Making sense of vibrations

So what’s the connection between the measured body stiffness of a high performance car, the ride quality of a limousine – and designing rubber isolation mounts that actually work?

The answer is that each relates to vibrations – their frequency, amplitude and damping. 

Understanding vibration is central to many concepts in engineering – and automotive engineering is no different. 

So let’s take a look at these things. 

Terms

Vibration can be defined as “a rapid oscillation of something back and forth across a central position”. 

That ‘back and forth’ can also mean ‘up and down’ – as is the case when a wheel of a car is travelling along a bumpy road. The wheel is moving up and down; when it is not being subjected to the bumpy road, it takes up a (roughly) middle position. 

Another ‘back and forth’ is the cone of a sub-woofer.

The speed with which the speaker cone is moving is its frequency, expressed in cycles per second – or Hertz. That is, how many movements back and forth are made each second. 

The distance the speaker cone is moving back and forth is called the amplitude. In a speaker, the amplitude might be expressed in millimetres. In a car suspension, it might be expressed in inches.

Let’s apply those terms. 

We take a coil spring out of a car and mount it vertically on a surface. We then put a 400lb weight on top of it and push down firmly on the weight. When released, the weight bounces up and down with an amplitude of (say) 2 inches, and at three times per second – so at a frequency of 3 Hertz (Hz).

Now the interesting thing about the spring and weight that we’ve just described is this: it doesn’t matter how far we push the weight down before releasing it, this combination of spring and weight “likes” bouncing at 3Hz. This is called the system’s natural or resonant frequency. 

If we were to keep the spring the same but put on a heavier weight, the resonant frequency would get lower – for example, it might change from 3Hz to 2Hz. So when pushed down and released, the heavier weight would bounce at 2Hz, not 3Hz. If we were to stiffen the spring, the resonant frequency would get higher – say rise to 5Hz. 

An easy way to remember this is to think: the greater the static deflection (the more the spring squashes down when the weight is first placed on it), the lower the resonant frequency of the system. 

This is directly applicable to car suspensions. Think of a car lifted above the ground on a jack. The more the car sinks on its suspension as it is placed back down onto the road, the lower is the natural frequency of the suspension.  

So integral to any discussion of resonant frequency is the relationship between stiffness and mass. 

Let’s go back to the weight bobbing up and down on the spring. It’s moving at its resonant frequency, but does it keep on moving in this way forever? Nope – the movement gradually dies away. The speed with which this ‘dying away’ occurs is dependent on damping.  

A resonant system that is highly damped will have vibrations that quickly abate. This is easily seen if there is a ‘step’ input to the resonant system – like when you push the spring down and release it, touching it no more. Or the wheel of a car that drives through a pothole – how quickly do the up/down wheel movements then die away? A bell that you strike with a hammer will ring for a long time - the metal has little internal damping. 

Note that the presence of damping does not change the resonant frequency of the system.

Recap

Let's summarise those terms.

Frequency – the number of movements per second, expressed in Hz.

Amplitude – the magnitude of the movement, expressed in inches or millimetres.

Resonant frequency – the frequency that the system naturally ‘likes’ vibrating at, with the value determined by the relationship between stiffness and mass. Resonant frequency is measured in Hz.

Damping – influences how quickly the vibrations die away after a single input. A highly damped system will stop vibrating more quickly.

Let’s look at an example that ties all this together – then we’ll get a little more complicated. 

Car Suspension

Let’s say that we have a car that has static deflection of 3.3 inches. That is, when the car’s weight settles on its springs, they decrease in length by 3.3 inches. 

The resonant 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. 

This suspension, with a static deflection of 3.3 inches, therefore has a resonant frequency of 1.7Hz. 

Damping? Well, in this story we’re not going to quantify that, but if the wheel isn’t to be moving up and down a lot (especially at 1.7Hz), we’re going to need to fit a damper. 

More terms

Now let’s do some logging. We’re going to put a sensor inside the cabin of the car that logs the up/down accelerations the driver experiences as a result of the up/down wheel movements.

So what is causing those up and down movements of the wheel? The bumpiness of the road! And is the road just a ‘step’ input to the suspension – like the hammer was a step input when it hit the bell? No – the road is a constant input of differing amplitudes (up and down distances) and differing frequencies (how many of those up/down movements are occurring each second). This is called a 'broadband' exciting input.  

By looking at the data log trace, we can easily measure the maximum vertical acceleration we feel in the car – it’s just the highest value recorded on the trace. 

But how do we measure the frequencies with which we’ve been moving up and down? We could try looking really closely at the trace and counting how many up/down accelerations occurred each second – but because so many different frequencies are occurring, that would be very difficult. 

Instead, wouldn’t it be great if there were a way of telling the logger: “What are the dominant frequencies that are occurring in these accelerations I am feeling?” In fact there is such a way, and it’s given a very fancy name – Fast Fourier Transformation, or FFT.  In loggers so-equipped, you just hit the FFT key and there they are.

So what do you think will be the dominant frequency measured in this car? 

Here’s a hint: we said above that the natural frequency of the suspension was 1.7Hz. If you said that the dominant frequency of up/down movements would therefore be 1.7Hz – you’re really close. (Actually it would be a little lower because we need to add the driver’s weight to the equation.)

The lower the resonant frequency of the suspension, the softer the suspension, and the slower will be the up/down movements of the car’s occupants. And that makes perfect sense, doesn’t it – limousines have suspension systems with a low resonant frequency.

Vibrating driving lights

Now let’s look at another system. We’ve fitted some very powerful long-range driving lights but they vibrate, giving a distracting and flickering beam.

We inspect the lights and find that if we bump a light with the end of a closed fist (a ‘step’ input), the light can be seen to vibrate. 

We log the vibration and find that it’s at 21Hz (i.e. the natural frequency of the system comprising the mass of the lights and the ‘spring’ of the mounting brackets is 21Hz). We also note that the vibration takes a long time to die away (i.e. it is poorly damped).

Some dampers are made and added and the logging shows that after each ‘step’ input, the vibration dies away much more quickly (bottom trace at right). Problem solved!

Other systems

So what was that at the beginning about designing rubber isolation mounts that actually work?

Imagine a vibrating machine sitting on rubber mounts. These rubber mounts compress by 1 inch when the machine’s weight is on them. A 1 inch static deflection indicates a natural resonance of 3.1Hz. The particular machine has a reciprocating mechanism that vibrates at 3Hz. 

The machine is vibrating at 3Hz – and remember, the whole springy system wants to vibrate at nearly this value! 

In this case, the mounts will make the machine vibrate more, not less. Ooops. 

To reduce vibration, we need a system resonant frequency that is no more than one-third the ‘forcing frequency’ of the machine. This is shown in the graph - click on it to enlarge. 

And finally, there’s the body stiffness of the high performance car. 

Let’s take it as the body-in-white – the bare shell. We mount it on soft isolation mounts and then attach vibrating shakers to the body. These shakers vibrate with equal energy over a wide range of frequencies. We measure the accelerations that are occurring of the body, then do an FFT to find the dominant frequencies. 

There might be a few different resonant frequencies – a car body is a complex item. But the higher the lowest resonant frequency, the stiffer is the body. In this sort of test, 24Hz is stiff – but 30Hz is stiffer again! This BMW graphic shows the change in stiffness over time with BMW bodies.

Conclusion

Vibrations are very important in understanding how many engineered systems behave. Next time you see anything vibrating, consider its frequency, amplitude and damping. Consider also what its natural frequency tells you about the relationship between its springiness and mass - all food for thought!