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.But, when the car is hooked up, getting the most out of the car means simulating the response of the various DHOs in the system with steering, braking, and throttle inputs.Now we know the physics behind it.Let's do some math!The frequency turns out to be, as we show below.k is the spring constant,typically measured in pounds per inch, and m is the mass of the sprung weight, typically measured in pound-masses.Suppose our springs were 1,000 lb/in, supporting about 800lb of weight on one corner of the car.First, we note that a pound force is roughly (1/32) slug - ft/s2 and that a pound weight is (1/32) slug.So, we're looking at Notice that we've used the back-of-the-envelope style of computation discussed in part 3 of this series.We've found that the resonance frequency of one corner of a car is about 4 bounces per second! This matches our intuitions and experiences: if one pushes down on the corner of a car with broken shocks, it will bounce up and down a few times a second, not very quickly, not very slowly.We can also see that the frequency varies as the square root of the spring constant.That means that to double the frequency, say, to 858bounces per second, we must quadruple the spring strength to 4,000 lb/in or quarter the sprung weight to 200 lb.[Note added in proof: My friend, Brad Haase, has pointed out that 4 Hz, while in the "ballpark", is much too fast for a real car.Now, this series of articles is only about fundamental theory and ballpark estimates.Nonetheless, he wrote convincingly "can you imagine a 4-Hz slalom?" I have to admit that 4 Hz seemed too fast to me when I first wrote this article, but I was unable to account for the discrepancy.Brad pointed out that the suspension linkages supply leverage that reduces the effective spring rate and cited the topic "installation ratio" in Milliken's book Race Car Vehicle Dynamics.Since I have not peeked at that book, on purpose, as stated in the opening of this entire series and reiterated in this article, I can only confidently refer you there.Nonetheless, intuition says that 1 Hz is more like it, which would argue for an effective spring rate of 1000 / 16 = 62 lb/in.]How do we derive the frequency formula? Let's work up a sequence of approximations in stages.By improving the approximations gradually, we can check the more advanced approximations for mistakes: they shouldn't be too far off the simple approximations.In the first approximation, ignore the damper, giving us a mass block of sprung weight resting on a spring.This model should act like a corner of a car with a broken shock.Let the mass of the block be m.The force of gravitation acts downwards on the block with a magnitude mg, where g = 32.1 ft/s2 is the acceleration of Earth's gravity.The force of the spring acts upward on the mass with a magnitude k( y0 - y), where k is the spring constant and ( y - y0) is the height of the spring above its resting height y0 (the force term is positive--that is, upward--when y - y0 is negative--that is, when the mass has compressed the spring and the spring pushes back upwards).We can avoidschlepping y0 around our math by simply defining our coordinate system so that y0 = 0.This sort of trick is very useful in all kinds of physics, even the most advanced.It's worth noting that the model so far ignores not only the damper, but the weight of the wheel and tyre and the spring itself.The weight of the wheel and tyre is called the unsprung weight.The weight of the spring itself is partially sprung.We don't add these effects in the current article.Today, we stop with just adding the damper back in, below.Newton's first law guides us from this point on.The total force on the mass is -ky - mg.The mass times the acceleration is m ( dvy / dt) = m ( d 2 y / dt 2), where vy is the up-and-down velocity of the mass and dvy / dt is the rate of change of that velocity.That velocity is, in turn, the rate of change of the y coordinate of the mass block, that is, vy = ( dy / dt).So, the acceleration is the second rate of change of y, and we write it as d 2 y / dt 2 because that's the way Newton and Leibniz first wrote it 350 years ago.We have the following dynamic equation for the motion of our mass block [ Pobierz całość w formacie PDF ]