AMPING CROSS-REFERENCE There are at least eleven parameters commonly used to express damping. Cross-reference formulas are given in Tables A through C. The formulas are taken from Reference. Let be the natural frequency in units of radians per second. Note that is in units of Hertz. n π f n, where f n Table A. amping Reference Parameter 3 db Bandwidth (rad/sec 3 db Bandwidth (Hz (rad/sec π amping Frequency fd (Hz 4πfd Loss Factor (Hz amping Frequency fd (Hz Loss Factor Fraction of Critical amping π 4 π n n Quality Factor Q Q ecay Constant σ σ (/sec Time Constant τ (sec Reverberation Time RT 60 (sec ecay Rate (db/sec Logarithmic ecrement δ 3.8 RT 60 4. 34 π δ n π π Q π 4πfd πfd Q 4πfd π 4 π Q σ n σ π σ πfd π πfd.. 3.8 RT 60 RT 60 RT 60 fd 7.3 54.6fd 4.34 δ π f n δ 4 π fd n δπ
Table B. amping Reference Parameter Fraction of Critical amping (rad/sec (Hz amping Frequency fd (Hz Loss Factor Fraction of Critical amping Quality Factor Q ecay Constant σ (/sec π π Quality Factor Q Q πq 4πQ Q Q Q σ σ Q ecay Constant σ (/sec σ σ π σ π σ σ n Q σ Time Constant τ (sec τ πτ πτ τ τ Q n τ σ τ Time Constant τ (sec Reverberation Time RT 60 (sec ecay Rate (db/sec Logarithmic ecrement δ 6.9 RT 60 8.68 δ π Q 3.8Q RT 60 4.34 Q π δ Q σ 6. 9 RT 60 σ 8. 68σ πσ δ RT 60 6. 9τ 8.68 τ π δ τ
Table C. amping Reference Parameter Reverberation (rad/sec (Hz amping Frequency fd (Hz Loss Factor Fraction of Critical amping Quality Factor Q ecay Constant σ (/sec Time RT 60 (sec 3.8.. fd 3.8 6.90 Q n 3.8 6.90 σ 6. Time Constant τ (sec 90 Reverberation Time RT 60 (sec ecay Rate (db/sec Logarithmic ecrement δ 60 43.4 δ ecay Rate (db/sec 4.34 7.3 54.5 4.34 8.68 4.34 Q σ 8.68 8.68 RT 60 60 π δ 4.34 Logarithmic ecrement δ δ π δ π δ 4π δ π δ π π Q δ δ σ π π δ 43.4 RT 60 δ.38δ Reference. Svend Gade and Henrik Herlufsen, "igital Filter versus FFT Techniques for amping Measurement," Sound and Vibration, Bay Village, Ohio, March 990.
AMPING PROPERTIES OF MATERIALS The purpose of this tutorial is to give typical damping values for various materials and systems. The data in Tables and is taken from Reference. Table. Static Properties of Materials under Standard Conditions (approx. 0 C. Material ensity (kg/m 3 Elastic Modulus (N/m Shear Modulus (N/m Poisson s Ratio Aluminum 700 7 (0 9 7 (0 9 0.34 Lead,300 7 (0 9 6 (0 9 0.43 Iron 7800 00 (0 9 77 (0 9 0.30 Steel 7800 0 (0 9 77 (0 9 0.3 Gold 9,300 80 (0 9 8 (0 9 0.43 Copper 8900 5 (0 9 46 (0 9 0.35 Magnesium 740 43 (0 9 7 (0 9 0.9 Brass 8500 95 (0 9 36 (0 9 0.33 Nickel 8900 05 (0 9 77 (0 9 0.30 Silver 0,500 80 (0 9 9 (0 9 0.37 Bismuth 9800 3.3 (0 9.3 (0 9 0.38 Zinc 730 3.(0 9 5 (0 9 0.33 Tin 780 4.4 (0 9.6 (0 9 0.39
Table. ynamic Properties of Materials under Standard Conditions (approx. 0 C Material Propagation Velocity of Longitudinal Wave in a Rod Propagation Velocity of Torsional Wave Longitudinal Loss Factor Aluminum (meters/sec (meters/sec 500 300 0.3 to 0 ( 0 5 Lead 5 to 30 0 Flexural Loss Factor 0 4 ( ( 0 (pure 50 730 Lead (including to 4 ( 0 3 antimony Iron 5050 300 to 4 ( 0 4 to 6( 0 4 Steel 500 300 0. to 3 ( 0 4 Gold 000 00 ( 4 3 0 Copper ( 0 3 ( 0 3 (polycrystalline 3700 300 Copper to 7 ( 0 4 (single crystal Magnesium 5000 300 0 4 Brass 300 00 ( 3 0. to 0 < 0 3 Nickel 4800 900 < 0 3 Silver 700 600 4 0 4 < 3 ( 0 3 Bismuth 580 360 ( 4 8 0 Zinc 350 850 ( 4 3 0 Tin 780 470 ( 4 0 0 Notes:. Some loss factors are unavailable.. The relationship between the loss factor and the viscous damping ratio ξ is: ξ.
The data in Table 3 is taken from Reference. Table 3. Representative amping Ratios System Viscous amping Ratio ξ Metals (in elastic range <0.0 Continuous Metal Structures 0.0 to 0.04 Metal Structure with Joints 0.03 to 0.07 Aluminum / Steel Transmission Lines 0.0004 Small iameter Piping Systems 0.0 to 0.0 Large iameter Piping Systems 0.0 to 0.03 Auto Shock Absorbers 0.30 Rubber 0.05 Large Buildings during Earthquakes 0.0 to 0.05 Prestressed Concrete Structures 0.0 to 0.05 Reinforced Concrete Structures 0.04 to 0.07 The data in Tables 4 through 6 is taken from Reference 3. Table 4. Material amping Ratios (Bare Structure System Reinforced Concrete Small Stress Intensity (uncracked Medium Stress Intensity (fully cracked High Stress Intensity (fully cracked but no yielding of reinforcement Viscous amping Ratio ξ 0.007 to 0.00 0.00 to 0.040 0.005 to 0.008 Prestressed Concrete (uncracked 0.04 to 0.07 Partially Prestressed Concrete (slightly cracked 0.008 to 0.0 Composite 0.00 to 0.003 Steel 0.00 to 0.00 C. 3
Table 5. Footbridge amping Construction Type Viscous amping Ratio ξ Min. Mean Max. Reinforced Concrete 0.008 0.03 0.00 Prestressed Concrete 0.005 0.00 0.07 Composite 0.003 0.006 - Steel 0.00 0.004 -. Table 6. Building amping Construction Type Tall Buildings ( h > ~00 m Viscous amping Ratio ξ Min. Mean Max. Reinforced concrete 0.00 0.05 0.00 Steel 0.007 0.00 0.03 Buildings ( h ~ 50 m Reinforced concrete 0.00 0.05 0.030 Steel 0.05 0.00 0.05 3. The damping values in the tables should be used with caution. There are many types of damping, such as viscous, hysteresis, acoustic coupling, air pumping at joints, energy radiation to the soil, etc. Also, boundaries and bearings contribute damping. Furthermore, structures have many modes. Each mode may have a unique damping value. References. L. Cremer and M. Heckl, Structure-Borne Sound, Springer-Verlag, New York, 988. 4
. V. Adams and A. Askenazi, Building Better Products with Finite Element Analysis, OnWord Press, Santa Fe, N.M., 999. 3. H. Bachmann, et al., Vibration Problems in Structures, Birkhauser Verlag, Berlin, 995. 5