Back in Springs and Natural Frequencies we looked at one of the fundamentals of suspension systems – the natural (or ‘resonant’) bounce frequency of a suspension system. It’s been a popular article – probably because many people are after a more than just a basic understanding of how suspension systems work.
But can we take that article further? The answer to that is yes: not only can we look at how a car bounces, but also how it rolls and pitches.
If you are working with a lightweight car that has a wheel at each corner (the hardest combination from which to obtain a good ride quality), you should be aware of bounce, pitch and roll natural frequencies.
The Bounce
Let’s briefly recapitulate that previous article before moving on to further things.
Imagine a coil spring out of the car sitting vertically on the ground. You place a weight on it and then you carefully push the weight downwards and release it. The weight will bounce up and down at the spring’s natural frequency. Over time the bounces will die away (they’ll get smaller) but they’ll continue to occur at a certain number of cycles per second, or Hertz (Hz), until the spring is still.
The same applies to springs in a car suspension system – the suspension has a natural frequency of up/down bounces. Importantly, the fact that you might have dampers in the system makes very little difference to the natural frequency.
When thinking about suspension systems, natural frequency is almost never considered. However, it is vital for two reasons. Firstly, in terms of ride comfort, the human body likes to sit on a system that has a natural frequency of about 1Hz, or 1 up/down cycle per second. Secondly, as the natural frequency of the suspension numerically increases, it gets excited more easily by road bumps, making control of the spring more difficult. In short, a lot more of the vertical accelerations of bumps get through to the cabin.
So what determines natural frequency? It depends on how much the springs compress when the static weight of the car is lowered onto them. And is the natural frequency number easy to work out? Yep! The natural frequency in cycles per minute (divide by 60 to get Hertz) can be found by: 188 divided by the square root of the static deflection, measured in inches. So if we have a static deflection of 3.3 inches, the suspension has a natural frequency of 1.7Hz.
(That 1.7Hz figure is quite a long way above our optimalforride natural frequency of 1Hz, but the only way of getting a lower number is to increase the static deflection, which in turn means we need to have a greater total suspension travel if we’re still to be able to absorb bumps.)
But what if we don’t care much about ride – we want the stiffer suspension used to control pitch and roll? Then we’ll be in the range of 2 – 2.5Hz for sporting cars. (Typical sedans are in the range of 1 – 1.5Hz.)
Pitch and Roll
The above section mentioned pitch and roll. Let’s define these terms: roll is the rotation around a longitudinal axis, and pitch is rotation around a transverse axis.
As described in
Ride Quality, Part 1, for best ride comfort, minimising all types of accelerations transmitted to the occupants is important. However, it is accelerations in pitch and roll (not bounce) that have the greatest impact on occupants’ ride comfort.
Minimising vertical accelerations can be achieved by using soft springs and soft bump damping  but the picture gets much more complex when you start to think about pitch and roll.
Take pitch for example. To minimise pitch, what is needed is that the front and rear springs always compress and extend at the same time – that is, the front and rear suspension systems are in phase. But how can that be achieved?
One approach is to use softer front springs than rear springs. This means that the natural frequency of the front suspension is lower than the rear.
As this diagram shows, if softer front springing is used, it takes only a few motions of the suspension after a bump has been passed for the front/rear springing movements to be in phase. That is, there is an initial pitch but it very rapidly dies away.
Another approach is to use damping that is so firm that the oscillations of the springs are reduced to nothing very quickly. This approach is currently fashionable – and to my way of thinking, the result is a much overdamped ride.
Natural frequencies of bounce, pitch and roll
A way of examining how the suspension behaves is to look not just at the bounce (vertical) natural frequency, but also the pitch and roll natural frequencies. This brings into the picture:
· Natural frequency in bounce – determined by front and rear spring stiffness
· Natural frequency in roll – determined by spring stiffness on one side of the car, plus the spring effects of the antiroll bar(s)
· Natural frequency in pitch – determined by front and rear spring stiffness
In addition, the distribution of mass within the car is also important to each of these.
Imagine someone bouncing the front of a softly suspended car up and down. How many times it goes up (or down) each second is the front suspension’s natural frequency in Hz.
Do the same at the back and you can see the rear suspension’s natural frequency.
Now what about roll? Alternately step on and off a sill panel until the car starts rolling back and forth. Now the car’s natural frequency in roll can be seen.
To see pitch natural frequency, you’ll need two people. One pushes down on the front as the other lifts the rear. By working in concert, they get the car pitching. The natural frequency of pitch can then be determined.

Example cars
Now all this can start getting complicated waaay fast, so let’s make a few simplifications.
To start us off, let’s say that a car has equal front and rear natural frequencies and no antiroll bars. This means that:
· The front and rear natural bounce frequencies are the same
· The roll frequency is the same as the bounce frequency
· The pitch frequency is the same as the bounce (and roll) frequencies
See if you can picture in your head why these are so. If the front and rear natural frequencies are the same, when the car pitches (and so compresses and extends the front and rear springs) the number of times it pitches per second will be the same as the number of times the front and rear springs compress (or extend) per second. Therefore, in this case, the pitch frequency = the bounce frequency.
When the car rolls (and so compresses the springs on one side), the number of times it rolls per second will be the same as the number of times the springs on one side compress (or extend) per second. Therefore, in this case, the roll frequency = the bounce frequency.
But obviously, most cars are not like this.
In Automobile Suspensions (Campbell; 1981) the suspensions of both a frontwheel drive Ford Fiesta S and the rear wheel drive Porsche 928 are examined. When loaded with one person, the cars have the following characteristics:

Rear natural frequency 
Front natural frequency 
Pitch frequency 
Fiesta 
2.17Hz 
1.55Hz 
1.96Hz 
Porsche 
1.28Hz 
1.18Hz 
1.24Hz 
Notice two things:
· both cars have stiffer rear springs than front springs
· the pitch frequency is numerically between the bounce frequencies of the front and rear suspensions.
(Also note how the Porsche uses softer suspension than the Fiesta – as a sports car, it probably has much firmer damping.)
Another way of describing the relationship between these different frequencies is in terms of ratios or percentages. Let’s arbitrarily call the rear natural frequency 100 per cent – that way, we can easily compare the other frequencies to it.

Rear natural frequency = 100% 
Front natural frequency 
Pitch frequency 
Fiesta 
100% 
71% 
90% 
Porsche 
100% 
92% 
97% 
So in the case of the Fiesta, the front natural frequency is 71 per cent of the rear natural frequency, and the pitch natural frequency is 90 per cent of the rear natural frequency.
For the Porsche, the front natural frequency is 92 per cent of the rear natural frequency, and the pitch natural frequency is 97 per cent of the rear natural frequency.
Let’s take another example, derived from data contained in Hydragas Suspension (Moulton and Best, SAE 790374, 1979). This time we’ll compare the pitch and roll frequencies with the overall bounce frequency. To allow easy comparisons, we’ll say the overall bounce frequency = 100 per cent.
“Car A” is a light weight car with a front antiroll bar.

Overall Bounce frequency = 100% 
Pitch frequency 
Roll frequency 
Car A 
100% 
100% 
109% 
Two things to note.
Firstly, for the pitch frequency to be the same as the bounce frequency, the natural frequencies front and rear must be the same. Secondly, note how the roll frequency is higher than the bounce frequency because the antiroll bar (ie another spring) comes into effect – that is, the car is stiffer in roll than in bounce or pitch. (To put this another way, the suspension is stiffer if it meets a bump that attempts to make it roll, ie a onewheel bump or a bump that influences the wheels on only one side of the car.)
Now let’s put the cat among the pigeons. Have a look at the data for Car B, below. Car B has an unusual suspension system that we’ll get to in a moment.

Overall Bounce frequency = 100% 
Pitch frequency 
Roll frequency 
Car B 
100% 
83% 
100% 
This data is unlike anything we have seen. That’s especially the case because of pitch: how can we have a pitch frequency that is less than the bounce frequency? To put this another way, how can the front and rear springs be softer when the car is being pitched than when it is bouncing? It simply doesn’t seem possible – the springs would have to ‘know’ that the car is pitching rather than bouncing, and then soften themselves to suit.
In fact, Car B seems to throw the rule book out of the window – especially the rule that says that the pitch frequency has to be numerically between the front and rear bounce frequencies. To look at this further:

Rear natural frequency = 100% 
Front natural frequency 
Pitch frequency 
Car B 
100% 
104% 
85% 
So not only is the front suspension stiffer than the rear (and we said above that it should always be the other way around), but the pitch frequency is only 85 per cent of the rear natural frequency! To be numerically between the front and rear natural frequencies, we’d have expected it to be something like 102 per cent of the rear natural frequency – not just 85 per cent.
So how come Car B can break all the ‘rules’ we just established?
The answer is that the in car B, the front and rear springs are interconnected. This interconnection is arranged so that when a front wheel rises over a bump, the rear wheel on that side of the car pushes down, so keeping the car level. Single wheel bumps, and bumps that simultaneously affect the wheels on the front or back axle, are compensated for in terms of pitch.
Car B uses an interconnected suspension system called Hydragas. (The older Hydrolastic suspension is similar in its interconnectedness.) These systems use a liquid interconnection between the front and back suspensions systems, where the leftfront is connected to the leftrear, and the rightfront is connected to the rightrear. The systems were widely used on BMC cars in the 1960s including the Austin 1100, the Mini and the Austin 1800.
Other cars that have used front/rear interconnected suspension systems include the famous 1955 Packard (it used longitudinal torsion bars to make the front/rear link)…
..and the 1948 Citroen 2CV, that used two horizontal canisters containing the front and rear springs. The canisters were able to move, so connecting the action of the front and rear suspensions.
Note that none of these system approaches requires hydraulic pumps, electronic control systems, etc.
Significantly, most of these cars did not use antiroll bars – the roll stiffness was provided by the interconnected suspension.
Each of these suspension systems was acclaimed at the time for their high quality ride. In fact, I would argue that their ride quality is much higher than we typically experience in current cars.
Summary
In this article it’s likely that plenty of new concepts have been introduced. Here’s a summary:
All cars:
1. Any spring/mass system has a natural frequency: this is the number of movements per second that will occur when the system is excited by a single input.
2. For passenger cars, suspension bounce natural frequencies are typically in the 1.2 – 1.8Hz range.
3. The higher the natural frequency, the stiffer the suspension (and the stronger the ride accelerations transmitted to the occupants).
4. In addition to bounce, natural frequencies of roll and pitch can also be analysed.
Cars with conventional suspension:
1. The use of antiroll bars (ie an additional spring that acts only in roll) gives roll frequencies that are are higher than bounce frequencies. As a result, onewheel bumps are harsher than bumps that affect both wheels on the axle.
2. In cars with identical front/rear natural bounce frequencies, the natural pitch frequency is the same as the bounce frequency.
3. In cars with differing front/rear natural frequencies, the pitch frequency will numerically fall between the front and rear natural frequencies.
Cars with interconnected front/rear suspension:
1. The pitch frequency can be set independently of the bounce frequency, and thus can be significantly lower than the bounce frequency.
2. Roll can be controlled by setting the bounce frequency sufficiently high, so antiroll bars may not be needed. Even with this high bounce frequency, the low frequencies of pitch and soft suspension rates in onewheel bumps results in good ride quality.
Conclusion
It’s fascinating to watch the behaviour of cars on the road in terms of bounce, pitch and roll.
One standard model of car that I often see has a clear pitch problem: once you recognise its behaviour, you can see these cars porpoising along on all sorts of road surfaces! (No wonder I felt ill when I rode in the back of one.)
When following cars with modified suspension, very high roll frequencies can often be observed: you can clearly see the occupants’ heads whipping from side to side over bumps.
Natural frequencies of bounce, pitch and roll… they’re very interesting ideas.
References
Bastow, D., Car Suspension and Handling, Pentech Press, 1980
Campbell, C., Automobile Suspensions, Chapman and Hall, 1981
Coker, A.J. (Ed), Automobile Engineer’s Reference Book, 1959
Daniels, J., Car Suspension at Work, Motor Racing Publications, 1998
Gillespie, T., Fundamentals of Vehicle Dynamics, SAE, 1992
Howard, G., Chassis and Suspension Engineering, Osprey, 1987
Milliken, W. & Milliken, D., Chassis Design – Principles and Analysis, SAE, 2002
Moulton, A. & Best, A., Hydragas Suspension, SAE 790374, 1979
Moulton, A.,
Bristol to BradfordonAvon – a Lifetime in Engineering, Rolls Royce Heritage Trust, 2009
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