ABSTRACT
A hardware / software package is presented that is used to determine the dynamic mechanical properties of viscoelastic materials as a function of temperature and frequency. The material properties can then be used to aid in the design of surface damping treatments for various engineering structures. INTRODUCTION Damping Technologies, Inc. has developed a hardware and software package to conduct Vibrating Beam Technique (VBT) measurements and associated data processing for characterization of the dynamic mechanical material properties of viscoelastic materials. In broad terms, the package includes:
Although many test techniques have been used to measure dynamic mechanical properties of viscoelastic materials, the vibrating beam technique is often utilized because it is conducive to routine data acquisition over wide temperature and frequency ranges. This technique is based upon analysis developed by Ross, Kerwin, and Ungar (RKU) in which the dynamic material properties of viscoelastic materials can be extracted from measurements made on homogeneous as well as ‘composite’ beams. The ‘composite' beam is a generalized term referring to a beam that consists of viscoelastic material in combination with a metal beam (or beams). These can be configured to yield properties in shear or extensional states of stress. Evaluating material properties for a number of bending modes of the beam can monitor the effects of frequency on the material properties.The effects of temperature can be established by placing the test fixture inside an environmental chamber. The technique has been adopted by the American Standards for Testing of Materials as the preferred method for determination of the damping properties of materials. The details of the procedure can be found in ASTM E75698. SAE J1637 Type Tests In addition, it is common in industry today, particularly the automotive industry, to use the vibrating beam technique for evaluation of system loss factor performance of materials and treatments applied to “standard” vibrating beam samples. Using this approach, performace of treatments and materials can be compared to one another in terms of standard beam geometry in a very strightforward manner. The degree to which the standard beam specimen represents the actual application is often debated. However, the test does result in data that can be very useful for comparison purposes, particularly in quality control applications. Ford, Chrysler, Honda, and others specify these tests for evaluation of damping treatments. The VBT Measurement System Damping Technologies VBT Measurement System was created with the emphasis on flexibility and ease of use. The package has been thoughtfully created by structural dynamics engineers who have logged extensive experience using the Vibrating Beam Technique. As knowledgeable individuals realize, there quite simply is no substitute for experience regarding these measurements. Experience is crucial to the success of any material property measurement capability and is certainly a key ingredient of this particular effort. The VBT Measurement System consists of:
Composite Property Measurement – VBT Measurement ModuleThe VBT Measurement software interfaces with the data acquisition hardware and environmental chamber to automate testing of viscoelastic material specimens. Composite property data (i.e. mode number, frequency, and loss factor) is acquired for a specified temperature profile. The acquired composite properties are then applied to the post processor for material property extraction and data display. Four types of beams can be used to measure the dynamic properties of viscoelastic materials: 1. Homogeneous Beam – Single beam coonsisting of the material to be evaluated. 2. Sandwich Beam – Viscoelastic material sandwiched between two metal beams. 3. Oberst Beam – Viscoelastic material on one side of a metal beam in a freelayer configuration. 4. Modified Oberst Beam – Viscoelastic Material on both sides of steel beam in a freelayer configuration. The beams are usually mounted in a cantilevered configuration, however, freefree, pinpin, and clamped – clamped boundary conditions are also accommodated in the software. Figure 2 decribes the typical VBT System hardware test configuration for measuring the composite beam properties. In this configuration, a random noise signal is supplied to the free end of the beam via magnetic exciter. The response to this excitation is measured with a piezoelectric crystal mounted near the root of the beam. This configuration is demonstrated in Figure 3. With the known input and response of the beam, a frequency response function can be generated to determine the natural frequency and damping for each mode of the beam.
Measurement Methodology The following steps are performed to acquire composite beam data as a function of temperature: 1. Set environmental chamber to desired temperature. 2. Allow time for beam and fixture to stabilize at temperature. 3. Acquire broadband frequency response and identify modes and corresponding halfpower band width. 4. Acquire narrowband (zoom FFT) frequency response on each mode. Determine natural frequency and loss factor from this measurement. 5. Repeat steps 3 and 4 for each additional fixture. 6. Proceed to next selected temperature and repeat steps 2 through 5. An important process in the composite frequency and loss factor measurements is the proper identification of bending modes for the beam specimen. This information is determined from the broadband frequency response measurement. Observing the relative differences in the local maxima and minima identifies peaks in the response. A peak in the frequency response function could be a result of a bending mode, torsion mode or noise. A modetracking algorithm was developed to separate these three possibilities since we are only interested in the bending modes of the test beam. The algorithm observes each peak in the response and determines the ‘bestfit’ based on eigenvalue spacing. The eigenvalue spacing is determined from the boundary conditions of the beam specimen. Assuming a constant bending stiffness, the spacing between each frequency should be proportional to the eigenvalues of the beam bending modes. The advantage to this technique is ‘hands free’ operation. Once the test is started, the operator can walk away without further interaction with the data acquisition system. Figure 4, Variation of the modal Resonance Frequency with Temperature Figure 5, Variation of the Modal Loss Factor with Temperature
Figures 4 and 5 display the measured composite frequency and damping values from a typical damped beam specimen. The composite modal loss factor values are extracted from the narrow band frequency response measurements using the “n dB” down technique as specified in ASTM E756 and SAE J1637 which is governed by the following equation: where, x = 10^(n/20) n = “n dB” down point f_{n} = frequency bandwidth for “n dB” down f_{c} = resonance frequency Data Post Processing and Data Presentation – VBT Data Processing ModuleComposite beam frequency and loss factor only provides limited information regarding the characterization of a measured viscoelastic material (VEM).These measured values are specific to the particular beam material and geometry. To determine how a given VEM would perform on a specific engineering structure, we must first extract the raw material properties from the composite properties. Equations developed by Ross, Kerwin and Ungar (RKU) extract the viscoelastic material shear or Young’s modulus and material loss factor from these layered test configurations. The RKU equations utilize the measured composite data in combination with the particular beam geometry and material densities. These equations are dependent on the type of beam specimen used in the test, i.e. homogeneous, sandwich, oberst, or modified oberst beam. The equations can be found in ASTM E 75680. Temperature/Frequency Nomogram We can further reduce the data to a more functional method of data display and analysis using the concept of temperature / frequency superposition. Using temperature / frequency equivalence, we can display the material properties as a function of temperature and frequency simultaneously.Since the material properties vary trmendously as a function of temperature and to a high degree as a function of frequency, this is a very useful format for manipulating and displaying the properties. For example, for a given viscoelastic material, the modulus and loss factor might be the same at 10 Hz, 25 C as it is at 700 Hz, 50 C. Viscoelastic materials behave “colder” at high frequency and “warmer” at low frequencies. There is a relationship between temperature and frequency. In fact, there are several ways to generate properties of a given viscoelastic material. The temperature could be held constant and the material tested over a very wide frequency range. Or, the temperature could be varied and the material tested over a much narrower frequency range. Then, temperature / frequency equivalence is applied to “reduce” the data. Both techniques yield the same “master curves” for the material. For the case of the limited frequency range and wide temperature range, frequency data for each temperature is “shifted” an appropriate amount until the modulus and loss factor curves collapse to form smooth lines. f*a_{t} is the frequency of each measurement data point multiplied by the “shift factor” at the particular temperature.a_{t} is known as the “shift factor”. The shift factor defines the relationship between frequency and temperature for the particular material. There are several equations that define this relationship. However, one of the most commonly applied shift factor (and our favorite) is the Arrhenius equation: where, m = Activation Temperature T = Temperature T_{ref} = Reference Temperature The reference temperature, T_{ref}, is traditionally defined as the temperature at which peak loss factor occurs at a frequency of 1 kHz, however, this value is arbitrary. The activation temperature parameter defines the relationship between temperature and frequency for the particular material. Figure 6, Jones Temperature/Frequency Nomogram
Figure 6 demonstrates a typical Jones Reduced Frequency Nomogram with a curve fit applied. Curve fitting the data in this format yields dynamic mechanical properties (complex modulus properties) of the given material as a simultaneous function of temperature and frequency described by (2) relatively simple equations. One equation for modulus. One equation for loss factor. This is our goal for quantifying the dynamic properties. We now have them conveniently expressed by only (2) equations. Equations for various materials can be catalogued in a library of viscoelastic materials which can be made available to the the VBT Design Module or particular finite element analysis models. Using the reduced frequency nomogram format and curve fits, we can determine the modulus and loss factor of viscoelastic material for virtually any temperature and frequency. In theory, if the temperature / frequency superposition is adequately applied to the data set, the resulting data can be interpolated and extrapolated to any temperature or frequency. However, caution must be used when extrapolating properties far from the limits of data acquisition. Figure 7 describes a plot of variation of the dynamic mechanical properties with temperature for several constant frequencies. This data was derived from the curve fits of the data of Figure 6. Figure 7, Variation of the Dynamic Mechanical Properties with Temperature for the Indicated Constant Frequencies
Damping Treatment Design – VBT Design ModuleWith the viscoelastic material modulus and loss factor defined by (2) mathematical equations, we can determine the effect of a given viscoelastic material in damping treatment configurations other than those tested. This process is actually the reverse of the process used to extract the complex modulus data from the “composite” measurement beams. Using the RKU equations^{(3)}, we can observe effects of numerous design parameters to optimize damping in “realworld” engineering structures. This is achieved where the target structural dynamics can be approximated via one or more modes of a cantilever beam, pinnedpinned beam, clampedclamped beam, or pinnedpinnedpinnedpinned plate analogy. This modeling process is often useful on it’s own, or can be used as a preprocessor prior to a investigation of a damping system design using a fullblown FEA model. Figures 8, 9, and 10 describe several damping system designs using this methodology. Note: The VBT Design Module software has the ability to predict damping treatment performance as a function of constraining layer or viscoelastic layer thickness. Singlelayer or multilayer treatments may be designed. Damping systems may easily be optimized for weight, performance, and cost. The methodology allows the user to analytically design and optimize the damping system thereby avoiding the costly (and often unfruitful) matrix of test iterations which is often the common approach. The dynamic mechanical properties generated with VBT are obviously very useful in combination with finite element models to predict damping using a modal strain energy or complex eigenvalue type analysis. Figure 8, Predicted Modal Resonance Frequency and Modal Loss Factor as a Function of Temperature
Figure 9, Predicted Modal Resonance Frequency and Modal Loss Factor as a Function of Temperature
Figure 10, Predicted Modal Resonance Frequency and Modal Loss Factor as a Function of Temperature
CONCLUSION  VBT System Characterization of viscoelastic material properties is the key ingredient to creating an effective surface damping treatment. The VBT Measurement System was developed to measure viscoelastic material properties and utilize these characteristics to design effective surface damping treatments. Three separate software programs were developed to perform these tasks with the following features: VBT Measurement Data acquisition hardware and software system that automates the measurement of composite beam properties as a function of temperature.
VBT Post Processor Extraction of dynamic material properties, data display, Jones Temperature / Frequency Nomogram, and curvefitting.
VBT Damping Treatment Design
