Characterisation of dynamic loading by PSD (Power Spectrum Density)

Example for the vibration behaviour of a spring-mass-system located on a rigid plate which is stimulated by the PSD shown.


In many technical fields random loads occur, which, however, can be described with the aid of statistical analyses, because there are certain regularities.

Tests on aircraft, ships or road vehicles showed that the stimulation at certain frequencies are particularly intense and that it is possible to describe them with the help of power spectrum density. Using the FE method this is relatively simple to determine the behaviour of structures by stimulating them with a power spectrum density.

The structure analysis is relatively simple to carry out and compared to the static analysis, it has the advantage that the dynamic system characteristics are considered in relation to the motivating frequencies.

The following example describes the simplest case of stochastic stimulation, just in one direction. Simultaneous stimulations in more than one direction are basically the same and will be treated later.

The following diagram shows an example of a typical case. The Power Spectrum Density of a system motivation xin  (e.g. stimulation of a vibrating bench, input) and a system response of the acceleration (movement of one point of the measured structure, response) in g²/Hz over the frequency shown. In reality the structure being tested is not a point-mass. It is fixed to a test bench.

Real structures have – unlike point masses – many points with different movements and a power spectrum density of the acceleration can be measured and also calculated for each of them. The occurring stresses can also be shown for each point as PSD and a fatigue life calculation can be carried out accordingly.

The flow chart below shows the procedure of a fatigue life calculation. It is presumed here that the component to be tested is a rigid body on a vibrating test bench.

Fatigue life calculation of a component under stochastic loading

Experimental Solution

A power spectrum density is defined as stimulation for the vibrating test bench and the structure being tested is fixed on to this bench. The junctures of the component and the vibrating test bench are then all stimulated with the power spectrum density of the acceleration.

In most cases the power spectrum density is given as a step-curve in which, for example, the value of the power density is given as 1 Hz for each step width.

Such power spectrum density can be found in standards for building ships or aircraft.

For the system the Eigen-frequencies and natural mode is calculated with the FE-system. This shows important provisional results which enable a plausibility control. It is important that all masses and stiffness is entered correctly.

Each structure node leads to oscillations and resulting stresses which are shown as a power spectrum density of the stress.

A Gaussian distribution of the stress amplitude is calculated for each node from the power spectrum density of the stress.

Characteristic properties of a dynamic system

An important value is the RMS (Root Means Square)

This complies with the area below the curve which can be shown in a simple diagram. You can show the data versus the frequency in Hz or the circle frequency in 1/sec. The relation between both is:

A PSD is shown in the next figure and there are also shown characteristic properties of a PSD. Important are the moments of different order n. n=0, 1,…, 4. The corresponding Moments you get by a multiplication of the area F(f) with fn. Based on these Moments characteristic properties can be got like the following.

  • Effective value:
  • Number of zero crossings:
  • Number of peak values
  • Irregularity factor
Power Spectral Density of the stress and important properties got from spectral Moments


The Irregularity factor enables to divide the system in following four cases:

Dividing stochastic systems using the irregularity factor I to select a suitable calculation procedure

Under the assumption that the system is stochastic, ergodic and stationary a suitable procedure to calculate adequate amplitude collective can be found.

Pure Sinus-function with constant amplitude und frequency (I=1)

This case is not a stochastic process and the following methods are not suitable. But simple deterministic procedure exists and a constant amplitude with a cycle number calculated from the duration time and the frequency can calculated easily.

Narrowband-process (I~1)

Calculation of the distribution is done by the following equation:


Note: S is the stress range (double of amplitude).

Wideband-process (0<I<1)

The distribution uses Dirlik-equation [  ] :


Z =             I =                 

Note: S is the stress range (double of amplitude).

Flow of the Calculation

Defining the excitation of the test rig table

Calculation of Eigenfrequencies and Eigenmodes of the system (Modalanalyse) using FEM

Calculation of system answer (PSD of stress for each (interested) node.

Calculation of a damage equivalent collective

Calculation of damage 

An example is shown using FEMAP.