); while it has as inputs as many as six joint torques. Such systems are called multiple-input, multiple- output systems (MIMO, or in short multivariable systems). Furthermore, the system may also be large, in which case, it may present a lot of difficulties in modeling and simulation. One approach may be to divide the system into smaller subsystems and study the behavior of each subsystem separately. This approach works very well if the interaction between the subsystems is small and can always be predicted. The overall behavior of the system can then be studied by combining the behavior of the subsystems:

Another alternative, is to study the effect of each input on the overall system outputs separately, and then sum up these effects for all the inputs to get the overall response of the system.

It is therefore the purpose of this experiment, to introduce the student to the modeling and simulation of MIMO systems.
BACKGROUND
Figure 1. shows the model of the system. It is basically a two mass-spring system coupled side-by-side together. The equations governing the behavior of the system are given by:

where all the terms are explained in the previous experiment and in Figure 1. Note that the two subsystems are coupled together by the terms , .
EQUIPMENT

GP-6 Analog Computer
X-Y Plotter

PROCEDURE
(i) Rewrite the dynamical equations governing the behavior of the system in the form suitable for simulation. For the values of parameters given in Table 1 below, draw the analog simulation block for the system.

Table 1. B3=1

M₁	M₂	B₁	B₂	k₁	k₂	f₁	f₂	T(sec)
1	1	1	1	4	4	10	0	10
1	1	1	1	3	4	0	10	10

Now simulate the system for each set of values of the parameters and plot the response of the system for both x₁, x₂ , using the X-Y plotter as in the previous experiments.

(ii) Find analytically, x₁(t) and x₂(t) by assuming f₁(t)=0 and f₂(t) = 10u(t).

Find for this case, and
Compare the values you got in (iii) with the values you got from the simulation.

Report
In your report, include the following:

Free-body diagram for the different masses.
Analog simulation diagrams with labels.
Plots of the displacements of the variables x and x
Discussion of results and conclusions.

Figure 1

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