Committee Draft No. 99 To be combined with T-150 as a method B. Determination of Natural Frequency and Flexural Modulus by Experimental Modal Analysis

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1 Committee Draft No. 99 To be combined with T-150 as a method B CCTI Standard Testing Procedure T-148 rev. special August 2002 Determination of Natural Frequency and Flexural Modulus by Experimental Modal Analysis Objective This test method is used to determine a fiber core s flexural modulus in bending by using the experimentally measured first natural frequency of the core in the free-free mode of transverse vibration. This flexural modulus is then used by T-150 part 2 to calculate critical speed. Note 1: Flexural Modulus is a material property of the core and together with the cores dimensions relates to the resistance of the core to bending deflection. For wide paper widths, stiff cores are needed to minimize unwanted vibrations which can lead to poor roll structure during winding and web flutter and breaks during printing. Note 2: This test method is applicable to cores having a: 2 14 (51mm 355mm) inside diameter minimum wall thickness of 0.02 times the inside diameter and greater than 0.1 (2.5mm). minimum length of eight times the inside diameter. Material and Equipment (1) Light weight clamps and wire from which you hang the core. The clamps should not weigh more than 0.01 times the weight of the core. (2) An accelerometer 1 which weighs no more than 0.01 times the weight of the core, has an mv/g sensitivity and a frequency range of 0.1 to 10,000 Hz. (3) A coupler 2 which also supplies the controlled current and voltage necessary for driving the accelerometer. (4) Two accelerometer/coupler cables are required 3. It should be noted that these cables tend to be extremely fragile. (5) The hammer is used to tap the core and cause it to vibrate. Any hard rubber, wooden, or plastic tipped hammer is preferred to avoid damaging the core. (6) A signal analyzer 4 is used to read the output from the coupler. This device is sometimes called an FFT analyzer for the Fast Fourier Transform mathematical calculations used to convert amplitude signal (or time domain) information into the frequency spectrum that reveals the natural frequency. Procedure (1) Condition the sample in accordance with CT-142 (2) To ensure that the core satisfies the free-free boundary condition, it must be hung in such a way that it is free to vibrate in its transverse direction, (figure 1). One recommended technique is to suspend it from lightweight clamps, (see

2 figure 2). In addition, the angle (α in figure 2) should be more than 45 and the distance from the screw from the end of the core should be approximately 3/8. The length of the support wire should not be less than 12. It can also be horizontal and resting on supports at each end with the accelerometer attached to the center. (3) Attach the accelerometer to the lower end of the tube with its long axis in the Z-direction. Electrical connections re made as shown in figure 1. The accelerometer-to-coupler connection should allow free movement of the core. (4) Set the signal analyzer to trigger on the input signal. Tap the core at mid-length with the hammer (figure 1) and adjust the coupler gain factor, high-pas filter cutoff frequency, analyzer input gain, trigger level/sensitivity and offset to get a clearly defined signal. Note 3: The actual adjustments to optimize the signal will vary with the coupler/analyzer chosen. When the settings are optimized, again tap the core with the hammer to initiate data acquisition. (5) The frequency spectrum displayed on the analyzer is a plot of amplitude versus frequency. The dominant peak should be the cores fundamental or first natural frequency in bending. Overtones or harmonics may also be present. These are peaks at multiples of the fundamental frequency. Repeat the measurement for a total of five times. Record the fundamental frequency (ƒ) for each measurement. This value is used in Equation 1 to calculate the flexural modulus of the tested sample core. Calculation (1) The Flexural Modulus in bending, E(N/mm 2 ), is calculated according to the following formula 2 4 f m L Q E = I Where ƒ(hz) is the first natural frequency, L(mm) is the length of the specimen and m(kg/m) is the mass per unit length of the specimen. Q is dimensionless and I (mm 4 ) is the second moment of inertia. (2) The second moment of inertia, I(mm 4 ), of the core cross section is calculated according to the following equation: π 4 4 I = ( D d ), 64 where D(mm) is the outside diameter of the core and d(mm) is the inside diameter of the core. (3) The area, A(mm 2 ), of the core cross section is calculated according to the following equation: π 2 2 A = ( D d ). 4 (4) Q is a dimensionless coefficient, which is calculated according to the following equation: 1. Test date I Q = AL Report of Results 2. Temperature, and relative humidity of the testing lab.

3 3. Sample identification number and other pertinent sample information. 4. Core dimensions - - inside diameter (d, mm), outside diameter (D, mm), and length, (l, mm). 5. Moisture content of the core. 6. The five individual fundamental frequencies (ƒ, Hz). 7. The five individual values and the mean for flexural modulus (E, N/mm 2 ) along with the standard deviation. 8. Frequency spectrum of the sample core if desired. 9. Any deviation from this method. Reference document ISO/TC6/SC#/WG4/N97/REV. C 1. One accelerometer is model #8730A500, Kistler Instrument Corporation, 75 John Glen Drive, Amherst, NY tel(716) One Coupler is model #5118A2, Kistler Instrument Corporation, 75 John Glen Drive, Amherst, NY (716) Two workable cables are the model #1631C and #1511, Kistler Instrument Corporation, 75 John Glen Drive, Amherst, NY ,(716) The NI-4552 is an FFT card which can be installed in a standard PC. National Instruments, 6504 Bridgepoint Parkway, Austin, TX , (512)

4 Hanging Cable Z Y Resonance Frequency X Tube Clamps Paper Tube Impact Hammer Frequency Analyzer Accelerometer Accelerometer Amplifier or Coupler Figure 1. Experimental Setup

5 support Light weight wire Clamp or drilled hole α cone point screw Core section Figure 2. Schematic Drawing of the Specimen Support