Why a coffee cup sometimes reveals more about dynamics than any textbook
What happens when dynamics are underestimated in everyday life?
It's early morning in production. The machine in your factory is running, the control cabinet next to it is humming quietly, and there's a coffee cup on top of it. Nothing unusual. This is exactly how dynamics begin in everyday life: inconspicuous, seemingly harmless, almost mundane. And yet everyone who regularly spends time in production halls knows the moment when this scene takes a turn. The cup begins to shake, slowly moves across the metal sheet, and finally tips over. Not spectacular, but unpleasant enough to raise questions.
When static safety is deceptive
From a static point of view, everything is fine. The coffee cup is stable, its weight is supported by the surface, the stresses are low, and the safety factors are comfortable. In a purely static calculation, there would be no cause for concern. And this is precisely where the problem lies. Because reality is rarely static. Machines do not simply run; they accelerate, they brake, they generate periodic forces, random vibrations, and occasionally even hard impacts.
Where vibrations occur: Understanding natural frequencies
When rotational speeds meet: resonance during operation. The coffee cup is therefore less an everyday object than an excellent indicator of dynamic behavior. The first look at dynamics almost always leads to modal analysis. Even before talking about specific loads, time sequences, or operating conditions, the fundamental question arises as to how a system can actually vibrate. Every structure has natural frequencies and associated natural modes. This applies to solid machine frames as well as thin-walled switch cabinet sheets or table tops. If operational excitations happen to coincide with one of these natural frequencies, the system responds with significantly amplified vibrations. The coffee cup then wobbles not because it is poorly positioned, but because its base does exactly what it is physically inclined to do. Modal analysis thus explains the fundamental “why” without worrying about the ‘when’ or “how strong.” The next step is more specific.
When speeds collide: resonance during operation
Many machines do not excite their surroundings randomly, but rather in a very targeted and periodic manner. Rotating masses, imbalances, gear meshing, or electromagnetic forces generate vibrations with a clearly defined frequency. This is precisely where harmonic analysis comes in. It answers the question of how a structure reacts to continuous sinusoidal excitation. For the coffee cup, this means that if the machine runs at a certain speed, the vibration of the surface can build up continuously. Small forces are then sufficient to generate large movements. What is particularly treacherous is that the problem often only occurs in a narrow speed range. Below and above this frequency, everything seems calm, but at this one setting, the cup begins to dance. Harmonic analyses make these resonance effects visible long before they become annoying or cause damage during operation.
Starting, braking, transitions
But even machines that are unremarkable in stationary operation often reveal their true dynamic nature when starting up or braking. At these moments, speeds, forces, and frequencies change continuously. The structure goes through a whole series of possible resonance states, albeit only for a short time in each case. This is precisely the realm of transient analysis. It considers time-dependent processes in their actual sequence and shows how vibrations build up and decay again. The coffee cup often reacts with a short jolt, a tremor, or a small jump before returning to rest. Nothing happened statically, nor permanently, and yet the dynamic load was real and measurable.
In practice, however, excitations cannot always be described so clearly. Many machines do not generate clear sinusoidal vibrations, but rather a complex mixture of different frequencies that constantly changes over time. Manufacturing processes, material transport, bearing noises, or changing load conditions lead to excitations that appear random but are statistically very easy to detect.
When suggestions appear by chance
This is where PSD (power spectral density) analysis comes into play. It describes the dynamic behavior under the assumption of a linearized system by considering not the exact time course, but rather the energy distribution of the excitation across the frequency spectrum. This allows conclusions to be drawn about the frequencies at which a system reacts particularly sensitively and the vibration levels that can be expected in the long term.
For the coffee cup, this means that it does not tip over suddenly, but the structure beneath it can be repeatedly stimulated over a long period of time. Such effects are less spectacular, but often decisive for fatigue, wear, and comfort—even if nonlinear phenomena such as actual wandering or tipping cannot be directly represented with them.
The extreme case: short bursts, big impact
And then there are those moments when everything happens very quickly. A tool falls onto the table, a drive jams, an emergency stop is triggered. The load is short, hard, and intense.
In such cases, the detailed temporal course of the excitation is less important than the question of what maximum response a linearized system can experience in principle. This is precisely where shock analysis in the form of a shock response spectrum is often used.
The SRS describes how strongly an ideal single-mass oscillator system would respond to an impulse – regardless of the details of that impulse. However, in real-world situations, it cannot be used to determine whether the coffee cup will actually remain standing or fly off.
As soon as contact losses, tilting, or other nonlinearities come into play, there is no way around a transient time domain calculation. The shock response spectrum then only provides an initial estimate of the potential load, but not the actual movement behavior.
One situation – many answers
In the end, the coffee cup reveals something astonishing. One and the same object, in one and the same situation, can be viewed using completely different dynamic methods, depending on the question being asked. Modal analysis explains where vibrations are fundamentally possible. Harmonic analysis shows what happens during periodic excitation. Transient analysis illuminates start and stop processes. PSD analysis evaluates random loads over time. Shock analysis answers the question of survival in extreme cases.
Dynamism begins in everyday life
Perhaps the most important insight here is this: dynamics do not begin in the calculation model, but in everyday life. If you only calculate statically, you see the coffee cup standing still. If you look at it dynamically, you understand why it sometimes falls. And it is precisely this understanding that determines quiet machines, long service life, and robust designs in practice.
If things are moving in your system that should actually be stationary, if noises occur that no one can explain, or if problems only happen “sometimes,” then this is almost always an indication of dynamics. And that is exactly where it is worth taking a closer look.
I promise you that next time you spill coffee from your overturned cup onto your computer keyboard, you'll think of us 😊
And then remember that although we can't help you clean your keyboard this time, if a machine is acting up, making loud noises, and starting to vibrate instead of purring like a kitten, we're here to help you get the dynamics under control.
Yours Stefan Merkle
PS: To give you a better understanding of the world of dynamics, I have written a white paper that I would be happy to send you. Just send me a short email with the subject line “White paper on dynamics.”