Mechanical Vibration and Shock Analysis, Sinusoidal Vibration

Christian Lalanne

• Publisher: Wiley-ISTE
• Year: 2014
• ISBN:
  9781118931097
• Edition: 3
• Topic: Ciencias físicas
• Subtopic: Mecánica

Everything engineers need to know about mechanical vibration and shock...in one authoritative reference work!

This fully updated and revised 3rd edition addresses the entire field of mechanical vibration and shock as one of the most important types of load and stress applied to structures, machines and components in the real world. Examples include everything from the regular and predictable loads applied to turbines, motors or helicopters by the spinning of their constituent parts to the ability of buildings to withstand damage from wind loads or explosions, and the need for cars to maintain structural integrity in the event of a crash. There are detailed examinations of underlying theory, models developed for specific applications, performance of materials under test conditions and in real-world settings, and case studies and discussions of how the relationships between these affect design for actual products.

Invaluable to engineers specializing in mechanical, aeronautical, civil, electrical and transportation engineering, this reference work, in five volumes is a crucial resource for the solution of shock and vibration problems.

The relative and absolute response of a mechanical system with a single degree of freedom is considered for an arbitrary excitation, and its transfer function is defined in various forms. The characteristics of sinusoidal vibration are examined in the context both of the real world and of laboratory tests, and for both transient and steady state response of the one-degree-of-freedom system. Viscous damping and then non-linear damping are considered. The various types of swept sine perturbations and their properties are described and, for the one-degree-of-freedom system, the consequence of an inappropriate choice of sweep rate are considered. From the latter, rules governing the choice of suitable sweep rates are then developed.

Chapter 1

The Need

1.1. The need to carry out studies into vibrations and mechanical shocks

During their service life, many materials are subjected to vibratory environments, during their transport [OST 65], [OST 67], because they are intended to equip themselves with means of transport (airplanes, road vehicles, etc.) or because they are placed beside vibratory sources (engines, wind mills, roads, etc.). These vibratory environments (vibrations and shocks) create dynamic strains and stresses in the structures which can, for example, produce intermittent or permanent breakdowns in electrical equipment, plastic deformations or fractures by up-crossing an ultimate stress of the material (yield limit, rupture limit), optical misalignments of systems or may contribute to the fatigue and the wear of the machine elements.

It is therefore necessary to take all of these points into consideration during the design phase of structures and of mechanical equipment. The approach is normally made up of several steps:

– measuring the vibration phenomena;

– analyzing the results of the measurements, bearing in mind that this analysis will be used for different objectives, including:

- the characterization of the frequency contents of the vibration (the search for predominant frequencies, amplitudes, etc.), for example, to compare the natural frequencies of the structures,

- comparing the relative severity of several different vibratory environments (transport on various vehicles) or comparing the severity of such vibration environments with a standard,

- confirming a posteriori the validity of a dimensioning or test specification which is established starting from fallback level values, from data collected at the time of a preceding project or starting from values resulting from normative documents;

– the transformation of measurements into dimensioning specifications for research departments; these are presented in the simplest possible form requiring a synthesis of all the measured data;

– during and at the end of the design phase, at the time of the qualification, realization of tests intended to validate the behavior of the materials developed from these environments.

The vibrations most frequently encountered in the real environment are of a random nature. Along with shocks, they constitute the main part of mechanical excitations. These two environments can be severe, shocks by their amplitude and random vibrations by their duration.

In certain situations, however (near turning machines), it is possible to observe sinusoidal vibrations which are often polluted by noise. This is especially the case for vibrations which are produced by propeller airplanes and helicopters. In these cases, the random noise which is produced is significantly important compared to the sinusoidal lines (fundamental and harmonics).

Whenever such rotating machines are switched on and off, their frequency varies, in a continuous way, generating a vibration similar to a swept sine. This type of environment is primarily used in laboratory tests in order to carry out research into the resonance frequency of structures.

The mechanical excitations which are then analyzed, resulting from measurements of the environment or test laboratory, belong to one of the following groups:

– sinusoidal vibrations;

– swept sine vibrations;

– random vibrations;

– mechanical shocks;

or a combination of these vibrations:

– sine on random (one or several lines);

– a swept sine on random (with a sweeping on one or several frequency bands);

– a narrowband random vibration swept on a wideband noise, etc.

The vibrations which are produced in the real world have quite different frequency domains:

– between approximately 1 and 500 Hz for road vehicles;

– between approximately 10 and 2,000 Hz for airplanes and spacecraft;

– between approximately 1 and 35 Hz for earthquakes;

– more than 10,000 Hz for shocks which are created by metal–metal impacts, several tens of thousands of Hz for shocks which are created by pyrotechnic devices.

Vibrations are often classed into three different categories, depending on their frequency. The different categories are as follows:

– very low frequency for frequency values between 0 and 2 Hz;

– medium frequency for frequency values between 2 and 20 Hz;

– high frequency for frequency values between 20 and 2,000 Hz.

These values in conventional matter are given only as an indication and do not have any theoretical legitimacy. The low frequency concept can in fact be definite only according to the natural frequency of the system which undergoes the vibration. The frequency of a vibration will be low for a mechanical system if it induces any dynamic response (no attenuation and no amplification).

1.2. Some real environments

1.2.1. Sea transport

The sources of vibrations on board ships have various origins and natures. They are primarily due to:

– the propeller (periodic vibrations);

– the propelling unit and the auxiliary groups (periodic vibrations);

– the equipment used on board (for example, winches);

– the effects of the sea (random vibrations).

The measured levels are in general the lowest amongst all the means of surface transport.

1.2.1.1. Vibrations produced by the ship's propeller

The rotation of the propeller can excite the modes of the ship's frame in different ways:

– the accelerations transmitted to the hull via the line shafts;

– forces exerted on the ship's rudder;

– hydroelastic coupling between the propeller and the shafts' line;

– fluctuations in pressure distributed on all parts of the back hull, having as an origin the wake in which the propeller works. These fluctuations in pressure are dependent on:

- the variations of propeller's push. When the propeller provides a push, the back of each blade is subjected to a “negative pressure” (suction) compared to the environmental pressure, and the front face is subjected to an overpressure,

- the number, area and thickness of the blades. The fluctuations in pressure are a linear function of the average thickness of the blades and decrease very quickly when the number of blades increases,

- the presence of a variable vapor pocket on the surface of the blade and in its slipstream, as a consequence of cavitation.

Around the propeller is formed a cavity filled with vapor within the liquid, due to a local pressure lower than the saturating steam pressure. When the vapor bubbles reach higher pressure zones, they condense brutally. This phenomenon, known as cavitation, involves very strong mechanical actions (vibrations, noises, etc.).

Cavitation is the source of the majority of vibration problems encountered on ships. It is equivalent to an increase of the thickness of blades and, as a result, increases the pressure fluctuations. The variation of the volume of the cavitation pocket over time is a second source of pressure fluctuation. The fundamental frequency is around 20 Hz for fixed blade propellers from 5 to 6 m in diameter and 10 Hz for propellers from 8 to 10 m in diameter. The natural frequencies of the blades decrease when the diameter increases.

1.2.1.2. Vibrations produced by the ship's engine

The vibrations which are produced by a ship's engine are caused by the alternate movements of the piston, connecting rod and crankshaft systems.

They can excite the modes of the ship's frame, especially for medium-sized ships. Their vibratory frequency generally lies between 3 and 30 Hz.

1.2.1.3. Vibrations produced by the state of the sea

Vibrations due to the swell

The swell heave leads to the creation of vibrations of a long duration and of very low frequency (less than 2 Hz) in both the longitudinal (pitching) and transverse direction (rolling). These random oscillations are always of a seismic nature.

Their frequency varies between 0.01 Hz (when the sea is very calm) and 1.5 Hz (during bad weather). Their associated accelerations range from approximately 0.1 m/s² to 9 m/s².

Vibrations of the whole of the ship due to the state of the sea

In general, two types of vibrations are considered:

– hydrodynamic shocks applied to the front of the ship lead to the vibration of the whole of the ship, which works like a beam. This phenomenon occurs whenever the ship navigates the sea with its front first, with relative movements of the stem sufficiently significant to create impacts. These impacts can be distinguished as follows:

- shocks which are produced on the flat part at the bottom of the ship, when the ship makes contact with the sea, after it emerges from the water,

- shocks on planking of the stem, without emergence, without the ship resurfacing from the sea,

- areas of seawater;

– excitations which are caused by the swell's variable hydrodynamic forces, which lead to a steady state free vibration of the entire ship.

These vibrations generally have low or very low frequencies and, to a lesser extent, some can have high frequencies [VIB 06]. The frequencies range from 0.01 Hz to 80 Hz, with a maximum value of between 3 Hz and 30 Hz. The vibrations are periodic or random.

1.2.2. Earthquakes

The rapid release of the deformation energy which is accumulated in the Earth's crust or mantle (the underlying layer) is felt as a vibration on the Earth's surface: an earthquake. The vibration (the tremor) lasts in general for a few tens of a second. Their amplitude on the ground level can reach several m/s².

The shock response spectrum was created in the 1930s in order to group together the different effects that earthquakes of different amplitudes have on buildings. The amplitudes are taken from actual acceleration signals which were measured from real earthquakes (see Volume 2)....

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Foreword
• Introduction
• List of symbols
• Chapter 1: The Need
• Chapter 2: Basic Mechanics
• Chapter 3: Response of a Linear One-Degree-of-Freedom Mechanical System to an Arbitrary Excitation
• Chapter 4: Impulse and Step Responses
• Chapter 5: Sinusoidal Vibration
• Chapter 6: Response of a Linear One-Degree-of-Freedom Mechanical System to a Sinusoidal Excitation
• Chapter 7: Non-viscous Damping
• Chapter 8: Swept Sine
• Chapter 9: Response of a Linear One-Degree-of-Freedom System to a Swept Sine Vibration
• Appendix: Laplace Transformations
• Vibration Tests: a Brief Historical Background
• Bibliography
• Index
• Summary of Other Volumes in the Series