EQUIVALENT STATIC LOADS FOR RANDOM VIBRATION
By Tom Irvine
October 8, 2010
The following approach in the main text is intended primarily for single-degree-of-freedom
systems. Some consideration is also given for multi-degree-of-freedom systems.
A particular engineering design problem is to determine the equivalent static load for equipment
subjected to base excitation random vibration. The goal is to determine peak response values.
The resulting peak values may be used in a quasi-static analysis, or perhaps in a fatigue
calculation. The response levels could be used to analyze the stress in brackets and mounting
hardware, for example.
Limitations of this approach are discussed in Appendices F through K.
A particular concern for either a multi-degree-of-freedom system or a continuous system is that
the static deflection shape may not properly simulate the predominant dynamic mode shape. In
this case, the equivalent static load may be as much as one order of magnitude more conservative
than the true dynamic load in terms of the resulting stress levels.
Ideally, the dynamics engineer and the static stress engineer would mutually understand, agree
upon, and document the following parameters for the given component.
1. Mass, center-of-gravity, and inertia properties
2. Effective modal mass and participation factors
5. Natural frequencies
6. Dynamic mode shapes
7. Static deflection shape
8. Response acceleration
9. Modal velocity
10. Relative displacement
11. Transmitted force from the base to the component in each
of three axes
12. Bending moment at the base interface about each of three
13. The manner in which the equivalent static loads and
moments will be applied to the component, such as point
load, body load, distributed load, etc.
14. Dynamic stress and strain at critical locations if the
component is best represented as a continuous system
15. Response limit criteria, such as yield stress, ultimate stress,
fatigue, or loss of clearance
Each of the response parameters should be given in terms of frequency response function, power
spectral density, and an overall response level.
Furthermore, assumptions must be documented, including a discussion of conservatism.
Again, this list is very idealistic.
Importance of Modal Velocity
Bateman wrote in Reference 24:
Of the three motion parameters (displacement, velocity, and acceleration) describing a
shock spectrum, velocity is the parameter of greatest interest from the viewpoint of
damage potential. This is because the maximum stresses in a structure subjected to a
dynamic load typically are due to the responses of the normal modes of the structure,
that is, the responses at natural frequencies. At any given natural frequency, stress is
proportional to the modal (relative) response velocity. Specifically,
= Maximum modal stress in the structure
= Maximum modal velocity of the structural response
= Elastic modulus
= Mass density of the structural material
= Constant of proportionality dependent upon the geometry of
the structure (often assumed for complex equipment to be
4 < C < 8 )
Some additional research is needed to further develop equation 1 so that it can be used for
equivalent quasi-static loads for random vibration. Its fundamental principle is valid, however.
Further information on the relationship between stress and velocity is given in Reference 25.
Importance of Relative Displacement
Relative displacement is needed for the spring force calculation. Note that the transmitted force
for an SDOF system is simply the mass times the response acceleration.
Specifying the relative displacement for an SDOF system may seem redundant because the
relative displacement can be calculated from the response acceleration and the natural frequency
per equation (7) given later in this paper.
But specifying the relative displacement for an SDOF system is a good habit.
The reason is that the relationship between the relative displacement and the response
acceleration for a multi-degree-of-freedom (MDOF) or continuous system is complex. Any
offset of the component’s center-of-gravity (CG) further complicates the calculation due to
coupling between translational and rotational motion in the modal responses.
The relative displacement calculation for an MDOF system is beyond the scope of a hand
calculation, but the calculation can be made via a suitable Matlab script. A dynamic model is
required as shown in Appendices H and I.
Furthermore, examples of continuous structures are shown in Appendices J & K. The structures
are beams. The bending stress for the equivalent static analysis of each beam correlates better
with relative displacement than with response acceleration.
The first step is to determine the acceleration response of the component.
Model the component as an SDOF system, if appropriate, as shown in Figure 1.
is the mass
is the viscous damping coefficient
is the stiffness
is the absolute displacement of the mass
is the base input displacement
Furthermore, the relative displacement z is
z = x – y (2)
The natural frequency of the system fn is
The Miles’ equation is a simplified method of calculating the response of a single-degree-of-
freedom system to a random vibration base input, where the input is in the form of a power
The overall acceleration response
Fn is the natural frequency
P is the base input acceleration power spectral density at the natural frequency
is the damping ratio
Note that the damping is often represented in terms of the quality factor Q.
Equation (4), or an equivalent form, is given in numerous references, including those listed in
Table 1. Miles’ equation References
Reference Author Equation
1 Himelblau (10.3)
2 Fackler (4-7)
3 Steinberg (8-36)
4 Luhrs -
5 Mil-Std-810G -
6 Caruso (1)
Furthermore, the Miles’ equation is an approximate formula that assumes a flat power spectral
density from zero to infinity Hz. As a rule-of-thumb, it may be used if the power spectral density
is flat over at least two octaves centered at the natural frequency.
An alternate response equation that allows for a shaped power spectral density input is given in
Relative Displacement & Spring Force
Consider a single-degree-of-freedom (SDOF) system subject to a white noise base input and with
constant damping. The Miles’ equation set shows the following with respect to the natural
5 . 1
= Response Acceleration