Mechanical Vibration and Shock Analysis, Specification Development

Chapter 1

Extreme Response Spectrum of a Sinusoidal Vibration

1.1. The effects of vibration

Vibrations can damage a mechanical system as a result of several processes, among which are:

– the exceeding of characteristic instantaneous stress limits (yield stress, ultimate stress etc.);

– the damage by fatigue following the application of a large number of cycles.

In what follows we will consider the case of a single degree-of-freedom linear system only. This model will be used to characterize the relative severity of numerous vibrations. It will be assumed that, if the greatest stresses and damage due to fatigue generated in the system are equal, then these excitations are of the same severity in the model and, by extension, in a real structure undergoing such excitations.

Since it is only the largest stresses in a single degree-of-freedom standard model with mass-spring-damping that are of interest here, this is equivalent to consideration of extreme stress or extreme relative displacement, these two parameters being linked, for a linear system, by a constant:

[1.1]

1.2. Extreme response spectrum of a sinusoidal vibration

1.2.1. Definition

The extreme response spectrum (ERS) [LAL 84] (or maximum response spectrum (MRS)) is defined as a curve giving the value of the highest peak zsup of the response of a linear one-degree-of-freedom system to vibration, according to its natural frequency f0, for a given damping ratio ξ. The response is described here by the relative movement z(t) of the mass in relation to its support, and the coordinate axis refers to the quantity (2 πf0)² zsup, by analogy with the shock response spectrum (Figure 1.1).

Figure 1.1. ERS calculation model

1.2.2. Case of a single sinusoid

A sinusoidal vibration can be defined in terms of a force, a displacement, a velocity or an acceleration.

1.2.2.1. Excitation defined by acceleration

Given an excitation defined by a sinusoidal acceleration of frequency f and amplitude

where Ω = 2 π f. The response of a single degree-of-freedom linear system, characterized by the relative displacement z(t) of the mass m with respect to the support, is expressed by:

[1.2]

(ω0 = 2 π f0) and the highest response displacement (extremum) by

[1.3]

The extreme response spectrum (ERS) is defined as the curve that represents variations of the quantity as a function of the natural frequency f0 of the system subjected to the sinusoid, for a given damping ratio ξ (or Q = 1/2 ξ).

[1.4]

NOTE.– The relative displacement is multiplied by in order to obtain a homogeneous parameter compatible with an acceleration (as with the shock response spectra; see Volume 2). The quantity zm is actually a relative acceleration ( m) only when Ω = ω0 (in sinusoidal mode) or more generally an absolute acceleration ( m) when damping is zero.

Figure 1.2. Relative displacement response of a single degree-of-freedom linear system

The spectrum corresponding to the largest negative values may also be considered. The positive and negative spectra are symmetrical. The positive spectrum has a maximum when the denominator has a minimum, i.e. for:

yielding

[1.5]

and

[1.6]

Figure 1.3. Maximum of reduced ERS versus damping ratio ξ

Table 1.1. Reduced ERS for values of Q factor

At a first approximation, it is completely reasonable to assume that R ≈ Q m. When f0 tends towards zero, R tends towards zero. When f0 tends towards infinity, R tends towards m.

Figure 1.4. Example of ERS of a sinusoid

Figure 1.5. ERS peak co-ordinates of a sinusoid

1.2.2.2. Reduced spectrum

It is possible to trace this spectrum with reduced coordinates by considering the variations in the ratio as a function of the dimensionless parameter .

NOTE.–

The reduced transfer function of a single degree-of-freedom linear system is defined by

where H ( ) is plotted as a function of the ratio f/f0, whereas the extreme response spectrum shows the variations in the same expression as a function of f0/f.

1.2.3. General case

The cases of an excitation defined by an acceleration, a velocity and a displacement can be brought together in the following general expression:

[1.7]

where

If the excitation is a force,

[1.8]

1.2.4. Case of a periodic signal

If the stress can be represented by the sum of several sinusoids

[1.9]

the response of a linear system to only one degree-of-freedom is equal to

[1.10]

and the extreme response spectrum is given by

[1.11]

[1.12]

1.2.5. Case of n harmonic sinusoids

Let us take an acceleration signal composed of a sum of n harmonic sinusoids:

[1.13]

ω1 = 2πf1

f1 = fundamental

φk = phase of sinusoid k

The relative displacement response of a linear system to one degree of freedom (f0, Q) is equal to

[1.14]

where

[1.15]

Q = Quality factor

[1.16]

The maxima of displacement z(t) are calculated by looking for values of time that cancel the velocity:

[1.17]

which can be written:

[1.18]

φk = φk – ψk

[1.19]

where

[1.20]

Let us look at the general case of a function f (periodic with period 2 π) developed in the Fourier series and write:

[1.21]

θ = ωt ∈ [0; 2π[ and n ≥ 1.

If θ is a solution:

This hypothesis is verified since, in our application, a0 = 0.

Let us write with (we assume that cn ≠ 0).

Thus . Let us write

[1.22]

We call Pfn the polynomial associated with f (polynomial of degree 2n with coefficients on C). Pfn has complex coefficients.

Equation fn(θ) = 0 has, at most, 2n roots on Pfn [0;2π[. In fact, fn(θ) = e–jn θ Pfn (ejθ)

According to d’Alembert’s theorem, Pfn has exactly 2 n roots (with their order of multiplicity) on C. For each root X of modulus 1, we solve the equation ejθ = X, θ [0;2π[ which gives a single solution.

The search for the roots of this polynomial can be carried out digitally for example by using the Laguerre algorithm. The roots retained must have a modulus equal to 1. They make it possible to calculate the amplitudes of peaks from [1.14]. The largest peak gives the ERS produced by .

NOTE.– The ERS can also be obtained by numerical calculation of the amplitude of the peaks of the sum of responses of one-degree-of-freedom systems to each sinusoidal component (Volume 1, relations [6.112], [6.68] and [6.120]).

1.2.6. Influence of the dephasing between the sinusoids

The ERS is sensitive to phase differences which can exist between the sinusoids making up the vibratory signal [COL 94]. When they are known, it is thus important to take them into account.

Example 1.1.

Let us take a vibration made up of 4 sinusoids all having an amplitude equal to 50 m/s² and frequencies equal to 20 Hz, 40 Hz, 60 Hz and 80 Hz.

Figure 1.6 makes it possible to compare the ERS of this vibration when the dephasings are zero and when they are equal to 0°, 130°, 30° and 90° respectively for each of the sinusoids.

The difference between the ERS can be significant, more than 20% in this example (Figure 1.7).

Figure 1.6. Comparison of ERS calculated from a sum of 4 non-dephased and dephased sinusoids

Figure 1.7. Difference between the ERS calculated from the dephased and non-dephased sinusoids

1.3. Extreme response spectrum of a swept sine vibration

1.3.1. Sinusoid of constant amplitude throughout the sweeping process

1.3.1.1. General case

The extreme response spectrum is the curve giving the highest value (or lowest value) for the response u(t) of a single degree-of-freedom linear system (f0, Q) when f0 varies. The upper value and lower value curves being symmetrical for a sine wave excitation, only one of them need be traced.

Given a sine wave excitation whose frequency is swept according to an arbitrary law, we will assume that the sweep rate is sufficiently slow that the response reaches a value very close to the steady-state response. If the amplitude of the sinusoid remains constant and equal to m during sweeping, the response of the system is, in the swept frequency interval f1, f2, equal to Q m [GER 61], [SCH 81].

For frequencies f0 located outside the swept range, at its maximum the response is equal to (Volume 1, [9.29]):

[1.23]

for f0 ≤ f1 and to (Volume 1, [9.30]):

[1.24]

for f0 ≥ f2. These values are obtained for an extremely slow sweep.

Figure 1.8. Construction of the ERS for a swept sine vibration

The value of um calculated in this way for f0 = f1 is the largest of all those which may be calculated for f0 ≤ f1 ranging between 0 and f1. In the same way, for f0 ≥ f2, it is the limit f2 which gives the greatest value of um (Figure 1.8).

Figure 1.9. ERS of swept sine vibration of constant amplitude between two frequencies

The type of spectrum obtained in this way is shown in Figure 1.9 for a sweep between f1 and f2 and a sinusoid of amplitude m. The spectrum increases from 0 to Q m at f1 (if the sweep is sufficiently slow), remains at this value between f1 and f2, then decreases and tends towards the value m [CRO 68], [STU 67].

1.3.1.2. Sweep with constant acceleration

In this case, the generalized co-ordinates are equal to and um = zm. Between f1 and f2, the spectrum has as an ordinate

[1.25]

For f0 ≤ f1

[1.26]

For f0 ≥ f2

[1.27]

1.3.1.3. Sweep with constant displacement

Here, yielding

for f1 ≤ f0 ≤ f2

[1.28]

for f0 ≤ f1

[1.29]

(Ω1 = 2πf1) and for f0 ≥ f2

[1.30]

(Ω2 = 2πf2).

1.3.1.4. General expression for extreme response

All these relationships may be represented by the following expressions:

if f1 ≤ f0 ≤ f2

[1.31]

if f0 < f1

[1.32]

and if f0 > f2

[1.33]

where Em and a characterize the vibration as indicated in the following table:

Table 1.2. Parameters Em and a according to the nature of the excitation

1.3.2. Swept sine composed of several constant levels

In the case of a swept sine composed of several constant levels of amplitude mj, the extreme response spectrum is defined as the envelope of the separately plotted spectra of each sweep corresponding to a single level.

mj represents accelerations, velocities, displacements or a combination of the three.

It should be noted that, in the range (fa, f2) in Figure 1.10, the largest response occurs in the band (f2, f3) and not in (f1, f2), a range in which the resonators are excited at resonance (um1 = Q m1).

Figure 1.10. ERS of a swept sine comprising several levels

Example 1.2.

1. Figure 1.11 shows the extreme response spectrum of a swept sine vibration defined as follow:

Constant acceleration

The spectrum is plotted from 1 Hz to 2,000 Hz in steps of 5 Hz.

Figure 1.11. Example of ERS of a swept sine vibration

2. Let us consider a swept sine vibration with constant displacement:

The ERS of this vibration, calculated for Q = 10 between 1 Hz and 200 Hz with 200 points (logarithmic step), is plotted in Figure 1.12.

Figure 1.12. ERS of a swept sine vibration at constant displacement

Chapter 2

Extreme Response Spectrum of a Random Vibration

The extreme response spectrum (ERS) (or maximum response spectrum, MRS) was defined as a curve that gives the value of the largest peak zsup of the response of a single-degree-of-freedom linear system to any given vibration (a random acceleration (t) in this chapter), according to its natural frequency f0, for a given ξ damping ratio. The response is described here in terms of the relative displacement z(t) of the mass with respect to its support. By analogy with the shock response spectrum, the y-axis refers to the quantity (2 π f0)² zsup [BON 77], [LAL 84]. The negative spectrum consisting of the smallest negative peak (2 π f0)² zinf is also often plotted.

Due to the random nature of the signal, the choice of parameter used to characterize the largest response is not as simple as it is for a sinusoidal vibration. The definitions most commonly used are (for any given f0):

– the largest value of the response on average over a given time T;

– the amplitude of the response equal to k times its rms value;

– the peak having a given probability of not being exceeded;

– the amplitude equal to k times the rms value of the response.

In the following sections we shall calculate the ERS in the above four cases. These four cases will be supplemented by some other relations from the statistical study of extreme values (Volume 3), as they can be useful for the sizing of the structures.

The cases should initially be distinguished as cases in which the vibration is characterized either by a time history signal or by a power spectral density.

2.1. Unspecified vibratory signal

When the signal is unspecified, and in particular when it is not stationary or Gaussian, it is not possible to determine a power spectral density. In such cases, each point of the ERS can only be obtained by direct numerical calculation of the response displacement z(t) of a single-degree-of-freedom linear system to the excitation (t), and by noting the largest peak response observed (positive zsup and/or negative zinf or the greater of the two in absolute value) over the considered duration T (Figure 2.1).

Figure 2.1. Principle of ERS calculation

This method is similar to that used to obtain a primary shock response spectrum, since the system’s residual response at the end of the vibration is not considered.

If the duration is lengthy, the calculation is limited to a sample that is considered representative and of a reasonable duration for such a purpose. However, there is always a risk of a significant error being made, as the probability of finding the largest peak in another sample is not negligible (a risk related to the duration of the selected sample).

As far as possible, it is preferable to use the procedures set out in the following sections to limit the costs, the computing time being much longer when the duration T is greater.

Example 2.1.

Figure 2.2. ERS of a random vibration measured on a truck calculated from the time history signal

NOTE.– Shock response spectra (Volume 2), extreme or with up-crossing risk response spectra and fatigue damage spectra defined in the following chapters must be calculated from a signal with a much greater sampling frequency than the one supported by Shannon theorem, approximately 10 times the maximum frequency of the desired spectrum. However, a sampled signal based on this theorem can be reconstructed to respect this requirement (Volume 1).

2.2. Gaussian stationary random signal

2.2.1. Calculation from peak distribution

2.2.1.1. General case

When the distribution of instantaneous values of the stationary signal follows a Gaussian law, the response instantaneous values distribution is Gaussian itself. We can then calculate directly from its power spectral density (PSD) the probability density of the response maxima of each single-degree-of-freedom linear system, as well as the corresponding distribution functions of peaks ([6.67], Volume 3):

where Q(u0) is the probability that u > u0 where , z0 is a peak amplitude of a relative movement of response to the single-degree-of-freedom system considered and zrms is the rms value of this relative movement.

In order to determine the ERS from this expression, we must calculate:

– the irregularity factor (Volume 3 [6.48]);

– the mean number of positive maxima per second of the relative displacement:

– the mean frequency of the response (Volume 3 [5.43] [5.53]):

– the mean total number of response peaks higher than a threshold u0 over chosen T duration:

[2.1]

– the probability of the largest peak on average during T (or 1 / N);

– the amplitude z0 of this peak, by consecutive iterations.

The method consists of setting a value of Q(u0) and determining the corresponding value of u0. The largest peak during T (on average) corresponds roughly to the level u0 which is only exceeded once (N = 1), yielding

[2.2]

The level u0 is determined by successive iterations. The distribution function Q(u) being a decreasing function of u, two values for u are given such that:

[2.3]

and, for each iteration, the interval (u1, u2) is reduced until, for example,

yielding, by interpolation,

[2.4]

and

[2.5]

The peak obtained here is the largest peak, on average. It is therefore the average of results that would be obtained from the numeric calculation of the system’s response and the histogram of peaks considering several signal samples.

Example 2.2.

Random vibration defined by:

The extreme response spectrum is plotted on Figure 2.3 for 5 ≤ f0 ≤ 1500 Hz and Q = 10.

Figure 2.3. ERS of a random vibration defined by its PSD

NOTE 1.– Difference between extreme response spectrum and shock response spectrum

Extreme response and shock response spectra both offer the greatest response of a single degree-of-freedom linear system according to its natural frequency, for a given Q factor, when submitted to vibration or the shock studied (we do not use the definition of ERSs from three times the rms value of the response). The calculation algorithm (Volume 2, Chapter 2) is the same.

In the case of long duration vibrations, this response occurs during the vibration: we only focus here on the primary spectrum.

In the case of shocks, on the other hand, the largest response peak can occur during or after the shock. We generally use the envelope spectrum of primary and residual spectra. It is the only difference between ERSs and SRSs.

NOTE 2.– Difference between extreme response spectra calculated from a signal and its PSD.

When the random vibration is Gaussian, the ERS can be calculated from a signal of acceleration or from its PSD. With random vibration, the ERS has a statistic character: calculated from a PSD, it gives the largest peak on average over T duration. When it is directly obtained from a signal based on time, it represents the largest peak for this signal sample and duration.

In addition, the ERS is obtained from the peak probability density of response, when the PSD is used, whereas we establish a range histogram (from which we could deduct a peak histogram) with a signal according to time. In order to standardize the methods, the distribution law of Dirlik’s ranges (Volume 4, Chapter 4) could be used.

Examples 2.3.

1. The theoretical PSD in Figure 2.4 is a case in which there is significant difference between Rice’s density probability of peaks and Dirlik’s probability density of half-ranges; see Example 4.2, Volume 4). And yet, ERSs calculated from Rice and Dirlik hypotheses are similar (Figure 2.5).

Figure 2.4. Power spectral density of Example 4.2, Volume 4

Figure 2.5. Extreme response spectra calculated from the PSD of Figure 2.4

2. Vibration measured in a plane

Consider a vibration measured in a plane, where the PSD is given in Figure 2.6. The comparison of ERSs calculated from Rice’s peak probability density and the density of Dirlik ranges shows that these spectra are very close (Figure 2.7).

Figure 2.6. PSD of a vibration measured in a plane

Figure 2.7. ERS of the plane vibration calculated from Rice and Dirlik probability densities

2.2.1.2. Case of a narrowband response

We consider the assumption where the vibration (t) is Gaussian and of zero mean. If the relative displacement response z(t) and its derivative (t) are independent functions, the mean number per second of crossing a given level a with positive slope can be written (Volume 3)

or, over duration T:

The largest level during this length of time T is that which is only exceeded once:

yielding level a:

[2.6]

At the frequency f0 (and for the selected value of Q), the extreme response spectrum has as amplitude:

[2.7]

This result can also be obtained from the distribution function [6.67] in Volume 3.

The approximation is acceptable when the irregularity factor r is higher than 0.6. We can then consider that the distribution of maxima follows Rayleigh’s law approximately.

NOTE.–

The extreme response spectrum can be plotted separately for a positive and a negative, as a function of f0 (as the shock response spectrum).

One curve could also be plotted by considering up-crossings of threshold |a|. In this case, must be replaced by na = 2 in the above relations (and by n0 = 2 ). Having selected level a, uncertainty related to the random nature of the phenomenon does not relate to the amplitude a which can be exceeded over duration T, but to the possibility of obtaining level a in a time shorter than T. The duration T being given, level a is that which is observed on average over this duration.

2.2.1.3. ERS for a duration larger than that of the analyzed signal

The duration of the vibratory signal used to calculate the ERS is generally very short. To estimate the ERS in the case where the vibration is of a larger duration, we can use relation [2.7]. Although established for a narrow-band noise, it gives a good approximation of the ERS in the case of a wideband vibration.

The ERS calculated from a signal of duration T is thus given by:

[2.8]

If the vibration in reality has a duration TR > T, the ERS would be given by

[2.9]

By eliminating between these two expressions, we obtain a relation which enables us to estimate the ERS relative to a duration TR from that calculated for a signal sample of duration T:

[2.10]

2.2.2. Use of the largest peak distribution law

The relations described in Chapter 7, Volume 3 can be used. Although more complex, they lead to results close to the preceding ones for the higher levels, in general.

The probability density of the largest peak can thus be considered for one length of time T, which has as its expression ([7.25], Volume 3):

for mode ([7.43], Volume 3):

(this is equation [2.6]) and for mean [7.30] (Volume 3):

[2.11]

where ε is Euler’s constant, equal to 0.577 215 665…

NOTE.–

1. Expression [2.11] is an asymptotic limit of the relation [7.29] (Volume 3):

(Np = T). But the error is acceptable: lower than 3% for Np > 2 and lower than 1% for Np > 50.

2. Furthermore, it was established ([7.37], Volume 3) that the standard deviation of this distribution, equal to

is all the weaker when T is larger: a slight error is consequently made by taking the mean as an estimate of the largest peak. Figure 2.8 shows that at a first approximation can be replaced by m, the variation being equal to and the error e, given by

[2.12]

This error decreases quickly when T increases (Figure 2.9).

Figure 2.8. Mean and mode of the largest peak over duration T

Figure 2.9. Error made by considering the mode of the law of the largest peaks instead of the mean

3. Relation [2.7] is an approximation of [2.11].

4. The mean and median of the largest peak distribution law are not identical, except at