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Discrete and Continuous Models of
Critical Damping for a Simply Supported Beam

presented by

Julia Ann Gray

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_Master's Mechanical Engineering

 

 

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DISCRETE AND CONTINUOUS MODELS OF
CRITICAL DAMPING FOR A SIMPLY SUPPORTED BEAM

BY

Julia Ann Gray

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1981

ABSTRACT

DISCRETE AND CONTINUOUS MODELS OF
CRITICAL DAMPING FOR A SIMPLY SUPPORTED BEAM

3)’

Julia Ann Gray

The concept of "critical damping" for both multidegree of freedom,
discrete linear systems and distributed parameter systems, has been
investigated by a number of researchers. Conditions under which a
system is critically damped, which depend on certain combinations of
the coefficient operators, have been obtained. Conditions under which
a damping mechanism provides underdamping, overdamping, or mixed damping
have also been described.

In this thesis, specific results are applied to a particular
continuous system, a simply supported beam. Results are obtained by
considering both the analytic model of the beam and an approximate,
discrete version of this model obtained by the finite element method.
A variety of damping mechanisms are included in a number of example

problems.

TABLE OF CONTENTS

LIST OF TABLES . .............................................
LIST OF FIGURES .............................................
I. INTRODUCTION ............................................
II. A DISCRETE MODEL OF CRITICAL DAMPING ...................
2.0 Introduction ..................................
2.1 Definitions ...................................
2.2 The Finite Element Model ......................
2.3 A Simple Method for Calculating Ccr ...........
2.4 A Note On The Comparison Test .................
2.5 Results .......................................
III. AN ANALYTIC MODEL OF CRITICAL DAMPING .................
3.0 Introduction ..................................
3.1 The Undamped Simply Supported Beam ............
3.2 The Damped Simply Supported Beam ..............
3.3 A Result on Discrete Damping ..................
3.4 Examples ......................................
IV. CONCLUSION .............................................
APPENDIX A - FINITE ELEMENT MATRICES AND THE UNDAMPED
NASTRAN MODEL ..................................
APPENDIX B - EIGENVALUES, AND MATRICES FROM CHAPTER II ......
APPENDIX C - FINITE ELEMENT DAMPING MATRICES FOR DISCRETE
AND CONTINUOUS DAMPING .........................
APPENDIX D - EIGENVALUES FOR EXAMPLES 5-8 ...................
LIST OF REFERENCES ..... . ....................................

iii

H
#00qu

15
16

20
20
21
24
26
29

34

38

49

60

65

7O

LIST OF TABLES

Table Page
1 Comparison of Three Nastran Models with Theoretical

Model ................................................ 17

2 Summary of Data for Examples 1-8 ..................... 20

3 Comparison of NASTRAN Model with Analytical Model .... 33

iv

Figure

2

LIST OF FIGURES

The Simply Supported Beam The Beam
With Finite Element Grid Cross Section ......
A Typical Element of the Beam ........................

I. INTRODUCTION

The damped, free vibrations of both multidegree of freedom discrete
linear systems and continuous systems have received considerable atten-
tion. The first matter to receive attention concerned the definitions
of the various possible types of damping. For the simplest systems,
the various types of damping, which result in systems that are under-
damped (oscillatory), overdamped (nonoscillatory), or critically damped
(also nonoscillatory), are easily defined in terms of the parameters of
the system. Defining similar criteria for more complex systems has been
an area of current interest.

Another area of current interest concerns the qualitative nature
of the motion of a system, under a given type of damping. It is desired
to determine, without resorting to lengthy calculations, the type of
damping a particular damping mechanism provides. Ideally, this problem
is to be attacked by examining the parameters of the system, not by
solving the equations of motion of the system, or, alternatively, re-
sorting to experiment.

The inverse of this problem has also been addressed. Given a
system with a predetermined desired motion, it is necessary to identify
a damping mechanism, or mechanisms, which will insure that the desired
motion is attained. Again, this problem is to be attacked without
resorting to lengthy calculations or experiment.

In this paper, recent results on each of these topics will be
applied to a particular continuous system, a simply supported beam.

The last of the above topics will be addressed by considering an

1

2

analytic model of the beam. The first two of the above problems will
be attacked by considering an approximate, discrete version of this
model obtained by the finite element method. First, however, a brief
summary of other results on these topics will be presented.

It is well known that the qualitative nature of the solution of one
of the simplest vibratory systems, a single degree of freedom spring-
mass-damper system (spring and damper in parallel) can be determined by
examining the coefficients of the ordinary differential equation which

describes its motion. This equation is
mx + cx + kx = 0 (1.1)

where m, c, and k are the positive values of the mass, damper, and spring,
respectively, and x(t) is the "displacement" from static equilibrium

of the system. Equation (1.1) can be rewritten as

SE + 2ng< + mzx = o (1.2)

where the natural frequently of the undamped system is m 2 /§-, and

the damping ratio is c = c/Z/Em: The system is said to be overdamped
if c > 1, critically damped if c = 1, and underdamped if c < 1. For
more complicated single degree of freedom systems, the analysis is still
straightforward and simple.

The equation of motion for the more complex, multidegree of freedom,
discrete, damped linear system is an analogous matrix differential

equation

M§(t) + Gin) + 1620;) = o (1.3)

3
where M and K are the positive definite, symmetric, mass and stiffness
matrices respectively, and C is the positive semi-definite symmetric
damping matrix. For this problem, the concepts of over, under, and
critical damping are more subtle and require closer scrutiny than the
single degree of freedom case. In fact, a number of results addressing
the qualitative nature of damping of solutions of this equation have been
published, each shedding some light in different areas.

In previous work Duffin [1], in considering (1.3) defined an over-
damped system in terms of a function of the quadratic ferms of the M, C,
and K matrices. More recently, Nicholson [2] defined an underdamped
system in terms of the eigenvalues of the mass, damping, and stiffness
matrices. Maller [3], upon considering Nicholson's work, defined an
underdamped system similar to that of Duffin and derived a sufficient con-
dition for underdamping in terms of the definiteness of the coefficient
matrices.

Recently, additional results have been obtained by Beskos and
Boley [4]. Techniques used in [4] are different from those found in
matrix theory which were used in [1,2,3]. Essentially Beskos and Boley
1‘“ different damping

k
k=1
coefficients, find the locus of combinations of the {ck} leading to

address the fbllowing question: Given {c

critically damped motion. Their approach requires calculating critical
damping surfaces in a parameter space (this surface corresponds to the
concept of finding the "minimum value of damping required for non-
oscillation"). By calculating critical damping surfaces in parameter
space, one is led to the following observations: A combination of
damping coefficient which correspond to a point in the parameter space

"above" the critical damping surface yields a solution to (1.3) which

4

is overdamped (nonoscillatory); a combination of damping coefficients
yields an umderdamped (oscillatory) solution of (1.3) if the point in
parameter space corresponding to the combination lies "below" the
critical surface.

The conditions derived by Duffin, Nicholson, and Beskos and Boley
require substantial calculations. In the cases of Duffin and Nicholson,
the calculations are used for verification of their respective con-
ditions; there are different reasons for the computational load found
in [4]. It appears that the technique of Beskos and Boley works well
when the damping is Rayleigh (C = aK + 3M) and the order of the system
is small. This stems from the fact that the technique requires dif-
ferentiations for which there are presently no efficient numerical
schemes available.

More complete results, which do not require substantial calcula-
tions, have recently been obtained by Inman and Andry [S]. In this
paper, the definition of critical damping is stated in terms of the
coefficient matrices in a manner analogous to the single degree of
freedom case. A critical damping matrix is defined in terms of the
unique positive definite square roots of the matrices M and K. Having
obtained Ccr’ a system is defined as overdamped, critically damped, or
underdamped if the matrix (C - Ccr) is positive definite, zero, or
negative definite, respectively. A fourth possibility, mixed damping,
is defined if (C — Ccr) is indefinite.

The computational load for these results is nominal and does not
encounter any special problems due to high order. In the special case
that the damping matrix is diagonalized by the undamped modal matrix,

all the conditions presented become necessary and sufficient; for the

general case (C not diagonalized by the undamped modal matrix), suf-
ficient conditions are presented which are easily verified. For these

reasons, the results in [S] are more general than MGller's and reduce to

Muller's in the special case.

II. A DISCRETE MODEL OF CRITICAL DAMPING

2.0 Introduction

 

In this chapter the first two problems outlined in the general
introduction are considered for the special case of a simply supported
beam. The beam is discretized using a finite element model and the
definitions of the various types of damping in [S], for multidegree of
freedom discrete systems, are directly applied to the discretized
continuous system.

The definitions of the various types of damping, including critical
damping, are presented in the first section. A discussion of the
physical implications of these definitions is also included. The con-
sistent finite element model of the simply supported beam is presented
in the second section. The coefficient matrices necessary to apply the
definitions from section one are obtained from this model.

The next two sections present derivations which make the definition
of section one easier to apply. In the first, a simple method for
calculating the critical damping matrix with respect to the original
coordinate system is derived. In the second, it is shown that the test
for the type of damping a given damping matrix provides can be carried
out in the original coordinate system. Working in the original coordin-
ate system eliminates the need to calculate square roots and inverses of
matrices. Numerical accuracy is increased and computing costs are
decreased.

In the last section, the critical damping matrix is calculated and

introduced into the equations of motion of the beam. These equations are

solved directly using MSC/NASTRAN finite element code, and the resulting
motion is shown to be indeed critically damped. Other example damping
matrices are compared to the critical damping matrix and the results are

presented.

2.1 Definitions

 

Consider a multidegree of freedom, discrete, damped linear system

described by the matrix differential equation

M§+c§+ {i=0 . (2.1.1)
where M and K are the positive definite, symmetric mass and stiffness
matrices, respectively, and C is the positive semi-definite damping

matrix. In previous work, Inman and Andry [S] rewrite (2.1.1) as

37+W+W=0 (2.1.2)
-% -1 -i -i i
where C: M CM , R = M KM , and M denotes the unique positive
definite square root of the positive definite matrix.M.

The critical damping matrix fer this system is defined to be

c = 2M5 m_i)i= 2? (2.1.3)

(21‘

in direct analogy to the single degree of freedom case. The motion of
the system is defined to be critically damped if’e = Ccr’ to be over-
damped if (B - Ccr) is positive definite, and to be underdamped if
(C - Ccr) is negative definite. A fburth possibility, mixed damping, is
defined if (2: - cc?) is indefinite.

Defining the various types of damping in this way results in

physically reasonable eigenvalues. As usual, the eigenvalues of (2.1.2)

are taken to be the Zn roots of the polynomial equation
IAZI + A8 + kl = 0 (2.1.4)

where -l denotes the determinant. In particular, Inman and Andry show

 

that (i) if (2.1.1) is critically damped then there are at most n dis-
tinct negative real eigenvalues and no complex eigenvalues, (ii) if
(2.1.1) is underdamped the eigenvalues are all complex and appear in
complex conjugate pairs with negative real parts, which corresponds to
all modes of the system oscillating in damped harmonic motion, and (iii)
if (2.1.1) is overdamped, then the eigenvalues are all negative real
numbers, and none of the modes oscillates. For the special case that

C is diagonalized by the undamped modal matrix (2.1.1) exhibits mixed
damping if and only if there is at least one real eigenvalue and at least
one complex conjugate pair of eigenvalues. One of the modes of the

damped system will oscillate and at least one will not.

2.2 The Finite Element Model

 

The partial differential equation of motion for the damped, trans-

verse, free vibrations of a simply supported beam is

3“v x t a 2v x,t
_I_le . c 3? v(x,t) + g—I—Lr—la at = 0 (2.2.1)
ax

with boundary conditions

v(0,t) = v(L,t) = 0 (2.2.2)

V"(O,t) = V"(L,t) = 0 (2.2.3)

where v(x,t) is the transverse deflection of the beam. This equation is

valid for the case in which damping of constant magnitude c, is applied

over the entire length of the beam, and the mass per unit length p,
modulus of elasticity E, and moment of inertia I, are all assumed to be
constant.

This equation will be approximated by a five element, consistent
finite element model [6]. The finite element model for the simply sup-

ported steel beam being considered is shown in Figure l.

 

 

5m 1 ‘k-fihn'fi

 

” (1) (l (2) (3) ), (4) , (5) 4,___,

.2m
1 2 4 5 6 _JL

 

 

 

 

 

 

 

The simply supported beam The beam cross
with finite element grid. section.
Figure 1

The physical quantities assumed are

E = 2.0 x 1011 PA 111 = 7.852 x 103 Elf-é
I = 6.67 x 10-5 m“ A = .02m2 (cross sectional area)

The node number for each node is located to the left and slightly below
each grid point, and the element number is enclosed by parentheses on
each element.

Consider a typical element of the beam, (e), shown in Figure 2.

It I
u. 1“! uj

6i *3 in (e) j) «3) 9J.

0+____.’+X 1

Figure 2. A Typical Element of the Beam

 

 

 

 

 

 

10

Each node has two degrees of freedom associated with it; one vertical
translation u and one rotation 6. Four unknowns must be determined for
each element. A cubic displacement function, whose four coefficients
will be determined in terms of the four unknown displacements, will be

assumed of the form
(e) _ 2 3
v (x,t) - ao(t) + a1(t) x + a2(t) x + a3(t) x (2.2.4)

Substituting the known boundary conditions for each element, and making

the small angle assumption, tan 6 = 6, gives

v(e)(0,t) ao(t) = ui(t)

v(e)(l,t)

ao(t) + a1(t) + a2(t) + a3(t) = uj(t)

v(e)'(0,t) a1(t) = -Bi(t)

v(e)'(1,t)

a,(t) + 2a.(t) + 3a.(t) = ej(t)

where the superscript (e) denotes a typical element and ( )' denotes
differentiation of ( ) with respect to the spatial coordinate x.
Solving for the unknown coefficients ai(t), collecting terms, and
rearranging gives

v‘e)(x,t) = Nfe)(x)ui(t) + N§°)(x)ei(t) + N§e)(x)uj(t) + N£°)(x)ej(t)

or in vector notation

v(e) (x,t) = 55(6) (x) 37(6) (t)

where N1(x) = l - 3x2 + 2x3 w(8)(t) ="ui(t)'
N2(x) = -x(1 - 2x + x2) 61(t)
N3(x) a 3x2 - 2x3 uj(t)
Nu(x) = x(x - x2) he). (t)J

 

 

11

The interpolating polynomials chosen are by definition Hermitian
polynomials since both the function and its first derivative are
specified at nodes i and j. The vector N(e)(x) is composed of the shape
functions for the element. As an approximation to equation (2.2.1),
the kinetic and potential energy of the beam and the Rayleigh dissipa-
tion function will be calculated from the finite element model. These
quantities will then be introduced into Lagrange's equations of motion
and the approximate equations of motion will be obtained.

The kinetic energy of a typical element, dropping the superscript
(e) for convenience, is

T =-% J m GT 0 d vol
vol

1 I O
%é-J WT NT NI; dx

I
NIH
t
,....,
B
hu-l
H
(D
V
S

1

where [m](e) = mA I NT (x) N (x) dx is the element mass matrix

0
156 -22 54 13
(e) =.Efl_ -
[m] 420 22 4 -13 -3
54 -13 156 22
13 -3 22 4

The Rayleigh dissipation function for a typical element is

1 . O
F = %] c(x) IT fiToc) N(x) i3 dx
0
C 1 .
=%cflj flu)Mndx$
0

12

where [c](e) is the element damping matrix

[c](e) c [1 Firm N(x) dx

0

156 -22 S4 13
~22 4 -13 -3
S4 -13 156 22
13 -3 22 4

.2.
420

The potential energy of a typical element is

EI 1 32
“TL (3??de

2 2 +, + 2 +
But %;;-= 3x [NT(x) w(t)] = BT(x) w(t) where B(x) =-%;y (N(x))

Substituting gives

1
V = El-WT J BT‘B dx'w
2 o
.1. 2T [1.] (e) z

1
where [k](e) = EI I BT‘B dx is the element stiffness matrix
0

[k](e)=EI -3 2
-6 3
-3 1 3 2

O)!»
oar-I

Using the usual direct stiffness method of assembly [7] one obtains
the mass matrix ”0’ the viscous damping matrix Co’ and the stiffness
matrix K0, for the entire beam. Hence, the kinetic and potential energy

and the dissipation function for the entire beam is

l +T 3 1 +T + 1 4T 4
T =-§ Mo w, V =-§ K0 w, F - E-w Co w
+T
where w = [111 61 u2 ... 116 66]

Substituting into Lagrange's equations of motion.

 

%— (ET— -§T +£+§—Y_‘=O, i=1, 2, .00, 12
t at} ”"i at} 3“'1
‘ i i
gives
M0 ; 4' CO I: 4' K0 I: = 0

This is a system of 12 equations in 12 unknowns. A usual occur-
rence in constructing a finite element model is that the system does
not satisfy the vertical translation boundary conditions. It also con-
tains spurious equations which arise by differentiating with respect to
known constant quantities. The overall analysis is not affected by
eliminating the spurious equations and satisfying the boundary condi-
tions. In particular, notice u1 a 1111 = 0. Substituting this fact
into the system will eliminate the first and eleventh columns of Mo,
Co, and K0. Also notice the first and eleventh equations were obtained

by differentiating with respect to the known quantities u and “11’ and

1
hence must be eliminated from the final system.

The final equations of motion fer the simply supported, steel

beam, with the boundary conditions satisfied, are

a
+

M; + Cw + K: = O

For the final mass, stiffness, and damping matrices see Appendix A.

14

2.3 A Simple Method fer Calculating Ccr

 

Consider again the matrix differential equation of motion for a

continuous system obtained from a finite element model of the system

MS? + c3? + )6? = 0 (2.3.1)

1
Proceeding in the same manner as Inman and Andry in [5],let i M 2 y
_1
and premultiply (2.3.1) by M 2 to give
- -190 -1 -0 - -1
Mimzy+MiCMiy+Mimiy=0 (2.3.2)
Let S be the orthogonal, undamped modal matrix (STS = I), let y = SE,

and premultiply (2.3.2) by ST to obtain

- _1 " _ _1 ’ _1 _1
STim2$2+STMiCMZSE+STM1Wis-2:0 (2.3.3)
1
Notice ST i MM 5 S = I and ST -% KM.2 S = A, a diagonal matrix with the

eigenvalues of the undamped system along the diagonal. Let o = M'i S.

T T }T

‘ .
Then ¢ 8 S M But(M 3F 2 M 5, since M is symmetric. Substituting

into (2.3.3) gives

T

oM¢E+¢T T

(:3? + a K43 = 0 (2.3.4)

Notice m is orthogonal with weightll(¢TM¢ = I) and ¢TK¢ = A. Hence,

from vibration theory [8], ¢ is the undamped modal matrix normalized
with respect to the mass matrix M.

For critical damping, M.i CM'% = Ccr in (2.3.3). But, from

matrix theory [9], ST Ccr S = 2A3. Thus (2.3.3) becomes
3; g;
z 4- 2A 2 + A? = 0 (2.3.5)

Comparing the coefficient of 2 in (2.3.4) and (2.3.5) one concludes

15

that for critical damping

T 1

4 C o = 2A
cr

01‘

3 T -1 i -1
Ccr 2(¢ ) A O (2.3.6)

-1 T T -1 . . .
Note, however, that o = t M, and (4 ) = M0. Subst1tut1ng into

(2.3.6) gives

c = zuonitTM (2.3.7)
CT

Thus, the critical damping matrix in the original coordinates is obtained

by simply perfbrming the indicated matrix multiplications.

2.4 A Note On The Comparison Test

 

Given an arbitrary damping matrix C, with respect to the original
coordinates, the test for the type of damping provided by C can be
carried out in the original coordinates. By definition, C - Ccr is

positive definite if and only if
*T 8 - c ) 3 > o 2 4 1
x ( - cr x ( . . )

+ .
for x an arb1trary nonzero vector. Denoting the critical damping matrix

A

A - 1
in ori inal ' = -$ -1 ' '
g coord1nates as ccr’ we have Ccr M Ccr M . Subst1tut1ng

.'.-~ I '. ~ -i 91 .
into (2.4.1) w1th C a M CM; gives

SETau‘i CM'1 - M‘i écr n‘i)§ > o
01'
ET ‘3(c - ccr) u'i I > 0 (2.4.2)

Letting ; = n‘5 I in (2.4.2) gives

16

'y’T(c - cm)? > 0 (2.4.3)

. + . . . + .
But, Since x is an arbitrary nonzero vector, so 15 y. Hence 8 = Ccr 15
positive definite if and only if C - Ccr is positive definite. A similar

argument also works for the negative definite and indefinite cases.

2.5 Results

The critical damping matrix for the simply supported steel beam
under consideration is calculated using (2.3.7). Both the eigenvalues
and the eigenvectors, normalized with respect to the mass matrix, of
the undamped system are needed to calculate the matrices in (2.3.7).
They are obtained by using MSC/NASTRAN finite element code. The
NASTRAN model (see Appendix A) of the beam consists of S CBAR elements.
As a model check, the consistent mass and stiffness matrices generated
by NASTRAN are printed out. These matrices are exactly the same as the
mass and stiffness matrices calculated by hand.

As a further model check, the eigenvalues generated by NASTRAN,
using the Given's method, are compared to the eigenvalues obtained by
solving the separated partial differential equation of motion of the
beam, see Table 1. Good agreement in the first fbur modes indicates
that the NASTRAN model is working properly. Table 1 also shows that
better numerical accuracy in the higher modes can be obtained by
modeling the beam with a larger number of elements. An examination of
the eigenvectors generated by NASTRAN shows the expected sinusoidal
behavior. As a final check, all calculations were perfbrmed using IBM
double precision scientific subroutines, with nearly identical “501115-

1

The matrix A is formed by replacing the diagonal of the 10 x 10

identity matrix with the square root of the eigenvalues obtained from

seem.Hq<ocem¢om=e\ns<umemmom=e-m smeozvv ...
woo“.fle<ocemmom=e\As<ueemmom=e-~ smaozcv ..
secst~S<u~emzom=e\As<uHemaom=e-H smoozvv .

17

mm.o mm.o~ es.h~ ec+mw~smma.c ec+memmmme.c ee+mem~no~.o ec+mmmw~m~.c ca
e~.e wm.m me.nm eo+memmme~.o eo+mm~nwee.o eo+msmmmma.o eo+mme~eo~.o m
ne.o om.~ Ne.m~ eo+m~o~n-.e eo+memma-.c eo+mfioam-.e eo+memefim~.o m
oe.o ~e.~ co.m~ mo+mm~mnmw.o mo+mnemmmw.c mc+mmeec~m.c eo+meaeeo~.e A
Hm.m an.c e~.~H nc+m~e~mme.o ne+mmHemme.o mo+memeeee.o mo+mmmemnn.o e
mo.o mm.o ma.o~ no+msmmame.e mo+mmmmnme.e mc+mm~emme.o mo+moe~mom.o m
He.o a~.o om.~ mc+mmoome~.c mo+memomm~.e no+m¢eemm~.c no+mmmnmm~.e e
oo.o mo.o ma.o mo+me~eees.o no+me~meea.c ne+mnemee~.o mo+mmmeee~.o m
co.c He.o e~.c ~o+me~m~nn.e ~o+mn~m~mn.c ~e+mnem~mn.o ~o+m-nmma.o N
oe.o- o.e so.e ~o+mwmfinm~.o ~e+mmwanm~.o Ne+me-nm~.c ~o+mneame~.o a
A~=c mm_uzm:ommm sues: ammo: smaoz .oz
... e. . zo~e<30m ezmzmmm em ezmzmsm a“ ezmzmem m mac:
mmuzmmmmea ESE: .ZEzmzmaeS 3:: mmuuzmaomi z<Em<zBm=

ammo: q<um%mmomzh Ibaz mamooz z<mhm<z mmmzk mo zomHm<mzou

H mam<k

18

NASTRAN, ordered from smallest to largest. The modal matrix of the
undamped system, o, is formed by placing the 1th eigenvector (which
NASTRAN has already normalized with respect to the mass matrix) in the
ith column of o. The critical damping matrix, with respect to the
original coordinates, is calculated using (2.3.7), see Appendix B.

The accuracy of this procedure is checked by performing the matrix
multiplication in (2.3.4). As expected, the result is the decoupled
system (2.3.5).

The matrix differential equation of motion (2.3.1) is solved
directly on the computer using NASTRAN with C = ch. A solution of
(2.3.1) of the form ; a Jext is assumed, where $ is a complex vector
and A is a complex number. The resulting complex eigenvalue problem is
solved by NASTRAN using the upper Hessenberg method. Very little
alteration. of the original NASTRAN model is necessary to perform these
calculations. The real eigenvalue extraction technique is replaced with
a complex eigenvalue extraction technique and the critical damping
matrix is entered using direct matrix input bulk data cards.

Ten complex conjugate pairs of eigenvalues and eigenvectors are
calculated. An examination of the eigenvalues (see Appendix B) shows
that the motion of the system is indeed critically damped. The fre-
quency of oscillation of any mode, fj = IIm(Aj)|/2n, where Xj is the
eigenvalue of the jth mode, is less than .38 Hertz. These frequencies
are nonzero due to round off error in the above calculations. The
eigenvalues appear in complex conjugate pairs, with imaginary parts
that approach zero. Thus, there are only ten distinct negative, real

eigenvalues, as expected for the critically damped case.

As an example of a damping matrix that theory predicts will

19

A

overdamp the system, let G = 2Ccr. Then G - Ccr = Ccr' From matrix

theory [10], a matrix A is positive definite if and only if there exists

A

a nonsingular matrix W such that A = WTW. Letting A = CCr and

W = 15Ai d-l, it is easily seen that Ccr is positive definite. Note
that W is nonsingular since both /'2_.Ai and 4-1 are nonsingular. The
frequency of oscillation of all modes, as calculated directly by
NASTRAN (see Appendix B) using C = Zécr in (2.3.1) is zero, as expected
for an overdamped system.

As an example of a damping matrix that theory predicts will under-
damp the system, let G = .l Ccr‘ fiThen C - écr = -.9Ccr, and hence is
negative definite. Ten complex conjugate pairs of eigenvalues, all
with nonzero imaginary parts (see Appendix B) are calculated using
NASTRAN. Thus the eigenvalues are of the expected type for an under-

damped system and all modes oscillate.

As a final example, let C in (2.3.1) represent damping of constant

NT/m

magnitude, C 3 W3468.04 m

distributed over the entire beam. The
theory of the next chapter predicts that the resulting system will dis-
play mixed damping, with the first five modes being nonoscillatory and
the last five modes being oscillatory. See example 8, Appendix C for
the actual finite element damping matrix associated with this type of
damping.

The theory of Inman and Andry also predicts that the resulting

A

system will display mixed damping. The matrix C - C (see Appendix

cr’
B) is clearly indefinite. Calculating the eigenvalues of the system
with this damping matrix in place, it is clear that the first four modes

are nonoscillatory with zero frequency and the other modes are oscilla-

tory with frequencies between 220 and 2300 Hertz, (see Appendix B).

III. AN ANALYTIC MODEL OF CRITICAL DAMPING

3.0 Introduction

 

In this chapter, the last of the problems outlined in the general
introduction is considered for the special case of a simply supported
beam. This problem is approached by considering an analytic model of
the beam, consisting of a partial differential equation with boundary
conditions. Both discrete damping, centered at a finite number of
points on the beam, and damping distributed over the entire beam are
considered.

The analytic model of both the undamped and damped simply supported
beam is presented in the first two sections of this chapter. Although
this information is well known and readily available (for instance, see
[11]), it is presented here to aid in the understanding of the defini-
tions and derivations presented in the remainder of this chapter.

In the next section, an expression for the amount of damping
necessary to insure a particular mode to be critically damped or over-
damped is derived, for a particular configuration of discrete damping.
This expression is obtained by applying a direct extension to continuous
systems of the definitions presented in Chapter 2. In particular, it
is shown that, for a system possessing classical normal modes, the
qualitative nature of the motion of the system can be determined by
examining the definiteness of certain combinations of the coefficient
operators.

In the final section, two sets of examples, which illustrate these
results, are presented. A finite element model is obtained for each

example and the resulting damping matrices are introduced into the

20

21

equations of motion for the beam. The equations are solved directly

using MSG/NASTRAN finite element code and the results are discussed.

3.1 The Undamped Simply Supported Beam

 

The equation of motion for the undamped, transverse free vibrations
of a uniform beam, with constant mass per unit length p, modulus of

elasticity E, and moment of inertia I is

1. 2
3 v(x,t) +.g_ 3 V(X,t) = 0 (3.1.1)
3x, EI at2

where v(x,t) is the transverse deflection of the beam. For the simply
supported beam being considered, the deflection and the bending moment

are zero at each end. This leads to the two boundary conditions

V(0,t) = V(L,t) = 0 (3.1.2)

V"(0,t) = V"(L,t) = 0 (3.1.3)

where ( )' denotes differentiation of (.) with respect to the spatial
coordinate x.
Assume that the solution to the above system is separable in time

and space, and is of the ferm
v(x,t) = V(X) ftt) (3.1.4)

Introducing the assumed solution into (3.1.1) and simplifying yields

.§; 1 d2V(t) a -1 d2f(t)
p VTx) dx. f(t) dtT “‘1'”

 

 

The left side of (3.1.5) depends only on x, and the right side depends
only on t. Both x and t are independent variables, so (3.1.5) has a

solution only if both sides are constant. Choosing a positive value oz,

22

for this constant, called a characteristic value or eigenvalue, leads to

two ordinary differential equations

1.

93159- - 5-1-62 v(x) = 0 (3.1.6)
X
2

1%}? + (n2 f(t) = 0 (3.1.7)

The choice of a positive value for the eigenvalues of the problem leads

to the harmonic, rather than exponential, solution of (3.1.7)
f(t) = C1 cos mt + C2 sin wt (3.1.8)

which is consistent with the fact that a conservative system has con-
stant total energy.

Equation (3.1.6) is a fourth order, homogeneous, ordinary differ-
ential equation. Four boundary conditions are required for its solution.
Substituting the assumed solution (3.1.4) into the boundary conditions

(3.1.2) and (3.1.3) yields the four required boundary conditions

V(O) I V(L) = 0 (3.1.9)

‘V"(O) = V"(L) = 0. (3.1.10)

The general solution of (3.1.6), with B“ =‘%f'w2, is

V(x) = C sin Bx + C2 cos Bx + C sinh Bx + Cucosh Bx (3.1.11)

1 3

Using boundary condition (3.1.9) and (3.1.10), we obtain the frequency

equation
sin BL = 0 (3.1.12)

with solution BL a nu, n = l, 2, 3, ... or

23

Bi. 8 (ml p—

2 3
L EI w n 1, 2, 3, ...

Thus, this system possesses a countably infinite sequence of eigen-

values

EI nn “
2 a - — = ... . .
(on p [L n 1, 2, 3, (3 1 13)

Corresponding to the eigenvalues on, one obtains the eigenfunctions or

natural modes of the system

nnx

L (3.1.14)

V“(x) = an Sin

which is the solution of the ordinary differential equation (3.1.6)
with boundary conditions (3.1.9) and (3.1.10).
The final solution of (3.1.1) is obtained from the expansion theorem

of modal analysis and is of the form

v(x,t) = 2 sin 1‘13 nn(t) (3.1.15)

n=1 L

where nn(t) are time dependent generalized coordinates- Introducting

I'TI’X

(3.1.15) into (3.1.1), multiplying by sin L

and integrating from

zero to L gives

L rnx a“ nnx
I n (t) sin ---—- (sin —) dx
1 n "

i

n 0 L ax L
-- EL. . nnx . rnx =
+ 2 nn(t)EI $1n-—E— s1n-—r— dx 0
n=1
Simplifying yields
" 2 = =
nn(t) + on nn(t) 0 n 1, 2, 3, ... (3.1.16)

This is a set of uncoupled ordinary differential equations with solution

24

 

nnCO) .
nn(t) a nn(0) cos out + n sin mnt
L nnx
where nn(0) = J p Si“"if’ v(x,0) dx (3.1.17)
0
. L nnx -
n (0) = I p sin -——- v(x,0) dx (3.1.18)
n o L

Thus knowledge of the initial deflection and velocity is required to

proceed further.

3.2 The Damped Simply Supported Beam
The equation of motion for the damped, transverse free vibrations

of a uniform beam, in operator notation, is

L[v(x,t)] + gz-C[v(x,t)] + M(x)-gixég-fi-El = 0 (3.2.1)

Where L a EI 31:- and M(x) a p. C is a linear homogeneous differential
operator consigting of derivatives with respect to the spatial coordi-
nate, x, but not with respect to time. C represents the effects of
continuous or discrete viscous damping. The boundary conditions fer this
system are (3.1.2) and (3.1.3), unchanged from the undamped system.
Proceeding in the same manner as in the undamped case, assume the
solution of (3.2.1) to be a superposition of the product of the eigen-
functions of the undamped system times time dependent generalized

coordinates, nn(t)

v(x,t) = 2 sinBE—x nn(t) (3.2.2)

n=1

Substituting into (3.2.1), multiplying by sin-3%5 , integrating from

zero to L, and simplifying gives

ZS

Hn(t) + Z Cn

O 2 8 =
r=1 nnCt) + wn nn(t) 0 n 1,2,3,... (3.2.3)

1'

where

L
. DTI'X . 1‘1TX
Cnr In 511'! T C[51n T ] dX (3.2.4)

Equation (3.2.3) represents an infinite set of coupled ordinary differ-
ential equations. In the special case where the operator C represents

constant damping distributed over the entire beam

L2
C J sin32%§ dx = gk. n = r
Cnr = { 0 (3.2.5)

0 ' n x r
where C is a viscous damping constant with units of distributed force per
unit velocity. This type of damping leads to an infinite uncoupled set

of ordinary differential equations.

fin(t) +.§ nn(t).+ w; n(t) = o n = 1, 2, 3, ... (3.2.6)
01‘

Hn(t) + 2;n w nn(t) + m2 n (t) = o n = 1,2,3,... (3.2.7)

n n n

The notation in (3.2.7) was chosen to render it similar in structure
to the equation of a single degree-of-freedom system. For critical

damping of the nth mode, Cu 2 1.

 

* m

c
Hence 5 = p

n 2pm

n
n - 1 or c a prn - 2p I70

= 2(11“-"l)2 m. (3.2.8)

We note from equation (3.2.7) that an infinite amount of damping would
be required in order to overdamp every mode of the system; the best

one could obtain would be the overdamping of a finite number of modes.

26

3.3 A Result on Discrete Damping

 

Consider a simply supported beam with discrete damping centered
at a finite number of points, m, on the beam. For this case, the

operator C in equation (3.2.1) is

m
c a 2 ciCx) XE (x) (3.3.1)
i=1 i
0 x t Ei
where XE (x) = { is the characteristic function for
i 1 x e E1

the set Bi' The set E1 is an open interval (ai, bi) where 0 S ai < b1
5 L, and, for any pair of open intervals, Eiiw‘Ej = 0, for i 2 j. De-
fining C in this way reflects the fact that the damping is not applied
to a set of discrete points, but instead, is distributed over a collection
of discrete open intervals.

Define the linear operator L1, to be

m c.(x)

le = 10 xE_(x) <l>(x) (3.3.2)
1

 

and the linear differential operator L2 to be

L2H] = % LN] (3.3.3)
at
where L a El BEF' as in equation (3.2.1). Also define the operator
L3 to be
L3[¢] = (4L2 - L§)[¢] (3.3.4)

Inman and Andry have shown in [12] that the type of damping
provided by the damping mechanism described by (3.3.1) depends on the
definiteness of the operator L3. Recall, an operator L is said to be

positive definite if and only if, fer any comparison function u, the

27

following holds,

I

I uL[u]dD Z 0 (3.3.5)
D

and equality holds only for u a 0. A function is a comparison function if
it is sufficiently differentiable over the domain D and it satisfies all
the boundary conditions of the eigenvalue problem. One set of comparison
functions for the simply supported beam being considered is the eigen-
functions of the undamped system, {sin 1F}::1 .
Utilizing the above definitons, the definiteness of the operator

L3 is seen to depend on the sign of the quantity dn’ defined by

L

L
d = 4 J vn(x) L2[vn(x)]dx - J

I‘ll m
n o I 2 x5. (xlej (20¢i (x)cj (x)v;(x)

0 i=1 j=1 1

L m L
= 4 J vn(X) Lzlvn(X) dX - 11 J C; (X) xE (X) v; (X) dX (3.3-6)
0 is o i

where V“(X) is the nth eigenfunction of the undamped system. It can be

shown that the nth mode is

underdamped if dn > 0,.
critically damped if dn = 0 and

overdamped if dn < 0.

Notice the similarity of these results to the results obtained fer the
discrete, multidegree of freedom systems discussed in section 2.1.
These results can be used to determine either the placement of damping
or the magnitude of damping necessary for the system to be critically

damped when one or the other of the parameters is in some way con-

'strained.

28

Consider the special case of damping of constant magnitude ci
placed on the beam such that the intervals over which the damping is
distributed are centered about the maximum and minimum values of the

eigenfunctions. For this case, the open intervals B1 are of the form

E.- Z—iL—l-5L,Zi‘l+s L i=l,...,n
1 2n 2n

with 2nd 5 1. Evaluating the first integral in (3.3.6) with

vn (x) a sin E%§-and L2 as defined in (3. 3. 3) gives

L 4
4f sin “—L" (131) 3., (sin ms)ax= ZE—I—L(— n) . (3.3.7)
0 3x L

 

Evaluating the second integral gives

21-1

 

 

 

 

m L c? (x) m C? ( 2n + ”L
zMI-g—z—x. (x) de' —. . ....
p 21 1
n: i=1 ( 2 - 6)L
n
m C2 o 0
i 1 . 21 - 1 . 21-1
121 L 37-[6 —'za;|(51n 2nu( 2n + 6) - s1n Znn ( 2n - 6)]
m L C?
a if -2——l,mp [2nn8 + sin 2nn8 ] . (3.3.8)

For the nth mode of the beam to be critically damped or over-

damped dn 5 0. Substituting (3.3.7,8) into (3.3.6) gives

2E1 V(fln)u "TL“’ g c2 [Znnd + sin Znnd] S 0

01'

 

I. Ill
32. nn 2

If the magnitude of damping is the same over each interval 51’ then for

critical damping‘we have

29

 

 

- 111 2 ‘ 081 (1r)
(ccr)n - 2(L ) IanS 4- sin Znné (3'3°10)

Notice, if 2nd a 1, that is, the damping is of constant magnitude,

continuously distributed along the entire beam, there results

"n 2
(ccr)n = 2 (E-) “051 (3.3.11)

This is the same value as was obtained by considering the partial

differential equation for this type of damping.

3.4 Examples

Two sets of example problems, consisting of four related examples
each, are considered. In the first set, the amount of damping necessary
to critically damp the first mode of vibration is calculated. In each
example, the damping is distributed over a single interval whose center
coincides with the geometric center of the beam. The width of this
interval is varied from .04m to 5m; a width of .04m being a physical
model of damping applied at a single point, and a width of Sm repre-
senting damping applied over the entire beam.) The intermediate cases
considered include damping distributed over the entire center element
and center three elements. See Table 2 for the amounts of damping
necessary to critically damp the first mode using these configurations
of dampers.

In the second set of example problems, the amount of damping
necessary to critically damp the fifth mode of vibration is calculated.
In each example, the damping is distributed over five intervals, each
of whose centers coincides with the geometric center of an element.

The widths of each of these intervals is varied from .04m to 1m; the

first case representing damping applied at five discrete points, and

30

the last case representing damping applied over the entire beam. The
intermediate cases considered include damping applied over widths of
.2m and .6m. See Table 2 for the amounts of damping necessary to

critically damp the fifth mode using these configurations of dampers.

Table 2
Summary of Data fbr Examples 1-8

Example Damping NT/m Complex Eigenvalues Frequency
Number Magnitude 57333. (real) (imag) (Hertz)

1 127853.760 -6.30172 3114.9086 18.28827

2 58084.188 -72.7073S t 90.34698 14.37917

3 38035.899 -109.5956 t 35.55132 5.658167

4 36138.722 -115.0625 1 1.682878 .2678384

5 3196344.000

6 l452119.700 Examples 5 through 8

7 950897.460 see Appendix

8 903468.040 for this information

The original finite element damping model (see Chapter 2) can be
modified to represent the various cases of discrete damping described
above by recognizing a characteristic each of these cases has in com-
mon. In each case, the damping is applied over one or more regions, each
of whose center coincides with the center of an element. Hence an
element damping matrix fer this general type of damping is calculated,
and the final finite element damping matrix is obtained using the same
procedures employed in the original model.

The general element damping matrix for discrete damping is obtained
in a manner analogous to the original model. Assume the damping is dis-
tributed over the intervals Bi 8 (ai, bi)’ where 0 5 a1 < b1 5 1, and

i - number of elements which experience damping. Then the Rayleigh

31

dissipation function for a typical element experiencing damping is

1 O O
F=%J c(x)xE (x)-wTNTN-\zdx
o i
.. b- '
a é- wT[c J 1 NT(x) N(x) dx '15
£11
= 2%- :T [c](e) .1:
where [c](e) is the element damping matrix

b. +
[c] (e) = c I 1 w(x) N(x) dx

a.
1

156011 ai) 22(bi ai) 54(bi ai) 13(bi ai)

= 4:0 4(bi ai) 13(bi ai) 3(bi ai)
- 2- 2

156(bi ,ai) 22 (b1 ai)
s

_ SYM «bi-aim

 

 

The Rayleigh dissipation function for elements which do not experience

damping is identically zero, and hence the element damping matrix for

this type of damping is the 4 x 4 zero matrix. The final finite

element damping matrix for each case is obtained by proceeding as in the

original model, using the appropriate elements damping matrices. See

Appendix C for the damping matrices fer each example described above.
The discretized matrix differential equations of motion (2.1.1)

is solved directly on the computer using NASTRAN with the various

damping matrices in place. The eigenvalues and frequency of the first

mode of the first four examples are listed in Table 2. Notice, as the

width of the region over which the damping is distributed is increased,

32

the magnitude of the real part of the eigenvalue also increases. This
indicates that the motion of the first mode decays faster as the width
of the region over which the damping is applied is increased. Also
notice, as the width of the region over which the damping is applied is
increased, the imaginary part of the eigenvalues and the frequency
decreases. Critical damping is attained only in the last example,
while the first three examples exhibit underdamped motion. This could
be due to the finite element model itself, which consistently over-
estimates these quantities. Indeed, the imaginary part of the eigen-
value for the last example can be shown to be zero, by examining the
decoupled ordinary differential equations which govern the motion of this
example.

The eigenvalues and frequencies of all ten modes of the last four
examples can be found in Appendix D. The first four modes should be
critically damped; Again, as the width of the regions over which the
damping is applied is increased, the theoretically predicted behavior
is more closely achieved. Indeed, the fifth example shows only the
first mode overdamped, the sixth example shows only the first two modes
overdamped, the seventh shows the first three modes overdamped, and
the last example shows the first four modes overdamped, with the
remainder underdamped. Again, this behavior could be due to the finite
element model. This model results in accurate eigenvalues for only the
first feur modes of the last example. This can be seen by examining
Table 3 where the theoretically predicted eigenvalues are compared to

the eigenvalues obtained from the finite element model.

Mode

No.

10

Table 3

Comparison of NASTRAN Model with Analytical Model

THEORETICAL
.Eigenvalues
(Real) (Imago)
-2.3021644
-5750.9155
~37.058599
-5716.1591
-l92.86618
-SS60.351S
_ -666.28015
-5086.9375
-2864.0435
-2889.1742
-2876.6088 15961.0677
—2876.6088 29698.1044
-2876.6088 213557.925
-2876.6088 t17730.18l
-2876.6088 122281.889

33

NASTRAN
Eigenvalues
(Real) (Ima8-)
-2.302657

-S750.801

-37.18228
-S715.922

-196.0541
-5557.051

-702.3069
~5050.801
-2876.552 -1385.286
-2876.554 +138S.286
-2876.SS4 t3600.778
-2876.554 :6062.379
-2876.554 19085.953
-2876.556 112500.37
-2876.558 114345.40

IV. CONCLUSION

The damped transverse free vibrations of a simply supported beam
have been considered. The question of critical damping for this beam
has been addressed in two different ways. Both a continuous model and
a discretized approximation to this model have provided infermation.

The continuous model of the beam was discretized using a five
element, consistent finite element model. A critical damping matrix
for this model was calculated, and subsequently shown to critically
damp the first ten modes of vibration simultaneously. Various other
damping matrices were also shown to provide the predicted type of
damping.

It should be emphasized that the critical damping matrix which has
been calculated is very heavily model dependent. A different finite
element model will result in a different, although possibly similar (in
structure), critical damping matrix. Alternatives to the finite element
model presented in this paper include a model consisting of a larger
number of elements and a lumped parameter finite element model. Note
that the latter of these alternatives would not be appropriate for this
analysis. The mass matrix in a lumped parameter model is singular, not
positive definite. Thus the theory on which these calculations are
based would not apply.

It has already been demonstrated that a model consisting of a
larger number of elements provides more accurate results. A critical
damping matrix developed for such a model would critically damp a

larger number of modes. Naturally, the physical effort involved in

34

35

computing such a critical damping matrix would also increase. However,
the theory upon which these calculations are based does not increase
in complexity as the order of the system is increased. Thus, the
number of modes which are critically damped is limited only by the
amount of physical effort one is willing to expend in the computa-
tion of the critical damping matrix.

The critical damping matrix which has been developed was used
primarily for comparison. A given damping matrix was compared to the
critical damping matrix to determine the type of damping it would pro-
vide. Although this is a very immediate and important use for this
matrix, a more satisfying use would be the direct application of this
matrix to the system. This, however, requires that the critical damping
matrix be interpreted physically, i.e., what magnitude of damping, placed
where on the beam would result in this damping matrix.

This question is difficult to answer because the critical damping
matrix is fully populated. A typical finite element damping matrix
possesses a very orderly "block diagonal" structure. A closer look
at the critical damping matrix reveals an approximate block diagonal
structure. One possible cause of the nonzero "off-block" terms could be
that the colunns of the orthogonal modal matrix used to calculate the
critical damping matrix are not quite orthogonal. This is a matter
which requires further investigation.

The question of critical damping fer the beam was addressed in a
second way, utilizing a continuous model of the beam. An expression
for critical damping of a given mode was developed for a particular
configuration of discrete regions on the been over which the damping

was applied. The model was discretized using the same finite element

36

model as in the first approach, with a damping matrix modified to
describe discrete damping. Several examples were considered.

The results from these examples were mixed. The model worked well
for the cases in which the damping was distributed over the entire beam,
or a large fraction of the entire beam. However, the model worked less
well for the cases in which the damping was more truly discrete. This
is not surprising, since the original, consistent finite element model
was designed to model continously distributed parameter system. Perhaps
a different finite element model, such as a lumped model, would give
improved results in the discrete cases.

The above calculations were carried out for a particular configur-
ation of discrete damping. This configuration was chosen for a mmber
of reasons. It was symmetric about the center of the beam, it simplified
the analysis, and it simplified the finite element model. .However, at
this time, it is impossible to say if this particular configuration
was physically the best choice for this problem. Determining the best
possible configuration of discrete damping fer this problem is another

matter which requires further study.

APPENDICES

APPENDIX A - FINITE ELEMENT MATRICES AND
THE UNDAMPED NASTRAN MODEL

39

fl

 

fi.)666.v

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6.6
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50

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APPENDIX C - FINITE ELEMENT DAMPING MATRICES FOR
DISCRETE AND CONTINUOUS DAMPING

61

3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3

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APPENDIX D - EIGENVALUES FOR EXAMPLES 5-8

66

 

 

 

 

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LIST OF REFERENCES

10.

11.

12.

LIST OF REFERENCES

Duffin, R. J., "A Minimax Theory for Overdamped Networks," Journal
of Rational Mechanics and Analysis, Vol. 4, 1955, pp 221-233.

Nicholson, D. N., "Eigenvalue Bounds for Damped Linear Systems,"
Mechanics Research Communications, Vol. 5, No. 3, 1978, pp.
147-152.

Muller, P. C., "Oscillatory Damped Linear Systems," Mechanics
Research Communications, Vol. 6, No. 2, 1979, pp. 81-85.

Beskos, D. E. and Boley, B. A., "Critical Damping in Linear Discrete
Dynamic Systems," Technical Report, Grant / Contract #N00014-
75-C-1042.

Inman, D. J., and Andry, A. N., "Some Results on the Nature of
Eigenvalues of Discrete Damped Linear Systems," Journal of
Applied Mechanics, Dec. 1980.

Cook, R. 0., "Concepts and Applications of Finite Element Analysis."
John Wiley and Sons, Inc., New York, 1974, pp 237-239.

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