Waves-Vibratory Analysis

Abstract

This chapter presents the theoretical basis of the vibration analysis method applied to the field of tie rods. The elements obtained during acquisition are then used to obtain results concerning free and total lengths, the diameters of the elements tested and the dynamic stiffness used to calculate the internal force at the tie-rod.

Radio waves are electromagnetic waves of the same nature as light, which result from the disturbance of electric and magnetic fields.

In addition to visible light, there are radar waves, microwaves, infrared, ultraviolet, X and gamma rays, and radio waves.

Our area of intervention is sound. Of sound waves need material support to spread, electromagnetic waves, spread better in a vacuum, and much faster: sound only goes at 300 m/s, while electromagnetic waves go at about 300,000 km/s. You see lightning before you hear thunder.

1.1 Definitions—Period—Frequency

In addition to visible light, there are radar waves, microwaves, infrared, ultraviolet, X-rays, gamma rays and radio waves. Our operating range is generally below 2000 Hz.

A wave is the modification of the physical state of a physical environment. In our case, it's a reinforcement, usually steel, contained in a cylinder of cement grout. The wave propagates as a result of local action, with a velocity determined by the characteristics of the material through which it passes.

A periodic function is a function which, when applied to a variable, takes on the same value as a certain fixed quantity called period is added to that variable (Fig. 1.1).

Fig. 1.1

Wave / period “T” example

The period T is the time between the passage of two maxima.

Frequency f is a measure of the number of times a periodic phenomenon repeats per unit time f = 1/T (Fig. 1.2).

Fig. 1.2

Sinusoidal waves of different frequencies. − the lower one has the high frequency. − the upper one has the lower one.

The frequency is the number of periods per unit time, which is the inverse of the period: f = 1/T where f is the frequency in Hertz (Fig. 1.3).

Fig. 1.3

Types of frequency

1.2 Principle of the Method

The principle of the method is described in:

• European Patent Office PCT/FR2005/001597 (Appendix A)

• WIPO Patent No. WO 2006/010830 A1, 2006 (Appendix B)

The method consists of:

• Delivering a mechanical shock to the head of the tested element by means of a hammer equipped with a force sensor.

• Measure the particle velocity at the head of the test item.

• Do the acquisition in time mode.

• And analyze the signal in frequency mode.

The figure below visualizes the compression wave emitted by the impact of the hammer and which propagates inside the tested element to go up at its extremity (Fig. 1.4).

Fig. 1.4

Diagram of the compression wave

1.3 Mechanical Impedance

Impedance in a circuit fed by an alternating current is equivalent to resistance.

It is the factor that absorbs the energy used.

Whatever it is:

• Electrical

• Mechanical

• Acoustics

Mechanical impedance is a measure of the resistance to motion of a structure subjected to a given periodic force. It refers to the velocity forces acting on a mechanical system. The mechanical impedance of a point relative to a structure is the relationship between the force applied at a point and the resulting velocity at that point. Mechanical impedance is the inverse of mechanical admittance or mobility.

Example of a curve resulting from a mechanical impedance test:

F in Newton V in m/s V/F in m/sN, frequency in Hz (Figs. 1.5 and 1.6).

Fig. 1.5

V/F curve as a function of frequency

Fig. 1.6

Vibratory responses

The blue test curve (1) is the geometric sum of the three green (3) and two red curves (2 and 4). These curves correspond to different vibration responses of the tested element.

1.4 Ties Rods

A tie rod has

• A high strength steel bar or strand wire

• A free part and a sealed part

• A cement grout to protect the tie rod and ensure the bond with the soil (Figs. 1.7 and 1.8).

Fig. 1.7

Source Rincent BTP—Recife

Cleaning the head and fixing the geophone 3D.

Fig. 1.8

Example: free length and sealed length

Equipment—Force and Velocity (Fig. 1.9).

Fig. 1.9

Source Rincent BTP—France

Hammer with different ends.

The force sensor is piezoelectric.

Piezoelectricity is the property of certain materials to polarize electrically under the action of a mechanical stress and, conversely, to deform when an electric field is applied.

Equipment measures millivolts as a function of time (Fig. 1.10).

Fig. 1.10

Force versus time

Geophone velocity measurement.

The principle of operation of the geophone is that the movement of the support on which it is fixed causes the magnet in the solenoid to move, which creates an electric current linked to the movement. Without movement there is no current measurement (Fig. 1.11).

Fig. 1.11

Source Rincent BTP—France

Geophone, working principle.

For velocity measurement equipment is the geophone 3D or accelerometer (Figs. 1.12, 1.13 and 1.14).

Fig. 1.12

Velocity as a function of time

Fig. 1.13

Realization of the impact. Source Rincent BTP—Recife

Fig. 1.14

Force and velocity as a function of time test results

Changing from temporal mode of acquisition to frequency mode for analysis of results. The Fourier transform is an operation that allows non-periodic signals to be represented in frequency.

The calculation method that allows you to pass in a reversible way from a function to the corresponding trigonometric series is the Fourier transform. This very fruitful method has become a must-have in signal theory, with important applications for the processing and compression of sound and digital images, JPEG image compression, or the 3G and 4G telephony standards are derived directly.

Fourier Transform.

Any curve can be decomposed into a sum of sinusoidal curves of different frequencies with a different weighting coefficient. This operation allows you to switch from time mode to frequency mode (Figs. 1.15 and 1.16).

Fig. 1.15

Fourier transform

Fig. 1.16

Non-periodic curve and sum of periodic functions

Fourier Transform for Velocity (Fig. 1.17).

Fig. 1.17

Fourier transform for velocity

The next step is to establish V/F as a function of frequency (Fig. 1.18).

Fig. 11.18

V/F curve as a function of frequency

After 2000 Hz the force function of frequency is close to zero. V/F curve shows two parts one before 2000 Hz and one after 2000 Hz (Fig. 1.19).

Fig. 1.19

V/F curve function of frequency filtered curve

Time x Frequency

A short phenomenon in time covers wide frequency ranges. In the case of thunderstorms and lightning, the electrical discharge is a short phenomenon that emits over a wide frequency range, causing disruption to radio and TV signals.

For an earthquake the phenomenon is long in a limited frequency range of 2 to 4 Hz.

A hammer with a metal end will produce in a frequency range of 0 to 5000 Hz, and with a “matte” impact elastomer end from 0 to 2000 Hz.

For us, the latter option we used.

Evaluating the V/F curve as a function of frequency

The first step is to calculate the length of the tested element (Fig. 1.20).

Fig. 1.20

Calculating tie rod length

The value of the plane wave propagation velocity in concrete for calculation is 4000 m/s. This theoretical value is given by the testing standard.

There can be two vibration responses that will correspond to the total part of the tie and the part of soils with low mechanical characteristics found in the first meters of the tie (Fig. 1.21).

Fig. 1.21

Two vibration regimes

Example of the tie rod: Free part and total length of the tie rod (Figs. 1.22 and 1.23).

Fig. 1.22

Tie rod, free part and total tie rod

Fig. 1.23

Analyzing test results

Overall length:

• 23,1m with a velocity assumption of 3500m/s

• 26,0m with a velocity assumption of 4000m/s (Fig. 1.24)

  Fig. 1.24

  

  Same test, second vibration regime

Length free part:

• 8.4m with a velocity assumption of 3500m/s

• 9.5m with a velocity assumption of 4000m/s

Mobility.

The mobility read directly on the y-axis allows you to calculate the diameter of the tested element (Fig. 1.25).

Fig. 1.25

Mobility as a function of diameter of the tested element

V/F = 1 / ρb V b A

Concrete volume mass ρb in kg/m³

Circular cross section (m²)

Dynamic stiffness (Fig. 1.26).

Fig. 1.26

Dynamic stiffness calculation

Dynamic stiffness equals 2πb/a and is a complex number.

A complex number is of the type z = x + iy, X is real, Y is imaginary (Fig. 1.27).

Fig. 1.27

Source Rincent BTP—France

Simultaneous static and dynamic testing.

Simultaneous static testing and non-destructive testing allow the following figure to be drawn (Figs. 1.28 and 1.29).

Fig. 1.28

Dynamic strength and stiffness

Fig. 1.29

Example of a stiffness versus force curve (Tons)

Real and imaginary part of stiffness.

A thesis “Measurements and Modelling of Resilient Rubber Rail-pads” by Klaus Knothe, MinyiYu, Mrs. HeikeIlias shows that stiffness part real and part imaginary is constant before 80 Hz (Fig. 1.30).

Fig. 1.30

Dynamic stiffness real part and imaginary part

Static stiffness.

Static stiffness of an element under a load is the slope of the calculated curve under this load (Fig. 1.31).

Fig. 1.31

Static stiffness under 3000 kN load

Philippe Guillermain's thesis shows that there is a direct link between dynamic stiffness and static stiffness, dynamic stiffness is higher than static stiffness, there is a constant relationship between the two stiffnesses for an identical situation.

Stiffness has the same units as modulus.

The Rd/Rs calculations were done from tests on tie rods, the relationship between dynamic and static stiffness and 31 by following example (Fig. 1.32).

Fig. 1.32

Rd/Rs for different loads (tons)

This rule can be checked for other elements, e.g., micro piles in tension (Fig. 1.33).

Fig. 1.33

Rd/Rs for different loads (kN)