Page 1

1

EQUIVALENT STATIC LOADS FOR RANDOM VIBRATION

Revision M

By Tom Irvine

Email: tomirvine@aol.com

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.

Introduction

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

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.

Load Specification

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

3. Stiffness

4. Damping

5. Natural frequencies

6. Dynamic mode shapes

7. Static deflection shape

8. Response acceleration

9. Modal velocity

10. Relative displacement

Page 2

2

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

axes

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,

EVC maxmax

(1)

where

max

= Maximum modal stress in the structure

max

V

= Maximum modal velocity of the structural response

E

= 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 )

C

Some additional research is needed to further develop equation 1 so that it can be used for

Page 3

3

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.

Model

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.

Figure 1.

m

k

c

x

y

Page 4

4

where

M

C

K

X

Y

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

1

k

m

fn

2

(3)

Acceleration Response

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

spectral density.

The overall acceleration response

GRMS

x

is

fn

xf ,

GRMSn

22

where

Fn is the natural frequency

P is the base input acceleration power spectral density at the natural frequency

is the damping ratio

P

(4)

Page 5

5

Note that the damping is often represented in terms of the quality factor Q.

1

Q

2

Equation (4), or an equivalent form, is given in numerous references, including those listed in

Table 1.

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

Appendix A.

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

frequency fn:

(5)

Page

246

76

225

59

516.6-12

28

Response Acceleration

n

f

(6)

Relative Displacement

5 . 1

n

f /1

(7)

Relative Displacement

= Response Acceleration

2

n

/

(8)