This work formulates a method for the modeling of material damping characteristics in distributed parameter models which may be easily applied to models such as rod, plate, and beam equations. The general linear boundary value vibration equation is modified to incorporate hysteresis effects represented by complex stiffness using the transfer function approach proposed by Golla and Hughes. The governing characteristic equations are decoupled through separation of variables yielding solutions similar to those of undamped classical theory, allowing solution of the steady state as well as transient response. Example problems and solutions are provided demonstrating the similarity of the solutions to those of the classical theories and transient responses of nonviscous systems

Inman, D. J.

Slater, J. C.

English

NASA Technical Reports Server

N94-35895
TRANSFER FUNCTION MODELING OF DAMPING MECHANISMS IN
DISTRIBUTED PARAMETER MODELS
J.C. Slater* and D.J. Inmant
Mechanical Systems Lab, Department of Mechanical and Aerospace Engineering
State University of New York at Buffalo, Buffalo, NY 14260
(716) 636-2733
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INTRODUCTION
This work formulates a method for the modeling of material damping characteristics in
distributed parameter models which may be easily applied to models such as rod, plate and
beam equations. The general linear boundary value vibration equation is modified to
incorporate hysteresis effects represented by complex stiffness using the transfer function
approach proposed by Golla and Hughes. The governing characteristic equations are
decoupled through separation of variables yielding solutions similar to those of undamped
classical theory, allowing solution of the steady state as well as transient response.
Example problems and solutions are provided demonstrating the similarity of the solutions
to those of the classical theories and transient responses of non-viscous systems.
BACKGROUND
Classical damping models assume a simple damping term proportional to velocity
and/or strain rate such that the equations of motion decouple readily. (See for instance
Reismann 1 and Soedel2). This modeling technique is more often an equivalent viscous
damping representation of non-viscous damping than a representation of the real structural
damping 3. Banks et.al. 4 have shown experimentally that hysteretic effects dominate the
damping mechanisms in a cantilevered composite beam. However, analysis of hysteretic
models is quite difficult. Inman 5 proposed using the Golla and Hughes viscoelastic model
for modeling damping in composite beams, and Slater 6 extended this theory to plates. A
simple method of representing damping mechanisms over a wide frequency range which
can be included in the general linear boundary value vibration equation to arrive at a closed
form solution for the damped response is presented in the following.
The time hysteretic stress-strain relation is given by
I
or(x, t) = e(x, t)E- f g(t - s)e(x,s)ds
o
(1)
where o(x, t) is the stress, x is the vector of spacial coordinates, e(x,t) is the strain, and
the kernel g(t-s) describes the hysteresis as developed by Christensen 7 or Banks 8, for
example. The general linear boundary value vibration equation incorporating viscous and
hysteretic damping may then be written as
w(x, t) + dw(x, t) + L_[L_ (x)w(x, t) + Law(x, t)] = f(x, t) (2)
=Graduate Research Assistant, Mechanical Systems Lab, Department of Mechanical and Aerospace
Engineering
]'Professor and Department Chair, Mechanical Systems Lab, Department of Mechanical and Aerospace
Engineering
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547
https://ntrs.nasa.gov/search.jsp?R=19940031388 2020-06-16T11:12:01+00:00Z
where w(x,t) is the displacement, d is theviscous damping coefficient, andf(x,t) is the
forcing function. The operators L 2 and L3 are stiffness operators; LI represents the
integral part of (l) and () denotes partial differentiation with respect to time.
In modeling damping using finite element models, Golla and Hughes 9 have shown that
the variation of material damping and stiffness properties as a function of frequency (i.e.
time hysteresis) can be represented by
-+:,-+],+,., , (3)
where E* (s) is the complex modulus of elasticity, E is the static modulus of elasticity, and
s is the Laplace domain operator. The symbols an,/_n, _'n, and tln represent constants for
curve fitting to match complex modulus data. Ideally, k would be infinite in order to
perfectly match the material properties over all frequencies. However, it is up to the analyst
to determine the value of k in order to obtain sufficient accuracy. Although this transfer
function has been developed for finite element models, it is easily extendable to distributed
parameter theory.
Using (3) to model the time hysteretic stress-strain effects of (1) gives
o(_s) ffiE'(s)c(_s)-- E 1 + a, _,s_,.1 + y s + _1+
Incorporating the GHM model into the Laplace domain representation of (2) results in
s2W(x, s) - sw( x, O) - w, (x, 0) + sdl_ x,s)
-dw(x, 0)+ la[E" (s)N(x)W(x,s)]ffi F(x,s)
(5)
where the operator Li has been replaced by
fl, s 2 + y,sI_ = E or, 2
,-I fl, S +y,s+ tl,
(6)
and the operator L3 has been factored into
Ls ffi EN(x) (7)
to yield a linear partial differential equation. The purpose of the factorization of (7) is for
the case of the beam or rod of varying cross section. For example, for a beam with varying
cross section,
N(x) = I(x) o_2 (8)
pA a:x
where x is the distance along the beam, l(x) is the moment of area, p is the material
density, and A it the cross sectional area. Note that when the force or displacement is
represented by capital letters it denotes the Laplace transformed variable.
548
ANALYSIS
Consider the solution of (5) by separation of variables/modal expansion with associated
permissible boundary conditions. The solutions are then in the form
m 0o
n-I n-I
Substituting (9) into (5) yields
0o
E[(s: + sd + E'(s)L_ N(x))A (s)](I)(x)- (s + d)w(x, O) - (v(x, O)= F(x, s)
n-I
(10)
If ¢,(x) are chosen to be the eigenfunctions of the operator L2N(x ) such that the
boundary conditions are satisfied and
/_N(x)(I_ = A_ (11)
then (10) becomes
E[(s2 + sd+E'(s)A)A(s)]dP(x)-(s+d)w(x,O)-¢v(x,O)ffi-F(x,s) (12)
n=l
where A are the eigenvalues of the associated with the eigenfunctions _ (x). Note that
(11) is the usual modal assumption for the undamped equations of motion.
The functions are assumed to be orthonormal such that
f_, (x)_ (x)dQ = 6,. (13)
Q
where Q is the spacial domain of the equation, i.e. length for a beam and area for a plate.
Multiplying (12) by _,_ (x), integrating over the region Q and applying (13) yields the
n temporal equations
[(sZ+ sd + E'(s)A)A(s)-(s + d)a (0)-/t (0)]= f.(s) (14)
where
and
f. (s) =fF(x,s) (x) dQ (15)
Q
a (0) = fw(, (x)dQ
I:1
, a (0) =fg,(x.0) (x)dQ (16)
Q
549
The solution of the temporal equation is then
A.(s) ffi
k
[I.+(s
j-I
where (3) has been substituted into the solution and d, .. ¢l,s 2 + y,s + rl,.
(17)
EXAMPLE
Consider a problem involving decay from initial conditions of a simply supported 1 cm
thick 1.5m x 2m plate given an assumed complex modulus. A complex modulus
represented by a single expansion term has been arbitrarily chosen in order to obtain a
simple solution. The parameters are as labeled.
xA
1.5
2
v
Y
Figure 1
p ffi 2700 k-_g3 , E ffi6.89 × 10 '° Pa, v _.34, h _. 01m
m
The viscoelastic properties are a _.6,/_ - 1,y ffi 1.5× 10",/u ffi1.5× 10 6 .
The initial conditions are given as w(x, y, 0)_.01sin(X_)sin(y_) which is the first1.5 2 '
mode shape, with an amplitude of .01. The following time derivatives are also set:
_,(x, y,0)-- w(x,y,O) ,,. i;;,(x,y,O) ffiO. The need for this will become apparent. The
equation of motion for a plate is
550
s :W(x, y, s) + D'(s__.._)V4W(x, Y, s) ,. F(x,y,s) (El)
ph ph
From eqns. (3) and (5),
( s2+lsxlo4s1ph - ph 12(1 - v 2) -- 240. 5 1+. 6 s2 + 1.5 × 104s + 1.5 × 106 (E2)
where m and n are integers from 1 to oo. The natural frequencies of an undamped simply
supported plate are given by (See Reismann 1 for instance)
,jill/71]_/Re (D(s) / ph m_mn m + (E3)
Using eqns. (E2) and (E3), Q11 = 124.14 rad/sec, and K(124.14 i) = 328.3 (1+ .217
i). From eqn. (17)
A(s) =
s _ + 1.5 x 104s + 1.5 × 10 6
4
s + 1.5 × 104 s _ + 1.5247 × 106 s 2 + 3.6986 x 108 s + 2.3116 x 101° (E4)
which can be represented in state space form as
Iz.,] o 1 o o,,z.0 0 1 0 //_0 10 l° 0 0 1
-1.69x -2.71×108 -1.52x10 _ -1.5×104.1LZ4
(ES)
and
a(t)=[56x104 5.6x10" 3.7x10-2 0 z: (E6)
where the states z are related to the displacement by equation (E6) and the initial conditions
are a(0)=.01 and _i(0) -- ki(0) -- d(0) -, 0.
From eqn. (E6), its time derivatives, and the initial conditions,
56XLO237XlOo][zl]l5.6x 104 5.6x 102 3.7x 10 -5 Z2-1.0 x 107 -6.7 xl0 s 0 Z3 "
-6"3x108 -l'0xl0_ -6.7x102 Z4 -o
i l (E7)
the solution of which is
551
izl[rn8x107,l-t6 ×10"1=/-4.3 × 10-'1
Z4 "0 [ 64×103 ]
(ES)
The results of a digital simulation of the plate response are shown in figure 2. The decay
envelope for an equivalent viscous damped plate (_ = _1/2 where r I is the loss factor of the
material at 124.14 rad/sec) is shown for comparison.
0.01
0.005
i 0
-0.005
Displacement versus Time
i i i t i i
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time (sec)
Figure 2
DISCUSSION
A method has been presented solving viscoelastic plate problems which accounts for
frequency dependent modulus of elasticity. The solution yields the same orthogonal
eigenfunctions/modes as classical plate theory while producing decaying temporal functions
representing viscoelastic effects. This method provides a simple approximation for
modelling viscoelastic plates. The solution reduces to the classical Sophie Germain
solution when the hysteresis terms are dropped, as can be seen in equation (17) when a n is
set equal to zero.
552
ACKNOWLEDGEMENTS
This work was supported by NASA Langley Research Center, Grant No. NGT50541,
under the direction of Dr. Garnett Homer.
REFERENCES
1) Reismann, H., Elastic Plates, Theory and Application, John Wiley & Sons, New
York, 1988.
2) Soedel, W., Vibration of Shells and Plates, Marcel Dekker, Inc., New York, 1981.
3) Cudney, H.H. and Inman, D.J., "Determining Damping Mechanisms in a Composite
Beam by Experimental Modal Analysis," International Journal of Analytical and
Experimental Modal Analysis, Vol. 4, No. 4, Oct. 1989, pp. 138-143.
4) Banks, H.T., Fabiano, R.H., Wang, Y., Inman, D.J., and Cudney, H.H., "Spatial
Versus Time Hysteresis in Damping Mechanisms," Proceedings of the 27th IEEE CDC,
1988, pp. 1674-1677.
5) Inman, D.J., "Vibration Analysis of Viscoelastic Beams by Separation of Variables
and Model Analysis," Mechanics Research Communications, Voi. 16, Apr. 1989, pp.
213-218.
6) Slater, J.C. and Inman, D.J., "Transfer Function Modeling of Damping Mechanisms
in Viscoelastic Plates," 32nd Structures, Structural Dynamics, and Materials Conference,
Apr. 1991, pp. 2381-2383.
7) Christensen, R.M., Theory of Viscoelasticity: An Introduction, Academic Press, New
York, 1982.
8) Banks, H.T. and Inman, D.J., "On Damping Mechanisms in Beams," No. #89-3,
University of Southern California, Sep. 1989.
9) Golla, D.F. and Hughes, P.C., "Dynamics of Viscoelastic Structures--a Time
Domain Finite Element Formulation," ASME Journal of Applied Mechanics, Vol. 52, Dec.
1985, pp. 897-906.
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Transfer function modeling of damping mechanisms in distributed parameter models

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