Miles' Equation in Random Vibrations

A general introduction about the book ‘Miles Equation in Random Vibrations’ is given, and the contents of chapters and appendices are summarized.

The root mean square (r.m.s.) response of a single degree of freedom (SDOF) system when exposed to white noise random excitation can be approximated applying Miles’ equation. In this chapter, Miles’ equation is derived for a) a random enforced acceleration and b) a random applied force. A further approximation was introduced assuming a more or less flat PSD of acceleration and force in the vicinity of the natural frequency of the SDOF system. The accuracy of the approximation with Miles’ equation is investigated, and examples are worked out. Problems with solutions are provided to get more understanding of the limitations of Miles’ equation.

The replacement of dynamic random response analysis by a simple approximate quasi-static analysis in combination with Miles’ equation is discussed in this chapter. Equivalent static acceleration and force fields are considered. A procedure to perform an equivalent finite element (stress) analysis is presented. Examples are given to illustrate this equivalent quasi-static approach. Problems with solutions are provided to gain more insight in using Miles’ equation in quasi-static applications.

In general, the structural design of spacecraft structures, instruments, and structural elements is based on the static load factors specified in the launch vehicle (L/V) user’s guide(s) or by other quasi-static load (QSL) specifications. In this chapter, the calculated random load factors are based on the \(3\sigma \) values of the interface loads of a particular system and depend on the random load levels, natural frequencies, and associated modal effective masses. If the random load factors are beyond the specified quasi-static load factors, adaptation (notching) of the random dynamic loads may be considered. Examples and problems with answers included are provided.

The mass participation is one of the approaches for notching analysis, in which the modal effective mass and apparent mass in conjunction with Miles’ equation are the basic elements to determine notched random acceleration input comparing \(3 \sigma \) reaction loads with the reaction loads caused by the quasi-static design loads (QSL). The quasi-static design limit load applied for the design of equipment and instruments is mostly based on experience from previous spacecraft projects and is defined, in general, using the mass acceleration curve (MAC). Procedures to calculate random load factors and the estimation of the depth of the notch (modification of input spectrum) using the mass participation are discussed. Examples and problems with answers are given.

When developing a qualification test program for spacecraft, it is necessary to determine whether there should be an acoustic or random vibration test for each instrument, component, etc. The decision (key) factor is the area/mass ratio of an instrument, component, etc., which is very helpful to make a choice to perform either a random vibration on a shaker table or an acoustic test in a acoustic reverberant chamber. The calculation of the decision factor is completely based on random response analyses applying Miles’ equation.

A simple manner to estimate structural responses for simple structures exposed to acoustic loads is discussed. Peak pressures are obtained using Miles’ equation. Shape factors are introduced to include the effect of boundary conditions. Spann’s prediction method of the vibrational environment of components mounted on plates or panels is discussed too. Examples and problems with answers are provided.

Three approximate methods to calculate the responses of shell structures when exposed to an acoustic pressure field using a single degree of freedom (SDOF) system approach are discussed in this chapter. The most straightforward method is the approximation applying the second approach, when the acoustic pressure field is proportional to the assumed or vibration mode of the shell structure. To gain more insight examples and problems with answers are provided.

In general, the sinusoidal vibration (sweep) test is in the frequency range between 5 and 100 Hz and the random vibration test has a frequency range between 20–2000 Hz. Therefore, it is not so straightforward to replace a random vibration test by an equivalent sinusoidal vibration test or vice versa. Damaging and fatigue aspects have to be considered transferring a specific sinusoidal specification into equivalent random vibration specification. Examples and problems are provided.

A number of damage spectra to characterise measured or computed random acceleration vibration spectra are discussed in this chapter. The characterisation is based on equivalent damage caused by extreme peaks (SRS, ERS, VRS) and Rayleigh distribution of peaks or cumulative damage (FDS), using relative displacements and pseudo-velocities. The response spectra are all based on the response of single degree of freedom (SDOF) systems exited to random accelerations, both in the time and frequency domain. The principles to compute the response spectra are illustrated and discussed. One general practical example is discussed in very detail. Miles’ equation fulfills a key role in the synthesis process to generate equivalent random acceleration vibration specifications. An envelope of the damage response spectrum is achieved by dividing the spectrum into a number of fields or regions. The lower the number of field more severe and smoother equivalent random acceleration vibration specification is obtained.

A number of typical spacecraft structure-related applications of Miles’ equation are worked out and explained. To gain more understanding out of the problems favorite finite element software packages may be applied to compare the answers.