abstract.
A simple model of car suspension was simulated using an analog computer. The analog computer itself was first simulated on the computer with Multisim 7. Then, using discrete and integrated components, the analog simulator circuit was constructed on breadboards, and verified using a function generator to provide the road surface as input, and the output was observed using an oscilloscope. The results obtained clearly verify that the analog computer is a capable tool for accurate modeling of second order (and higher order) systems.
modeling suspension.
A car's suspension system can be modeled by a simple mass-damper-spring system. The mass represents the weight of the car sitting atop the suspension components. The viscous friction element representing the shock absorber and the standard spring comprise the remaining elements of the system. A free-body diagram of the aforedescribed system is shown.
Note that this is a somewhat simplified model. A car in real life rides on a tire that has some spring constant, damping effect, and mass itself. For discussion and analysis of a more complicated (but more accurate) model of suspension, refer to the Speed Bump Laboratory Assignment #4. For our purposes here, we will simply assume that the simpler model is "good enough" and we will model it using an analog computer. To simulate the car going over a bump in the road, we let the road surface "move" under the car, and the height of the road at any time t is the input function to the system, x(t). The output of the system, y(t) is the movement of the mass (the car) as a function of the input. Writing the differential equation for the system, we get:
Reorganizing terms, we get that
Basically, the vertical acceleration of the car (and its passengers) is a function of the height of the road, how fast the height of the road changes, as well as the suspension parameters. Based on this model, we can attain a sense of what a passenger feels when riding in a car over a variety of different bumps. For example, driving straight over a sidewalk curb could be modeled by a step function input to the system. The derivative term of the input x(t) at t=0 is essentially an impulse function, meaning that a large vertical acceleration will be transferred to the car body. That is why when driving up onto a curb does a passenger feel a large jolt. Making some back-of-the-envelope calculations, we realize that even the smallest downtown Philadelphia pothole (a 2cm x 10cm bump), when driven over by an average Philadelphia policeman at 55 miles per hour, gives about a 35 mph straight vertical velocity transferred to the policemen at the impact point.
simulink simulations.
In Lab Assignment 4, we modeled this system using Matlab Simulink. Simulink is a simulated analog computer with integrator, differentiator, gain, and summation components. We can reorganize the differential equation so that only the positive second derivative of y is present on the left hand side.
This way, we can see that dy^2/dt^2 is simply the sum of (-b/m) * dy/dt, (-k/m) * y(t), (b/m)* dx/dt, and (k/m) * x(t). The idea behind designing an analog computer to simulate a differential equation is to start out with an integrator whose input is the highest order derivative of the output. From there, the output of that first integrator is the next highest order derivative, and so on, until we have integrated our way to the zeroth derivative term. Then, we take those outputs, multiply them by the appropriate gains indicated by our differential equation, sum them, and send them back as the input to the first integrator. The schematic for the Simulink circuit is given below.
analog circuit elements.
Operational amplifier integrated circuits can be used to create integrators, differentiators, gains, and summation units. Then, wiring the correct components together, we can model the car suspension differential equation analogously as in Simulink. The basic opamp differentiator configuration is given below.
Summing the currents into the v- terminal of the opamp, we get
Using the properties of an ideal opamp that state that v+ = v- and i+ = i-, we get that the output voltage is given by -RC*dv/dt. Thus, we have obtained the negative of the derivative of the input voltage. This circuit works because the current through a capacitor with a steady voltage across its terminals is 0, and is only non-zero when the voltage across the terminal is changing (hence the derivative).
Likewise, we can develop the equations governing the integrator circuit shown below.
As a result, the output of the integrator circuit is proportional to the integral of the input voltage.
The standard opamp gain circuit is governed by the equation
and the circuit is given below. Note that if Rf = Ri, we simply have an inverter that gives out the negative of the input voltage. As will be seen later, we only this simple gain circuit with unity gain to create two inverters.
A sample summing configuration is given below.
Each input can be given its own gain, which is determined by the feedback resistor value divided by the input resistor value for the respective input. Thus, the feedback resistor should be the lowest common multiple of all the input gains in order for all the respective gains to be achieved. Note also that this configuration is still an inverting configuration, so the negative sum is the output.
computer circuit schematics.
Following the Simulink schematic, it is a simple matter to generate an analogous opamp circuit that models the car suspension electronically. The trickiest aspect is to ensure that all the negative and positive signs are correctly maintained, and to insert unity-gain inverters where necessary to achieve that aim. The circuit schematic is given below.
Download the Multisim circuit file.
W>e were forced to use two inverters to invert the plain road input x(t), and the dY/dt terms entering the summation circuit. Following the wires in the circuit, we verified that indeed the correct signs were achieved for all the values.
After a successful simulation, we took a break and ate some fresh cantaloupe. That having been as it may, we took to the task of building the circuit on breadboards. Pictures from the event are included. Using a function generator to create the same road surface used in simulation, we were able, after some debugging, to get a real output on the oscilloscope that was identical to the simulation results.
 
circuit simulation.
This circuit was simulated in Multisim 7. The 411 operational amplifier element was used because of its good electrical characteristics. We discovered by trial and error that the "virtual opamp" and even the standard 741 did not give acceptable results. Each opamp was supplied +/- 12 volts. For our road input, we attempted to generate something akin to the average Philadelphia road surface. Subsequently, we used a 30 Hz square wave with a 5% duty cycle to generate a stream of rather high but short bumps. Using the virtual oscilloscope, we obtained the following output
We can observe immediately that our simulated analog computer gives as output a nice second order response that is strangely identical to our Simulink results.
 
 
analysis and conclusions.
The Simulink model allows us to express our differential equation by a system. When the input is a square wave of low enough frequency, Simulink responds to each pulse with a slightly underdamped output before returning to equilibrium. In terms of our model car, this translates into a minor bounce in the suspension, which is reasonable because from practical experience we know that small bumps have this same effect on real cars.
The Multisim system represents the actual circuitry that our theoretical Simulink model simplifies into more basic elements. When the Multisim circuit is given a square wave, the output is again slightly underdamped as expected. Multisim was a very useful and convenient tool because it provided us a way to construct and test a clean version of our desired circuit before we undertook the real implementation.
The breadboard circuit is the physical manifestation of our theoretical Multisim system. When the breadboard circuit is presented with a square wave, we once again observe slightly underdamped behavior when the pulses are distributed at a low enough frequency. However, when the frequency is reduced, the output changes significantly. In terms of our car, this corresponds to a continuous stream of bumps where the suspension doesn't have time to fully react to the first before hitting the second. The observed output makes sense for this input. There is initially a large peak similar to that for a single bump, but the return to equilibrium is clearly interrupted by an upward jump from the next bump. This same process repeats for each consecutive pair of bumps. A car's suspension would wear out quickly from continuous exposure to such roads because of the perpetual need for sudden compression and extension.
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