Anthony E. Prombonas,
1
Maria A. Paralika,
2
Michael P. Sotiriou,
3
Dimitris S. Vlissidis
3
1 Department of Dental Technology, Technological and Educational Institute of Athens
2 Department of Civil Works Technology, Technological and Educational Institute of Athens
3 Department of Dentistry, University of Athens
Received 4 June 2001; revised 5 March 2002; accepted 21 March 2002
Published online 00 Month 2002 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jbm.10350
Abstract: The aim of this work was to estimate the internal friction of dental gypsum products by measuring their loss factor with the use of the peak-amplitude method of vibration analysis. Cylindrical specimens for the measurement of compressive strength and rectangular specimens for the measurement of loss factor were constructed from plaster, dental stone, and high-strength dental stone. The loss factor was measured by vibrating specimens up to their resonance frequency, where the peak-amplitude method was applied. It was found that the peak-amplitude method is a simple, quick, and reliable method for measuring the loss factor of dental gypsum products. Furthermore, it was also found that the drying process of these materials reduces their internal friction and hence lowers the loss factor, whereas the use of excess water in their formulation has the reverse result. The correlation between strength and loss factor was found to be strongly negative. The results of this study are in agreement with the results from research on the microstructure of dental gypsum.
© 2002 Wiley Periodicals, Inc. J Biomed Mater Res (Appl Biomater) 63: 605– 609, 2002
INTRODUCTION
The study of the motion of physical systems resulting from the forces acting on them is referred to as dynamics. One type of dynamic behavior of physical systems is vibratory motion, in which the system oscillates about a certain equilibrium position. The surfaces of a solid body are able to vibrate because of the ability of materials to store potential energy via their elastic properties.
1,2
Forced vibration takes place when a system oscillates under the action of external forces. When the excitation force is oscillatory, the system is forced to vibrate at the excitation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, resonance is encountered, a phenomenon in which the amplitude can build up to dangerously high levels, limited only by the degree of damping.
1,3
When materials are set into vibration, imperfections within an otherwise perfectly elastic material give rise to damping of the vibration and cause some energy to be dissipated as
* This research has been carried out by the Technological and Educational Institute
(TEI) of Athens in cooperation with the Department of Dentistry, University of Athens.
Correspondence to: Anthony E. Prombonas, Skiathou 44, Ano Glyfada, Athens
16561, Greece (e-mail: aprob@teiath.gr)
© 2002 Wiley Periodicals, Inc.
heat.
4,5
Damping can be produced externally or internally by internal friction within the vibrating body. In an ideal elastic solid there is a one-to-one relationship between stress and strain, and the vibrational energy is conserved. In practice there are usually small deviations from this one-to-one relation— called inelastic effects. Actually there are a variety of deviations, like internal friction, plasticity, and microcrack damage. The stress–strain diagram of a cyclically loaded specimen then forms a closed loop, the area of which gives the energy dissipated per cycle. Internal friction is directly related to the damping, and its magnitude is described by various terms such as the loss factor or specific damping capacity .
2
Obviously any attempt to estimate the time course of changes in the strength properties of dental gypsum products during setting (aging effects) cannot involve destructive tests.
It is therefore necessary to assess some physical property of the gypsum that is related to the strength but can be measured nondestructively.
It is well known from structural research that changes occur in the intercrystalline boundaries in polycrystalline gypsum specimens as they change from the wet to the dry condition. As the last traces of water leave such specimens during drying, fine crystals of gypsum, which anchor the largest crystals, are precipitated. If excess water is present, or
605

606 PROMBONAS ET AL.
TABLE I. The Normal and Increased Ratios of (water/powder) for the Three Types of Gypsum
Normal Ratios Increased Ratios
Plaster
Dental stone
High strength dental stone
45 ml/100 g
35 ml/100 g
25 ml/100 g
60 ml/100 g
47 ml/100 g
33.25 ml/100 g if water is then added, these small crystals are the first to dissolve, and thus the reinforcing anchors are lost.
6,7
The dynamic properties of some dental materials have been measured by several researchers. The nondestructive determination of Young’s modulus of dental composites at the resonance frequency has been found to be reliable and accurate. Postpolymerization effects could clearly be detected, and the regression analysis showed a correlation coefficient of 0.92 with the amount of volumetric filler present.
8
The resonance frequency of resin-based materials increases during polymerization, due to increases in the elastic modulus.
9 The measurement of dynamic parameters of chemical- and light-cured composites has shown that the dynamic viscoelastic properties of these materials are temperature dependent, but not to a significant degree.
10 The torsion pendulum method has been used for the determination of dynamic mechanical properties, and it has been shown that composite materials have high moduli and low damping compared to unfilled resin.
11
The authors are not aware of any research into the vibration properties of dental gypsum products. The main purpose of this research was to introduce the peak-amplitude method of vibration analysis as a way of nondestructively studying the vibration properties of these products. A further aim of this study was to estimate whether the determination of the loss factor could be a useful control tool for estimating the structural quality of dental gypsum products. In addition, there was an investigation of whether the loss factor was correlated with the compressive strength of these materials, as well as with the results from research into their microstructure.
MATERIALS AND METHODS
Three types of dental gypsum were used in this study: (a) plaster (Greek Plaster), (b) dental stone (Giludur, Giulini
Figure 1.
The experimental apparatus.
Figure 2.
The rectangular specimen mounted on the vibrator. The accelerometer is distinguished on the right end of the specimen.
Chemie, Ludwigshafen, Germany), and (c) high-strength dental stone (Giludur, Giulini Chemie, Ludwigshafen, Germany).
The specimens for compressive-strength testing were made according to the specification No. 25 recommended by the American Dental Association (ADA), using a two-piece metallic mold with five cylindrical holes of specified dimensions (1 cm in diameter
⫻
2 cm deep). A strength-testing machine (Schenck Trebel, Darmstadt, Germany) was used with the following maximum-load scales: 60 kN, 30 kN, 12 kN, and 6 kN.
12,13
For each type of dental gypsum, two ratios of water to powder (w/p) were used. The first one was called the normal ratio (NR), which was that recommended by the ADA. The second one was called the increased ratio (IR), having the same amount of powder but 33% more water than the NR.
Table I lists the amounts of water and powder that were used to achieve the NRs and IRs that were used in the construction of the specimens.
For each type of dental gypsum and for every ratio, 10 cylindrical specimens were constructed for the measurement of their compressive strength.
10
Five rectangular specimens were constructed for each type of dental gypsum and for every ratio for the measurement of the vibration magnitudes of the material. These specimens were constructed with the use of five special split molds made from plexiglass. The molds consisted of five pieces, namely the floor and the four walls, and had dimensions of 9.5
⫻
0.5
⫻
1.9 cm. Two measurements of vibration magnitude were made on each specimen, one on the day of casting (first day) and one five days later (fifth day). During the intervening period they were kept under controlled environmental conditions (temperature
25 °C and relative humidity 60%).
Several pieces of equipment were used for the measurement of damping: (a) a digital signal generator (TFG-811,
Topward Instruments, Taipei, Taiwan), (b) a vibrator (4810
PM Mini Shaker, Bruel & Kjaer, Denmark), (c) a twochannel oscilloscope (3132, Topward), (d) a digital voltmeter

PEAK-AMPLITUDE METHOD OF VIBRATION ANALYSIS 607
TABLE II. The Means and Standard Deviations of Compressive Strength and Loss Factor
NR
Plaster
IR NR
Dental Stone
IR
M
⫽
10.56
Compressive
Strength
(MPa)
M
⫽
11.48
M
⫽
8.94
M
⫽
20.827
Loss Factor First day
SD
⫽
0.2502
M
⫽
0.06308
SD
⫽
3.2
⫻
10
⫺
4
Fifth day M
⫽
0.02842
SD
⫽
8.7
⫻
10
⫺
5
SD
⫽
0.3419
M
⫽
0.06704
SD
⫽
6.5
⫻
10
⫺
4
SD
⫽
0.2026
M
⫽
0.0333
SD
⫽
1.3
⫻
10
M
⫽
0.02118
SD
⫽
7.4
⫻
10
Note: NR ⫽ normal ratio, IR ⫽ increased ratio, M ⫽ mean value, SD ⫽ standard deviation
⫺
4
⫺
5
SD
⫽
0.4544
M
⫽
0.0525
SD
⫽
1
⫻
10
⫺
4
High-Strength Dental Stone
NR
M
⫽
33.583
SD
⫽
0.3356
M
⫽
0.0275
SD
⫽
1.6
⫻
10
⫺
4
M
⫽
0.01807
SD
⫽
4.1
⫻
10
⫺
5
M
SD
⫽
⫽
IR
19.87
0.5437
M
⫽
0.0372
SD
⫽
9.2
⫻
10
⫺
5
(HC-777, Topward), and (e) an accelerometer (4374, Bruel &
Kjaer). Figure 1 provides the path of the signals through the above apparatus. In this figure the arrows on the solid lines indicate the path of the signal, while the arrows on the interrupted lines are for the identification of accelerometer, vibrator and specimen.
A hole was drilled through the center of gravity (the point of intersection of the diagonals) in each rectangular specimen, through which a screw was attached in order to mount the specimen on the vibrator. The accelerometer was mounted on one end of the specimen (beam) with sticky wax
(Figure 2).
The excitation amplitude of the generator was increased until the sinusoidal response of the specimen was detected by the accelerometer. The signal from the accelerometer was simultaneously fed to the oscilloscope and the voltmeter, in order to obtain the graphical and numerical representations of the response. After the specimen was mounted on the vibrator, the excitation frequency was increased up to resonance frequency of the specimen, where the damping was calculated.
Damping is determined from the sharpness of the response peak at resonance, and is normally measured in terms of the loss factor. The most common method of determining loss factor is from the bandwidth of the resonance curve. This merely entails taking the reading of the voltmeter ( Ad , in mV), which is proportional to the displacement of the end of the beam at the resonance frequency (
␻
0
), and finding the frequencies (
␻
1 and
␻
2
) above and below resonance at which the displacement falls to Ad /(
公
2), that is Ad
⫻
0.707.
loss factor
␩ is given by the following equation:
1,4
The
␩ ⫽
␻
2
⫺ ␻
1
␻
0
.
The loss factor can also be calculated from other parameters:
␩ ⫽
E
2
␲
W
, where E is the energy dissipated per cycle at resonance and W is the energy stored in the system. This equation is a measure of the amount of work dissipated as heat by internal friction of a unit volume of a material during one cycle of vibration.
1,5
For statistical analysis, the means and standard deviations were calculated for the loss factor of the groups. An ANOVA was used to examine the variation between the means, and a
TABLE III. Statistical Analysis between the Means of Loss Factor
Normal Ratio on first day
Increased ratio
Normal ratio on fifth day
ANOVA between first and fifth day
Plaster
M
⫽
0.06308
SD ⫽ 3.2
⫻ 10
⫺
4
V
⫽
1.3
⫻
10
⫺
7
M ⫽ 0.06704
SD
⫽
6.5
⫻
10
⫺
4
V ⫽ 4.3
⫻ 10
⫺
7
M
⫽
0.02842
SD ⫽ 8.7
⫻ 10
⫺
5
V
⫽
7.7
⫻
10
⫺
9
F ⫽ 42716.17*
P
⫽
3.3
⫻
10
⫺
16
Note : M ⫽ Mean, SD ⫽ Standard Deviation, V ⫽ Variance
* The means are significantly different at the level 0.001
Dental Stone
M
⫽
0.0333
SD ⫽ 1.3
⫻ 10
⫺
4
V
⫽
1.7
⫻
10
⫺
8
M ⫽ 0.0525
SD
⫽
1
⫻
10
⫺
4
V ⫽ 1 ⫻ 10
⫺
8
M
⫽
0.02118
SD ⫽ 7.4
⫻ 10
⫺
5
V
⫽
5.5
⫻
10
⫺
9
F ⫽ 31246.79*
P
⫽
1.2
⫻
10
⫺
15
High-Strength
Dental Stone
M
⫽
0.0275
SD ⫽ 1.6
⫻ 10
⫺
4
V
⫽
2.6
⫻
10
⫺
8
M ⫽ 0.0372
SD
⫽
9.2
⫻
10
⫺
5
V ⫽ 8.5
⫻ 10
⫺
9
M
⫽
0.01807
SD ⫽ 4.1
⫻ 10
⫺
5
V
⫽
1.7
⫻
10
⫺
9
F ⫽ 15949.35*
P
⫽
1.7
⫻
10
⫺
14
ANOVA between
Materials
F
⫽
30784.67*
P ⫽ 0
F ⫽ 7373.8*
P
⫽
0
F
⫽
28537.8*
P ⫽ 0

608 PROMBONAS ET AL.
TABLE IV. The Correlation Coefficient between Loss Factor and Compressive Strength in Normal and Increased Ratio of w/p
Loss Factor Compressive Strength
Normal ratio
Increased ratio
Plaster
Dental stone
High-strength dental stone
Plaster dental stone
High-strength dental stone
0.06308
0.0333
0.0275
0.06704
0.0525
0.0372
11.48
20.827
33.583
8.94
10.56
19.87
r r
⫽ ⫺
⫽ ⫺
0.97289
0.93191
correlation coefficient between the loss factor and the compressive strength for the NR and the IR was also calculated.
RESULTS
Table II shows the means and standard deviations of the loss factor
␩ for the NRs and IRs at the first day for the three types of dental gypsum. Tables III and IV present the data of the statistical analysis of the results.
Figure 3 is a bar graph showing the means of the loss factors of NR against IR for the three types of dental gypsum, and Figure 4 is a similar graph of the means of loss factors of first and fifth day for the same materials. Figure 5 plots the means of the loss factor against the corresponding means of the strength, which demonstrates the strong negative correlation between these two quantities.
DISCUSSION
The peak-amplitude method is a simple and quick method for measuring the loss factor of a material. The use of a very light accelerometer (0.65 g) and hard sticky wax for mounting minimizes, respectively, mass loading effects and vibration energy losses during measurements. Furthermore, the use of a coaxial cable with a special noise-reduction treatment eliminates the local changes in capacitance and the influence of electromagnetic fields.
14
The differences between the means of the loss factors in the first- and fifth-day groups was found to be statistically significant in plaster, dental stone, and high-strength dental stone ( P
⬍
0.001). This indicates that as dental gypsum products dry, their internal friction decreases, which consequently results in a decrease in their loss factor. The development of secondary microcrystals in dental gypsum products seems to be a better explanation of the decrease in their loss factor, after the drying process. These crystals act as anchors between the primary crystals, leading to the minimization of internal friction in the gypsum material.
The differences between the means of the loss factors in the NR and IR groups were also statistically significant ( P
⬍
0.001), which indicates that increasing the w/p ratio causes a significant increase in internal friction.
There was a strong negative correlation between the strength of the dental gypsum materials and their loss factor.
The correlation coefficients of NR and IR with the strength were
⫺
0.973 and
⫺
0.932, respectively.
The very low standard deviations of the loss factors (Table
III) reflects the accuracy and reliability of measurements made with the peak-amplitude method.
The results of the experimental study lead to the following conclusions:
CONCLUSIONS
Figure 3.
The loss factor of normal ratio ( Norm.Rat.
) and increased ratio ( Incr.Rat.
) of dental gypsum products. (PL
⫽
Plaster, D.S.
⫽
Dental Stone and H.S.D.S.
⫽
High Strength Dental Stone).
Figure 4.
The loss factor of first and fifth day of the three gypsum products.

PEAK-AMPLITUDE METHOD OF VIBRATION ANALYSIS
Figure 5.
The loss factor and the corresponding values of compressive strength of several gypsum products and water/powder ratios.
1. The peak-amplitude method is a simple and quick method for determining the loss factor of dental gypsum products.
Furthermore, the method is reliable.
2. The drying process of dental gypsum products causes a decrease in their internal friction (and hence their loss factor).
3. Increasing the water/powder ratio in dental gypsum products increases the internal friction.
4. The results of this research agree with the existing research on the microstructure of dental gypsum products, because the proposed mechanism of secondary microcrystals can explain the alteration of internal friction that is associated with the drying process.
5. In dental gypsum products there is a very strong negative correlation between compressive strength and loss factor.
609
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