The Extreme Response Spectrum (ERS) (or Maximum Response Spectrum (MRS)) is defined as a curve giving the value of the highest peak of the response of a linear Single Degree of Freedom System (SDOF system) to vibration, according to its natural frequency

Fundamental frequency

The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...

, for a given damping ratio

Damping ratio

[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

. The response is described here by the relative movement

Kinematics

Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....

 of the mass of this system in relation to its support. The x-axis refers to the natural frequency and the y-axis to the highest peak multiplied by the square of the quantity (2 π x natural frequency), by analogy with the relative displacement shock response spectrum

Shock response spectrum

A Shock Response Spectrum is a graphical representation of an arbitrary transient acceleration input, such as shock in terms of how a Single Degree Of Freedom system responds to that input. Actually, it shows the peak acceleration response of an infinite number of SDOFs, each of which have...

.

The severity of a vibration can be evaluated by calculating the stresses on a mathematical or finite element model of the structure and, for example, comparison with the ultimate stress of the material. This is the method used to dimension the structure. Generally, however, the problem is instead to evaluate the relative severity of several vibrations (vibrations measured in the real environment, measured vibrations with respect to standards, establishment of a specification etc.). This comparison would be difficult to carry out if one used a fine model of the structure and besides, this is not always available, in particular at the stage of the development of the specification of dimensioning.

A solution consists of applying the vibration under consideration to a “standard” mechanical system, which thus does not claim to be a model of the real structure, composed of a support and N linear one-degree-of-freedom resonators, each one comprising a mass, a spring and a damping device.

A vibration A is considered as more severe than a vibration B of it produces a highest relative displacement (i.e. a highest stress) on this SDOF system than the vibration B.

An ERS is generated from a vibration signal using the following process:

1. Choose a damping ratio for the ERS to be based on;

2. Assume a hypothetical Single Degree of Freedom System, with a given natural frequency (Hz);

3. Calculate (by time base simulation or from a Power Spectral Density

Spectral density

In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

 (PSD) of the vibratory signal) the highest instantaneous relative displacement experienced by the mass element of this SDOFs at any time during exposure to the vibration in question. Plot this value multiplied by the square of (2 π x natural frequency) against the natural frequency of the hypothetical system;

4. Repeat steps 2 and 3 for other values of the natural frequency.

The resulting plot is called an Extreme response spectrum

Response spectrum

A response spectrum is simply a plot of the peak or steady-state response of a series of oscillators of varying natural frequency, that are forced into motion by the same base vibration or shock. The resulting plot can then be used to pick off the response of any linear system, given its natural...

.

Note

Vibrations can damage a mechanical system as a result of several processes, among which are:

• the exceeding of characteristic instantaneous stress limits (yield stress
  Yield (engineering)
  The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed...
  
  , ultimate stress etc.)
• the damage by fatigue following the application of a large number of cycles.

ERS is used according to the first criterion. The second is considered with the fatigue damage spectrum

Fatigue damage spectrum

The Fatigue Damage Spectrum of a vibration is obtained by tracing the fatigue damage experienced by a linear Single Degree of Freedom System according to its natural frequency, for given damping ratio and for a given value of parameter b .Regardless of the...

(FDS).

Reference

• Lalanne, C., Mechanical Vibration and Shock Analysis. Volume 5: Specification Development, Second Edition, ISTE - Wiley, 2009.