Sound and Vibration Measurements of Metal Plates

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Sound and Vibration Measurements of Metal Plates

Forooz Shahbazi

This thesis is presented as part of Degree of Master of Science in Electrical Engineering

Blekinge Institute of Technology February 2009

Blekinge Institute of Technology School of Engineering

Department of Signal Processing Supervisor: Tatiana Smirnova Examiner: Lars Håkansson

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Abstract

The purpose in this project is to classify different metal plates according to their size and material. The plates which are studied are small plates with the maximum size of: radius=25.75 mm and thickness=2.20 mm. The quantities of interest are the natural frequencies of the plate which may be estimated experimentally based on the plate's frequency response function (FRF) estimate produced using the measured input force signal applied to the plate which had been recorded simultaneously with the output vibrations of the plate. The small size of the studied plates makes the measurements different from the ordinary sound and vibration measurements.

The difference was basically in the sensor size and excitation method due to the small size of the studied objects. As in the case of small plates, it was necessary to find a suitable excitation method which could excite the natural frequencies of the plates.

Plates different in geometrical dimensions as well as having different material properties are likely to be distinguished by their natural frequencies. Plate vibrations were excited using two methods: impact excitation by means of an impulse hammer and shaker excitation.

In the studied structures in this project, the plate is excited by a random or transient input signal.

The vibrations of the plate are measured using an accelerometer or a laser vibrometer.

Repeatability of the measurements was one of the important interests which we tried to fulfill.

But the first and most important goal was exciting and measuring the natural frequencies of the plate apart from the natural frequencies of the other components of the system. In most of the structures it was difficult to excite the natural frequencies of the plates due to the small size of them. To solve this problem, different boundary conditions of the plates are studied.

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Acknowledgment

I would like to express my gratitude to all those who gave me the possibility to complete this thesis. I want to thank the Department of Signal Processing for giving me this possibility to work on this project and to use equipments and facilities for my measurements. I am deeply indebted to my supervisor Dr. Lars Håkansson whose help, suggestions and encouragement helped me in all the time of research for and writing of this thesis. I have furthermore to thank my other supervisor Tatiana Smirnova who really supported me in all the steps of the project. I am also thankful to Istvan Gersner for his supervision.

I also want to thank Mansoor Mojtahedi for his kind support during the project in accessing the research lab and using the equipments.

I would like to give my special thanks to my parents who always have supported and encouraged me with their love. I dedicate this final step in my Master degree to them, who made it possible for me to study and achieve my goals with their constant and kind support.

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Chapter 1 Introduction

Characteristics of an object can be useful in detecting a specific object or in classifying different groups of them. In this project, the metal plates which are very small in size are studied to be classified according to their size and material based on a nondestructive method.

One possible method is the analysis of metal plates by the X-ray analysis data which typically bounce radiation or subatomic particles off the surface of a plate to produce an x-ray signature, with each element that makes up the plate having its own distinct signature. In this way, one is able to determine the percentage of each element that makes up the plate. But this method is very expensive and still not infallible [13].

Another method in clustering the metal plates is based on the electrical conductivity of the plates. In this method, a plate discriminator measures both the surface and average electrical conductivity of plates in order to classify them or distinguish between the plates of a predetermined type. The conductivities are measured using a coil to induce eddy currents within the plate. The high frequency components of the eddy current are monitored to measure the surface conductivity. The low frequency components are measured to monitor the bulk or average conductivity. And by comparing the conductivities of the plates under the test and those of the predetermined plates, one can classify them [10].

Another plate detector circuit which is discussed here comprises a constant frequency and voltage maxima and minima square wave oscillator input to a tank circuit including an inductive probe and an adjustable capacitance. The tank circuit is driven at a fixed and stable frequency and has variable peak to peak AC output voltage amplitude which is a function of the characteristics of the plate under test. The AC output of the tank circuit is converted to a DC voltage correlated to the peak to peak AC output from the tank circuit. A group of window detector circuits are employed to determine whether the DC voltage output falls within certain preset ranges corresponding to the presence of particular plates. The presence or absence of a window detect can be used to accept or reject the plate or to otherwise take appropriate action with respect to the plate under test [16].

The purpose in this project is to classify metal plates based on their sound and vibration characteristics based on e.g. experimentally estimated frequency response functions.

Different plates have different dynamic characteristics according to their size and material. Based on this fact, the plates being excited by an input excitation force are expected to have special natural frequencies. But the problem is with the size of these plates as they are small in thickness and radius.

The maximum radius in this project is r= 25.75mm and the maximum thickness is d= 2.20 mm, which makes the measurements more complicated in the sense of excitation. Efficient methods are necessary to excite the natural frequencies of the small and light plates. Due to the small size, the usual excitation methods may end in rigid body motion of the plate only.

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Page 9 of 38 In different steps of this project the plate is excited using hammer or shaker in different boundary conditions trying to measure the valid natural frequencies. And the vibrations are measured using an accelerometer or a laser vibrometer.

The signals which are measured from the vibrations of the plate are processed using different mathematical and signal processing tools which are explained further.

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Chapter 2 Materials and Methods

Dynamic properties of the metal plates were investigated based on the experimental estimates of the frequency response function produced based on the excitation force signal and the plate's response acceleration recorded simultaneously.

The following equipment was used to experimentally estimate plate's frequency response function 1. Signal analyser, HP 35670A

2. Electrodynamic shaker, Gearing& Watson v4 3. Impulse Force Hammer, Kistler 9722A500 4. Charge Accelerometer, B&K Type 4374 5. Laser vibrometer, VS 100

6. PC with I-DEAS 10 NX Series

7. Hewlett Packard VXI Mainframe E8408A.

8. Hewlett Packard E1432A4-16 Channel l51.2kSa/sDigitizer.

9. Power amplifier PA 30/2, Gearing & Watson Electronics LTD

Excitation or identification signals was produced by the signal source of the HP 35670A and was input signal to the Gearing & Watson Power amplifier which in turn powered the Gearing& Watson Electronics shaker v4. Also impact excitation was investigated and a Kistler 9722A500 Impulse Force Hammer was used as the impact excitation source. In the different experimental setups response measurements were carried out using either a Bruel&Kjaer Charge Accelerometer Type 4374 or a laser vibrometer of type VS 100. In fact, the excitation signal and the vibration sensor output signal were connected to two of the input channels of the 16-channel digitizer in the HP Mainframe. In all of the setups; the measured signals were analyzed using IDEAS.

The main purpose of investigating different setups and excitation methods and signals is to find a method that enables measurement of dynamic properties of the metal plates, e.g. parts of their modal properties. Different excitation methods and boundary conditions of the plate are considered in the different experimental structures and setups.

In the first setup which is described in details in the section that follows, the metal plate is suspended in a string hanging vertically to simulate the free boundary condition of the circular plates. The plate is excited using both a hammer and a shaker and the vibration signals are measured using the charge accelerometer and a laser vibrometer. The details are described in sections 2.1.1,2.1.2 and 2.1.3.

In the next setup, the boundary conditions of the plate are changed. The plate is fixed from two sides using two needles, and excited by the shaker at the same time. This setup is described in detail in section 2.1.2.

Another setup which is studied is measuring the vibrations of the plate, while it is suspended at 3 points using 3 elastic ropes. This setup is also described in detail in 2.1.4.

And in the last setup, the plate is fixed from two sides, but the fixing points are thicker than the needles which were tried before. This structure is described in details in 2.1.5.

The characteristics of the plate which the measurements were carried out on are as follows:

Diameter : 0.025 m

Thickness h=0.0019 m Material: Nickel Radius: a=0.025/2=0.0125 m

Poisson’s ratio (Nickel steel) v=0.291

γ = mass per unit area of plate( hρ ), density of the plate 8.092 10³ ₃ m

× Kg

ρ =

E: Modulus of elasticity= 1.8961*10¹¹ (Pa) at 26.7 degrees Celsius

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2.1 Experimental Setups

2.1.1 F ree boundary condition and charge accelerometer

The first experimental set-up which is implemented in this project is the structure in which the plate has the free boundary condition. To have the free vibration boundary conditions of the plate, it is suspended as shown in the figure 2.1 using a rope. The Accelerometer is attached on the plate using wax to measure the vibrations. In this case, transient hammer excitation force is used.

Figure 2.1 The suspended plate using the setup free boundary condition and charge accelerometer.

2.1.2 Clamped boundary conditions using needles

In this structure, the plate is suspended using a rope and the accelerometer is attached on the plate.

The excitation force is the random noise.

Due to the small size and thickness of the plates in comparison to the surface of the shaker, it was not possible to attach the plate directly to the shaker, as we do in most of the vibration measurements dealing with substantially larger structures.

Attaching a plate directly to the shaker’s load table is not a good strategy for exciting the eigen modes of a plate.

A solution to this problem was to excite two small points on the plate. These two small points are the tips of two needles which touch the surface of the plate from two sides. The set-up is shown in the figure 2.2. The frequency range of interest in this setup is 0-10 kHz.

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Figure 2.2: The plate in the clamped boundary conditions using needles.

2.1.3 Free boundary condition and laser vibrometer

Due to the small geometric dimensions and light weight of the plate, measurements of the plate's response using an accelerometer are likely to have a non negligible influence on the dynamic properties of the plate. E.g. it may result in the mass loading of the plate and the measured resonance frequencies will be shifted in frequency compared to the actual resonance frequencies of the plat without mass load.

Also, the wax between the plate and the accelerometer may be viewed as a spring which may introduce an extra resonance frequency and a 180 degrees phase shift above it in the measured response. To avoid such problems, the vibration of the plate was measured using a laser vibrometer.

A Laser Doppler Vibrometer (LDV) is a scientific instrument that is used to make non-contact vibration measurements of a surface. The laser beam from the LDV is directed at the surface of interest, and the vibration is extracted from the Doppler shift of the laser beam frequency due to the motion of the surface. The output of an LDV is generally a continuous analog voltage that is directly proportional to the target velocity component along the direction of the laser beam.

One measurement with this setup is done by hanging the plate using a rope, exciting it using a hammer and measuring the velocity by means of the Laser Vibrometer. Figure 2.3 shows this structure.

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Figure 2.3: The laser vibrometer and the suspended plate representing the free boundary condition and laser vibrometer.

2.1.4 Elastic ropes boundary conditions

In this setup plate is supported using 3 elastic ropes each with approximately equal distance and angle from the other two ropes (figure 2.4). The plate is excited using the impact hammer. The velocity of the plate is measured using the laser vibrometer.

Before measuring the vibration of the plate due to the excitation force, the rigid body motion of the plate is also measured. By doing that, it would be possible to measure the low frequency rigid body motion of the plates.

To do the rigid body vibration measurement, the plate is not excited by a hammer but just stretched a little (2-3 mm) by hand and then its vibration is measured by a laser vibrometer.

Figure 2.4: The elastic ropes boundary condition of the plate.

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2.1.5 Adjustable Structure and Laser Vibrometer:

Another setup that was investigated is an adjustable structure which can be used for plates with different radiuses. This setup is shown in figure 2.5. The plate is fixed between two pieces of aluminium with the dimensions given in the figure 2.6. And is excited using a hammer. The velocity is measured by the use of the laser vibrometer.

Figure 2.5: The adjustable structure and laser vibrometer.

Figure 2.6: Cross section and front view of the adjustable structure used in the adjustable structure and laser vibrometer setup, see section 2.1.5.

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2.3 Vibration of Plates

Dynamic properties of a system may be estimated based on spectrum estimates of the system's response and excitation signals. Therefore, selection of an appropriate scaling of the spectrum estimator is of great importance [5, 15].

The spectrum estimates may be scaled either for the tonal components of a signal –Power Spectrum (PS) estimates or the random part of a signal –Power Spectral Density (PSD) estimates [1].

In the following sections, a brief introduction to different estimators of power and energy of signals is presented.

2.3.1 Power Spectra (PS)

In this section signals sampled in time domain or discrete time signals and their corresponding power spectrum will be discussed. To describe a periodic signal, either a linear spectrum or a power spectrum is used. The linear spectrum of periodic signals is shown in Figure 2.7. A periodic continuous time signal with period T₀has the property that

) ( )

(t X t T₀

X = + (2.1) Then according to Fourier series, X (t) can be defined as

∑

^∞

∑

−∞

=

∞

−∞

=

k k

T t jk k t

jkw

ke C e

C t

X ⁰ ⁰

2

) (

π

(2.2)

Power Spectrum

A ---

…………. Frequency

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Page 16 of 38 Figure 2.7: Single Sided Power Spectrum of a periodic signal. This spectrum contains only the discrete frequencies , …etc. where is the signal period.

Where A is

A=2* (2.3) Thus, assume that we have carried out measurements of vibration x(t), the sampled version of this measured signal may be written

) ( ) ( )

(n w n x n

x_m = (2.4) Where w(n)is a rectangular window, defined as



 ≤ ≤ −

= otherwise N n n

w 0,

1 0

, ) 1

( (2.5)

And after some calculations, the single sided power spectrum for the measured signal x_m(n)is [1]

{ }

∑

₌

= ^M

m PS w

x

x DFT x n w n

N A k M

P m 1

2

) ( ) 2 (

)

( (2.6)

∑

₌⁻

= 1

0 ( )

N n

w w n

A N

Where, M is the number of segments and ^DFT

{}

^. stands for the discrete Fourier transform.

2.3.2 Power Spectral Density (PSD)

As opposed to periodic signals, random signals have continuous spectra. In this case they contain all the frequencies in the range of measurement and not only at discrete frequencies. One can obtain power of the signal within a specific frequency range by integrating PSD within that frequency range. Computation of PSD is done directly by FFT method or computing autocorrelation function and then FFT transforming it. The autocorrelation function for a stochastic ergodic time signal is defined as[1,4]

[

⁽ ⁾ ⁽ ⁾

]

)

(τ =E x t x^* t+τ

R_xx (2.7) The autocorrelation of a signal shows the measure of dependence of the signal with a time-shifted version of itself (shifted by timeτ . )

Then the two sided power spectral density of a weakly stationary signal is defined as [4]

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∞

∫

∞

−

= R τ e− ^π^τdτ f

S_xx( ) _xx( ) ^j² ^f (2.8)

The power spectral density (PSD) of a sampled stationary random signal x(n) may be defined as the discrete time Fourier transform of the autocorrelation function ^R_xx⁽^k⁾⁼^E

[

^x⁽ⁿ⁺^k⁾^x⁽ⁿ⁾

]

^,i.e.[4]

∑

^∞

∞

−

= _xx −^j ^FK

xx F R k e

S ( ) ( ) ²^π (2.9)

The Welch spectrum estimate is obtained by averaging a number of periodograms. Each periodogram is based on segment of a time sequence x(n), each segment consisting of N samples.

Thus the original time sequence of data must be divided into data segments as [12]





=

− + =

= l M

N where n

lD n x n x_l

,...., 1

1 ,...., 1 , ) 0

( )

( (2.10)

Where lD is the starting point for each periodogram and D is is the overlapping increment. After doing some calculations the PSD Welch power spectral density estimator ( )

^

k PSD

xx f

P is given by [12]

∑ ∑ ∑

⁼

−

=

−

=

≤

<

=

M

l

s k

N

n

N nk j N l

n s

k PSD xx

N F f k N k e

n w n x n

w M F

f P

1

2 1

0

/ 2 1

0

2

^

, 2 / 0

,

| )

( ) (

| )) ( ( 2

) (

π (2.11)

∑ ∑ ∑

⁼

−

=

−

=

M

l

s k

N

n

N nk j N l

n s

k PSD xx

N F f k k e

n w n x n

w M F

f P

1

2 1

0

/ 2 1

0

2

^

, 0 ,

| )

( ) (

| )) ( ( 1

) (

π

Where M is the number of periodograms, N is the length of the periodogram, F_s is the sampling frequency.

2.3.3 Energy Spectral Density (ESD)

As opposed to random signals, transient signals do not continue indefinitely. It is therefore not possible to scale their spectra to represent the power in the signal. Instead spectrum of transient signals is scaled to represent the energy distributed over frequencies and is called Energy Spectral Density (ESD). The spectrum of a transient signal is also continuous as in the case of random signals. When estimating a ESD of a signal the signal has generally a limit time duration and frequently it is not required to use a time window [4, 1].

) ( .

)

( PSDxx _k

k ESD

xx f T P f

P = Where T =NT_s (2.12)

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∑ ∑ ∑

⁼

−

=

−

=

≤

<

=

M

l

s k

N

n

N nk j N l

n s

k ESD xx

N F f k N k e

n w n x T

n w M F

f P

1

2 1

0

/ 2 1

0

2

^

, 2 / 0

,

| )

( ) (

| )) ( ( 2

) (

π (2.13)

∑ ∑ ∑

⁼

−

=

−

=

≤

<

=

M

l

s k

N

n

N nk j N l

n s

k ESD xx

N F f k N k e

n w n x T

n w M F

f P

1

2 1

0

/ 2 1

0

2

^

, 2 / 0

,

| )

( ) (

| )) ( ( 1

) (

π

2.4 Frequency Response Function (FRF)

The frequency response is defined as the ratio between the spectrum of the output (response) and the spectrum of the input (force). In noise and vibration analysis it is common that different types of frequency response are measured experimentally. In some cases, frequency response functions between signals that are naturally occurring are measured, for example when measuring transmissibility, where the relevant signals are random. In more common cases, frequency response functions between an applied force and the resulting response are measured, which can then be used to identify the mechanical properties of the structure under test [4].

Frequency response functions are complex functions, with real and imaginary components. They may also be represented in terms of magnitude and phase.

2.4.1 Experimental estimation of a Frequency Response Function

We shall now see how the frequency response can be estimated experimentally, under the assumption that the systemH( f) is linear and time-invariant.The frequency response function for a linear time-invariant system can be estimated experimentally based on the simultaneous measurement records of the excitation force and response signals. It was assumed that noise was corrupting only measurements of the response signal. The block diagram of the system with the noise only on the output is shown in Figure 2.9. It is common to use H1 estimator in this case [4].

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Page 19 of 38 Figure 2.9. Linear system with noise only at the output.

) (

) ) (

(

^

1

f P

f f P

H

xx

= yx (2.14)

Where ( )

^

f P_xx and

^

) ( f

P_yx are respectively auto-power spectral density estimate of the excitation force signal and cross-power spectral density estimate between the excitation force signal and response i.e. acceleration or velocity signal. For transient signals, these quantities may be replaced with corresponding energy spectral densities to form a frequency response function estimate.

Equation (2.14) is the formula used for FFT analyzers for estimating H (f) from measurements of x (t) and y (t) if it is assumed that the contaminating noise in the input signal is negligible [4]. We use the symbol ^ (hat) to denote that we are dealing with estimated functions. This experimental estimation of FRF was used in the measurements in this project.

2.4.2 Time Windows

Depending on the class of the signal, different time windows are used in the spectrum analysis.

Flattop window is the best choice for periodic signal [4]. In the case of random signals, Hanning window is recommended [4]. And in the case of transient signals either no window or an exponential window may be selected [4].

2.4.3 Coherence Function

Coherence directly yields a measure on the quality of an estimated FRF and is described by

^

^2

) ( ) (

) ( )

(

f P f P

f P f

xx yy yx

yx =

γ (2.15)

Where ( )

^

f P_xx ,

^

) ( f P_yy and

^

) ( f

P_yx are respectively auto-power spectral density estimate of the excitation force signal, auto-power spectral density estimate of the response i.e. acceleration or velocity signal and cross-power spectral density estimate between the excitation force signal and response (acceleration or velocity) signal. For transient signals, these quantities may be replaced with corresponding energy spectral densities to form a coherence function estimate.

Coherence has a value between zero and one and if it is one, then the measurement is not affected by noise and the output signal can be explained linearly from the input signal.

2. 5 Calculation of natural frequencies based on the distributed-parameter model of a plate

The classical differential equation of motion for the transverse displacement w of a plate in the z-

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Page 20 of 38 direction (the direction normal to the plate surface) is given by [2]:

) 0 , , ) (

, ,

( 2

2

4 =

∂ + ∂

∇ t

t y x t w

y x w

D γ (2.16) Here x,yare spatial coordinates.

Where D is flexural rigidity and is defined by [2]

) 1 (

12 ²

3

ν

= Eh−

D (2.17)

E is Young’s modulus; h is the plate thickness,ν is the Poisson’s ratio, γ is mass density per unit area of the plate and∇⁴ =∇²∇², where ∇²is the Laplacian operator.

For circular plates the free plate vibration in terms of polar coordinates may be written [2]:

t r

W t r

w( ,θ, )= ( ,θ)cosω (2.18) Where, ω is the circular angular frequency(ω =2πf )and W(r,θ) is a function of polar coordinates in the r−θ plane, ris the radius from the centre of the plate and θ is the angle of that radius.

Substituting equation (2.18) in equation (2.16) results in

(∇⁴ −k⁴)W(r,θ)=0 (2.19) Where k is defined as [2]

k D

2

4 γω

= and ² ² k ²a²

a D _ij

ij = ij γ =

ω

λ (2.20) Here a is the radius of the plate. The natural frequencies of the plate may now be calculated as [2]

2 1

2 3 2

2 2 1

2 2 3

) 1 ( 12 2

) 1 ( 12

2 ^_

 





= −



 





= −

ν γ π

λ ν

π γ

Eh a

k Eh

f_ij ^ij ^ij (2.21)

Chapter 3 Results

3.1 Theoretical Calculations of Natural Frequencies for a Plate

Table 1 represents values of λij 2

corresponding to values of i and j where i is the number of nodal diameters and j is the number of nodal circles, not counting the boundary [2].

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j i 0 1 2 3

0 * * 0.0053*10³ 0.0122*10³

1 9.0840 0.0205*10³ 0.0352*10³ 0.0529*10³

2 38.5500 0.0598*10³ 0.0839*10³ 0.1113*10³

3 0.0878*10³ 0.119*10³ 0.1540*10³ 0.1922*10³

Table 1: Values of λ_ij²[2]

The natural frequencies for two plates were calculated based on the equation 2.21. Natural frequency estimates for one of the plates were calculated using the plate dimensions and material properties given in section 2. For the other plate the natural frequency estimates were calculated using the plate material properties and plate diameter given in section 2 however with the plate thickness, h=0.0015 m.

In table 2 natural frequency estimates for a plate with the characteristics given in section 2 and in table 3 natural frequency estimates for a plate with the plate material properties and plate diameter given in section 2 but with plate thickness, h=0.0015 m. The expressions for the corresponding mode shapes can be found in [2].

j i 0 1 2 3

0 * * 0.1414*10⁵ 0.3291*10⁵

1 0.2444*10⁵ 0.5522*10⁵ 0.9486*10⁵ 1.4238*10⁵

2 1.0374*10⁵ 1.6108*10⁵ 2.2577*10⁵ 2.9951*10⁵

3 2.3627*10⁵ 3.2023*10⁵ 4.1441*10⁵ 5.1721*10⁵

Table 2: Natural frequencies f_ij in Hz for a plate with the characteristics given in section 2.

j i 0 1 2 3

0 * * 0.1087*10⁵ 0.2530*10⁵

1 0.1879*10⁵ 0.4245*10⁵ 0.7293*10⁵ 1.0946*10⁵

2 0.7975*10⁵ 1.2384*10⁵ 1.7357*10⁵ 2.3026*10⁵

3 1.8164*10⁵ 2.4619*10⁵ 3.1859*10⁵ 3.9742*10⁵

Table 3: Natural frequencies f_ijin Hz for a plate with the plate material properties and plate diameter given in section 2 but with plate thickness, h=0.0015 m.

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Page 22 of 38 Figure 3.1: Mode shape corresponding to natural frequency f₂₀.

Figure 3.2: Mode shape corresponding to natural frequency f₀₁.

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Page 23 of 38 The theoretical calculations are followed by real measurements. It is expected that the estimates of natural frequencies for the plates will indicate where approximately in frequency the actual natural frequencies of the plates are.

3.2 Experimental Results Concerning Natural Frequencies for a Plate

A number of different experimental setups have been investigated for vibration measurements of circular plates, see section 2.1. The aim has been to measure plate vibration enabling identification natural frequencies of circular plates. A limited and representative selection of the results is given in this section.

In Fig. 3.3 Energy Spectral Densities of plate acceleration measured on three identical plates of version 2 are shown. The setup with free boundary condition, accelerometer and impact hammer (see section 2.1.1.) was utilized.

Figure 3.3: Energy Spectral Density of plate acceleration measured on three identical plates of version 2. Free boundary condition, accelerometer and impact hammer, see section 2.1.1.

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Page 24 of 38 For the same setup accelerance function estimates for three identical plates of version 2 are shown in Figs. 3.4 and 3.5. Observe, the peaks close 18 kHz in both the Energy Spectral Densities of plate acceleration in Fig. 3.3 and in the accelerance magnitude functions in Fig. 3.5 b).

Figure 3.4: Accelerance function estimates for identical plates of version 2, a) real part of accelerance and in b) imaginary part of accelerance. Free boundary condition, accelerometer and impact hammer, see section 2.1.1.

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Page 25 of 38 Figure 3.5: Accelerance function estimates for three identical plates of version 2, a) magnitude function and in b) phase function. Free boundary condition, accelerometer and impact hammer, see section 2.1.1.

In Fig. 3.6 typical Energy Spectral Density of the hammer excitation force when exciting plates is shown.

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Page 26 of 38 Figure 3.6: Energy Spectral Density of the hammer excitation force measured for three identical plates of version reference. Free boundary condition, accelerometer and impact hammer, see section 2.1.1.

In Fig. 3.7 Energy Spectral Densities of plate acceleration measured on three identical plates of the reference version are shown. The setup with free boundary condition, accelerometer and impact hammer ( see section 2.1.1. ) was utilized.

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Page 27 of 38 Figure 3.7: Energy Spectral Density of plate acceleration measured on three identical reference plates.

Free boundary conditions, accelerometer and impact hammer, see section 2.1.1.

For the same setup accelerance function estimates for three identical plates of reference version are shown in Figs. 3.8 and 3.9. Observe, the peaks close 18 kHz in both the Energy Spectral Densities of plate acceleration in Fig. 3.7 and in the accelerance magnitude functions in Fig. 3.9 b). Also, peaks close to 13 kHz may be observed in these figures_.

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Page 28 of 38 Figure 3.8: Accelerance function estimates for identical plates of version reference, a) real part of accelerance and in b) imaginary part of accelerance. Free boundary condition, accelerometer and impact hammer, see section 2.1.1.

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Page 29 of 38 Figure 3.9: Accelerance function estimates for three identical plates of version reference, a) magnitude function and in b) phase function. Free boundary condition, accelerometer and impact hammer, see section 2.1.1.

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Page 30 of 38

Chapter 4 Summary and Conclusion

This project intended to investigate the possibility to experimentally identify natural frequencies of small thin circular metal plates. The studied metal plate’s dimensions are substantially smaller than objects usually studied using traditional experimental vibration analysis. The dynamic properties of a plate are related to its physical dimensions, the material it’s made of, the clamping conditions of it, etc. For instance, two circular plates with different diameters with identical thickness, material and clamping conditions would not have the same dynamic properties. Theoretically, plates with both different geometrical properties and material properties are likely to have different sets of natural frequencies. Information concerning plates natural frequencies may for instance enable to classify different plates with identical clamping conditions into different groups. From the theoretical plate model it followed that the lowest natural frequency of the plate was estimated at approx. 11 kHz.

This is a challenge for the available measurement equipment, particularly the excitation sources available. They may usually be considered for measurements up to 10 kHz. Basically the usable frequency range of the impact hammer is exceeded. Thus, energy in the impulse force exciting the plate over 10 kHz will be low and a low signal-to-noise ratio of force sensor signal may also be a challenge. In the measurements of the plate response both a miniature accelerometer and a laser vibrometer were utilized. Using an accelerometer will cause the measured dynamic properties of the plate to differ with the actual dynamic properties of the plate without an accelerometer attached to it.

For instance, the size and weight of the accelerometer may cause some changes in resonance frequencies measured as compared to the actual resonance frequencies of the plate without an accelerometer attached to it. Another challenge when using an accelerometer at high frequencies is the mounting of it. In the experiments wax was used as mounting method and is usually only recommended up to 10 kHz. Using wax to mount an accelerometer introduces a “spring” connection between the plate and the accelerometer and this accelerometer spring system has a resonance frequency that might be substantially lower than the so called mounted resonance frequency of the accelerometer given its specifications. By using a laser vibrometer contact free vibration measurement of the plate response is obtained. The laser will measure the vibration velocity of the plate in the light beam direction. The plate that has to stay in focus by the vibrometer to provide high signal-to-noise ratio in the measured vibration signal as well as the vibration has to be within the selected velocity and frequency ranges. Furthermore, vibration velocity will decay with 6 dB per octave compared to the corresponding vibration acceleration.

In this case the usage of hammer excitation provide broadband excitation of the plate and high vibration levels below eigenfrequencies of the plate in the plate - plate suspension system are likely to cause low signal-to-noise ratio in the measured vibration signal the high frequency range over 10 kHz.

Thus, a significant challenge in this project has been to find an experimental setup enabling reliable and robust excitation and vibration measurement of small circular plates. Different plate boundary conditions have been investigated in the different setups and a substantial amount of measurements have been carried out.

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Page 31 of 38 Two natural frequencies f20 and f01 of the investigated plates seems to be observable in the experimental results (see Fig. 3.9) that may be correlated to the theoretical plate model. Variations in frequency of the natural frequency f₀₁ for the reference plates are observable (see Fig. 3.9). This may also be observed for the plates of version 2 (see Fig. 3.5). However, is the observed variation in natural frequency between the plates because of variation in dimensions and/or material properties of the plates or because of inconsistencies between the measurements, i.e. different accelerometer positions, different boundary conditions, etc.

Future work might be to develop an experimental setup that provides minimal variation in boundary conditions of a mounted plate and to design an excitation source that provides controllable excitation energy in the frequency range between e.g. 10 - 20 kHz. It should be suitable for small plates as well as it should introduce negligible influence on the plate’s dynamic properties. During such circumstances laser vibrometer measurements of plate response may be adequate. Also an accelerometer with higher sensitivity and is designed to measure acceleration of

“tiny” structures should be utilized. Methods for easy mounting of accelerometer on a plate providing repeatable similar influence on a plate’s dynamic properties should be developed.

.

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Page 32 of 38

Appendix

IDEAS Setups in 2.1.1

Transducers Sensitivity

Hammer: 10 mv/ EU Accelerometer

:

1 mv/ EU

Data Type

Hammer: Excitation Force Accelerometer

:

Acceleration

Overall Setup Sampling

Spectral Lines: 6401

Frequency Range: 0- 10 KHZ

Trigger

Method: Every Frame

Time Window

Reference Response: No Window

Averaging

Frames per average

: 5

Frame Acceptance: Manual

Measurements

Acquisition Results: FRF

Normalization: Units Square Sec/HZ (ESD)

IDEAS Setups in 2.1.4

Transducers Sensitivity

Shaker: 1 mv/ EU

Accelerometer

:

1 mv/ EU

Data Type

Shaker: Excitation Force

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Page 33 of 38 Accelerometer

:

Acceleration

Overall Setup Sampling

Spectral Lines 6401

Frequency Range: 0- 10 KHZ

Trigger

Method: Free Run

Window

Reference Response: Hanning Broad

Averaging

Frames per average

: 30

Frame Acceptance: Accept all

Measurements

Acquisition Results: FRF

Normalization: Units Square /HZ (PSD)

IDEAS Setups in 2.1.2, 2.1.3 and 2.1.5

Transducers :

Sensitivity

Hammer: 10 mv/ EU

Laser vibrometer

:

100 mv/ EU

Data Type

Hammer: Excitation Force Laser vibrometer

:

Velocity

Overall Setup Sampling

Spectral Lines: 3201

Frequency Range: 0- 1 KHZ Sampling Frequency: 2560 Hz

Trigger

Method: Every Frame

Method: Free run (for the rigid body motion measurements)

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Page 34 of 38

Window

Reference Response: Exponential Decay Window

Averaging

Frames per average

: 5

Frame Acceptance: Manual

Measurements

Acquisition Results: FRF

Normalization: Units Square Sec/HZ

Laser Vibrometer Setup AGC/LIM

Selection of ‘’AGC’’ is recommended for measurements on fixed structures executing low level of vibration. AGC (Automatic Gain Control) acts on the Doppler signals to provide more consistent, linear operation of the electronic circuits inside VS 100 and can result in a lower noise floor (wide dynamic range) on the analogue velocity output. However, AGC is not recommended for measurements on surfaces which are rotating, or vibrating with moderate to large amplitudes; in such applications, the AGC option is disadvantaged by its finite response time so it is recommended that

’’LIM’’ is used instead[6].

In this setup the LIM is chosen.

Velocity Range and Frequency Response

The combination of the velocity range and the frequency response results in the required frequency range. In this setup the Velocity is equal to 100 mm/s and the frequency response is equal to 3, then the frequency range is equal to 5000 Hz ( according to table 3)

Frequency Response Velocity Range ,mm/s 10

Velocity Range ,mm/s 100

Velocity Range ,mm/s 1000

1 2 3 4

5 HZ 50 Hz 500 HZ 5000 HZ

50 HZ 500 HZ 5000 HZ 50000 HZ

500 HZ 5000 HZ 50 000 HZ 400 000 HZ

Table 4: Frequency range of the laser vibrometer

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Page 35 of 38

Matlab Codes

Matlab code for calculating natural frequencies of the plate based on the values of lambda (tables 1 and 2)

h=0.0019;

E=1.89*10^11;

density=8.092*1000;

A=E*h^3;

nu=density*h;

mu=0.31;

B=12*nu*(1-mu^2);

C=sqrt(A/B);

a=12.5*10^-3;

D=C/(2*pi*a^2);

lambda=[9.084;38.55;87.80;20.52;59.86;119;5.253;35.25;83.9;154;12.23;52.91;111.3;

192.2];

for i=1:14;

lambdaa=lambda(i);

f(i)=lambdaa*D;

end

Matlab code for converting UNV file to Matlab file

clear all close all

unv_r(‘file name.unv’);

load (' file name.mat');

X1=Data;

f=linspace(0,10000,6401);

subplot(2,1,1);

hold on

plot(f,real((X1(:,4))),'g');

xlabel('Frequency Hz');ylabel('Real part of FRF m/s^2/N');title('FRF') grid on;

subplot(2,1,2);

plot(f,imag(X1(:,4)),'g');xlabel('Frequency Hz');ylabel('imaginary part of FRF m/s^2/N');title('FRF')

grid on figure

subplot(2,1,1);

plot(f,20*log10(abs(X1(:,4))),'g');

xlabel('Frequency Hz');ylabel('FRF dB rel 1(m/s^2/N)');title('FRF') grid on;

subplot(2,1,2);

plot(f,(angle(X1(:,4))*180)/pi,'g');

ylabel('Phase degrees');xlabel('frequency Hz');

grid on;

figure

f=linspace(0,10000,6401);

plot(f,20*log10(abs(X1(:,1))),'g');

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Page 36 of 38

grid on

xlabel('Frequency Hz');ylabel('Power Spectral Density dB rel N^2 /HZ ');

figure hold on

plot(f,(20*log10(abs(X1(:,2)))),'g');

ylabel('Power Spectral Density dB rel (m/s^2)^2 /HZ ');xlabel('frequency Hz');

figure

plot(f,X1(:,3),'g');xlabel('Frequency Hz');xlabel('Frequency Hz');ylabel(' Coherence Function');

function unv_r(fname);

% function unv_r(fname);

%

% This function reads an SDRC Universal File, and puts the information into a MATLAB

% Mat-file with the same name as the universal file but with extension ".mat"

%

% Data are stored in columns. Header values are stored in two matrices, numerical

% header values in matrix NumHead, and string header values in CharHead.

% Use the function Unv_split to split header values into variables.

%

% Restrictions: All frames (blocks) in the universal file MUST have the same length.

% If uneven data, ALL data must be uneven (with x-axis values).

% The filename must include the file extension (usually 'unv' or 'uff').

%--- ----

% Anders Brandt, Saven Utbildning AB, Sweden

%

% Modified November 1998 to cope with any extension

%--- ----

%

% fname = 'frfs.unv'; % For test runs fid=fopen(fname); % Open file if fid == -1

disp('Error opening file ');

error(fname);

end

dummy=fgetl(fid); % Loop until First -1 while (~(strcmp(dummy(1:6),' -1')) & ~feof(fid)),

disp('Looping to find a -1...') dummy=fgetl(fid);

end

if feof(fid)

error('File Format Error. Not a Universal File.');

end

FuncId=fscanf(fid,'%i',1); % Function identifier

col=1; % Index into columns in result matrices skip=0;

%--- ---

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while ~feof(fid), if (FuncId == 58)

% Read one dataset type 58

S=['Fileset ' num2str(FuncId) '. Reading...'];

disp(S)

[a b c d]=unv_r58(fid);

XData(:,col)=a;

Data(:,col)=b;

NumHead(:,col)=c;

if col == 1

CharHead=str2mat(d);

else

CharHead=str2mat(CharHead,d);

end

col=col+1; % Counter for next row number in matrix

% elseif FuncId == % Add here for more fileset types else

S=['Fileset ' num2str(FuncId) '. Skipping...'];

disp(S)

n=unv_skip(fid);

skip=skip+1;

end

if ~feof(fid)

dummy=fgetl(fid); % Read until next fileset, if not EOF if (length(dummy) < 6),

dummy='dummy ';

end

while (~feof(fid) & ~(strcmp(dummy(1:6),' -1'))), % Find -1 of next dataset

dummy=fgetl(fid); % Read down to next dataset identifier

if (length(dummy) < 6), dummy='dummy ';

end end

if ~(feof(fid)),

FuncId=fscanf(fid,'%i',1); % Identifier of next fileset end

end end

fclose(fid);

% Find the dot before the extension in filename dotpos=findstr(fname,'.');

S=['Done. Saving data to file ' fname(1:dotpos) 'mat...'];

disp(S)

S=[num2str(col-1) ' datasets were read and accepted.'];

disp(S)

S=[num2str(skip) ' datasets were read and skipped.'];

disp(S)

Save_string=['save ' fname(1:dotpos-1)];

clear S FuncId a b c d dummy fid col skip fname dotpos eval(Save_string)

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Page 38 of 38

References

1. Lars Håkansson, Signal analysis: Power Spectra and Power Spectral Density of time Limit Signal records, and stationary test methods, Blekinge Institute of Technology, (2008).

2. W. Leissa, Vibration of plates, U.S. Government Printing Office, first edition, (1969).

3. T.Y. Wu a, Y.Y. Wang a, G.R. Liu, Free vibration analysis of circular plates using generalized differential quadrature rule, Journal of sound and vibration, volume 264, pp.

883–891, (2003).

4. Anders Brandt, Introductory noise and vibration analysis, Saven Edutech AB and the department of telecommunications and signal processing Blekinge Institute of Technology, (2001).

5. J. S. Bendat and A. G. Piersol, Random Data Analysis and Measurement Procedures, John Wiley & Sons, third edition, (2000).

6. Laser Vibrometer VS 100, Manual.

7. Helmut F. Bauer, Werner Eidel, Determination of the lower natural frequencies of circular plates with mixed boundary conditions, Journal of Sound and Vibration, volume 292, pp.

742-764, (2006).

8. DING Hao-jiang, LEE Xiang-yu, CHEN Wei-qiu; Analytical solutions for a uniformly loaded circular plate with clamped edges, Journal of Zhejjang University, volume 6A(10), pp.

1163-1168, (2005) .

9. Roberrt Ray Goodrich, counterfeit Coin Separator, United States Patent No 5915520.

10. Howells, Geoffrey, Coin discriminators, European Patent Specification, Publication No 0300 781 B1.

11. Huadong Wu, Me1 Siege1, Correlation of Accelerometer and Microphone Data in the “Coin Tap Test”.

12. M. Hayes, Statistical Digital Signal Processing and Modeling, John Wiley and Sons INC, 2^nd edition, (1996).

13. Reid Goldsbrough, Counterfeit Coin Detection.

14. F. Harris. On the use of the windows for harmonic analysis with the discrete Fourier transform. In Proc. Of IEE, volume 66, (1978).

15. Thomas J.Quinlan, Counterfeit coin detector circuit, United States Patent No 4,845,994.

Sound and Vibration Measurements of Metal Plates