Non-linear Response Effects on Vibration Signatures

by Sidney Booksh


Since many systems are non-linear (Vibration response does not always increase proportionately with increased force and spectral components appear due to waveform clipping) one cannot be sure whether a spectral component occurs due to excitation of the system by a force of that frequency or due to clipping of the waveform. This article provides some insight into system vibration response where misalignment, looseness, and rubs are a factor. Some comments on beats and modulation are included.

The use of both linear & exponential curve fitting to vibration trend data is explained as well as using this data to estimate time to alarm. Determining the accuracy of curve fitting to actual data using both graphical and mathematical techniques is discussed. This paper is of value for someone wanting to improve the relevance of their machine alarm limits and learn how to quickly identify when these limits have been violated.


“The way a system responds to forces determines the vibration observed. System response varies with the frequency content, level, timing, and path of the force as it travels from the source to the transducer. Part of vibration analysis is to determine if the forces or the response or both have changed, by how much, and why. Frequency response can be determined by applying a known force, measuring the vibration, and dividing one by the other. The force is applied as an impact or through a shaker. The test procedure is chosen based on the purpose of the test, available instrumentation, how the result is being used, and economics.

The often unstated assumption is that the system is linear when frequency response is tested. That is, doubling and tripling the force will double and triple the response. Stated differently, the ratio between the response and the force (transfer function) is constant no matter the force. Also, for impact or broad frequency content shaker tests, the overall force level will not affect the transfer function at individual frequencies. The top plot is a frequency response plot from an impact test of a fixed beam (a long 2x4 in two blocks of concrete). The bottom one is a set of responses at increasing impact peak force. The transfer function clearly changes with increasing input force. It is the author’s opinion that most things are non-linear based on empirical observation of vibration spectra.

This is a re-plot of the response data on the previous slide. The peak amplitude of the transfer function changes similarly for the 1st and 2nd resonance as both the modal force (the force at the resonance frequency) and the peak force increase. However, they are not identical. It is possible that the peak force influences the modal response. A swept frequency shaker test would help define this influence. This needs further investigation. Perhaps this has already been done and a literature search will suffice.

The change in response with changing force will be discussed first. Also, equations describing the system response as a function of force will be introduced.

There are limited force generators in machinery — unbalance, vane/blade/fluid interaction, and instabilities, to name a few. Unbalance produces a single frequency force which has a single frequency response when acting on a linear system. If multiples of the forcing frequency are seen in a spectrum, the response can be judged to be non-linear.”

Free for members.

Click here to login and access the member's only Directory of all Technical Resources. All members have free access.

Special Offer!

Become a Vibration Institute Member today for only $95 and get access to the entire document library at no charge.

Buy it for $99.

Only interested in this article? You can make a one time purchase to download this content.

Vibration Analysis Certifications

In an increasingly competitive marketplace employers and clients seek the most qualified and knowledgeable professionals.

Get Expertise

Expand your knowledge of condition monitoring and vibration analysis.

Become A Member