Addendum-02Equations & Calculations My ASNT Level III UT Study Notes2014-June.

Speaker: Fion Zhang2014/July/31

http://en.wikipedia.org/wiki/Greek_alphabet

http://www.mathwarehouse.com/trigonometry/sine-cosine-tangent.php

Trigonometry

Contents:

1. Material Acoustic Properties2. Ultrasonic Formula3. Properties of Acoustic Wave4. Speed of Sound5. Attenuation6. What id dB7. Acoustic Impedance8. Snell’s Law9. S/N Ratio10. Near / Far Field11. Focusing & Focal Length12. Offsetting for Circular Specimen13. Quality “Q” Factors14. Inverse Law & Inverse Square Law

http://www.ndt-ed.org/GeneralResources/Calculator/calculator.htm

45.63.2515.842Steel 4340

45.43.2515.893Steel 1020

41.32.9975.385Steel - SS 400 Series

44.6 3.0485.613Steel - SS 300 Series

3.231.442.74Acrylic resin (Perspex)

Z Acoustic Impedence

Shear wavemm/μs

Logitudinal wavemm/μs

Material

http://www.ndtcalc.com/utvelocity.html

1.0 Material Acoustic Properties

2.0 Ultrasonic Formula

http://www.ndt-ed.org/GeneralResources/Calculator/calculator.htm

Ultrasonic Formula

Ultrasonic Formula

α = Transducer radius

3.0 Properties of Acoustic Plane WaveWavelength, Frequency and Velocity

Among the properties of waves propagating in isotropic solid materials are wavelength, frequency, and velocity. The wavelength is directly proportional to the velocity of the wave and inversely proportional to the frequency of the wave. This relationship is shown by the following equation.

4.0 The Speed of SoundHooke's Law, when used along with Newton's Second Law, can explain a few things about the speed of sound. The speed of sound within a material is a function of the properties of the material and is independent of the amplitude of the sound wave. Newton's Second Law says that the force applied to a particle will be balanced by the particle's mass and the acceleration of the theparticle. Mathematically, Newton's Second Law is written as F = ma. Hooke's Law then says that this force will be balanced by a force in the opposite direction that is dependent on the amount of displacement and the spring constant (F = -kx). Therefore, since the applied force and the restoring force are equal, ma = -kx can be written. The negative sign indicates that the force is in the opposite direction.

F= ma = -kx

What properties of material affect its speed of sound?

Of course, sound does travel at different speeds in different materials. This is because the (1) mass of the atomic particles and the (2) spring constants are different for different materials. The mass of the particles is related to the density of the material, and the spring constant is related to the elastic constants of a material. The general relationship between the speed of sound in a solid and its density and elastic constants is given by the following equation:

Density→ mass of the atomic particles

Eleatic constant→ spring constants

V is the speed of sound

Where V is the speed of sound, C is the elastic constant, and p is the material density. This equation may take a number of different forms depending on the type of wave (longitudinal or shear) and which of the elastic constants that are used. The typical elastic constants of a materials include:

Young's Modulus, E: a proportionality constant between uniaxial stress and strain.

Poisson's Ratio, n: the ratio of radial strain to axial strain Bulk modulus, K: a measure of the incompressibility of a body subjected to

hydrostatic pressure. Shear Modulus, G: also called rigidity, a measure of a substance's

resistance to shear. Lame's Constants, l and m: material constants that are derived from

Young's Modulus and Poisson's Ratio.

E/N/G

5.0 AttenuationThe amplitude change of a decaying plane wave can be expressed as:

In this expression Ao is the unattenuated amplitude of the propagating wave at some location. The amplitude A is the reduced amplitude after the wave has traveled a distance z from that initial location. The quantity α is the attenuation coefficient of the wave traveling in the z-direction. The αdimensions of are nepers/length, where a neper is a dimensionless quantity. The term e is the exponential (or Napier's constant) which is equal to approximately 2.71828.

http://www.ndt.net/article/v04n06/gin_ut2/gin_ut2.htm

Spreading/ Scattering/ adsorption (reflection is a form of scaterring)

Scaterring

Scaterrring

Adsoprtion

Spreading

Attenuation can be determined by evaluating the multiple backwall reflections seen in a typical A-scan display like the one shown in the image at the bottom. The number of decibels between two adjacent signals is measured and this value is divided by the time interval between them. This calculation produces a attenuation coefficient in decibels per unit time Ut. This value can be converted to nepers/length by the following equation.

Where v is the velocity of sound in meters per second and Ut is in decibels per second.

Amplitude at distance Z

Where v is the velocity of sound in meters per second and Ut is in decibels per second (attenuation coefficient).

α is the attenuation coefficient of the wave traveling in the z-direction. The α dimensions of are nepers/length (nepers constant).

Attenuation is generally proportional to the square of sound frequency.Quoted values of attenuation are often given for a single frequency, or an attenuation value averaged over many frequencies may be given. Also, the actual value of the attenuation coefficient for a given material is highly dependent on the way in which the material was manufactured. Thus, quoted values of attenuation only give a rough indication of the attenuation and should not be automatically trusted. Generally, a reliable value of attenuation can only be obtained by determining the attenuation experimentally for the particular material being used.

Attenuation ∝ Frequency2 (f )2

Which Ut?

U1t , A1

o , α1

U0t , A0

o

1 1

7.0 Acoustic ImpedanceSound travels through materials under the influence of sound pressure.

Because molecules or atoms of a solid are bound elastically to one another, the excess pressure results in a wave propagating through the solid.

The acoustic impedance (Z) of a material is defined as the product of its density (p) and acoustic velocity (V).

Z = pVAcoustic impedance is important in:

1. the determination of acoustic transmission and reflection at the boundary of two materials having different acoustic impedances.

2. the design of ultrasonic transducers.3. assessing absorption of sound in a medium.

The following applet can be used to calculate the acoustic impedance for any material, so long as its density (p) and acoustic velocity (V) are known. The applet also shows how a change in the impedance affects the amount of acoustic energy that is reflected and transmitted. The values of the reflected and transmitted energy are the fractional amounts of the total energy incident on the interface. Note that the fractional amount of transmitted sound energy plus the fractional amount of reflected sound energy equals one. The calculation used to arrive at these values will be discussed on the next page.

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/applet_2_6/applet_2_6.htm

Reflection/Transmission Energy as a function of Z

Reflection and Transmission Coefficients (Pressure)

This difference in Z is commonly referred to as the impedance mismatch.

The value produced is known as the reflection coefficient. Multiplying the reflection coefficient by 100 yields the amount of energy reflected as a percentage of the original energy.

the transmission coefficient is calculated by simply subtracting the reflection coefficient from one.

Ipedencemismatch

Reflection coefficient

Using the above applet, note that the energy reflected at a water-stainless steel interface is 0.88 or 88%. The amount of energy transmitted into the second material is 0.12 or 12%. The amount of reflection and transmission energy in dB terms are -1.1 dB and -18.2 dB respectively. The negative sign indicates that individually, the amount of reflected and transmitted energy is smaller than the incident energy.

If reflection and transmission at interfaces is followed through the component, only a small percentage of the original energy makes it back to the transducer, even when loss by attenuation is ignored. For example, consider an immersion inspection of a steel block. The sound energy leaves the transducer, travels through the water, encounters the front surface of the steel, encounters the back surface of the steel and reflects back through the front surface on its way back to the transducer. At the water steel interface (front surface), 12% of the energy is transmitted. At the back surface, 88% of the 12% that made it through the front surface is reflected. This is 10.6% of the intensity of the initial incident wave. As the wave exits the part back through the front surface, only 12% of 10.6 or 1.3% of the original energy is transmitted back to the transducer.

Practice Makes Perfect

Following are the data:

Q1: What is the percentage of initial incident sound wave that will reflected from the water/Aluminum interface when the sound first enter Aluminum?

R= (Z1-Z2)2 / (Z1+Z2)2 = (0.149-1.72)2/(0.149+1.72)2

R= 0.707, Answer= 70.7%

Q2: What is the percentage of sound energy that will finally reenter the water after reflected from the backwall of Aluminum? (Do not consider material attenuation and other factors)Answer: 6%

0.2934

0.706 – initial Back wall

0.2934x 0.706 = 0.207

0.207x 0.2934=0.0609Second Backwall echo

8.0 Snell’s LawSnell's Law holds true for shear waves as well as longitudinal waves and can be written as follows

=

Where:

VL1 is the longitudinal wave velocity in material 1.VL2 is the longitudinal wave velocity in material 2.VS1 is the shear wave velocity in material 1.VS2 is the shear wave velocity in material 2.

Snell’s Law

http://education-portal.com/academy/lesson/refraction-dispersion-definition-snells-law-index-of-refraction.html#lesson

Practice Makes Perfect

5. For an ultrasonic beam with normal incidence, the reflection coefficient is given by:

(a) [(Z1+Z2)2]/[(Z1-Z2)2](b) (Z1+Z2)/(Z1-Z2)(c) [(4) (Z1)(Z2)]/[(Z1+Z2)2](d) [(Z1-Z2)2]/[Z1+Z2)2]

6. For an ultrasonic beam with normal incidence the transmission coefficient is given by:

(a) [(Z1+Z2)2]/[(Z1-Z2)2](b) (Z1+Z2)/(Z1-Z2)(c) [(4) (Z1)(Z2)]/[(Z1+Z2)2](d) [(Z1-Z2)2]/[Z1+Z2)2]

Practice Made Perfect

7. Snell's law is given by which of the following:(a) (Sin A)/(Sin B) = VB/VA(b) (Sin A)/(Sin B) = VA/VB(c) (Sin A)/ VB = V(Sin B)/VA(d) (Sin A)[VA] = (Sin B)[ VB]

8. Snell's law is used to calculate:(a) Angle of beam divergence(b) Angle of diffraction(c) Angle of refraction(d) None of the above

Practice Makes Perfect

9. Calculate the refracted shear wave angle in steel [VS = 0.323cm/microsec] for an incident longitudinal wave of 37.9 degrees in Plexiglas [VL = 0.267cm/ microsec]

(a) 26 degrees(b) 45 degrees(c) 48 degrees(d) 64 degrees

10. Calculate the refracted shear wave angle in steel [VS = 0.323cm/microsec] for an incident longitudinal wave of 45.7 degrees in Plexiglas [VL = 0.267cm/ microsec]

(a) 64 degrees(b) 45.7 degrees(c) 60 degrees(d) 70 degrees

Practice Makes Perfect

11. Calculate the refracted shear wave angle in aluminium [VS = 0.31cm/ microsec] for an incident longitudinal wave of 43.5 degrees in Plexiglas [VL = 0.267cm/microsec]

(a) 53 degrees(b) 61 degrees(c) 42 degrees(d) 68 degrees

12. Calculate the refracted shear wave angle in aluminium [VS = 0.31cm/microsec] for an incident longitudinal wave of 53 degrees in Plexiglas [VL = 0.267cm/microsec]

(a) 53 degrees(b) 61 degrees(c) 42 degrees(d) 68 degrees

9.0 S/N RatioThe following formula relates some of the variables affecting the signal-to-noise ratio (S/N) of a defect:

FOM: Factor of merits at center frequency

The following formula relates some of the variables affecting the signal-to-noise ratio (S/N) of a defect:

Flaw geometry at center frequency: Figure of merit FOM and amplitudes responds

Sound Volume: Area x pulse length Δt

Material properties

10. Near/ Far Fields

http://miac.unibas.ch/PMI/05-UltrasoundImaging.html

where α is the radius of the transducer and λ the wavelength.

For beam edges at null condition K=1.22

Modified Near Zone

Modified ZfT Perspex

Example: Calculate the modified Near Zone for;

• 5 MHz shear wave transducer• 10mm crystal• 10 mm perspex wedge

Perspex L-wave: 2730 m/sSteel S-wave: 3250 m/sSteel L-wave: 5900 m/s

Modified NZ= (0.012 x f) / (4v) – 0.01(2730/3250)=0.0300m= 30mm

Apparent Near Zone distance

11.0 Focusing & Focal Length

http://www.olympus-ims.com/en/ndt-tutorials/flaw-detection/beam-characteristics/

The focal length F is determined by following equation;

Where:

F = Focal Length in waterR = Curvature of the focusing lenn = Ration of L-velocity of epoxy to L-velocity of water

12.0 Offset of Normal probe above circular object

θ1

θ2

θ1R

V1

V2

Calculate the offset for following conditions:

Aluminum rod being examined is 6" diameter, what is the off set needed for (a) 45 refracted shear wave (b) Longitudinal wave to be generated?(L-wave velocity for AL=6.3x105cm/s, T-wave velocity for AL=3.1x105 cm/s, Wave velocity in water=1.5X105 cm/s)

Question (a)

Question (b)

13.0 “Q” Factor

3dB down

14.0 Inverse Law and Inverse Square LawFor a small reflector where the size of reflector is smaller than the beam width, the echoes intensity from the same reflector varies inversely to the square of the distance.

5cm 7.5cm

75% FSH 33% FSH

Inverse Square Law

http://www.cyberphysics.co.uk/general_pages/inverse_square/inverse_square.htm

Inverse Law:

For large reflector, reflector greater than the beam width, e.g. backwall echoes from the same reflector at different depth; the reflected signal amplitude varies inversely with the distance.

10cm 7.5cm

DGS Distance Gain Sizing

Y-axis shows the Gain

X-axis shows the Distance from the probe in # of Near Field

size of reflector is given as a ratio between the size of the disc and the size of the crystal.

– Distance Gain Size is a method of setting sensitivity or assessing the signal from an unknown reflector based on the theoretical response of a flat-bottomed hole reflector perpendicular to the beam axis. (DGS does not size the flaw, but relate it with a equivalent reflector) The DGS system was introduced by Krautkramer in 1958 and is referred to in German as AVG. A schematic of a general DGS diagram is shown in the Figure. The Y-axis shows the Gain and X-axis shows the Distance from the probe. In a general DGS diagram the distance is shown in units of Near Field and the scale is logarithmic to cover a wide range.

The blue curves plotted show how the amplitudes obtained from different sizes of disc shaped reflector (equivalent to a FBH) decrease as the distance between the probe and the reflector increases.

In the general diagram the size of reflector is given as a ratio between the size of the disc and the size of the crystal. The red curve shows the response of a backwall reflection. The ratio of the backwall to the crystal is infinity (∞).

Specific DGS curves for individual probes can be produced and so both the distance axis and the reflector sizes can be in mm.

If the sensitivity for an inspection is specified to be a disc reflector of a given size, the sensitivity can be set by putting the reflection from the backwall of a calibration block or component to the stated %FSH. The gain to be added can be then obtained by the difference on the Y-axis between the backwall curve at the backwall range and the curve of the disc reflector of the given size at the test range. If the ranges of the backwall and the disc reflector are different, then attenuation shall be accounted for separately. Alternatively, the curves can be used to find the size of the disc shaped reflector which would give the same size echo as a response seen in the flaw detector screen.

20dB (measured)

20-4dB=16dB (deduced)

Δ Flaw =30-16=14dB

Data:Probe frequency: 5MHzDiameter: 10mm compression probePlate thickness: 100mm steelDefect depth: 60mm deepGain for flaw to FSH: 30dBBWE at 100mm: 20dB

Example: If you has a signal at a certain depth, you can compare the signal of the flaw to what the back wall echo (BWE) from the same depth and estimate the FBH that would give such a signal at the same depth. The defect can then be size according to a FBH equivalent.

Data:Probe frequency: 5MHzDiameter: 10mm compression probePlate thickness: 100mm steelDefect depth: 60mm deepGain for flaw to FSH: 30dBBWE at 100mm: 20dB-------------------------------------------------------------------------Near field: 21mm, flaw location= 3xNear FieldFrom the chart BWE at 60mm will be 20-4dB=16dBFlaw signal Gain is 30dB-16dB= 14dBUsed the flaw signal Gain and locate the equivalent reflector size is between 0.4 to 0.48 of the probe diameter, say 0.44 x10mm = 4.4mm equivalent reflector size.

http://www.olympus-ims.com/en/atlas/dgs/

More on DGS/AVG by Olympushttp://www.olympus-ims.com/en/ndt-tutorials/flaw-detection/dgs-avg/

DGS is a sizing technique that relates the amplitude of the echo from a reflector to that of a flat bottom hole at the same depth or distance. This is known as Equivalent Reflector Size or ERS. DGS is an acronym for Distance-Gain-Size and is also known as AVG from its German name, AbstandVerstarkung Grosse. Traditionally this technique involved manually comparing echo amplitudes with printed curves, however contemporary digital flaw detectors can draw the curves following a calibration routine and automatically calculate the ERS of a gated peak. The generated curves are derived from the calculated beam spreading pattern of a given transducer, based on its frequency and element diameter using a single calibration point. Material attenuation and coupling variation in the calibration block and test specimen can be accounted for.

DGS is a primarily mathematical technique originally based on the ratio of a circular probe’s calculated beam profile and measurable material properties to circular disk reflectors. The technique has since been further applied to square element and even dual element probes, although for the latter, curve sets are empirically derived. It is always up to the user to determine how the resultant DGS calculations relate to actual flaws in real test pieces.

An example of a typical DGS curve set is seen below. The uppermost curve represents the relative amplitude of the echo from a flat plate reflector in decibels, plotted at various distances from the transducer, and the curves below represent the relative amplitude of echoes from progressively smaller disk reflectors over the same distance scale.

As implemented in contemporary digital flaw detectors, DGS curves are typically plotted based on a reference calibration off a known target such as a backwall reflector or a flat bottom hole at a given depth. From that one calibration point, an entire curve set can be drawn based on probe and material characteristics. Rather than plotting the entire curve set, instruments will typically display one curve based on a selected reflector size (registration level) that can be adjusted by the user.

In the example below, the upper curve represents the DGS plot for a 2 mm disk reflector at depths from 10 mm to 50 mm. The lower curve is a reference that has been plotted 6 dB lower. In the screen at left (figure 1), the red gate marks the reflection from a 2 mm diameter flat bottom hole at approximately 20 mm depth. Since this reflector equals the selected registration level, the peak matches the curve at that depth. In the screen at right (Figure 2), a different reflector at a depth of approximately 26 mm has been gated. Based on its height and depth in relation to the curve the instrument calculated an ERS of 1.5 mm.

Figure1:

Figure2:

15.0 Pulse Repetitive Frequency/Rate and Maximum Testable Thickness

Clock interval = 1/PRR

Maximum testable length = ½ x Velocity x Clock interval

Note: The Clock interval has neglected the time occupied by each pulse.

16.0 Immersion Testing of Circular Rod

Q4-12 Answer:First calculate the principle offset d; ϴ = Sin-1(1483/3250 xSin45)=18.8 °d=R.Sin18.8= 0.323 (Assume R=1).Wobbling ±10%; d’=0.355 ~ 0.290d’=0.355, ϴ = Sin-1(0.355)=20.8 °giving inspection Φ = Sin-1(3250/1483xSin20.8)=51, 13.3% above 45 °d’=0.290, ϴ = Sin-1(0.290)=16.9 °giving inspection Φ = Sin-1(3250/1483xSin16.9)=39.6, 12% below 45 °

Maximum ϴ

ϴ max = Sin-1 (ID/OD)