Viscous vs. Structural Damping in Modal Analysis

It is shown that among the various damping mechanisms that are generally encountered in a mechanical structure, only the “viscous component” actually accounts for energy loss. The remaining portion of the damping force is due to non-linearities, which do not dissipate energy. Consequently, a linear model of a vibratory mechanical system involves only viscous damping, although the damping factor may depend upon the waveform or amplitude of the excitation signal. Introduction The motion of an elastic mechanical system is commonly modeled in the time domain by the equation ) t ( f ) t ( Kx ) t ( x C ) t ( x M = + + & & & (1) where f(t) is the driving force vector, and x(t) is the resulting displacement vector of a system with mass. Damping, and stiffness matrices denoted by M, C, and K, respectively. The dots indicate derivatives with respect to the time variable (t). If we pre-multiply this equation by the transposed velocity vector ) t ( x ) t ( v t t & = , we obtain an instantaneous power balance equation. ) t ( Kx ) t ( v ) t ( Cv ) t ( v ) t ( x M ) t ( v t t t + + & & ) t ( f ) t ( v = (2) We can integrate this equation over any time interval ) (τ of interest to obtain an energy balance equation for the particular time interval. The energy associated with the mass and stiffness matrices is stored energy that can always be recovered, but the portion given by ∫τ dt ) t ( Cv ) t ( v t , is dissipated, usually in the form of heat, and is lost from the system. In this mathematical formulation, the force ) t ( Cv is called a viscous damping force, since it is proportional to velocity. However, as we discuss next, this does not necessarily imply that the physical damping mechanism is viscous in nature. It is important to recognize that the physical damping mechanism and the mathematical model of the mechanism are two distinctly different concepts. The term “viscous” is commonly used indiscriminately to denote both a damping mechanism (i.e. fluid flow), and a mathematical representation of dissipated energy described by a force (i.e. ) t ( Cv ) that is proportional to velocity. In practice most mechanical structures exhibit rather complicated damping mechanisms, but we will show that all of these can be mathematically modeled by a force proportional to velocity, so that the mathematical usage of the term “viscous” is generally implied. We will show that there is always a viscous component of damping force (proportional to velocity), and that this viscous component accounts for all energy loss from the system. We will see that all remaining force terms are due to non-linearities, and do not cause energy dissipation. Thus, we only need to measure the viscous term to characterize the system using a linear model. In appendix A, we discuss the current technique of using an imaginary stiffness to model structural damping (for sinusoidal excitation), and in appendix B. we discuss the concept of hereditary damping introduced by Klosterman [4]. Reference [5] is also recommended for a general discussion of damping mechanisms and mathematical models. Damping Mechanisms Three of the most common damping mechanisms are: 1.) coulomb (sliding frictions [2] in which the force magnitude is independent of velocity.) 2.) viscous, where force is proportional to velocity, and 3.) structural (hysteretic, internal, material) [3], in which the force is proportional to the magnitude of the displacement from some quiescent position. From a microscopic point of view, most damping mechanisms involve frictional forces that oppose the motion (velocity) of some part of a physical system, resulting in heat loss. For example coulomb friction force is caused by two surfaces sliding with respect to one another, and this sliding force is independent of velocity, once the initial static friction (stiction) is overcome. Structural damping may be viewed as a sliding friction mechanism between molecular layers in a material, in which the friction force is proportional to the deformation or displacement from some quiescent or rest position. Imagine a rod made of a bundle of axial fibers. The sliding friction force between each fiber and its neighbor will increase as 46 Shock and Vibration Symposium October 1975 Page 2 of 7 the rod is bent and the fibers are pinched together. This pinching phenomenon occurs in most materials as the various molecular layers slide past one another. The result is a damping force that is proportional to the displacement from the undisturbed position. This mechanism was verified for a wide range of materials by Kimball and Lovell in 1927 [3]. Viscous damping occurs when molecules of a viscous fluid rub together, causing a resistive friction force that is proportional to, and opposing the velocity of an object moving through the fluid. We can conveniently characterize damping mechanisms of these types by the force equation. ) v sgn( ) t ( g c ) t ( f − = (3) Where ) t ( f is the damping force, c is a scalar damping coefficient, ) t ( g is some arbitrary magnitude function, and sgn(v) is the signum of velocity defined by 1 ) v sgn( = , for v > 0 0 = , for v = 0 1 − = , for v < 0 (4) We can catalog the three most common damping mechanisms by choosing the appropriate ) t ( g as follows: 1) Coulomb: c ) t ( f , 1 g c − = = ) v sgn( (5) 2) Viscous: cv ) v sgn( v c ) t ( f , v g v − = − = = (6) 3) Structural: ) v sgn( x c ) t ( f , x g s − = = (7) Notice that only the viscous damping force is a linear function of velocity, and that the other mechanisms are inherently non-linear in nature. As an example, let's assume that the displacement is sinusoidal so. , t sin ) t ( x ω = and (8) t cos ) t ( x ) t ( v ω ω = = & (9) The resulting coulomb damping force is obviously a square wave of period ω π 2 , with peak amplitude c. The viscous damping force is a cosine function, and the structural damping force is the product of the coulomb square wave force times t sin ω . Hysteresis Curves It is instructive to plot damping force vs. displacement for the three cases mentioned above. These plots are called “hysteresis” curves, and we will see that the areas enclosed by these curves represent the system energy loss per cycle of excitation. For viscous damping, we obtain the ellipse shown in Figure 1, having an area of c πω , units of energy loss per cycle. Coulomb damping yields a rectangular hysteresis curve, shown in Figure 2. Structural damping causes the “bow tie” curve shown in Figure 3. Figure 1. Hysteresis Curve for Viscous Damping Figure 2. Hysteresis Curve for Coulomb Damping 46 Shock and Vibration Symposium October 1975 Page 3 of 7 Figure 3. Hysteresis Curve for Structural Damping An “equivalent viscous” representation can always be obtained by constructing an elliptical hysteresis curve having the same area and corresponding to the same displacement as the actual hysteresis curve. The resulting force is simply the “equivalent viscous” force for that particular mechanism. The use of the term “hysteretic” damping is somewhat confusing, since all damping mechanisms involve a hysteresis curve of some sort. Thus, we prefer to use the word “structural” to describe this particular mechanism. It should be emphasized that the hysteresis curves, which have been illustrated, are for the case or sinusoidal displacement. The curves will be different if other displacements are used. By definition, the total energy loss over each cycle of the displacement is given by dx ) t ( f 2 dt ) t ( v ) t ( f 1 1 0 2 ∫ ∫ − = = ε ∆ ω π (10) which is simply the area enclosed by the appropriate hysteresis curve. Notice that his area is proportional to frequency for the viscous case, but independent of frequency for the other two mechanisms. This explains why the damping factor of structures having primarily non-viscous damping remains low. Even at high resonant frequencies. If all damping were viscous, then small, high frequency bells would react to a strike with a dull thud, instead of a clear tinkle. Fourier Series Analysis Each of these damping forces can be represented by a Fourier series. For the coulomb case, we have .. , 5 , 3 , 1 k , ) 1 ( k c 4 2 k F 2 1 k c = − π − =       π ω − (11) where k is the harmonic number, and c F ( π ω 2 k ) is the frequency spectrum amplitude of ) t ( fc . For structural damping, we can write [ ] 5 , 3 , 1 k , 1 2 ) 1 ( c 2 2 k F 4 k 2 1 k s = + − π − =       π ω −