Viscoelastic Damping 101 aarticlePaulr Macioce,ti Roush Industries,c Inc. le

As a technical professional, one is often faced with the loading. All the energy is lost as “pure damping” once challenge of reducing unwanted vibration in a structure. the load is removed. In this case, the stress is The effect of this vibration can range in severity from the proportional to the rate of the strain, and the ratio of actual failure of components due to high-cycle fatigue, to stress to strain rate is known as viscosity, µ. These the consumer’s perception of poor quality in a product that materials have no stiffness component, only damping. radiates too much noise or rattles excessively. Whether the failure mode is real or perceived, designers and product For all others that do not fall into one of the above development engineers from all industries are working extreme classifications, we call viscoelastic materials. harder than ever to solve these types of noise and vibration Some of the energy stored in a viscoelastic system is issues. recovered upon removal of the load, and the remainder is dissipated in the form of heat. The cyclic One of the most effective and widely used methods for noise and vibration control is passive damping. Damping is most beneficial when used to reduce the amplitude of σ(t) dynamic instabilities, or resonances, in a structure. ε(t) What is Damping? Damping is the conversion of mechanical energy of a ε σ 0 Time, t structure into thermal energy. A structure subject to 0 oscillatory deformation contains a combination of kinetic and potential energy. In the case of real structures, there is also an energy dissipation element per cycle of motion. The amount of energy dissipated is a measure of the structure’s damping level. 1a. Elastic Material Definition of Viscoelasticity σ(t) A viscoelastic material is characterized by possessing both ε(t) viscous and elastic behavior. What this means exactly is best illustrated in Figure 1, which shows how various types of materials behave in the time domain. For a slab of material ε σ 0 Time, t with a cross-sectional area, A, and a thickness, T, subject to 0 cyclic loading, F(t), the corresponding response is given by the displacement function, x(t). The cyclic stress on the sample material is found by dividing the input load by the cross-sectional area, and the resulting cyclic strain on the material is found by dividing the displacement by the 1b. Viscous Material thickness. σ(t) A purely elastic material is one in which all the energy ε(t) stored in the sample during loading is returned when the load is removed. As a result, the stress and strain curves for elastic materials move completely in phase. For elastic ε σ 0 Time, t materials, Hooke’s Law applies, where the stress is 0 proportional to the strain, and the modulus is defined at the φ/ω ratio of stress to strain. A complete opposite to an elastic material is a purely viscous material, also shown in Figure 1. This type of 1c. Viscoelastic Material material does not return any of the energy stored during Figure 1. Cyclic Stress and Strain Curves vs. Time for Various Materials stress at a loading frequency of Ͷ is out-of-phase with the strain by some angle φ, (where 0 < φ < π/2). The angle φ is a measure of the materials damping level; the larger the angle the greater the damping. For a viscoelastic material, the modulus is represented by a complex quantity. The real part of this complex term (storage modulus, E1) relates to the elastic behavior of the material, and defines the stiffness. The imaginary component (loss modulus, E2) relates to the material’s viscous behavior, and defines the energy 3a. Viscous Model 3b. Hysteretic Model dissipative ability of the material. Using Hooke’s Law Figure 3. Single-Degree-of-Freedom Damping Models to define the modulus for complex values, we can define the complex modulus, E* as: refers to the elbow of the temperature and σ storage modulus curve at frequency. 0 iφ the edge of the glassy E* = E1 + E2i = ε e Measurement of 0 region as it enters into the Damping transition region. Tg also Behavior of Viscoelastic Materials defines the peak of the The simplified single- degree-of-freedom (SDOF) One of the unique characteristics of viscoelastic loss modulus, Ez, curve. model shown in Figure 3a materials is that their properties are influenced by The transition region is so can represent many real many parameters. They can include: frequency, named because the world structures. In this temperature, dynamic strain rate, static pre-load, time material is transitioning analytical system, damping effects such as creep and relaxation, aging, and other from the glassy to the is represented using a irreversible effects. In working with this class of rubbery region. It is in this viscous model in which materials, we strive to define the materials complex area that the viscoelastic the damping element modulus (stiffness and damping properties) as a material goes through its (shown as a dashpot) is function of these parameters. Most important of these most rapid rate of change proportional to velocity, include temperature and frequency effects. Viscoelastic in stiffness and possesses dx/dt. When using a materials are typically characterized as having the type its highest level of viscous damping model, a of behavior as shown in Figure 2. damping performance. direct analytical solution of These materials exist in various unique states or The reference temperature the equations of motion is “phases” over the broad temperature and frequency of a material, To, is used available. ranges in which they are used. These regions are to define the peak of the The classic complimentary typically referred to as the Glassy, Transition, Rubbery, loss factor curve. In this solution to the transient, and Flow Regions. Viscoelastic materials behave region, the long molecular free vibration response of differently based on which region they exist in for a chains of the polymer are the system of Figure 3a specific application. in a semi-rigid and semi- flow state, and are able to yields the analytical In the glassy region the polymer chains are rigidly rub against adjacent representation of viscous ordered and crystalline in nature, possessing glass-like chains. These frictional damping as the damping ζ behavior. Stiffness, E1, is at its highest for the material effects result in the ratio, , shown in the in this region, and damping levels are typically low. mechanical damping following equation: c The glass transition temperature, Tg, of a material characteristic of ζ = Where: c = 2 √km c o viscoelastic materials. o ζ Where: %Cr =100% In the rubbery region, the The term c is the critical material reaches a lower o viscous damping plateau in stiffness. coefficient. The critical Damping is at a lower, but damping for a system is reasonable level. A defined as the smallest material selected to exist level of viscous damping in this region is ideally in which the mass of suited for such devices as Figure 3a will exhibit no isolators or tuned mass oscillation when displaced dampers because the from equilibrium. modulus varies only Figure 2. Variation of Complex Modulus with Temperature for a Typical Viscoelastic Material slightly with changes in continued on page 12 continued from page 11 dashpot, and representing the energy dissipation in The damping ratio is often the system by a complex presented as a percentage spring element, k*, of the fraction of critical resulting in a complex damping or percent critical modulus, E*. damping, %Cr. A system is classified as underdamped The hysteretic model can if ζ < 1, critically damped be used for damping if ζ = 1, and overdamped if estimates made from the ζ > 1. Vibratory motion response characteristics of will only exist for an steady-state vibrating underdamped system. In systems. The SDOF system all the cases above, the of Figure 3b is shown to response of a system set be subject to a harmonic Figure 5. Compliance Transfer Function of a SDOF System into motion will eventually forcing function, F(t), decay to zero with time, applied to the mass, m. Another representation of levels of damping, this except when ζ = 0. The equation of motion for this system can be damping for this SDOF relationship between Damping estimates can be solved to yield the transfer system is called material damping and loop made from this transient function of the nature Amplification Factor, Q. area can be defined by response. A SDOF shown in Figure 5. This is As illustrated in Figure 5, this equation: underdamped system a graph of compliance for Q is the ratio of the D subjected to an impulsive a SDOF system having a response amplitude at η = ω πσοεο force at t = 0 will exhibit single resonance at ωo. resonance, o, to the static ω the displacement response The level of damping can response at = 0. x(t) as shown in Figure 4. be subjectively determined Yet another estimate of Relationship Between A measure of damping can by noting the sharpness of damping can be achieved Measures of Damping be made from the rate of the peak — the more by calculating the energy For low levels of damping decay of the response for rounded the shape, the loss per cycle of oscillation (η < 0.2), and within the consecutive cycles of more damping present. due to steady state linear region of the vibration, referred to as the harmonic loading. For a viscoelastic material, the log decrement, δ, and A quantitative measure of viscoelastic material different measures of defined in this equation: damping is achieved by subject to the cyclic damping discussed above using the Half-Power loading as shown in Figure can be equated using the δ = 1 xn+m Bandwidth method, shown 1c, the hysteresis of the m In x following relationship as n graphically in Figure 5 and material can be defined by shown in Figure 7. The selection of a viscous given by the following plotting the input stress σ damping model is equation: (t) versus responding strain ε(t) for one cycle of primarily used for ease of ∆ω η = √ motion. The elliptical analysis. However, the ω For n= 2 behavior of a viscoelastic ο (Half-Power) shape shown in Figure 6 is material is better described defined as the hysteresis η through the use of the The material damping, , loop. The area captured hysteretic model, in which of the complex spring within the hysteresis loop, the damping is element can be D, is equal to the proportional to strain and determined by the ratio of dissipated energy per cycle ∆ω ω ∆ω Figure 6. Typical Hysteresis is independent of rate. to o with of harmonic motion by the Loop for a Viscoelastic This is achieved by determined from the half- material. For reasonable Material eliminating the viscous power point down from the resonant peak value, Amax. On a decibel scale, this corresponds to –3 dB down from the peak value. For that reason, this damping estimation is also Figure 4. Transient Response referred to as the 3 dB of a Classically method. Underdamped SDOF System Figure 7. Interrelationship of Damping Measures