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Springs are primarily responsible for keeping the tire in contact with the road surface over bumps and dips.
In the realm of physics, springs are noted as being efficient machines for storing mechanical energy. When a spring is compressed, the energy required to perform the compression is stored. When the compression force is removed, the spring returns to its original shape. No additional energy input is required. A spring can also be stretched (to a point), and it will return to its original shape. All this depends on the use of effective materials of course.
In a car, compression of the suspension spring is caused when the wheel travels across the front side of a raised bump. A portion of the energy used to cause forward motion of the vehicle is redirected causing the wheel to travel up, which compresses the spring. The spring stores the vertical energy, and as the wheel travels down the backside of the bump, the energy stored in the spring pushes the wheel back down. For safety and handling, this has the significant benefit of keeping the tire in contact with the road surface as the tire travels over a bump. A similar process occurs for dips, except that the spring elongates rather than compresses to start with.
Without springs, the wheels would transfer the redirection of the vertical energy into the vehicle chassis and cause the vehicle to bounce off the bumps. This would both be annoyingly uncomfortable to the passengers (a concern to street car manufacturers), and the driver would momentarily lose some or all of the ability to steer, accelerate or brake as traction from the tires would be lost (a safety concern for all, and a maximum performance concern for racing).
With springs, the vehicle body can maintain a relatively linear path (providing comfort for the passengers), while the wheel travels up and down over the bumps (allowing for continuous safe vehicle control, and continued traction for maximum racing performance).
Therefore, the purpose of the spring in an automobile is to isolate the wheel assembly from the body, and allow the tire to maintain contact with the road over surface imperfections.
Relative to its shock absorbing function, the spring must be stiff enough to prevent full compression or elongation in large bumps and potholes. However, it must also be soft enough maintain good contact with the road. The softer the spring the better the road contact over bumpy surfaces. However, the stiffer the spring, the better the resistance to bottoming out on large bumps. Somewhere between these extremes is a range of good spring rates (stiffness) to work for the expected environment.
In order to handle bumps and dips, the entire wheel assembly is designed to have a certain amount of vertical travel length from full extension to compression. The rougher the road, the more wheel travel is needed, and the longer the overall spring length needs to be. Factory passenger cars are designed to function well over a broad range of conditions, and the suspension system in particular must be prepared to compensate for potholes, freeway expansion joints, rutted gravel roads, and other less than ideal road surfaces. Therefore, a street car is designed with quite a bit of suspension travel length (think of how far you have to jack up a car's body to get the wheel off the ground--that's about half the wheel travel)). In a high performance sports car (i.e. of the Porsche, Ferrari, Viper, and NSX type), manufacturers assume a more limited range of road surfaces, and design in less wheel travel by a of couple inches. In the typical sports car or sports sedan of the Mustang, Camaro, Eclipse, Integra, and BMW 3 types, the suspension is a little better than the general sedan, but it's really not a great deal different.
In racing, we can assume a certain degree of ideal conditions, or at least more ideal than public roads. In a stock street car, even notoriously "bumpy" race courses feel glass smooth compared to most public roads. In these conditions, the purpose of the spring can be focused to maintain maximum and consistent contact of the tire with the relatively much smoother road surface. Under these conditions, very little wheel assembly travel is required. The spring can be optimized for smaller wheel travel conditions. For example, a CART or Formula 1 race car driven on smooth courses may only have 1/4 to 1/2" of total suspension travel!
How does wheel travel impact handling? Well, just from the CART example given above, we might assume that shorter wheel travel is better. And, of course, it is. Though the wheel assembly travels up and down, it does not do so on a linear path. The wheel assembly is at some point fixed, and the wheel assembly actually travels in an arc. Whether the body stays put, and the wheel travels (through bumps), or the wheel stays put and the body travels (body roll), this has impact on the camber angle of the wheel which changes the tire contact patch shape.
Therefore, for racing conditions, limiting the wheel travel distance is a desirable thing. For street cars, the use of lowering springs (shorter and stiffer) is one method to reduce wheel travel. In extreme cases, it will also be necessary to use shorter shocks.
So far we have used bumps in the road to illustrate how springs behave. Springs are also acted upon by the forces of acceleration, braking, and cornering. The momentum of the vehicle body in cornering, braking, and acceleration transfers into the springs causing compression and elongation. This is an easy to see effect of weight transfer as it results in visible body roll -- both the side-to-side roll we're mostly familiar with during cornering, but also front-to-back roll -- particularly the "nose dive" under hard braking.
Body roll by itself is not necessarily bad. If the four tires remain flat on the road surface with balanced downforce, who cares whether the car body is parallel to the road or not (aerodynamics aside). What body roll does though is change the angles of the suspension components to the wheel assembly (which we call suspension geometry).
This is pretty much the same thing as we discussed above with wheel travel, except this is from the opposite perspective. With the wheel on the fixed plane of a smooth road, the body now travels, and causes the wheel assembly to travel in the arc we described. This changes the camber and tire contact patch particularly of those tires which are unloaded and the suspension elongates.
Aside from bump absorption, the spring also contributes to the roll stiffness of the car--the ability to resist dive under braking, squat during acceleration, and body roll during corning. The anti-roll bars also play a roll in this, and the two combined create the total roll stiffness of the car. Stiffer springs will resist body roll more, reduce change in the suspension geometry, and maintain a more consistent tire patch size.
Note: many people are under the misconception that body roll causes weight transfer. This is not true. See the weight transfer article for details about this.
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(Part 2: Spring Stiffness, Lowering, Coil Overs)