The Evaluation of the Damping Characteristics of a Hard Coating on Titanium

Engine failures due to fatigue have cost the Air Force an estimated $400 million dollars per year over the past two decades (Garrison, 2001). Damping treatments capable of reducing the internal stresses of fan and turbine blades to levels where fatigue is less likely to occur have the potential for reducing cost while enhancing reliability. This research evaluates the damping characteristics of magnesium aluminate spinel, MgO+Al2O3, (mag spinel) on titanium plates. The material and aspect ratio were chosen to approximate the low aspect ratio blades found in military gas turbine fans. The plates were tested with a cantilevered boundary condition, using electrodynamic shaker excitation. The effective test area of each specimen was 4-1/2 in. x 4-1/2 in. The nominal plate thickness was 1/8 in. Mag spinel was applied to both sides of the plate, at a thickness of .01 in., and damping tests were run at room temperature. The effect of the coating was evaluated at the 2 bending mode (mode 3) and the chordwise bending mode (mode 4). A scanning laser vibrometer revealed the frequency and shape of each mode for the plates. Sine sweeps were used to characterize the damping of the coated and uncoated specimens for the modes tested. The coating increased damping nonlinearly for both modes tested. The test results are presented in this document.

For the wisdom of this world is foolishness with God. For it is written, He taketh the wise in their own craftiness.
1Corinthians 3:19 The fear of the LORD is the beginning of wisdom: a good understanding have all they that do his commandments: his praise endureth for ever.   x and high-cycle fatigue (HCF). LCF is characterized by higher loads and fewer cycles to failure than HCF. The reason for this is that LCF strains are dominated by larger plastic regions for each cycle, whereas HCF strains are predominately elastic for each cycle (Grady, 1999).

List of Tables
Over the years, LCF failures in aircraft engines have been greatly reduced through the use of fracture mechanics and a retirement-for-cause management philosophy, leaving HCF as the primary cause of engine failures (Nicholas, 1996). There are many different sources of mechanical vibration that lead to HCF damage in turbine engines.
Some of the se sources are aerodynamic excitation, airfoil flutter, and acoustic fatigue. As improvement s for increased performance and reduced weight have been implemented in engines, the operating temperatures, stresses, and stage loading have increased, making the HCF problem more acute (Cowles, 1996). Throughout the 1980s and 1990s, HCF was the single largest cause of aircraft engine failure, resulting in lost aircraft and the expenditure of many millions of maintenance man-hours and dollars. Estimates put the cost of high cycle fatigue at over $400 million per year (Garrison, 2001). Fatigue in engine blades, what this research is trying to minimize through damping, is the result of stresses generated by unsteady aerodynamic loading which cause vibrations at or near a resonance condition (Shen, 2002).

Damping
Damping is the dissipation of mechanical energy from a system while subjected to cyclic loading (Lazan, 1959 andUngar 2001). Damping materials are designed to maximize the energy dissipation in a system. Most damping materials must be combined with structural elements to be useful in engineering applications. When combined in this manner the damping material is generally called a 'damping treatment' (Ungar, 2001).
These treatments can be used to avoid premature failure by reducing the displacement amplitude of oscillations at resonance, which reduces the cyclic stresses. A system with relatively no damping will have a larger displacement amplitude at resonance than one with damping ( Figure 1) (Baz, 2001). In the figure, frequency is along the horizontal axis and vibration amplitude is along the vertical axis. The zero damping curve in the figure will theoretically extend to infinity. Damping is generally found to be one of the most structure-sensitive properties that can be measured (Lazan, 1968). There are two major categories of damping behavior; elastic and inelastic. For a material to be "perfectly elastic" its stress-strain curve must be linear and no rate or time dependence is present. "Inelasticity" is simply any deviation from the "perfectly elastic" condition.
There are several different types of damping mechanisms to include: dynamic hysteresis, static hysteresis, plastic strain damping, and internal friction. For reasons discussed later in this chapter internal friction is the mechanism of concern for this study. The effect of such test conditions as stress amplitude, frequency, and temperature can be significant and must therefore be well understood and documented. All testing was done at room temperature. The effects of stress amp litude and frequency are presented in Chapter IV.

Figure 1. Effect of Damping
There are two main approaches to damping design: passive and active. Active damping techniques incorporate the control of sensors and actuators and are generally used for low frequency excitations. The most common active dampers are made of piezoelectric films bonded to the specimen. Passive damping techniques include the application of damping coatings. Viscoelastic coatings are generally more effective against high frequency excitations, but are typically effective for only small frequency ranges because of the significant variation of the damping material properties with temperature and frequency (Baz, 2001). Hard coatings do not generally suffer from these constraints. The propulsion community, in its quest for a solution to the high cycle fatigue problem, has focused its efforts on passive damping, due to its simplicity of application and effectiveness at high frequency. Some hybrid approaches also exist; which are combinations of active and passive methods and can provide control over broader frequency ranges. (Baz, 2001)

Damping Treatments
Surface damping coatings are often used to solve resonant noise and vibration problems associated with structures of small cross-sectional area, such as beams, plates, or turbine engine airfoils. These coatings can be easily applied and provide high damping over wide temperature and frequency ranges (Nashif, 1985). Traditionally the materials used have been viscoelastic polymeric plastics or elastomers. A viscoelastic material has the properties of both viscous (energy dissipating) and elastic (energy storing) materials. The damping arises from relaxation and recovery of the polymer network after it has been deformed, and a strong dependence exists between frequency effects and temperature effects because of the direct relationship between material temperature and molecular motion. By proper tailoring, polymeric materials can be manufactured to possess a wide variety of damping, strength, durability, creep resistance, thermal stability, and other desirable properties, over selected temperature and frequency ranges (Nashif, 1985).
High-damping metal alloys have better damping properties than commo n metals but do not provide the same level of damping that viscoelastic materials do. However, viscoelastic materials are generally effective only for a small temperature range while the high-damping metal alloys can be effective over a greater temperature range (Ungar, 2001). These high damping alloys are not usually the best adapted to practical construction purposes, since the gain in damping is often at the expense of stiffness, strength, durability, corrosion resistance, cost, machinability, or long term stability (Nashif, 1985). In aircraft engines, even a small amount of damping can have a pronounced effect, and damping over an extended temperature range is crucial. Over the past few years the propulsion community has shifted focus from viscoelastic damping materials back to metallic or hard coatings.
The usefulness of hard coatings as dampers has been known to engineers since the early 1960's. The damping mechanism was initially assumed to be friction between the particles; which was recently supported experimentally (Green, 2002, Shipton, 2003. These coatings generally behave in a non-linear strain-dependent manner. Of particular interest is the air plasma sprayed oxide ceramic coating known as Magnesium Aluminate Spinel (mag spinel), MgO+Al 2 O 3 .

Mag Spinel
The value of mag spinel is that it has a higher damping capacity than other ceramic materials (Shipton, 2003 provides enough damping to be of interest to the propulsion community (Torvik, 2002).

Application of Mag Spinel
The two most common methods for applying hard coatings are air plasma spraying and physical vapor deposition. The mag spinel for this research was applied by air plasma spraying. Air plasma spray offers advantages in cost, lower application temperatures, and fewer limitations on component size (Patsias, 2001). Plasma spraying provides a denser, stronger coating than most other spray processes. The high temperature of the plasma allows materials with high melting points to be applied as a coating that canno t be applied by any other means (APS Materials, 2004).
Plasma spraying is the process of applying a coating to a substrate material by injecting the coating in a powder form into a high temperature plasma gas and spraying it at high velocity onto the target substrate. The plasma gas (typically air, argon, nitrogen, hydrogen, or helium) converts the powder to a molten state. As the spray impacts the substrate, it cools very quickly, forming a bond with the surface (Figure 2) (APS Materials, 2004). When the spray chamber contains the common atmosphere, the process is known as air plasma spraying, which is the most common form of plasma spraying.
Occasionally, the atmosphere can provide undesirable contaminants to the coating so the chamber is filled with an inert gas at low pressure. This process is called vacuum spraying (APS Materials, 2004). The thermal sprayed mag spinel has a more refined defect structure than most other thermal sprayed oxide coatings. The microstructure is similar to a massive array of parallel plates with an aspect ratio typical of any thermally sprayed oxide ceramic. The best performing coatings are those applied with a high power plasma, a fine powder, and a 90 o angle. (Shipton, 2003)

Objective of Thesis
The objective of this investigation is to determine the effect on damping of a square titanium plate by the application of a mag spinel hard coating. Ti-6Al-4V was chosen as the plate material for this investigation because of its extensive use in aircraft fan and compressor blades. The aspect ratio of the plates approximates the low aspect ratio blades found in military gas turbine fans. The effect of the coating will be evaluated at the 2 nd bending mode (mode 3) and the two stripe, or chordwise bending, mode (mode 4). A three dimensional representation of the first five mode shapes taken from laser vibrometry tests is shown in Figure 3.

Related Work
Several approaches to reducing turbine blade vibrations have been to introduce additional damping from blade dampers. Dry friction dampers, which include blade-toground, blade-to-blade, and shroud dampers are the most common in use today.
However, the gain in damping from dry friction dampers is negligible for high frequency vibration. This has motivated recent research of high frequency dampers. (Shen, 2002) Some experiments for reducing vibratory stresses on rotating blades have been done by inserting patches of viscoelastic damping materials into milled cavities of the airfoil, which is then sealed with a cover sheet to maintain structural integrity and airfoil shape (Kielb, 2000). This approach is limited by the temperature constraints of the viscoelastic damping treatment and the manufacturability and durability of the milled airfoils (Shen, 2002). Torvik, et al. (2002) examined the damping effect of mag spinel on Hastalloy X, a nickel-based alloy commonly found in aircraft engines, and found that the coating provided sufficient damping to be effective in reducing vibration amplitudes in aircraft engines. They observed that the response functions were not symmetric about the resonance frequency, and the resonance frequency decreased as the amplitude of the applied force was increased, indicating stiffness non-linearity, or softening. This was also observed by Shen (2002). The amount of energy dissipated by the coating was shown to be strain dependent. They further concluded that over the first four bending modes, the level of damping obtained was independent of frequency. (Ivansic, 2003)

Current Approach
This investigation compares the response of titanium plates before and after a mag spinel coating was applied to each side. The material and aspect ratio were chosen to approximate the low aspect ratio blades found in military gas turbine compressors. The plates were tested with a cantilevered boundary condition for modes three and four. This simulated the cantilevered condition of operational turbine blades and two of the more common mode shape families. The excitation was applied through the base. The specimens used for this study were 4-1/2 in. x 7 in. x 1/8 in. Ti-6Al-4V plates. The effective test area was 4-1/2 in. x 4-1/2 in. with a 2 in. clamped region and a 1/2 in. tail behind the clamp to help when removing the specimen from the fixture (Figure 4). For this research a two piece fixture was designed based on prior Turbine Engine Fatigue Facility (TEFF) experience to increase repeatability. Previous designs could not easily prevent the specimens from shifting in an in-plane direction while under load (Ivansic, 2003). This design, which sandwiches the plate between two steel blocks, has eliminated the tendency for the plates to shift by adding two guide shafts through the clamped area of the plate. Figure 5 shows the plate mounted on the bottom portion of the fixture with the two guide shafts through the clamped region. The top piece slides over the guide shafts and rests on the plate. Four bolts, one through each guide shaft and one beyond both ends of the plate are used to provide the clamping force on the plate.
These shafts also eliminated the variability of the effective plate geometry (4-1/2 in. x 4-1/2 in.) from one setup to the next. Without the guide shafts, it would have been necessary for the researcher to measure the test area before each experiment. A slight difference could influence the repeatability of the experiments. The guide shafts eliminated that problem and it can be seen from results presented in Chapter IV that repeatability was excellent.

Figure 5. Test Fixture
The mag spinel coating was applied to the test area on both sides of the plate by air plasma spray. All damping tests were conducted at room temperature. Damping was characterized by a series of sine sweeps for each of three plates before and after the mag spinel was applied. The excitation load was increased for each successive sweep. The "half-power bandwidth" method was used to determine the level of damping for each sine sweep and comparisons were made between the coated and uncoated configurations.
Test specimens were characterized by frequency, mode shape, damping, and stress pattern. Testing was conducted at the Turbine Engine Fatigue Facility, AFRL/PRTS, Wright-Patterson AFB, OH.

Test Specimen
Guide Shafts Top Clamp

Bottom Clamp
A scanning laser vibrometer measured the frequency for each mode and the displacement contour of the plate. These contours were compared with FEM models, which were used to determine the stress ratio of the stress at the strain gage location (discussed in Chapter III) to the maximum stress for the two resonant modes under consideration. Stress and strain are related through Hooke's Law.
FEM was used to determine the resonant frequencies, mode shapes, and stress ratios for the uncoated plates. Sine sweeps with strain gages attached were used to develop the strain/displacement relationship by comparing the peak laser vibrometer response to the peak strain gage response. Since strain is proportional to displacement for the levels reached in this research, it was necessary to convert the velocity measurements to displacement. Because strain gages will add some damping to the system, the strain/displacement relationship had to be established before the strain gages were removed and the damping could be measured for the uncoated and coated conditions. Damping ratios, for several strains up to a maximum of 500 micro-strain, were determined by conducting sine sweeps on a 6,000 lb electro-dynamic shaker and measuring the plate's dynamic response with a single point laser vibrometer. The purpose of this research was to focus on the damping characteristics of mag spinel at low strains; therefore 500 micro-strain was chosen as the upper limit. The same titanium plates were compared before and after the mag spinel coating was applied.

II: Prediction Methods
Two methods were used to predict the natural frequencies and mode shapes of the uncoated titanium plates: analytic calculation and finite element modeling. The mathematical calculations were used to validate the FEM.

Approximate Analytic Calculations for Natural Modes of Plates
When a system is subjected to an oscillatory load, vibration occurs. The displacement of the system from the vibration is related to the frequency and strength of the applied load. The amplitude of the displacement will reach peaks at points along the frequency bandwidth known as the natural frequencies. Classical Plate Theory can be used to derive the natural frequencies.
Leissa cites Young for the derivation of the natural frequency equation for a square plate in the clamped-free-free-free condition. In his derivation of the following equation he used the products of beam functions and the Rayleigh method to obtain the natural frequencies (Leissa, 1969). The nominal geometry was used for comparison to experimental values.
(3) (Leissa, 1969) where n ω = natural frequency C = modal constant first five modes: C 1 = 3.494 C 2 = 8.547 C 3 = 21.44 C 4 = 27.46 C 5 = 31.17 a = plate length ρ = mass density t = plate thickness D = flexural rigidity E = Young's modulus v = Poisson's ratio This method gives natural frequency solutions in terms of radians/sec. To convert to Hertz (cycles/sec) use the following equation: The results for the given modes of interest are presented in Table 1. The shape of the displacement pattern at a natural frequency is called the mode shape. The total motion at any point of the system is the sum of the motions resulting from the vibratio n of the respective modes. A completely undamped system excited at a natural frequency will continue to oscillate at that frequency. Such a condition is called resonance and if allowed to continue the vibration amplitudes will intensify until the system fails. However, there is always some degree of damping, which not only reduces the vibration amplitude but results in the superposition of all modes at each natural frequency with varying degrees of intensity. If damping is negligible then the intensity of the primary mode approaches unity and the others approach zero. If the only damping is a result of the plate's material properties and the geometry is simple, the modes will closely approximate the undamped natural modes (Soedel, 1993). The research presented here follows that assumption.
There are always nodal points, lines, or surfaces in each of the normal modes of vibration of any system. For the fundamental mode, which corresponds to the lowest natural frequency, the supported or fixed points of the system usually are the only nodal points; for other modes, there are additional nodes. In the modes of vibration corresponding to the higher natural frequencies of some systems, the nodes often assume complicated patterns. In certain problems involving forced vibrations, it may be necessary to know what the nodal patterns are, since a particular mode usually will not be excited by a force acting at a nodal point (Harris, 1996). Mode shapes, as defined by the nodal lines, for the first five modes of a cantilevered square plate are shown in Figure 6 ( Leissa, 1969). freedom to represent a cantilevered condition and all other nodes were unconstrained.
The material properties were obtained from MIL-HDBK-5CD-ROM, May 1997 for Ti-6Al-4V (Young's Modulus = 1.6*10^7, Poissan's Ratio = .31, and Yield Stress = 126 ksi). As the mesh density was increased, the frequency results converged to a single value; see Table 2 for mode 3 frequencies and The displacement and stress solutions were normalized to a maximum value of unity. This means that the actual stresses and displacements were unknown from the model, but the ratio of the displacement or stress at one point, usually the maximum location, to any other point on the plate is known. The second location is usually where the strain gage is placed. Using this ratio, the maximum stress will be known during experiments, even if the strain gage is not located at the point of maximum stress.
Hooke's law (Equation 1) was used to relate the stresses to strains. Figure 7 shows the plate out-of-plane displacement and relative stresses for mode 3. Figure 8 shows the plate out-of-plane displacement and relative stresses for mode 4.

III: Test Setup and Procedures
This chapter discusses equipment used and the procedures established to obtain test data. These include the test fixture, electro-dynamic shaker, and scanning laser vibrometer. The data collected is presented in Chapter IV, Results and Discussion.

Test Fixture
Essential to collecting accurate and repeatable data is the fixture or clamping device used to attach the plate to the excitation source. Previous work highlighted the difficulty of maintaining consistent boundary conditions from one data set to the next when using a simple sandwich design (Ivansic, 2003). Once installed, there was a tendency for the plate to shift in an in-plane direction while under load, causing the effective plate geometry to change. Even a slight change in geometry can have a significant effect on the results. The current design eliminated this deficienc y by adding two guide shafts through the clamped area of the plate. Figure 9 shows the plate mounted on the bottom portion of the fixture with the two guide shafts through the clamped region. The top piece slides over the guide shaft s and rests on the plate. Four bolts, one through each guide shaft and one beyond both ends of the plate are used to provide the clamping force on the plate. These shafts also eliminated the variability of the effective plate geometry (4-1/2 in. x 4-1/2 in.) from one setup to the next. Without the guide shafts, it would have been necessary for the researcher to measure the test area before each experiment. A slight difference could influence the repeatability of the experiments. The guide shafts eliminated that problem and it can be seen from results presented in Chapter IV that repeatability was excellent.

Uncoated Plates
The specimens used for this study were 4-1/2 in. x 7 in. x 1/8 in. Ti-6Al-4V annealed plates. The effective test area was 4-1/2 in. x 4-1/2 in. with a 2 in. clamped region and a 1/2 in. tail behind the clamp (Figure 10). The tail region served two purposes. The first was to provide a full diameter of material between the shaft holes and the edge of the plate. The second was to have a gripping surface to help remove the snug fitting plate from the clamp. Three plates were obtained from the same sheet of titanium and given a number from T1 to T3. The individual plates were cut from the sheet using a high powered water jet cutting system, which derives its efficiency and power by pressurizing water at up to 55,000 psi and focusing it through a nozzle as small as .003 in.
in diameter. Traveling at speeds up to 3 times the speed of sound the water stream cuts with negligible heat added and exerts little vertical or lateral force. Therefore, no internal stresses were added to the plates. The plates used for this research were cut in the same orientation.

Figure 10. Titanium Plate
Length measurements of the short sides for each plate were made using a Starrett hardened stainless caliper. The long side measurements were made using a Products Engineering 12 inch tempered ruler. Plate thicknesses were measured using a Mitutoyo SR44 micrometer. Each plate was weighed using an Acculab L series balance. Using the measurement averages, summarized in Table 5, the titanium density was determined from Equation 6.
where ρ = density This density is in very close agreement with the MIL-HDBK-5CD-ROM, May 1997 published density of .160 lbf/in 3 .

Coated Plates
The mag spinel coatings were applied by APS Materials, Dayton, OH. A coating of .01 in. was applied to both sides of plates T1 through T3. The coating was applied only to the 4 1/2 in. x 4 1/2 in. test surface. Each plate thickness was measured in the same manner used for the uncoated plates. The target thickness was .145 in. All measurements were within .003 in of the target. Table 6 provides a summary of the coated plate measurements. Figure 11 shows a sample of both the coated and uncoated plate.

Data Collection
Because of the non-linear characteristics of mag spinel, the standard material characterization techniques cannot be used. The strain-dependent effect must be considered. Previous work showed that when sine-sweep data from various strains or amplitudes are overlapped, the resonant peaks cannot be connected via a straight line (Ivansic, 2003). It can be seen from Figure 33 thru Figure 37 in Chapter IV that for the uncoated plates the peak resonant frequency is very nearly constant as the load is increased, but for the coated plates the peak frequency is moving nonlinearly down the frequency spectrum as the load is increased. This is known as strain softening. The strain softening effect requires that the sweep be done from high to low frequency; otherwise the peak may be missed.
Based on the scanning laser vibrometry results, presented in Chapter IV, gage one, for mode 3, was placed on the edge 2 ¼ in. from the tip on the side perpendicular to the clamp and gage two, for mode 4, was placed on the edge 2 ¼ in. from the tip on the fr ee edge parallel to the clamp (Figure 12). The effect of the strain gages on frequency response was measured using the scanning laser vibrometer. Because the gages may add damping to the plates and because they cannot be glued to the mag spinel coating without changing its properties, damping measurements were not made with strain gages attached.
The strain gages were only used on the uncoated plates to establish the strain/displacement relationship.

Modal Characterization
Scanning laser vibrometry was used to identify the mode shapes at each resonant frequency between 0 -2 kHz. Vibrometry is a non-contact procedure, where a laser is used to map the velocity or displacement of a vibrating specimen at discrete points. The system utilizes the concept of interferometry to measure the velocity or displacement of the vibrating specimen. Interferometry is the optical interference between two coinciding light beams. The two beams for this setup are the reference beam and the reflected beam.
The intensity of the coincident beam, a function of the phase difference between the two individual beams, is determined as follows: Clamped Region 2-1/4 in.
Gage 1 Gage 2 2-1/4 in. Where The phase shift is related to the path difference by: where ∆ L = path difference λ = wavelength of laser The wavelength ( λ ) and path difference ( ∆ L) are a function of time if one of the beams is scattered back from the vibrating specimen. The reflected beam also experiences a Doppler shift in its frequency, which is a function of the specimen's velocity. This frequency shift is determined as follows: where f D = Doppler frequency shift v = specimen velocity λ = wavelength of laser (Polytec Laser Doppler Vibrometer User Manual) The Polytec Laser Doppler Vibrometer system has two parts: the controller (OFV-3001) and the sensor head (OFV-056). Two types of sensor heads are available: single point and scanning laser. The scanning laser is used to find velocities at multiple points on the specimen, which is required for determining mode shapes. Test parameters are input to the controller which then directs the sensor head. The sensor head generates the laser beam, which is split into the reference beam and the specimen beam. The laser beam is reflected off the test specimen and returned to the sensor head as the reflected beam. It is combined with the reference signal and sent to the controller, which compares the frequencies and phases and calculates the velocities and displacements. The procedure is repeated for as many times as there are grid points on the plate surface.
When complete, the velocities of all the grid points are combined to create a ve locity map of the surface of the plate as a function of position for each frequency. If the velocities are converted to displacements a mode shape at that resonant frequency is created.
All three plates were tested with each possible configuration: uncoated without strain gages, uncoated with strain gages, and coated. Each plate was held using the mounting fixture designed for the sine sweeps. The torque on the two bolts that pass through the guide shafts was 125 ft-lbs and the torque on the two outer bolts was 100 ftlbs. The guide shaft bolts were set at a higher torque because they were directly above the plate and provided most of the clamping force. The fixture was then placed in a vise bolted to the table, which was floated to prevent outside vibrations from interfering with the experiment. An air horn placed at the free end corner provided the excitation source.
See Figure 13. Sweeps were done between 0 -2 kHz, which captured the first five resonant modes and displacement shapes. These shapes can be compared directly to the FEM modal results. The controller was set to take a data point at every 312.5 mHz, which provides a very high resolution output.

Strain-Velocity-Displacement Calibration
The strain/velocity relationship was determined for the 2 nd bend mode (mode 3) and the chordwise bend mode (mode 4) for all three plates using the 18,000 lb. shaker and verified on the 6,000 lb shaker. The verification was necessary because equipment failure on the 18,000 shaker made it necessary to complete the tests on the 6,000 lb shaker.
Each plate was clamped between two mounting blocks, which were bolted horizontally to the shaker (Figure 14). A laser, mounted to a rigid support, was used to measure the plate velocity at a single point. The target point was near one of the free corners opposite the clamped section, 0.1 in. from the free edge and 0.7 in. from the side

Air Excitation Source Plate
Mounting Fixture (Figure 15). The laser controller was not configured to take displacement measurements; therefore velocities were measured and converted to displacements. The velocity for maximum strain at this position for both modes was such that the laser controller would not have to be changed between mode testing. An accelerometer was placed at the base of the shaker to record the input load. It is acceptable to equate a change in acceleration as the equivalent change in applied force because the mass of the system is not changing and force changes linearly with acceleration. VibrationVIEW software, version 4.0.17, was used to control the data acquisition. The resonant frequency for each mode, determined from the modal characterization test, was swept through at a rate of 5 Hz/min.
The peaks from the laser and the appropriate strain gage establish the strain/velocity relationship at that excitation load. This process was repeated up to twenty times for each plate as the load was increased to create a complete strain/velocity curve. The strain/velocity relationship was determined for each mode and plate. Since strain is a function of displacement, the velocity measurements were converted to displacements.
Maximum displacement is correlated using the following relationship: where δ = displacement v = velocity ω = frequency (radians/sec) These curves represent the baseline curves for comparison to the coated plate.
Since strain gages were not applied directly to the mag spinel, velocity was the only measurement available. An equivalent displacement from the coated plates results in the same strain at the plate coating interface. Since the coating increases the damping of the plate, it is expected that a greater applied load will be required to produce the same displacement as for the uncoated plate.

Damping Characterization
Sine sweeps were used to determine the resonant frequencies and level of damping as a function of the maximum strain. The sweeps were conducted on a 6,000 lb shaker using the VWIN software, version 4.74. Each plate was clamped between the two mounting blocks, which were bolted to the shaker (Figure 14). An accelerometer was placed at the base of the shaker to record the input acceleration. The addition of strain gages may influence the damping of the system, therefore; it was necessary to measure damping with the strain gages removed. The damping levels of the uncoated plates without strain gages were established to provide a baseline for determining the damping levels provided by the mag spinel coating. A laser, mounted to a rigid support, was used to measure the plate velocity at a single point. The target point was near one of the free corners opposite the clamped section, 0.1 in. from the free edge and 0.7 in. from the side (Figure 15). Velocities were measured at this location for both modes -2 nd bend and chordwise bend -and converted to strain using the strain/displacement curve already established.
The frequency range for the sine sweeps was broad enough to capture response levels of 70% of the peak value. This provides sufficient data to perform the "half-power bandwidth" calculations, discussed later in this section. Each sweep was done at 5 Hz/min to ensure the peak was not missed. If the sweeps are made to quickly then the response does not have enough time to rise to the true peak amplitude. Six to ten different input accelerations representing a range of 10 -100 maximum strain and another six to eight representing a range of 100 -500 maximum strain were tested. The VWIN software takes the excitation signal and the plate response signal and conducts a fast Fourier transform to convert the data to the frequency domain. The output is an amplitude (velocity) versus frequency graph, where each peak represents a resonant frequency and mode.
Damping was determined by the "half-power bandwidth" calculation. The halfpower is calculated by measuring the bandwidth of the frequency curve 2 1 (or approximately 3dB) down from the resonant peak (Figure 16) (Meirovitch, 1986).

Figure 16. Half-power Bandwidth
The damping ratio, in percent, is then found using the following equation: A parameter commonly used in the damping community is the quality factor (Q).
The quality factor was developed by electrical engineers as a measure of the clarity of an electrical signal. The quality factor and damping ratio are inversely proportional. Q = ζ 2 1 (13) (Meirovitch, 1986) The VWIN software has the capability of determining the Quality factor by this method. It can be seen from this equation that as damping (ζ ) increases, Q decreases.
Therefore, a good damping material will dissipate more energy and have a lower Q value.
Another measure of damping is observed through the system loss factor (Nashif, 1985).
where η = system loss factor r ω = resonant frequency 1 ω = frequency to the left of r ω where amplitude is 2 1 A peak 2 ω = frequency to the right of r ω where amplitude is 2 1 A peak As damping increases so does the system loss factor. The relationship between ω ∆ / r ω and η is linear only for small ?, as shown in Figure 17 (Nashif, 1985). The loss factors for this research do not go above 0.01. Figure 17. Nonlinearity of η When comparing Equation 12, 13, and 14 the following relationship is observed.

Resonant Frequencies
Resonant Frequencies were experimentally determined from sine sweeps and laser vibrometry for both the uncoated and coated plates. The resonant frequencies were determined from the peaks on a frequency response curve. Theoretical predictions were also made using Classical Plate Theory and finite element modeling.

Comparison to Theoretical Predictions
Comparisons of the resonant frequencies between theoretical predictions and experimental results could only be made with the uncoated plates. If a good correlation exists between the theoretical and experimental results, the finite element model can be used as a benchmark for comparison of experimental values. The modes of interest for this research, mode 3 and 4, are compared in Table 7 and Table 8 respectively. The finite element model prediction is within 1.5% of the experimental results for mode 3 and wit hin 3.3% for mode 4. Therefo re, the finite element model closely approximates reality.
The theoretical predictions have even closer agreement. Because the measured frequencies are so very close to the theoretical predictions, it can be assumed the test fixture provided a very rigid boundary condition. A loose fixture would have resulted in measured frequencies much lower than predicted.

Experimental Resonant Frequencies
All three plate conditions; uncoated without strain gages, uncoated with strain gages, and coated, were tested for frequency response between 0 -2 kHz for all three plates using the scanning laser vibrometer. Five resonant frequencies can be identified within this range for each plate. Due to limitations of the test equipment the excitation load from the air horn was not measured. Therefore, just the location of the peak frequencies shown in Figure 18 thru Figure 20 are compared for the different plate configurations and not the amplitudes. Resultant strain levels do not exceed 2 microstrain for either mode 3 or 4. Configuration effects and repeatability were compared by repeating sine sweeps for the uncoated plates, with and without strain gages, and again when the plates were coated. It can be seen from Table 9 thru Table 14 that repeatability was excellent, with no more than a 0.3% difference between test runs of the same plate configuration. The effect on frequency response due the strain gages was another area of interest. It can be seen from Table 15 thru Table 17 that there was no significant influence to frequency response from the strain gages, with no more than a 1% difference in frequency response with strain gages added. It can also be seen from Table   18 thru Table 20 that the mag spinel coating caused a small increase in the resonant frequenc ies, with no more than 5.2% difference. This increase was expected based on the increased plate thickness from the mag spinel coating.

Mode Shapes
Scanning laser vibrometry yields not only the resonance frequencies, but also the mode shape at each resonant frequency. A three-dimensional representation of the first five mode shapes taken from laser vibrometry was shown in Figure 3 of Chapter I.
Results from the previous section revealed that the frequency response was virtually unaffected by the addition of strain gages and only a slight increase was observed when the mag spinel was applied. Before the strain/displacement relationship obtained from the uncoated plates with strain gages could be applied to the uncoated plates without strain gages and the coated plates, the mode shape, and hence the displacement pattern, had to be verified as the same for all three plate configurations. The comparison between the laser vibrometer data and the finite element model for plate T1, shown in Figure 21 and Figure 22, reveal that the mode shape is unaltered by the removal of strain gages or the addition of mag spinel. Only data for one plate is presented here, but the shapes are the same for all plates and configurations. (In Figure 22, missing data is shown as a discontinuity   Figure 12 in Chapter I. This is the maximum strain point for mode 4. This validation also allowed the strain/displacement relationship to be applied to all the plate configurations.

Strain/Displacement Relationship
Establishing the strain/displacement relationship with the single-point laser vibrometer was necessary because damping measurements were to be conducted on the plates without strain gages. However, comparisons of damping based on strain rather than ve locity were desired. Because strain gages would not be attached to the mag spinel coating they should also not be included in the baseline uncoated damping measurements.
To determine this relationship the velocity and strain at resonance were measured for each mode by conducting up to twenty slow (5 Hz/min) sine sweeps, with an increased excitation load for each sweep. The details of the test setup are discussed in Chapter III.
500 micro-strain was the upper limit for this research. Since strain is proportional to displacement and not velocity, the velocity measurements had to be converted to displacements. As indicated previously, this is done by dividing the velocity at peak resonance by the peak resonance frequency. Because there is very little shift in the resonant frequency from low to high load for the uncoated plate this step may seem unimportant. However, the strain softening effect of the mag spinel coating causes a noticeable shift in resonant frequency as excitation loads are increased and thus the frequency influence is more pronounced. The strain/displacement relationship for each plate can be seen in Figure 29  Results of the sweeps are shown in Figure 32 thru Figure 37. The non-linearity is immediately apparent when comparing the uncoated sweeps to the coated sweeps. The peaks for each sweep were fairly frequency stable as the load increased for the uncoated plates, but the peaks for the coated plates decreased by as much as 25 Hz as the load was increased. This phenomenon is known as strain softening. The small frequency decrease, less than 5 Hz, for the uncoated plates may be attributed to a less than perfect boundary condition at the fixture. For a perfectly linear system with a perfect boundary condition the peaks would occur at the same frequency for all loads. Another observation is the increased load needed for the coated plate to produce a sweep with an equivalent velocity to the uncoated plate. This is a legitimate method for quantifying Q, but it says nothing about the strain levels and is therefore not used in this report.
As was expected, the mag spinel coating caused an increase in damping; even at very low strain levels. The non-linear relationship between Q and strain is plotted in Figure 38 thru Figure 40. A best fit curve was applied to the experimental results and error bars of 5% were added to each data point to provide a perspective of how well the data fits the trendline (there is no correlation to any predicted precision of the measurements). The Q's show a rapid decrease, increased damping, up to about 100 micro-strain for both modes. The decreasing trend continues beyond 100 micro-strain for mode 3 but it levels off for mode 4. From the best fit curves the predicted improvement in damping over the uncoated plates were calculated for selected strains. The results are presented in Table 21 thru Table 23 for the individual plates. The average increase in damping across all three plates is presented in Table 24. Appendix B contains the experimental data for each point used to generate the curves in Figure 38 to Figure 40.
Mode 4 was tested first for each plate, with the first sweep at the high strain point and the last sweep at the low strain point. The same process was then repeated for mode 3.
Repeat strain points for both modes were measured in the same manner (high strain to low strain) and taken after the sweeps for both modes were complete.
At 10 micro-strain the difference in Q for mode 3 is 16% and for mode 4 is 63%.
At 100 micro-strain the difference in Q for mode 3 is 25% and for mode 4 is 76%. At 500 micro-strain the difference in Q for mode 3 is 31% and for mode 4 is 82%.

Conclusions
The clamp design used for this research showed excellent repeatability during the scanning laser vibrometer tests. This is partially because the excitation source provided by the air horn was very small. Repeatability became an issue when sine sweeps were conducted at much higher input loads on the shakers. Higher mode 3 stresses at the fixture than mode 4 made experimental results more difficult to repeat for mode 3.
Through trial and error, it was discovered that adding jack screws to the rear of the clamp made the data repeatable. The jack screws were threaded through the top clamp and impinged on the top surfa ce of the bottom clamp. This caused the front edge of the clamps to grip the plate tighter and prevented any back and forth vibration in the clamp.
Test results revealed that the resonant frequencies varied by less than 5% when the mag spinel coating was applied and that the mode shapes were unaffected. These measurements were made using the scanning laser vibrometer with an air horn used as the excitation source. The equipment used was not capable of quantifying the actual input force; however, it was very small compared to the levels used for the sine sweep tests. It was assumed for this research that the mode shape was unaltered by the mag spinel at the higher strain levels. By making this assumption the strain/displacement relationship could be used at all strain levels.
Strain softening was seen in the frequency response when the mag spinel was applied. The peaks for each sweep were fairly stable as the load was increased for the uncoated plates, but the peaks for the coated plates decreased by as much as 25 Hz as the load increased. The small frequency decrease, less than 5 Hz, for the uncoated plates may be attributed to a less than perfect boundary condition at the fixture. For a perfectly linear system with a perfect boundary condition the peaks would occur at the same frequency for all loads.
As was expected, the mag spinel coating caused an increase in damping; even at very low strain levels. For all three plates, damping appears to be a function of mode shape. The Q's show a rapid decrease, increased damping, up to about 100 micro-strain for both modes. The decreasing trend continues beyond 100 micro-strain for mode 3 but it levels off for mode 4.
The average increase in damping at 10 micro-strain for mode 3 is 16% and for mode 4 is 63%. At 100 micro-strain the average increase in damping for mode 3 is 25% and for mode 4 is 76%. At 500 micro-strain the average increase in damping for mode 3 is 31% and for mode 4 is 82%.

Recommendations
The width to length ratio of the clamp used for this research was 3.5. By increasing the width and bringing the ratio closer to 1, this researcher feels that the clamp will be more stable and clamping effects will be less likely to creep into the experimental results.
At this time it is uncertain how valid the strain/displacement curves are at higher strain levels than provided by the air horn. Future work should determine the effect of mag spinel on the mode shapes and strains at high strains.
Only one coating thickness was used in this research. The damping sensitivity to thicker and thinner coats should be evaluated. This research should also be repeated at temperatures characteristic of fan, compressor, and turbine blades, since that is the ultimate purpose of this research and the Air Force's interest in damping treatments in general.