Chapter 2

Shock Response Spectrum

2.1. Main principles

A shock is an excitation of short duration which induces transitory dynamic stress in structures. These stresses are a function of:

– the characteristics of the shock (amplitude, duration and form); – the dynamic properties of the structure (resonance frequencies and Q factors).

The severity of a shock can thus be estimated only according to the characteristics of the system which undergoes it. In addition, the evaluation of this severity requires the knowledge of the mechanism leading to a degradation of the structure. The two most common mechanisms are:

– exceeding a value threshold of the stress in a mechanical part, leading to either a permanent deformation (acceptable or not) or a fracture or, at any rate, a functional failure. – if the shock is repeated many times (e.g. shock recorded on the landing gear of an aircraft, operation of an electromechanical contactor, etc.), the fatigue damage accumulated in the structural elements can lead to fracture in the long term. We will deal with this aspect later on.

The severity of a shock 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 shocks (shocks measured in the real environment, measured shocks 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. One searches for a method of general nature, which leads to results which can be extrapolated to any structure.

A solution was proposed by M.A. Biot [BIO 32] in 1932 in a thesis on the study of the effects of earthquakes on buildings; this study was then generalized to analysis of all kinds of shocks.

The study consists of applying the shock 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 mi, a spring of stiffness ki and a damping device ci, chosen such that the fraction of critical damping is the same for all N resonators (Figure 2.1).

Figure 2.1. Model of the shock response spectrum (SRS)

When the support is subjected to the shock, each mass mi has a specific movement response according to its natural frequency and to the chosen damping ξ, while a stress σi is induced in the elastic element.

The analysis consists of seeking the largest stress σmi observed at each frequency in each spring. A shock A is regarded as more severe than a shock B if it induces a large extreme stress in each resonator. We then carry out an extrapolation, which is certainly open to criticism, by assuming that, if shock A is more severe than shock B when it is applied to all the standard resonators, then it is also more severe with respect to an arbitrary real structure (which can be non-linear or have a single degree of freedom).

NOTE.– A study was carried out in 1984 on a mechanical assembly composed of a circular plate on which one could place different masses and thus vary the number of degrees of freedom. The stresses generated by several shocks of the same spectra (in the frequency range including the principal resonance frequencies), but of different shapes [DEW 84], were measured and compared. It was noted that for this assembly, whatever the number of degrees of freedom:

– two pulses of simple form (with no velocity change) having the same spectrum induce similar stresses, the variation not exceeding approximately 20%. It is the same for two oscillatory shocks; – the relationship between the stresses measured for a simple shock and an oscillatory shock can reach 2.

These results were supplemented by numerical simulation intended to evaluate the influence of non-linearity. Even for very strong non-linearity we did not note, for the cases considered, a significant difference between the stresses induced by two shocks of the same spectrum, but of a different form.

A complementary study was carried out by B.B. Petersen [PET 81] in order to compare the stresses directly deduced from a SRS with those generated on an electronics component by a half-sine shock envelope of a shock measured in the environment, and by a shock of the same spectrum made up from WAVSIN signals (Chapter 8) added with various delays. The variation between the maximum responses measured at five points in the equipment and the stresses calculated starting from the shock response spectra does not exceed a factor of 3 in spite of the important theoretical differences between the model of the response spectrum and the real structure studied.

A more recent study by D.O. Smallwood [SMA 06] shows that we can find two signals with very close SRS which can lead to responses in a ratio of 1.4 on a system with several degrees of freedom.

For applications deviating from the assumptions of definition of the SRS (linearity, only one degree of freedom), it is desirable to observe a certain prudence if we want to quantitatively estimate the response of a system starting from the spectrum [BOR 89]. The response spectra are used more often to compare the severity of several shocks.

It is known that the tension static diagram of many materials comprises a more or less linear arc on which the stress is proportional to the deformation. In dynamics, this proportionality can be allowed within certain limits for the peaks of the deformation (Figure 2.2).

If the mass-spring-damper system is assumed to be linear, it is then appropriate to compare two shocks by the maximum response stress σm they induce or by the maximum relative displacement zm that they generate, since:

[2.1]

Figure 2.2. Stress–strain curve

zm is only a function of the dynamic properties of the system, whereas σm is also a function, via K, of the properties of the materials which constitute it.

The curve giving the largest relative displacement zsup multiplied by according to the natural frequency f0, for a given ξ, damping, is the SRS. The first work defining these spectra was published in 1933 and 1934 [BIO 33], [BIO 34], then in 1941 and 1943 [BIO 41], [BIO 43]. The SRS, then named the shock spectrum, was presented there in the current form.

This spectrum was used in the field of environmental tests from 1940 to 1950: J.M. Frankland [FRA 42] in 1942, J.P. Walsh and R.E. Blake in 1948 [WAL 48], R.E. Mindlin [MIN 45]. Since then, there have been many works which have used it as a tool for analysis and for the simulation of shocks [HIE 74] [KEL 69] [MAR 87] [MAT 77].

2.2. Response of a linear one-degree-of-freedom system

2.2.1. Shock defined by a force

Given a mass-spring-damping system subjected to a force F(t) applied to the mass, the differential equation of the movement is written as:

[2.2]

Figure 2.3. Linear one-degree-of-freedom system subjected to a force

where z(t) is the relative displacement of the mass m relative to the support in response to the shock F(t). This equation can be put in the form:

[2.3]

where:

[2.4]

(damping factor) and:

[2.5]

(natural pulsation).

2.2.2. Shock defined by an acceleration

Let us set as (t) an acceleration applied to the base of a linear one-degree-of-freedom mechanical system, with (t) the absolute acceleration response of the mass m and z(t) the relative displacement of the mass m with respect to the base. The equation of the movement is written as above:

[2.6]

Figure 2.4. Linear one-degree-of-freedom system subjected to acceleration

i.e.:

[2.7]

or, while setting z(t) = y(t) – x(t):

[2.8]

2.2.3. Generalization

Comparison of differential equations [2.3] and [2.8] shows that they are both of the form:

[2.9]

where (t) and u(t) are generalized functions of the excitation and response.

NOTE.– Generalized equation [2.9] can be written in the reduced form:

[2.10]

where:

[2.11]

[2.12]

[2.13]

m = maximum of (t).

Resolution

Differential equation [2.10] can be integrated by parts or by using the Laplace transformation. We obtain, for zero initial conditions, an integral called Duhamel’s integral:

[2.14]

where δ = variable of integration. In the generalized form, we deduce that:

[2.15]

where α is an integration variable homogeneous with time. If the excitation is an acceleration of the support, the response relative displacement is given by:

[2.16]

and the absolute acceleration of the mass by:

[2.17]

Application Let us consider a package intended to protect a material from mass m and comprising a suspension made up of two elastic elements of stiffness k and two dampers...