Contact us

What is the relationship between stiffness and vibration?

The relationship between stiffness and vibration is fundamental. Moreover; in most acoustic situations a mechanical vibration is the source of transmitted noise. For example, this is the case for loose floors, equipment supports or radiant panels. In reality, this vibration depends on two fundamental parameters: mass and stiffness of the system.

The topics covered in this article

Definitions

Let's start by defining stiffness of a system and vibrations.

Stiffness of a system

The stiffness of a system, denoted k and expressed in newtons per meter (N/m), measures its ability to resist deformation under the effect of a force. In acoustics, a distinction is made between static stiffness, measured at equilibrium or at low frequency, and dynamic stiffness, which characterizes the response of the system when subjected to harmonic vibration excitation. This dynamic stiffness depends on the frequency and the viscoelastic behavior of the materials.

Vibrations

The vibrations by their oscillatory mechanical movements are the origin of the generation or transmission of noise, whether through a fluid (air, water) or through a solid.

Indeed, with these two definitions, we see that the relationship between stiffness and vibration is at the heart of acoustic performance, because it conditions both the vibration transmission in the structures and their sound radiation.

The purpose of this article is to clarify the fundamental role of stiffness in acoustic vibration phenomena.

Vibrational foundations applied to acoustics

In acoustics, understanding vibration phenomena is essential to anticipate and control the structure-borne noise transmission. When a structural element vibrates under the effect of mechanical excitation, it begins to vibrate. These vibrations can propagate through adjacent elements, or radiate as sound waves into the air.

Consequently, three fundamental notions make it possible to model and control this behavior: the model spring masst, the resonance, and the notion of dynamic stiffness.

The mass–spring model

The first concept is the mass-spring model.
The mass-spring model expresses that the force F applied to an elastic system is proportional to its displacement via the stiffness.

This relationship is the basis for modeling vibrating systems. Thus, in vibro-acoustics, a component (floor, partition, support, etc.) is often assimilated to a mass-spring system (or several mass-springs), in which stiffness directly influences the system's ability to oscillate and transmit vibrations.

As a result, this model makes it possible to identify the natural frequencies, that is, the frequency at which the system amplifies the vibration.

Link to Acouvapp, vibration transmission calculation module

The resonance phenomenon

The second phenomenon is that of resonance frequency. When the frequency of an external excitation coincides with the natural frequency of a component, the phenomenon of resonance occurs. It manifests itself by a significant amplification of the vibration amplitude, and therefore an increase in transmitted noise.
As a result, this translates, in industry or construction, into a increased vibration and potentially noise around this frequency. Moreover, this phenomenon is particularly problematic for acoustic underlays or detached systems, whose natural frequencies can easily fall within the audible range if the dynamic stiffness is poorly calibrated.
In vibration isolation devices (suspensions under machines for example), the stiffness of the resilient elements is precisely adjusted to shift the resonance frequency out of the excitation frequencies of the machine.

The dynamic stiffness of structures

Finally, stiffness acts as a regulator of resonances: it allows them to be avoided by adjusting the dynamic response of the system. Unlike the static stiffness, which is measured under a slow or constant load, the dynamic stiffness reflects the behavior over a certain frequency range materials subjected to vibratory stresses.

Indeed, dynamic stiffness plays a fundamental role in la generation, transmission and control of structure-borne noise. It determines how a structure reacts to excitation. Too little stiffness can cause excessive vibrations (stiffness is related to the ratio of force and displacement), promoting noise propagation. Conversely, high stiffness will limit the amplitude of vibrations but can create unwanted resonances at higher frequencies. For example, equipment positioned on a structure with poorly dimensioned stiffness generates vibrations that can propagate in a building or vehicle, resulting in structure-borne noise.

Consequently, controlling the stiffness of the structural elements allows the system to be adjusted and vibration transmission to be reduced to limit the structure-borne noise.

spectrum relationship between stiffness and vibration

Dynamic behavior at the resonant frequency

Effects of stiffness on resonance

La stiffness directly influences the position of the natural frequency of a system: the more rigid the element, the higher its resonance frequency.

Moreover, in the acoustic domain, this frequency marks a critical threshold:

  • below this, the system behaves like a low-pass filter (low isolation efficiency at low frequencies),
  • beyond that, transmission is more important
But in the vicinity of the natural frequency, the system can enter into resonance, which generates un peak vibration and transmitted noise (therefore a reduction in sound insulation).

Link to Impactool, dynamic stiffness measurement solution

Influence of damping

If the stiffness fixes the niveau of rigidity of a system, it is theamortization who controls the dissipation of vibrational energyTogether they determine the shape of the coresponse form. ULow stiffness with adequate damping can limit low-frequency transmissions, while a poorly damped structure amplifies resonances.

Consequently, in vibration transmission – and therefore acoustic transmission – the right compromise between stiffness and damping allow :

  • to mitigate transmission peaks,
  • to smooth the frequency response
  • byimprove insulation in the useful band

Moreover, this principle is used in the design of many acoustic devices:

  •  flexible joints,
  • resilient fixings,
  • anti-vibration pads,
  • disconnected connections
However, unless the material is changed, it is often difficult to modify the damping of the same material. It is preferable to adjust the stiffness to avoid coupling between resonance and vibration excitation.

Practical applications

In building acoustics

First of all, good control of dynamic stiffness allows tooptimize sizing systems and avoid unwanted transmissions. Especially in low frequencies.

Indeed, the applications of good control of dynamic stiffness are numerous, here are some examples:

  • some floating screeds.
    Low dynamic stiffness of resilient layers achieves two things. The first is to lower the cutoff frequency of vibration isolation. The second is to improve sound insulation and impact noise performance.
  • some thermo-acoustic linings on insulation.
    The dynamic stiffness of the insulators determines the natural frequency of the mass-spring system. This natural frequency is the one from which the insulation increases sharply.
  • some anti-vibration mounts for technical equipment (ventilation units, heat pumps, etc.).
    For these supports, suitable stiffness prevents the direct transmission of vibrations to the supporting structure thanks to vibration filtering.

Note that for insulators, dynamic stiffness can be measured according to ISO 29052 with solutions such as Impactool.

dynamic stiffness measurement

In other words, poorly controlled stiffness can lead to significant vibrations that are detrimental to comfort, or even a reduction in the expected isolation.

For transport

Then, stiffness is crucial in the railway, automotive, and aeronautical sectors. Indeed, passenger comfort is an important criterion in specifications, and vibrations are the black spot of comfort!
Here are some examples where stiffness conditions the vibration response components interacting with comfort and structure-borne noise:
  • the suspensions must ensure a compromise between dynamic comfort, stability and vibration isolation,
  • the passenger seats are subject to strong requirements integrating their dynamic behavior,
  • the mechanical connections between structures, floors or on-board equipment are sources of solid-borne noise if their stiffness is not suitable

Conclusion

In conclusion, Stiffness has a direct relationship with vibration. Qwhether static or dynamic, stiffness is at the heart of vibrational phenomena. These phenomena condition la generation, the legacy and the noise perception.

Moreover, the direct relationship between stiffness and the natural frequency of structures makes it a key parameter in understanding the vibrational behavior of systems. In acoustics, this relationship is expressed in a tangible way through the resonance phenomena and the effectiveness of insulation devices.

Controlling stiffness is therefore an essential lever in the high-performance acoustic design, whether in construction, transport or industry. It is involved in the choice of materials, the dimensioning of assemblies, the behavior of interfaces and the control of solid transmissions.

In the future, the multiphysical modeling – coupling vibroacoustics, thermal, structural and material behavior – will allow for more detailed integration of stiffness effects into numerical simulations. It will pave the way for integrated design, driven by acoustic performance, with more accurate predictions and better targeted prototypes.

These articles may interest you

Acoustic training = performance lever

Reduce impact noise with acoustic underlays

Acoustic definitions, learn more about acoustics