analysis of damped waves using the boundary · formulation of damping models can be found in lazan...

Analysis of Damped Waves Using the Boundary

Element Method

Brian Vick & Robert L. West

Mechanical Engineering Department

Virginia Tech

g/acM,wrg, T/A, C/&4 24067

Email: [email protected] & [email protected]

Abstract

The subject of this paper is the one-dimensional, damped wave equationincluding both viscous damping, where resistance is proportional to velocity,and material damping, where resistance is proportional to displacement.'Disturbances propagating in a material of finite length subject to distributedforces, surface forces and initial conditions will be examined. The purpose ofthe paper is to develop a numerically accurate and computationally efficientsolution using the boundary element method (BEM) and to use this solution toexamine the physical behavior of systems subjected to a variety of forcingfunctions. The model equations are important in both mechanical and thermalwave propagation.

The BEM is particularly well suited for this class of problems sincediscontinuities at wave fronts could exist. The solution is based on theassociated full space Green's function (GF), which naturally possesses thepropagating wave front characteristics of the actual response.

Representative plots are presented to show the fundamental effects ofdamping on the GF since characteristic responses to actual disturbances can beseen in the GF. Results are then presented to show how various degrees ofdamping effect the propagation of waves caused by applied surface forces.

Introduction

The continuing need to operate machines and vehicles at higher speeds andhigher operating temperatures as well as better utilization of material requiresgreater understanding of the dynamic character of structures and the associateddesign trade-offs. As a result the engineering community at large has become

Transactions on Modelling and Simulation vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X

266 Boundary Element Technology XII

much more reliant on the solutions from today's computational models.Reducing the number of design/analysis scenarios based on model results alsoprovides a cost benefit to expensive and exhaustive prototype building andtesting. This puts a premium on formulating the problem with the appropriatephysics as well as being able to directly compare model results to experimentaldata.

The goal for developing newer models is to represent the relevant physics ofthe system with enough fidelity that the interaction of these processes results inbehavior that better represents the system under study. Validation ofcomputational models is a very active area of ongoing research which startswith representation of appropriate physics within the model and ends with themodel being reconciled with experimental data. The motivation for this work isthe inclusion of damping into the wave propagation process. Using the dampedGreen's function brings the representative physics of damping into theBoundary Element (BEM) formulation. Used within the BEM, the fidelity ofthe model is enhanced allowing direct comparison with experimental data aswell as extended post-processing of model results. This allows thecharacterization of dynamic processes in the elastodynamics framework basedon wave mechanics.

The area of classical elastodynamics has long been established. Establishedtreatises on the subject can be found by Achenbach [1] and Eringen and Suhubi[2]. Numerical treatment and solution of the theory of elastodynamics withinthe boundary element framework was first developed by Cruse and Rizzo [3]for transient and steady-state problems. Texts providing an introduction as wellas applications of the subject can be found by Manolis and Beskos [4] andmore recently by Dominguez [5]. It is noted here that material damping hasbeen documented in the literature prior to the development in this paper.However, it is typically treated in the linear viscoelastic representation of thematerial itself [4]. A comprehensive work and list of references on theformulation of damping models can be found in Lazan [6] as well as Crandall[7]. Application to damped waves in the area of heat transfer has beendeveloped by Vick and Ozisik [8] and Frankel et al [9].

This paper describes the development of the solution of the one-dimensionaldamped wave equation in a finite region. The research seeks to characterize thedynamic response of structures using the damped wave formulation for bothtransient and steady-state response. The damped free-space Green's functionincorporates the physics of both viscous and material type damping. Thisapproach fundamentally addresses 1) energy transport in solids propagated bywaves which travel with finite speed, and 2) the diffusive nature of damping instructures which seeks to distribute the energy throughout the structure. Theability of the approach to investigate both transient as well as steady-stateresponse is necessary for the broader application of the research as well as thefunctional comparison with experimental data.

The application of the damped wave solution is demonstrated relative to theundamped one-dimensional wave solution. The influence of both viscous and


Boundary Element Technology XII 267

material damping on the free-space Green's function is presented for variouslevels of damping. The effects of time interval selection on the solution is alsostudied. Results are presented for the one-dimensional fixed-free rod. Theeffects of viscous damping are investigated in the single pulse traction problemdemonstrating the influence of damping on both the incident and reflectedwaves. The case of multiple pulse tractions demonstrates the effects of dampingas well as its interaction with the period of the impulsive loading. The dynamicresponse of the rod is shown to be bounded by the undamped and heavydamping which results in diffusive type behavior.

Formulation

The motivation for the current work is to examine the effects of damping inone-dimensional wave propagation. The following physical model is used,

^u 1 3*u D 3u P,t). (1), , .

This equation is used to model wave propagation where u(x,t) represents adisplacement from equilibrium and C is the signal speed. The term involvingthe second derivative with x originates from the elasticity of the material whileS(x,t) is a distributed body force. Damping enters the formulation in two waysin the form of a viscous and a material damping term. The viscous dampingterm is proportional to the velocity and is characterized by coefficient D. Thematerial damping is proportional to the displacement itself and is modeled witha coefficient P. This formulation (with P=0) has also been used to describethermal wave propagation as described by Vick and Ozisik[8] and Frankel et al[9].

The material has length L and general boundary conditions can beformulated as;

9u ^-%i— + b,u = fi(t), atx = 0 (2a)

2 2, atx = L (2b)oX

These boundary conditions can be used to model a specified displacement atx=0 or x=L by setting aj=0 or ̂ =ft respectively. Similarly, an appliedboundary traction can be modeled by setting bj=0 or b2=0. Boundaryconditions of the mixed type can also be used to simulate a nonrigid support.

The mathematical formulation is complete once the following initialconditions are specified;

u = uj(t), att = 0 (3a)

3u— = V;(t), att = 03t (3b)



Next, the general solution to wave equation, Eq.(l), with applied boundaryconditions, Eqs.(2a, b) and initial conditions, Eqs.(3a, b) is developed using theBoundary Element Method.

Solution Using the BEM

Green' Function

The general solution is constructed from the full-space Green's function (GF),governed by the following equation;

9% 1 _ 1 a^G D 9G P

3x^ C^ ^ ^ C^9t^ C^3t C^ ' ^'^

This equation is simply the wave equation (1) subjected to a concentrated forceat location XQ impulsively applied at time to. There are no boundary restraintson this GF other than the implied condition that G —> 0 as x —> ±00 . Thecausality principle in time states that there is no response before the applicationof the impulsive force at to or ,

G- = 0, when t< to- (5)at

The solution for the GF can be obtained using the full-space Fouriertransform and inversion in the spatial variable. The solution is determined as,

G(x,tlxo,to) = —exp[-D(t-to)]lo

#H[C(t-to)-Abs(x-Xo)]

(6)where IQ is the modified Bessel function of the first kind and H is the unit stepfunction.

The physical characteristics of the wave propagation inside the material arerevealed in the Green's function, as displayed in the results section. However,since this is the full-space GF, interesting boundary interactions cannot berevealed directly.

Displacement Field Solution in Terms of the GF

The general solution for the displacement field in terms of the GF is obtainedby expressing the wave equation, Eq.(l), in terms of the cause variables XQ andto , multiplying by G(x,t I \o,to) and integrating over space and time to obtain;



J Jo=o x<,=o (7)

=0

After integrating by parts twice and applying boundary and initial conditions,the displacement field in terms of the GF can be expressed as;

( r)C/to=o+u;(xo) --- + 2DG

V c%0

(8)

A,

C?u(x,t) =

\i

il

du 3G | f 3u _ 9G-—G-u-— - G-u

9Xr

t Lf f c

Three types of forcing functions are clearly evident in this representation.The first two terms account for initial displacement and velocity, the secondtwo terms are a result of boundary interactions and the last term is due todistributed body forces. Other than for one term involving the viscous dampingD in the initial displacement term, this formulation is identical to the boundaryintegral formulation for undamped waves. Thus the damping enters t^formulation almost entirely through the GF itself.

Discretization

The terms involving the initial conditions and body force rrequire integrals of specified inputs and can be eva'appropriate quadrature. The boundary condition terrinvolve time integrals of the unknown solutiondiscretization. An implicit time discretization scher~

n=l

where f% is the time after the n^ time stethe current time is t=t̂ . The parametefor x inside the domain and 1/2 on theand gradients at x=0 are designated as



are the boundary values at x=L. The GF integrals over time interval n aredesignated as,

'*>'• A,

*

These quantities represent the influence coefficients. For instance, G"z/*,fj

can be interpreted as the influence at location x and time / of a boundary fluxq"i located at L applied over the time interval ln-i to in . The term F(x,t)accounts for the prescribed initial conditions and body forces and is defined as,

1 ^ ( r)GF(x,t)=— J Vj(xo)(G\=o+U;(xo) -—-+ 2DG

C "10 V #0

t L+ j J S(XQ,to)GdXodto

At the current time t=t^ , the discretized solution, Eq.(9), can be applied at the~0 boundary yielding,

(lOa)N-l

boundary to obtain,

(lOb)



expressions (lOa) and (lOb) are known since the summation from n-l to n=N-1 involves the already determined history of the solution while the F(x,t)function is the contribution of the specified initial conditions and body force.Once the time history of the solution on the boundary is determined, thecontinuous displacement field at internal locations can be determined fromEq.(9).

Results

The nature of the damped Greens' function and it impact on the dynamicresponse of a free-fixed rod will be investigated throughout this section. In thisexample, the rod is free on the left end, x=0, where a prescribed traction,du/dx=-fj, is applied. The right end of the rod, x=L, is a fixed displacementboundary condition, u - 0. The initial conditions and body forces are zero forthis case study.

The effects of viscous and material damping on the damped Green'sfunction will be demonstrated. The effect of the fully implicit time integrationscheme will also be shown. The solution for a single pulse traction is given aswell as the effect of viscous damping on the solution. Finally, the dynamicresponse of the rod due to a periodically applied pulsed traction will beinvestigated including damping effects.

The Damped Green's Function and its Behavior

The fundamental building block of the solution is the free-space Green'sfunction, which is generated from Eq.(4) and expressed analytically by Eq.(6).The nature of the wave propagation can be seen in this GF, as shown in Figures1-4.

Figures 1 and 2 display the effects of viscous damping on the signalpropagation. Figure 1 is for the case of C=7, D=2, P=0, representing asituation with zero material damping and a viscous damping term of about thesame order of magnitude as the acceleration term in Eq.(4). The zone ofinfluence is bounded by the characteristic lines x-xo=C(t-to) and x-xo=-C(t-to)where a sharp wave front can be seen. The magnitude of the disturbancedecays exponentially in time as the impact of the delta function propagates intothe medium. This behavior is further displayed in Fig. 2, where a timesequence of spatial distributions are displayed for various degrees of viscousdamping. The GF in the undamped case maintains the constant value 1/2Cover the entire zone of influence. At the other extreme, the highly damped case(D-10) shows a diffusive nature, since the viscous term involving a firstderivative in time dominates the acceleration term. Except for extremely shorttimes (Fig. 2 at t-to=0.25), the highly damped case is indistinguishable fromthe classical free-space diffusion GF which has no observable wave front.



x-xO

-i

0.

t-tO

Figure 1: Green's Function with viscous damping: C=7, D=2, P=0

G G

t-to-0.25

/-'X

D=0

D-lD=10

,.."

^

t-t̂ -0.5

-1 -0.5

Figure 2: Effect of viscous damping on the Green's Function

Figure 3 shows the effects of the material damping coefficient P on the GF.The material damping represents a restoring force directly proportional todisplacement which results in oscillatory behavior. The combined



contributions of the two types of damping can be seen in Fig. 4. For D*-P>0,the spatial distributions display an upward hump (overdamped) while for D*-P=0 the spatial distributions are flat (critically damped) and for D*-P<0 thespatial distributions show oscillations (underdamped). It is also interesting tonote that the magnitude at the wave front (exp[-D(t-to)]/2G) is independent ofthe presence of material damping, P.

Gp=iP=10P=100

t-to=0.25

t-to=0.75'

0\5 /''

:V

Figure 3: Effect of material damping on the Green's FunctionD=1,P=0D=1,P=1D=1,P=10D=1,P=100

G

t-to=0.25

t-to=0.75

t-to=0.5

Figure 4: Effect of both viscous and material damping on the Green's Function

Issues Associated with the Implicit Time Integration Scheme

The BEM used with the damped Green's function is a time domain basedapproach. As it is, there are issues associated with the discrete representation ofthe boundary values in time which affect the accuracy of the solution. Thedominant parameter for the time integration scheme is the size of the timeinterval. Figure 5 below shows the effects of refining the time interval to



obtain the solution for the case of a constant surface traction (9u/9x=-l) appliedto the free end of the rod. The accuracy of the resulting dynamic responseshown in Figure 5 is purely a function of the size of the time interval used forintegration.

*̂*̂ time interval = L/2c

— time interval = L/4c

— - time interval = L/8c

time interval = L/16c

0.2 0.4 0.6 0.!

Rod Position, x

Figure 5: Discretization error with the implicit time discretization

Given the boundary displacement and traction solution at any time, the axialdisplacement field is a continuous function given by Eq. (9). The displacementfield is computed over the entire domain of the rod for time, t=L/C. Figure 7was developed by plotting the displacement field over the domain and reducingthe time interval by halves starting from , At = L/2C. The stair-step phenomenashows the effect of the displacement field calculated using boundary valuescomputed over too large of a time interval. The is the direct result ofrepresenting the boundary displacement in an implicit fashion with a constantvalues over each time interval. There are two solutions to this, reduce the sizeof the time interval or represent the boundary values with a higher orderrepresentation.

The time interval in this example was reduced by halving the interval. Asthe time interval is halved, the constant boundary value representation is abetter estimation of the true boundary function as At—>0. This yields a betterestimate of the axial displacement field over the domain. As the time interval isreduce the boundary solution as well as the prediction of the displacement fieldover the domain converges to the exact solution. The following results are timestep independent.

Effects of Damping on the Single Pulse Response

The single pulse traction is a finite duration approximation to the Green'sfunction itself and key to understanding the problem. The free end of the rod



encounters a prescribed traction which launches the axial wave. The pulse isdefined in Figure 6 with a duration, tpuise = L/4C.

oSurface Traction at x=

ipul

^ —

;e

period

i

i

^

Time, t

Figure 6: Pulsed surface flux

Axial Displacement, t = L/2C Axial Displacement, t = L/Cu

D

U9UI3C

j9D.s

B

u

c1COD.5CgOQ

U0.25 v ^

\ D = 0.0 50.2 \ D=1.0 R

0.15 \\ D=10.0 5o . i \\ !_

0.05---__ A 0~"̂ -̂ . -\ c1

0 0.2 0.4 0.6 0.8 1 QRod Position, x

Axial Displacement, t = 3L/2C

i\ D = 0.0 ^

0'2 \ - D=1.0 50.15 \ D=10.0 |0.1--, _ _ ^ _ \ «

0.05 ""A 5

0 0.2 0.4 0.6 0.8 1 o

"•"" \ D = 0.00.2 \ D = 1.0

0.15 \ D=,0.0

\ \0.05 "\\

^ \\-i — , ̂ --^-, — .———., >\0 0.2 0.4 0.6 0.8 1

Rod Position, xAxial Displacement, t = 2 L/C

0 2\ 0=10\ D = 10.00.15 \01 \

0.05 ""̂ \

0 0.2 0.4 0.6 0.8 1Rod Position, x Rod Position, x

Figure 7: Propagation of a single pulse surface traction with viscous damping:C=l,P=0,tpuise=L/4C,fi = l

The period of the pulse is infinity, period = <*>. This case explores the nature ofincreasing the viscous damping component of the problem. Three dampingcases are explored in Fig. 7; undamped, D = 0.0, moderately damped, D=1.0,and heavily damped, D=10.0. The axial displacement field is plotted within thedomain of the rod at times t=L/2C, t=L/C, t=3L/2C and t=2L/C. The wave isfully developed and "just" encounters the boundary at t=L/C. The snapshots ofthe displacement at time t=3L/C and 2L/C show the interaction with the fixedwall and the subsequent developing wave as the wave "just" encounters the free



boundary on the left end. The overall effect of the damping is to reduce theamplitude of the displacement. With the undamped case, the classiccancellation effect is seen. The damped waves however are unable to maintaintheir signal strength and therefore do not cancel the forward propagatingdisplacement components after they encounter the boundary.

An important element to notice is the diffusive nature of the wave for thehighly damped case. Damping found in practice is bounded by the undampedwave nature and the diffusive case. The implication of this plot and this processis that the diffusive nature of damping makes it nearly impossible to reconstructthe upstream response from a sensor measuring the response downstream of thedisturbance. This brings up some very real issues as to how we might reconcilecomputational models with experimental data where damping is "significant".

The effects of material damping on the boundary surface displacement forthe free end of the rod is illustrated in Fig. 8. Various levels of materialdamping from P=0 (undamped) to P=10 (moderately damped) to P=100(highly damped). The decaying oscillatory nature of material damping isclearly evident in the response of the free end of the rod. The reflected wavereturns to the free end of the rod at t=2L/C, at which time the interactionbetween the inertial and material damping components is evident.

Time, t

Figure 8: Effect of material damping on surface displacement with a singlepulse surface traction

Effects of Viscous Damping on the Repetitive Pulsed Response

The surface displacement for a material subjected to a pulsed force at x=0 isshown in Fig. 9. For this case tperiod=L/C, thus the time between pulses isequal to the time for a signal to travel the length of the material. The responsein the rod with various degrees of viscous damping is displayed. Forcesoriginating at the x=0 boundary cause displacements which reflect off the fixedboundary at x=L with double the original amplitude. At times equal to 2, 3, ..,a new pulse is initiated while a previous signal is just returning to the x=0



boundary. For the undamped case, the returning signal causes the displacementto decrease. For the moderately damped case the returning signal is weak whilethe heavily damped case exhibits almost no noticeable returning signals.

Time, t

Figure 9: Effect of viscous damping on surface displacement with a pulsedsurface traction

Conclusions

The development of the damped Green's function and its use in a BoundaryElement Method was presented. In the paper the characteristic behavior of thedamped Green's function was investigated as well as the effect of various levelsof viscous and material damping. The material damping behaves as a restoringforce, opposing the displacement in the solid. The exponential behavior of theviscous damping term opposed the velocity field exhibiting a diffusivecharacter for large damping values relative to the wave speed.

The current implementation of the time domain formulation of the BEMrepresents the boundary values as constants over the time interval within in theimplicit integration scheme. The effect of using large time intervals with aconstant representation of the boundary values resultes in a stair-step predictionof the continuous displacement field in the domain. In all cases for all timeintervals the boundary values obtained from the solution of the BEM code wereaccurately computed. Refinement of the time interval in the integration schemeresulted in greater fidelity of the boundary values as a function of time andhence smoother prediction of the displacement field within the domain. It wasnoted that continued refinement of the time interval allows the displacementfield within the domain to converge to the analytic solution.

The effect of viscous damping levels and its interaction with a prescribedtraction pulse of finite duration was also explored. The boundary and domaindisplacement field was calculated and shown to be bounded by the undampedwave solution and the diffusive solution for the highly damped case. Interactionof waves with the boundary was also shown for the various damping values



resulting in wave cancellation for the undamped case and in a residualdisplacement field in the wake of the interacting waves for the damped cases.

The important case of the interaction of wave trains developed from periodictraction pulses was also investigated. The undamped case was used as thebaseline to evaluate both the viscous and material damped systems. Themoderately damped and highly damped viscous and material damping waveresulted in significant attenuation of the signal particularly for the reflectedwave front. The impulsive nature of the solution was evident for the dampedcases but significantly different than the undamped solution.

The solution of damped wave systems by the time domain BEM using thedamped free-space Green's Function allows a fundamental investigation intothe nature of the physical systems since the internal physics are inherentlymodeled in the damped free-space Green's function itself. Additionally, theapproach lends itself to the study of structures with physics described overmultiple dimensions. This approach provides prediction of system responsewith very high fidelity with the ability to accommodate extended post-processing without loss of continuity in the field variables. This approach hasthe very distinct advantage of providing a continuous field solution on theboundary of the system which naturally accommodates comparison withexperimental data.

References

1. Achenbach, J. D., Wave Propagation in Elastic Solids, North-Holland,Amsterdam, 1973.

2. Eringen, A. C. and Suhubi, S., Elastodynamics, Vol. 1 and Vol. 2, AcademicPress, New York, 1975.

3. Cruse, T. A., and Rizzo, F. J., A Direct Formulation and Numerical Solutionof the General Transient Elastodynamics Problem I, /. Math. Anal AppL, 22,244.

4. Manolis, G. D. and Beskos, D. E., Boundary Element Methods inElastodynamics, Unwin Hyman, London, 1988.

5. Dominguez, J., Boundary Element in Dynamics, Computational MechanicsInc. Billerica, MA and Elsevier Science Essex, UK, 1993.

6. Lazan, B. J., Damping of Materials and Members in Structural Mechanics,Pergamon Press, New York, 1968.

7. Crandall, S. H., The Role of Damping in Vibration Theory, /. Sound Vib.,11,3-18.

8. Vick, B., and M. N. Ozisik, Growth and Decay of a Thermal Pulse Predictedby the Hyperbolic Heat Equation, Journal of Heat Transfer, 1983, 105(4),902-907.

9. Frankel, J. I., B. Vick, and M. N. Ozisik, General Formulation and Analysis ofHyperbolic Heat Conduction in Composite Media, International Journal of

, 1987,30(7), 1293-1306.


analysis of damped waves using the boundary · formulation of damping models can be found in lazan...

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