167 Chapter 3 Optimal passive damping distribution 3.1 Introduction Damping is the process by which physical systems such as structures dissipate and absorb the energy input from external excitations. Damping reduces the build-up of strain energy and the system response, especially near resonance conditions where damping governs the response. Figure 3.1 illustrates the influence of damping on the time history of the strain energy for a system with mass of 1 kg and a period of 1 second subjected to the unscaled El Centro accelerogram shown in Fig. 2.21. The symbols in Fig. 3.1 refer to the energy input ( ), the energy stored ( ), and the energy dissipated ( ). During the early stage of the response, there is a rapid build-up of the input energy, similar to an impulsive loading. Damping dissipates energy over a response cycle, in this case, 1 second. For low damping ratio, the energy dissipated per cycle is small, and many cycles are required before the input energy is eventually dissipated. As is increased, the energy dissipated per cycle increases, and the stored energy build up is reduced. Shifting from = 0.02 to = 0.1 reduces the peak stored energy demand by a factor of 3.7 for this particular system and earthquake excitation. It should be noted that seismic accelerograms differ with respect to their frequency content and intensity, and therefore one needs to carry out energy time history studies for individual excitations applied to a specific system. For example, Fig 3.2 shows the response of the same system for a typical Northridge accelerogram. The input energy build up for the Northridge loading is quite different than for the El Centro loading. E I E s E D ξ ξξ 168 Chapter 3: Optimal Passive Damping Distribution Fig. 3.1: Energy Build Up, El Centro (S00E), Imperial Valley 1940 02 468101214161820 0.2 0.4 0.6 0.8 1.0 0 02 468101214 0.2 0.4 0.6 0.8 0 02 468101214161820 0.2 0.4 0.6 0.7 0 Time (s) Energy (J)Energy (J)Energy (J) ξ 2%= ξ 5%= ξ 10%= 16 18 20 E I E D E S 3.1 Introduction 169 Fig. 3.2: Energy Build Up, Arleta Station (90 DEG), Northridge 1994, = 2% Dissipation and absorption are attributed to a number of external and internal mechanisms, including the following: • Energy dissipation due to the viscosity of the material. This process depends on the time rate of change of the deformations, and is referred to as material damping. Viscoelastic materials belong to this category. • Energy dissipation and absorption caused by the material undergoing cyclic inelastic deformation and ending up with some residual deformation. The cyclic inelastic deformation path forms a hysteresis loop which correspond to energy dissipation; the residual deformation is a measure of the energy absorption. This process is generally termed hysteretic damping. • Energy dissipation associated with overcoming the friction between moving bodies in contact, such as flexible connections. Coulomb damping refers to the case where the magnitude of the friction force is constant. Structural damping is a more general friction damping mechanism which allows for a variable magnitude of the friction force. • Energy dissipation resulting from the interaction of the structure with its surrounding environment. Relative motion of the structure E I E D E S 0 2 4 6 8 101214161820 0 Time (s) 0.2 0.4 0.6 0.8 1.0 Energy (J) ξ 170 Chapter 3: Optimal Passive Damping Distribution generates forces which oppose the motion and extract energy from the structure. Fluid-structure interaction is a typical case. The fluid exerts a drag force which depends on the relative velocity and functions as an equivalent viscous damping force. • Damping devices installed at discrete locations in structures to supplement their natural energy dissipation/absorption capabilities. These mechanisms may be passive or active. Passive mechanisms require no external energy, whereas active mechanisms cannot function without an external source of energy. Passive devices include viscous, friction, tuned mass, and liquid sloshing dampers. Active damping is achieved by applying external control forces to the structure over discrete time intervals. The magnitudes of the control forces are adjusted at each time point according to a control algorithm. • Passive damping removes energy from the response, and therefore can not cause the response to become unstable. Since active control involves an external source of energy, there is the potential for introducing an instability in the system. The term “semi-active” refers to a particular class of active devices that require a relatively small amount of external energy and apply the control force in such a way that the resulting motion is always stable. Chapter 6 discusses active control devices. In this chapter, the response characteristics for material, hysteretic, and friction damping mechanisms are examined for a single degree-of-freedom (SDOF) system. The concept of equivalent viscous damping is introduced and is used to express viscoelastic, structural, and hysteretic damping in terms of their equivalent viscous counterpart. Numerical simulations are presented to demonstrate the validity of this concept for SDOF systems subjected to seismic excitation. This introductory material is followed by a discussion of the influence of distributed viscous damping on the deformation profiles of multi-degree-of- freedom (MDOF) systems. The damping distribution is initially taken to be proportional to the converged stiffness distribution generated in the previous chapter, and then modified to allow for non-proportional damping. Numerical results and deformation profiles for a range of structures subjected to seismic loading are presented, and the adequacy of this approach for distributing damping is assessed. Distributed passive damping can be supplemented with one or more discrete damping devices to improve the response profile. Discrete viscous 3.2 Viscous,Frictional, and Hysteretic Damping 171 dampers inserted in discrete shear beam type structures are considered in this chapter; the basic theory for tuned mass dampers is presented in the next chapter. Subsequent chapters deal with base isolation, a form of passive stiffness/ damping control, and active control. 3.2 Viscous, frictional, and hysteretic damping Viscous damping Viscous damping is defined as the energy dissipation mechanism where the damping force is a function of the time rate of change of the corresponding displacement measure: (3.1) where is the damping force and is the velocity in the direction of . The linearized form is written as: (3.2) where , the damping coefficient, is a property of the damping device. Linear viscous damping is convenient to deal with mathematically and therefore is the preferred way of representing energy dissipation. In general, the work done on the device during the time interval , is given by (3.3) Considering periodic excitation (3.4) and evaluating eqn (3.3) for one full cycle under linear viscous damping leads to (3.5) This term represents the energy dissipated per cycle by the damping device, as the system to which it is attached undergoes a periodic motion of amplitude and frequency . Figure 3.3 shows the force-displacement relationship for periodic excitation; the enclosed area represents . Ffu ˙ ()= Fu ˙ F Fcu ˙ = c W t 1 t 2 ,[] WFud ut 1 () ut 2 () ∫ Fu ˙ td t 1 t 2 ∫ == uu ˆ Ωtsin= W viscous cπΩu ˆ 2 = u ˆ Ω W 172 Chapter 3: Optimal Passive Damping Distribution Fig. 3.3: Viscous response - periodic excitation Example 3.1: Viscous damper Figure 3.4 shows a possible design for a viscous damping device. The gap between the plunger and external plates is filled with a linear viscous fluid characterized by (3.6) where and are the shearing stress and strain measures respectively and is the viscosity coefficient. Assuming no slip between the fluid and plunger, the shear strain is related to the plunger motion by (3.7) Fig. 3.4: Viscous damping device t0= t π Ω = uu ˆ Ωtsin= u u ˆ u ˆ – cΩu ˆ F τ G v γ ˙ = τγ G v γ u t d = t d t d L F , u Side View End View w fluid fluid plunger 3.2 Viscous,Frictional, and Hysteretic Damping 173 where is the thickness of the viscous layer. Letting and represent the initial wetted length and width of the plunger respectively, the damping force is equal to (3.8) Substituting for results in (3.9) Finally, eqn (3.9) is written as (3.10) where represents the viscous coefficient of the device, (3.11) The design parameters are the geometric measures and the fluid viscosity, . A schematic diagram of a typical viscous damper employed for structural applications is contained in Fig 3.5; an actual damper is shown in Fig 3.6. Fluid is forced through orifices located in the piston head as the piston rod position is changed, creating a resisting force which depends on the velocity of the rod. The damping coefficient can be varied by adjusting the control valve. Variable damping devices are useful for active control. Section 6.4 contains a description of a particular variable damping device that is used as a semi-active force actuator. This chapter considers only passive damping, i.e. a fixed damping coefficient. Fig. 3.5: Schematic diagram - viscous damper t d Lw F 2wLτ= τ F 2wLG v t d u ˙ = Fcu ˙ = c c 2wL t d G v = wLt d ,, G v 174 Chapter 3: Optimal Passive Damping Distribution Fig. 3.6: Viscous damper - 450 kN capacity (Taylor Devices Inc. http://www.taylordevices.com) Equation (3.5) shows that the energy loss per cycle for viscous damping depends on the frequency of the excitation. This dependency is at variance with observations for real structural systems which indicate that the energy loss per cycle tends to be independent of the frequency. In what follows, a number of damping models which exhibit the latter property are presented. Friction damping Coulomb damping is characterized by a damping force which is in phase with the deformation rate and has constant magnitude. Mathematically, the force can be expressed as (3.12) where denotes the sign of . Figure 3.7 shows the variation of with for periodic excitation. The work per cycle is the area enclosed by the response curve FF u ˙ ()sgn= u ˙ ()sgn u ˙ Fu 3.2 Viscous,Frictional, and Hysteretic Damping 175 Fig. 3.7: Coulomb damping force versus displacement (3.13) Figure 3.8 shows a coulomb friction damper used with diagonal X bracing in structures. Friction pads are inserted at the bolt-plate connections. Interstory displacement results in relative rotation at the connections, and the energy dissipated is equal to the work done by the frictional moments during this relative rotation. Fig. 3.8: Friction brace damper Structural damping removes the restriction on the magnitude of the damping force, and considers the force to be proportional to the displacement amplitude. The definition equation for this friction model has the form (3.14) where is a pseudo-stiffness factor. Figure 3.9 shows the corresponding cyclic response path. The energy dissipated per cycle is equal to (3.15) u u u– F F uu ˆ Ωtsin= uu ˆ = W coulomb 4Fu= Fk s uu ˙ ()sgn= k s W structural 4 k s u 2 2    2k s u 2 == 176 Chapter 3: Optimal Passive Damping Distribution Fig. 3.9: Structural damping force versus displacement Hysteretic damping Hysteretic damping is due to the inelastic deformation of the material composing the device. The form of the damping force-deformation relationship depends on the stress-strain relationship for the material and the make-up of the device. Figure 3.10 illustrates the response path for the case where the material force- deformation relationship is elastic-perfectly plastic. The limiting values are , the yield force, and , the displacement at which the material starts to yield; is the elastic damper stiffness. The ratio of the maximum displacement to the yield displacement is referred to as the ductility ratio and is denoted by . With these definitions, the work per cycle for hysteretic damping has the form (3.16) u u u– F t π 2Ω = k s uu ˆ Ωtsin= uu ˆ = F y u y k h µ W hysteretic 4k h u y 2 µ 1–[]4F y u µ 1– µ == [...]... -td (3. 36) Given γ , one evaluates τ with the stress-strain relation and then F using the equilibrium equation for the system F = 2wLτ (3. 37) Applying a periodic excitation ˆ u = u sin Ωt (3. 38) td F,u td w L L Side View Top View Fig 3. 16: Viscoelastic damper device and taking τ according to eqn (3. 29), one obtains F,u 3. 3 Viscoelastic Material Damping 1 83 ˆ F = f d G s u [ sin Ωt + η cos Ωt ] (3. 39)... intensity loading 3 6 x 10 4 Hysteretic damping versus Viscous damping T 1 = 5 .35 s ξ 1 = 2% γ - m/m 2 0 −2 −4 −6 0 Viscous damping Hysteretic d amping 10 20 30 40 50 Time t - s Fig 3. 20: Response of SDOF with hysteretic damping 60 3. 4 Equivalent Viscous Damping 189 5 4 x 10 3 F - N 2 Hysteretic damping T 1 = 5 .35 s ξ 1 = 2% Taft excitation 1 0 −1 −2 3 −6 −4 −2 0 2 4 6 3 γ - m/m x 10 Fig 3. 21: Hysteretic... 1 ≡ F y, 1 (3. 22) Since the force is the same for both devices, the total elastic displacement is the sum of the individual contributions 3. 3 Viscoelastic Material Damping 179 A1 , E1 A2 , E2 F F,u L1 L2 Fig 3. 13: Two rod hysteretic damping device L2 L1 1 (3. 23) u = - + - F = - F kh A1 E1 A2 E2 Specializing eqn (3. 23) for the onset of yielding, one obtains A1 L2 E1 F y, 1 (3. 24) u y =... the 3M Scotchdamp ISD110 material Using Fig 3. 15, data corresponding to five frequencies is generated Table 3. 1 contains this data Applying eqns (3. 74), (3. 75), and (3. 77), one obtains k eq = 5.7 f d α d = 0.104 c eq = 0.5 93 f d 192 Chapter 3: Optimal Passive Damping Distribution Table 3. 1: Data for ISD110 Scotchdamp material (from Fig 3. 15) Ω (r/s) G s (MPa) η 0.628 1.0 1.0 3. 14 2.5 1.0 6.28 3. 7 0. 93. .. Ωi Jk = (3. 72) N (3. 73) i=1 ∑ ) Minimizing eqn (3. 72) with respect to k eq yields N 1 Gs ( Ωi ) = f d G s k eq = f d -N (3. 74) i=1 Similarly, minimizing eqn (3. 73) with respect to c eq results in N G s ( Ω i )η ( Ω i ) 1 c eq = f d N Ωi ∑ (3. 75) i=1 The form of eqn (3. 75) suggests that c eq be expressed as c eq = α dNeq k Gs η Substituting for k eq -  c eq leads to... −4 −6 0 10 20 30 40 50 60 Time t - s Fig 3. 18: Response of SDOF with structural damping 5 4 x 10 Structural damping T 1 = 5 .35 s 3 ξ 1 = 2% F - N 2 Taft excitation 1 0 −1 −2 3 −4 −6 −4 −2 0 2 4 6 3 γ - m/m Fig 3. 19: Structural damping force versus deformation x 10 188 Chapter 3: Optimal Passive Damping Distribution The hysteretic model calibration defined by eqn (3. 58) is not as straight forward since... system is shown in Fig 3. 24 The structural elements are modelled as linear springs and the representation defined in Fig 3. 25 is used 3. 5 Damping Parameters - Discrete Shear Beam 1 93 ui θ ui – 1 (a) ui ui – 1 (b) Fig 3. 23: Damper placement schemes 194 Chapter 3: Optimal Passive Damping Distribution Fig 3. 24: Viscous dampers coupled with chevron bracing 3. 5 Damping Parameters - Discrete Shear Beam... - = = 34 6.6kN ⁄ m (3. 61) 2 2 Results for this model subjected to Taft excitation are compared with the corresponding results for the linear viscous model in Figs 3. 18 and 3. 19 Close agreement is observed 3. 4 Equivalent Viscous Damping 187 3 6 x 10 Structural damping versus Viscous damping T 1 = 5 .35 s 4 ξ 1 = 2% γ - m/m 2 0 −2 Viscous damping Structural damping −4 −6 0 10 20 30 ... relations are expressed as ˆ γ = γ sin Ωt (3. 28) ˆ τ = γ [ G s sin Ωt + G l cos Ωt ] (3. 29) where G s is the storage modulus and G l is the loss modulus The ratio of the loss modulus to the storage modulus is defined as the loss factor, η Gl (3. 30) η = - = tan δ Gs An alternate form for eqn (3. 29) is ˆˆ τ = γ G sin ( Ωt + δ ) ˆ G = 2 2 Gs + Gl = Gs 1 + η (3. 31) 2 (3. 32) The angle δ is the phase shift between... appropriate LEFT hand scale Fig 3. 15: Variation of 3M viscoelastic material, ISD110, with frequency and temperature The energy dissipated per unit volume of material for one cycle of deformation is determinedΩ 2π ⁄ from W viscoelastic = ∫ ˙ τγ dt (3. 33) 0 Substituting for τ and γ using eqns (3. 28) and (3. 29) results in ˆ W viscoelastic = πG l γ 2 (3. 34) 182 Chapter 3: Optimal Passive Damping Distribution . = A 2 σ y2, A 1 σ y1, F y1, ≡> 3. 3 Viscoelastic Material Damping 179 Fig. 3. 13: Two rod hysteretic damping device. (3. 23) Specializing eqn (3. 23) for the onset of yielding, one obtains (3. 24) (3. 25) When two. device. Then, (3. 19) (3. 20) (3. 21) Example 3. 3: Stiffness of two hysteretic dampers in series The device treated in Example 3. 2 is modified by adding a second rod in series, as shown in Fig. 3. 13. The. shearing strain is (3. 36) Given , one evaluates with the stress-strain relation and then using the equilibrium equation for the system (3. 37) Applying a periodic excitation (3. 38) Fig. 3. 16: Viscoelastic