Mechanical Vibration and Shock Analysis, Random Vibration

Christian Lalanne

• Publisher: Wiley-ISTE
• Year: 2014
• Print ISBN: 9781848216464
• eBook ISBN: 9781118931172
• Edition: 3
• Topic: Physical Sciences
• Subtopic: Mechanics

The vast majority of vibrations encountered in the real environment are random in nature. Such vibrations are intrinsically complicated and this volume describes the process that enables us to simplify the required analysis, along with the analysis of the signal in the frequency domain.
The power spectrum density is also defined, together with the requisite precautions to be taken in its calculations as well as the processes (windowing, overlapping) necessary to obtain improved results.
An additional complementary method – the analysis of statistical properties of the time signal – is also described. This enables the distribution law of the maxima of a random Gaussian signal to be determined and simplifies the calculation of fatigue damage by avoiding direct peak counting.

Chapter 1

Statistical Properties of a Random Process

1.1. Definitions

1.1.1. Random variable

A random variable is a quantity whose instantaneous value cannot be predicted. Knowledge of the values of the variable before time t does not make it possible to deduce the value at the time t from it.

Example: the Brownian movement of a particle.

If a vibration was perfectly random, its analysis would be impossible. The points that define the signal would have an amplitude that varied in a completely unpredictable way. Thankfully, in practice, it is possible to associate with all the points that characterize the signal a probability law which will enable a statistical analysis [AND 11].

The principal characteristic of a random vibration is to simultaneously excite all the frequencies of a structure [TUS 67]. In contrast to sinusoidal functions, random vibrations are made up of a continuous range of frequencies, the amplitude of the signal and its phase varying with respect to time in a random fashion [TIP 77] [TUS 79]. Thus, the random vibrations are also called noise.

Random functions are sometimes defined as a continuous distribution of sinusoids of all frequencies whose amplitudes and phases vary randomly with time [CUR 64], [CUR 88].

1.1.2. Random process

Let us consider, as an example, the acceleration recorded at a given point on the dial of a truck traveling on a good road between two cities A and B. For a journey, the recorded acceleration obeys the definition of a random variable. The vibration characterized by this acceleration is said to be random or stochastic.

Complexity of the analysis

Even in the most simple hypothesis where a vehicle runs at a constant speed on a straight road in the same state, each vibration measure ⁱ

(t) at one point of the vehicle is different from the other. An infinity of measures to completely characterize the trip should be completed a priori.

We define as a random process or stochastic process the ensemble of the time functions {ⁱ

(t)} for t included between – ∞ and +∞, this ensemble being able to be defined by statistical properties [JAM 47].

By their very nature, the study of vibrations would be intensive if we did not have the tools to limit the complete process analysis, made up of a large number of signals according to time, with a very long duration, to that of a very restricted number of samples of reasonable duration. Fortunately, random movements are not erratic in the common sense, but instead follow well-defined statistical laws. The study of statistical process properties, with averages in particular, will enable the simplification of the analysis from two very useful notions for this objective: stationarity and ergodicity.

1.2. Random vibration in real environments

By its nature, the real vibratory environment is random [BEN 61a]. These vibrations are encountered:

– on road vehicles (irregularities of the roads);

– on aircraft (noise of the engines, aerodynamic turbulent flow around the wings and fuselage, creating non-stationary pressures, etc.) [PRE 56a];

– on ships (engine, swell, etc.);

– on missiles. The majority of vibrations encountered by military equipment, and in particular by the internal components of guided missiles, are random with respect to time and have a continuous spectrum [MOR 55]: the gas jet emitted with a large velocity creates important turbulences resulting in acoustic noise which attacks the skin of the missile until its velocity exceeds Mach 1 approximately (or until it leaves the Earth’s atmosphere) [ELD 61], [RUB 64], [TUS 79];

– in mechanical assemblies (ball bearings, gears, etc.).

1.3. Random vibration in laboratory tests

Tests using random vibrations first appeared around 1955 as a result of the inability of sine tests to correctly excite equipment exhibiting several resonances [DUB 59], [TUS 73]. The tendency in standards is thus to replace the old swept sine tests which excite resona...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Foreword to Series
• Introduction
• List of Symbols
• Chapter 1: Statistical Properties of a Random Process
• Chapter 2: Random Vibration Properties in the Frequency Domain
• Chapter 3: Rms Value of Random Vibration
• Chapter 4: Practical Calculation of the Power Spectral Density
• Chapter 5: Statistical Properties of Random Vibration in the Time Domain
• Chapter 6: Probability Distribution of Maxima of Random Vibration
• Chapter 7: Statistics of Extreme Values
• Chapter 8: Response of a One-Degree-of-Freedom Linear System to Random Vibration
• Chapter 9: Characteristics of the Response of a One-Degree-of-Freedom Linear System to Random Vibration
• Chapter 10: First Passage at a Given Level of Response of a One-Degree-of-Freedom Linear System to a Random Vibration
• Appendix
• Bibliography
• Index
• Summary of Other Volumes in the Series