19 April 2002

Build a Simulated Bouncing Ball Analog Computer

By Craig Kendrick Sellen

Birch Hills Residence
25 Reservoir St.
Simpson PA 18407-1300

Click image to enlarge

Back in the days when digital computers were slow, cumbersome and very expensive, analog computers were smaller, faster, and cheaper than their digital counterparts. Thus in those early days, analog computers were extensively used to perform scientific calculations. Analog computers utilize electronic circuit models to mimic the mathematical behavior of real world phenomena. They produce calculation results that are less accurate than the results obtained from digital computers, but analog computers are (like slide rules) accurate enough for many purposes. Simulating the motion of a bouncing ball on an oscilloscope screen is one application where the accuracy of an analog computer is quite sufficient. This article shows the reader how to build and operate such an analog computer.

Consider the situation where a ball is suspended at a height (let's call it h) in a uniform gravity field above a hard, flat surface, which is perpendicular to the lines of force of the gravity field. For the moment we are assuming that we in close proximity to the planet Earth, where the acceleration of gravity is 32 feet per second per second. Also for the moment we are neglecting the effect of air resistance. The ball has potential energy with respect to the surface beneath it. The potential energy is given by the following equation.

Potential Energy = (Weight of the ball) times (Height above the Surface)

or

P.E. = wh

If the ball is released it will be accelerated at a uniform rate by the gravity field. As the ball falls, potential energy is being converted into kinetic energy. The kinetic energy at any point in the path of the ball is given by the following equation.

Kinetic Energy = 1/2 times (Mass of the Ball) times (Velocity Squared)

or

K.E.=mv²/2

At the instant before the ball hits the surface, all the potential energy has been converted into kinetic energy. The speed of the ball at this point is given by the equation

v = at

where v is the velocity, a is the acceleration of gravity, and t is the time that the ball required to fall to the surface.

When the ball hits the surface, kinetic energy is once again changed into potential energy as the ball is compressed by the force of the impact. If the ball is made of an elastic material, such as rubber, it will return to its spherical shape within a few milliseconds. The force of the ball returning to its spherical shape accelerates the ball upward (bounce up action). The ball will leave the surface with an upward velocity, v', which will be less than the velocity, v, at which the ball hit the surface. Because some of the energy of the ball has been converted into heat by the impact and deformation of the ball,

The acceleration of gravity will now slow the velocity of the ball until the ball stops instantaneously at a height, h', which is less than the original height, h. At this instant all the kinetic energy of the ball has been converted into potential energy once again.

The acceleration of gravity which brought the ball to an instantaneous stop continues to accelerate the ball in the direction of the underlying surface. velocity builds up at the rate of 32 feet per second per second toward this surface. The ball hits the surface once again, this time at the velocity, v', with which it left the surface seconds ago. The ball then compresses and expands, some energy is converted into heat, the ball is accelerated upward once again by the elasticity of the ball, and it leaves the surface with a velocity, v''. The v'' is less than each of the velocities, v and v'. The ball then attains a height, h'', which is less than each of the heights, h and h', falls again, bounces up again etc., until all the energy of the ball is converted into heat. At this point the ball stops bouncing.

The basic equation that gives the height above the flat surface for any given instant of flight is the following familiar equation from high school physics class.

y = v_Ot + (at²)/2

Here v is the initial velocity and t is the time since release, or (when bouncing up and then falling back) since impact. This equation is built into the circuit parameters of the analog computer, and the circuitry produces an output voltage that represents the height of the ball at any given instant of flight.

This varying DC voltage is used to position the image of the ball E on the oscilloscope screen. Other equations that are built into the circuitry of the bouncing ball computer modify this output DC : voltage so that the time required for deformation and recovery of the ball, as well as the loss of kinetic energy due to heat production are taken into consideration. Unfortunately the circuitry is not sophisticated enough to produce a display that : is in perfect agreement with the numerical values produced by the physical equations that govern the movement of the ball. Even though the display is not a perfect depiction of physical reality, it is close enough to give the feel of the real thing.

On the front panel of the computer are controls which allow the circuit elements representing the acceleration of gravity (Gravity), the mass of the ball (Mass), air viscosity (Damping), percentage energy loss at impact (Spring Rate), and the initial height (Initial Height), to be varied over wide ranges.

Up to this point in our discussion we have neglected the effect of air viscosity on the movement of the bouncing ball, but circuitry is also included in order to take air resistance into account. The effect of air resistance produces a force on the ball opposite to the direction of travel. This force is directly proportional to the speed of the ball. The greater the mass of the ball, the greater the force of gravity on the ball, and the 8 less the velocity of the ball will be slowed by the force of air resistance. Without the effect of air resistance (in a vacuum), changing the mass of the ball will have no effect (as Galileo a observed) on the velocity of fall of the ball.

HEADERS FOR THE BOUNCING BALL ANALOG COMPUTER WIRES COMING OFF BOARD TO VAR. COMPONETS OFF BOARD (REV.1) J1,J2-30-TO-36VAC IN FROM POWER TRANSFORMER J3-GND. FROM TRANSFORMER C.T. J39,J40-POWER LED F.P. J4,J5-REPETITIVE ACTION SWITCH F.P. J7,J8-DROP RELEASE SWITCH F.P. J6-JUNCTION OF R5, R66 J6-THRESH PIN 6 OF U4 J43,J44-TO RELAY STATUS LED F.P. J41,J42-RELAY STATUS LED F.P. J42=GND. J48,J49-BALL DROPING SOUND CAPACITOR C9 EXTRA CAP. J20,J21,J22-DAMPING ADJUST POT. F.P. J21=GND. (1OK) J12,J13,J14-SWEEP RATE ADJUST POT. F.P. J14=GND. (1OK) J17,J18,Jl9-GRAVITY ADJUST POT. F.P. Jl9=GND. (1OK) J23,J24-INITIAL HEIGHT POT. F.P. NO GND./JUMPER ON POT. (5K) J29,J30,J31-SPRING RATE POT. F.P. J31=GND. J30= IS WIPER (1K) J32,J33,J34-MASS ADJUST POT. F.P. J34=GND. (1K) J35,J36-OUTPUT TO SCOPE VERTICAL J36=GND. J37,J38-OUTPUT TO SCOPE HORIZONTAL J38=GND. J15,J16-SPEAKER OUTPUT BALL SOUND J16=GND. Figure 2

For any given fixed level of atmospheric viscosity, at a fixed level of gravitational acceleration, there is a maximum velocity that a falling ball of fixed mass can reach, which is called the : terminal velocity. At this velocity the force of resistance of the atmosphere to the movement of the object equals the force of gravity on the object. Increasing the mass of the ball increases the terminal velocity. In a vacuum there is no upper limit on the : velocity of a falling object, i.e. terminal velocity equals infinity.

There is a sixth control (Sweep Rate)on the front panel of the computer that controls the value of horizontal velocity of the ball. The horizontal velocity component is added to the motion of the ball so as to spread out the motion of the ball and thus to make an improved display. The above discussion of kinetic and potential energy was developed for the simpler case where the ball has no horizontal component of velocity. For the more general case where there are velocity components in both the vertical and horizontal directions, one must remember that the terms kinetic energy and velocity in the above refers to kinetic energy and velocity in

the vertical direction only. The components of velocity and kinetic energy of the ball in the horizontal direction remain constant throughout themotion of the ball.

ABOUT THE CIRCUIT

Op-amps U6 through U11, with the exception of U10 are used to compute the vertical coordinate of the ball. the physical equations that govern the vertical position of the ball are incorporated into the circuitry associated with these op-amp chips. U6 is an integrator. The rate of change of the output of U6 is determined (assuming a fixed voltage at pin 7 of U6) by the voltage setting of the "Gravity'' potentiometer. U7 integrates U6's output. Initial conditions are determined by the resistance setting of the "Initial Height" control. "Damping'' and "Spring Rate" are controlled by two feedback loops. U9 divides the sum of these two fluctuating DC voltages by the "Mass" voltage.

U10 generates the circle that represents the ball. U10 is used as an oscillator to produce a sine and a cosine output. These two outputs are summed to produce the circle.

U6 and U11 are used to compute the horizontal coordinate of the ball. U6 generates the horizontal sweep voltage across the oscilloscope screen. The sweep rate is determined by a 10K pot at headers J12, J13 and J14.

Initial conditions are set by one set of contacts of relay K2. When the drop switch contacts are closed, a comparator composed of U4 a 555 timer opens the initial contacts of both relays K1 and K2. When the repetition switch contacts are closed, the relays automatically reset initial conditions when the sweep voltage reaches approximately 2/3 of the supply voltage. The power to K2 runs through one set of contacts of relay K1. This is

to make sure that the correct initial height is reset before the next sweep cycle begins.

A sound generator (made up of components transistors Q1, Q2, transformer T1, trim pot TR1, capacitor C9, and resistors R10 and R7) makes an impact sound each time the simulated bouncing ball hits the floor, this adds realism to the action effect. The sound from the generator is amplified by power amplifier U12. Adjust trimpot TR1 for the most pleasing ball sound.

Figure 3. Here's the foil pattern for the solder side of the ball computer. You will need to use some method of connecting the traces between the two sides of the board. Click image to enlarge.

CONSTRUCTION

Probably the best way to assemble the main circuitry of the "bouncing-ball analog computer'' is to use a double-sided printed circuit board. You can make your own circuit board, or, alternately, you can buy a custom made board from a supplier (see the parts list for supplier listing). Another alternative is to assemble the main circuitry on a perforated board, using solder clips and IC sockets. Or you could use a solderless breadboard to do the construction. Which ever method you use, refer to the table in Fig. 2 for instructions on how to interconnect the main circuit board assembly with the components located off the main circuit board.

Figure 4. Here's the foil pattern for the component side of the ball computer. Click image to enlarge.

The author assembled the bouncing ball computer on a double-sided glass epoxy printed-circuit board measuring 4X6 inches. A full-size template of the unit's printed-circuit foil layout is shown.. The component side of the board is shown in Fig. 4 and the bottom foil layer is shown in Fig 3. These foil patterns give the actual-size etching, drilling and component placement guides. Those who are skilled in etching double-sided circuit boards can copy the patterns shown and use them on a blank board.

Once you have collected all of the components listed in the parts list, construction can begin. Please NOTE, before you go any further, that many of the components used in the construction of this board will have to be soldered from both sides, producing what is known as "plated through" holes. Alternatives to plated-through holes include small eyelets and short pieces of wires.

Where a component lead goes through the hole, you can use that lead for the connection to the other side. Keep in mind that in all of those suggestions, you will need to solder the connections on both sides of the board. The only ''hard and fast" rule is that the method you use should be the one that you are comfortable with and one that can reliably make a solid connection.

Figure 5. All of the components for the ball computer fit neatly on a small 4 x 6 PC board. Click image to enlarge.

Assemble the bouncing ball computer guided by the parts-placement diagram shown in Fig. 5 . Begin by first soldering all passive components to the board, followed by the capacitors and diodes and then the semiconductors. Be careful to get all the components orientations correct the first time. This is especially important for all polarized components such as solid-state devices and electrolytic capacitors into the board; be sure that they are properly oriented as shown. Just one part placed in backwards in the circuit will stop it from working and possibly destroy it or other components. While on the subject of polarized components, make careful note of the IC's used; make sure that they are oriented in the right way.

It's a good idea to use IC sockets for the integrated circuits; servicing is easier if you don't have to unsolder a multi-leaded device. Whether you decide to use integrated circuit sockets or not, do not insert the IC's at this time; we will do that later while testing the ball computer. Be extremely mindful that tiny solder splashes as well as the rosin used in the core of electronics solder is conductive and can in some cases lead to short circuits.

When the PC board is completed, examine it very carefully for proper parts placement, opens, or short circuits, and bad or cold solder connections that might look like dull blobs of solder. Any solder joint that is suspect should be redone by removing the old solder with a desoldering braid, cleaning the joint and carefully applying new solder. It is far easier to correct problems now rather than later on if you discover that your ball computer does not work as it should, after this is done it is time to insert the integrated circuits in there proper sockets.

FINAL ASSEMBLY

Select a cabinet for the "bouncing ball" analog computer that is large enough to accommodate all of the components without crowding. Next machine the front panel for the potentiometers, LED's, switches and binding posts or BNC connectors. Next drill the mounting holes for the printed circuit board, the line cord, and the fuse holder through the rear panel of the cabinet. Mount the off-the-board components in their respective holes, followed by the circuit board assembly inside the cabinet. Then using Fig. 2 as a guide, connect the potentiometers, switches, etc. to the board to complete the wiring of the unit.

SET-UP REQUIREMENTS

The computer has a built-in power supply that produces -+15 volts for the op-amps consisting of U1, U2,and U3. The required oscilloscope should either be a dual-trace scope with X-Y capability or a single trace scope that allows external input to produce the horizontal sweep of the trace, the scope should be DC coupled, because of the low sweep speeds produced by the computer. Full scale vertical and horizontal voltage output

OPERATION

The front panel potentiometers should be set to mid range. Turn on the power. First the ball display should be adjusted to produce a perfect circle of convenient size by manipulating the settings of trim-pots TR2 and TR3. The ball is then positioned to set on the floor with no distortion due to flattening by adjusting trim-pot TR4. The "Repetitive action" switch is now closed. The ball should immediately move to the upper left hand corner of the oscilloscope screen, and then fall, when the drop/release pushbutton is pressed, the ball should hit the floor and then go into a bouncing pattern.

The fall and bounce pattern of the ball may now be altered by adjusting the potentiometers on the front panel. Adjusting either the "Damping" or "Gravity" potentiometers will change the rates of both fall and rebound. Adjusting the "Spring Rate" potentiometer will change the character of the rebound. Adjusting the "Sweep Rate" potentiometer will change the horizontal velocity of the ball. Adjusting the "Mass" will determine the weight or heaviness of the ball. Adjusting the "Initial Height" will determine the height of the ball from which it will fall.