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Discrete damping effects on a coupled second order system

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Title:
Discrete damping effects on a coupled second order system
Creator:
Mette, Bernie D
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Language:
English
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vii, 48 leaves : illustrations ; 29 cm

Thesis/Dissertation Information

Degree:
Master's ( Master of science)
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University of Colorado Denver
Degree Divisions:
Department of Mechanical Engineering, CU Denver
Degree Disciplines:
Mechanical engineering

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Subjects / Keywords:
Damping (Mechanics) ( lcsh )
Damping (Mechanics) ( fast )
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theses ( marcgt )
non-fiction ( marcgt )

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General Note:
Submitted in partial fulfillment of the requirements for the degree of Master of Science, Department of Mechanical Engineering.
Statement of Responsibility:
by Bernie D. Mette.

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University of Colorado Denver
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Auraria Library
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ocm17882327
Classification:
LD1190.E55 1986m .M47 ( lcc )

Full Text
DISCRETE DAMPING EFFECTS ON A COUPLED SECOND ORDER SYSTEM
by
Bernie D. Mette
B.S., University of Colorado, 1984
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Mechanical Engineering


This thesis for the Master of Science degree by
Bernie D. Mette
has been approved for the
Department of
Mechanical Engineering
by
Steven W. Peterson
Date


i i i
Bernie D. Mette (M.S., Mechanical Engineering)
Discrete Damping Effects on a Coupled Second Order System
Thesis directed by Associate Professor John A. Trapp
Piecewise linear energy dissipative effects on a two
degree of freedom second order vibrational system are analyzed.
The analysis is limited to discrete changes in the damping co-
efficient which allows a piecewise continuous solution over each
subinterval. The subintervals are between periodic transition
times at which the damping values are changed discretely.
Solutions are determined for each of two damping cases, Position
and Velocity damped, in order to discern the character of a more
general nonlinear solution. Three independent means are used to
analyze the two damping cases. First, a linearized analytic
solution is determined for a single degree of freedom system to
gain an introductory insight. Second, a numerical approximation
is used to solve the coupled set of differential equations for
the two degree of freedom system. Finally, the displacements are
measured directly from an experimental model for the two degree
of freedom system, constructed to be an exercise in M.E. 316,
Measurements Laboratory. The numerical solution provides a tool
for analyzing the more practical data from the experimental
working model system. Being able to correlate the numerical
solution and experimental measurements provides an understanding
of the two discretely damped cases. These two solutions provide
a basic understanding to the nonlinear response of a more general
nonlinear energy dissipative system.


CONTENTS
CHAPTER
I. THE ANALYTIC SOLUTION .............................. 1
1.1 Problem Statement ................................. 1
1.2 The Position Damped Solution ...................... 5
1.3 The Velocity Damped Solution ...................... 7
1.4 A Solution......................................... 8
II. THE TWO DEGREE OF FREEDOM SYSTEM......................13
2.1 Governing Force Equations..........................13
2.2 Analysis of Energy Rates...........................16
2.3 Method of Solution.................................18
III. THE COMPUTERIZED SOLUTION.............................20
3.1 Input/Output Options ............................. 20
3.2 The Runge Kutta Numerical Scheme ................. 21
3.3 Reducing Numerical Algorithm Errors................25
3.4 A Numerical Example.............................. 26
3.5 The Energy Rate Plots..............................28
IV. THE CONSTANT DAMPED EXPERIMENTAL RESULTS..............33
4.1 Design Considerations..............................34
4.2 Estimating Parameters..............................36
4.3 Displacement Measurements..........................36
4.4 Plotting Measured Data.............................39
4.5 Analysis of the Data
39


V
CONTENTS (continued)
CHAPTER
V. DISCRETELY DAMPED EXPERIMENTAL RESULTS...................42
5.1 Position Damped Model Results.....................42
5.2 Velocity Damped Model Results.....................44
VI. SUMMARY.............'....................................47


vi
TABLES
Table
4.1 Typical Measured Data with Constant Damping .......... 38
5.1 Measured Displacement Offsets of Velocity Damping . 45


vii
FIGURES
Figure
1.1 Comparison of Position and Velocity Regulated
Damping Displacement Motions .......................... 12
2.1 Sketch of the Vibrational System........................14
2.2 Variable Natural Frequency Response.....................17
3.1 Comparison of Discretely Damped Displacement
Motions............................................... 27
3.2 Comparison of Forcing and Dissipative Powers .... 29
3.3 Position Damped Spring and Mass Powers . ............. 31
3.4 Velocity Damped Spring and Mass Powers .............. 32
4.1 Variable Natural Frequency Response Showing
Measured Data in Table 4.1..............................40


CHAPTER I
THE ANALYTIC SOLUTION
1.1 Problem Statement
In this section we consider a single degree of freedom
spring, mass, dashpot system under a sinusoidal forcing function.
The basic equation of motion in terms of x is
mx + cx + kx = P0 sin(wft) , (1.1)
where the mass m and spring stiffness k are constant. The
damping coefficient c is constant on intervals but switches from
a higher value ch to a lower value cl during the cycle. Two
prescribed values of c used are:
Position Regulated Damping Velocity Regulated Damping
c = ch if x > 0 c = ch if x > 0
c = cl if x < 0 c = cl if x < 0
The analytic solution for both Position and Velocity
regulated damping cases is subdivided into two intervals. One
subinterval has a higher damping constant, ch, while over the
remainder of the period the damping factor, cl, is a lesser


2
value. The solution in each case is piecewise continuous upon an
interval of about half the period.
The transition times, or rather the time of travel over
either interval, is dependent on the damping factors used. The
exact duration of each interval is unknown, yet it is reasonable
to assume that the interval of higher damping will be longer than
the lesser damped interval. The period at steady state must
equal the inverse of the forcing frequency.
The displacements will vary inversely with the damping
factor. This implies that a larger peak displacement is expected
for the less damped interval, although, as stated earlier, the
travel time over this interval is expected to be shorter. The
solution is complicated by both transition times and amplitudes
being adjusted over each continuously damped subinterval. Thus,
it may be helpful to first demonstrate the solution in a single
degree of freedom system.
Defining the displacement solution over the interval of
higher damping as H(t), and over the interval of lesser damping
as L(t), one obtains from Eq. (1.1),
H(t) = d0 eSot + di esit + D sin (wft + 0 L(t) = b0 er<)t + bi erit + B sin (wft + 0b) (1.2)
Both of these solutions are continuous over the
appropriate subinterval of the total period. Contained within


3
these two equations are six unknown constants for a given set of
model parameters. Solving for the six unknowns at steady state
will require six "boundary" conditions which for the Position
damped case are as follows:
a) H (t0) - = o ,
b) H(ti) = = o ,
c) L(ti) = = 0 ,
d) L(t0 + 2pi/wf) = 0 ,
e) dH(t0) dt 0 dL(t0 + 2pi/wf) at
f) dH(ti) dt 1 = dL(ti) dt 1
Likewise, the parallel set of "boundary" conditions for
the Velocity damped case is as follows:
(1.4)
e) H(t0) = L(t0 + 2pi/wf)
f) H(ti) = L(ti) .
The first four boundary conditions define the initial and
final position or velocity of each respective case to be zero at
the transition times. The remaining two equations prescribe

a, -
b, rij
o
spij -
0
0
0
0


4
continuous velocities or positions at the two transition times,
t0 and ti. Equations e) and f) are necessary to determine the
two transition times at which the solution changes from H(t) to
L(t) and from L(t) to H(t).
Using these boundary conditions and the solutions in Eq.
(1.2), we can solve for the six unknowns which are; d0, d]_, b0,
bis t0, and tj. Since the unknown times are contained in each
exponential term the system of equations to be solved is highly
nonlinear. To obtain an approximate solution the equations can
be linearized about the average damped case. This linearization
requires expansion of the exponential terms using a Taylor's
series approximation. Small angle approximations such as the
angle itself are used for the sine of small angles. This
linearization process allows us to arrive at a closed form
algebraic solution although it becomes rather cumbersome.
It is assumed that upon the higher damped interval the
damping ratio is greater than unity. This implies that the
/
exponential transient terms will contain real powers. The
damping ratio upon the lesser damped interval is three tenths.
On this interval the powers will be complex conjugate pairs.
This also requires that constants multiplying these terms be
complex. One should remember that this problem is solved at
steady state, but with every change in damping the transient
terms are reintroduced. It is the transient part of the solution
that allows each given set of boundary conditions to be


5
satisfied. Also, every previous change in damping remains summed
in the solution as a decaying exponential term indefinitely.
1.2 The Position Damped Solution
The damping ratio is chosen to be three tenths on the
lesser damped interval, L(t), resulting in the r0 and r^ terms
being complex conjugate pairs. This implies the constants, b0
and b]_, are also complex conjugate pairs as follows:
Using Euler's equations on the complex terms of the L(t)
equations, results in the following:
r0 = ra + 1 rc
n ra 1 rc
b0 = ba i bc
bi = ba + i bc
L(t) = b0 erot + bi erit + sin (wft + 0b)
= 2 efat [ baqi(t) + bcq2(t) ] + B sin (wft + 0b) ,
(1.5)
where q^ and q2 are defined as,
qi(t) = cos rct
q2(t) = sin rct .
Using Eq. (1.5), the velocity L(t) becomes
*atk
L(t) = 2 e
[ ba Qi(tk) + bc Q2(t|c) ] + Bwf cos (Wft|< + 0b) ,
(1.6)


6
where Qi and Q2 are defined as,
Ql(t) raqx rcq2
Q2(t) = rcq2 + raq2 .
Rewriting the boundary equations (1.3) for the Position damped
case and using Eq. (1.5) and (1.6) gives
d0 eSot + di eSlt + D sin (wft + 0d) = 0 (1.7)
1) evaluated at t = t0
2) evaluated at t = ti ,
2 erat [ baqi(t) + bcq2(t) ] + B sin (wft + 0b) = 0
(1.8)
3) evaluated at t = tj
4) evaluated at t = t0 + 2pi/wf ,
d0s0 eSot^ + disi esith + D Wf cos (wfth + 0d) =
2 eratk [ ba Qi(tk) + bc Q2(tk) ] + B Wf cos (wftk +b)
(1.9)
5) evaluated at th = t0 and tk = t0 + 2pi/wf
6) evaluated at t^ = ti and tk = ti .


7
Equations (1.7), (1.8), and (1.9) provide the six equations
needed to obtain the six unknowns d0, dj, ba, bc, t0, ti for the
Position damped case.
1.3 The Velocity Damped Solution
To aid in the comparison of the solutions for both cases
pi/2 is subtracted from each phase angle 0b and 0 Velocity damped case. By doing so the transition times for the
Velocity damped case occur at nearly the same time as in the
Position damped case. Since the peak value is nearly 90 from
the zero displacement value, this adjustment corrects the
Velocity damped case to begin in phase with the Position damped
case.
The linearized solution of the Velocity damped case
follows directly from the earlier case. Rewriting the boundary
equations (1.4) for the Velocity damped case using equations
(1.5) and (1.6) gives
d0s0 eSot + disi eSlt + D Wf cos (wft + 0(j) = 0 (1.10)
1) evaluated at t = t0
2) evaluated at t = ti ,
2 erat [ba Qi(t) + bc Q2(t) ] + B Wf cos (wft + 0b) = 0
(1.11)
3) evaluated at t = ti
4) evaluated at t = t0 + 2pi/wf ,


8
d0 esth + di eSlth + D sin (wfth + 0d) =
2 [ baqi(tk) + bcq2(tk) ] + B sin (wftk + 0b)
(1.12)
5) evaluated at tb = t0 and tk = t0 + 2pi/wf
6) evaluated at th = tj and tk = tj .
Equations (1.10), (1.11), and (1.12) provide the six equations
needed to obtain the six unknowns d0, di, ba, bc, t0, ti for the
Velocity damped case.
1.4 A Solution
System parameters representative of the working experi-
mental model are selected to illustrate the previous equations.
The parameters necessary to provide a solution are a spring
constant and mass value in addition to a value of the forcing
magnitude and frequency. As previously stated, the H(t)
interval is overdamped while the L(t) interval is underdamped
with a damping ratio of nearly three tenths. While the damping
factors were arbitrarily chosen, the following system parameters
are representative of the experimental model:
m = 2.268 Kgs k = 2,000 N sec/m
P0 = 10.0 Newtons wn = (k/m)*^
Wf = 5 cycles per second, a constant.


9
Using the following basic equations for a linear one
degree of freedom system,
c w.
0 = tan
and
m(wjj-w^)
Roots r0,ri =
-c (e? 2m
Amp =
C-m2Cw2 -
we have, for the underdamped interval, L(t),
cL = 40
0b = -10.74
B = .7818 cm
ra = -8.818
rc = 28.36
and for the overdamped interval, H(t),
cH= 200 0d = -2.17
D = .1590 cm
s0 = -76.68
si = -11.50
An amplitude is also calculated based upon an average
damping factor. As mentioned previously the highly nonlinear
equation is to be linearized about an average damped solution
defined as A(t). For this A(t) solution the "c" value used is


10
cavg = (CH + CL)/2 = (200 + 40)/2-
Using c = 120 the amplitude for the A(t) solution is
avg
ampaVg = .2647 cm
Linearizing the solutions about the A(t) constant damped
case solution with this example set of parameters yields the
following values for the six constants in each case.
The constants for the Position damped case are
Transition Times H(t) constants L(t) constants
t0 = .00038 sec d0 =-.1554 cm ba =-.1478 cm
t\ = .11024 sec di =+.1579 cm bc =-.9265 cm
The constants for the Velocity damped case are
Transition Times H(t) constants L(t) constants
t0 = .00043 sec d0 = -.4251 cm ba = .9811 cm
tj = .11010 sec dj = .0639 cm bc = .1371 cm
The corresponding displacement motions are plotted in
Figure 1.1. In Figure 1.1 the L(t) subinterval solution is shown
from .11 to .20 seconds. The H(t) subinterval solution begins at
zero then ends at .11 second and is shown again between .20 and


11
.24 seconds. Had the L(t) subinterval been chosen to begin this
plot the Velocity damped case would have been offset above the
zero axis correspondingly. One must remember when comparing the
transition times that the Velocity damped case is plotted 90
degrees out of phase. This helped in simplifying the solution
since only two transition times were required rather than three
had the phase not been shifted.
A predominant feature in Figure 1.1 is the offset of the
Velocity damped case downward from the zero axis. This offset
downward occurs as the lesser c value, q_, is used upon every
downward displacement motion until a balance is achieved by the
springs being extended. A second characteristic feature is the
longer duration of the subinterval in which c = ch is compared to
\
the subinterval in which c = q_. This skewing feature adjusts
the traversal times and is clearly shown relative to the averaged
damped case plotted with a dashed line in Figure 1.1. These
characteristic features are also displayed in the more complex
two degree of freedom system presented in the remainder of this
article.


12
VELOCITY DAMPED
0.0 0.04 0.08 0.12 0.16 0.20 0.24
Figure 1.1. Comparison of Position and Velocity PNegulated
Damping Displacement Motions


CHAPTER II
THE TWO DEGREE OF FREEDOM SYSTEM
In this chapter the force and power balance equations are
written for a two degree of freedom system representative of the
experimental model. It is a forced system where the forcing
frequency is constant at 5 hertz as sketched in Figure 2.1.
Amplitudes are determined by varying the mass of the smaller
system, thus varying it's natural frequency. This plot of
amplitudes versus the small mass system is termed a variable
natural frequency response which is explained later. Finally, in
this chapter a numerical solution is obtained providing a highly
accurate approximate solution to the coupled equations derived.
2.1 Governing Force Equations
Writing the equations of motion for the system sketched
in Figure 2.1, where Z is the relative coordinate, gives
MY = P0 sin Wft 4kZ cZ 4KY ,
mX = cZ + 4kZ ,
(2.1)
(2.2)


14
where: P0 = the forcing magnitude (8.84 Nt)
M = mass of large system (22.5 kgs)
4K = large spring constant (3,100 Nt/meter)
m = small mass
4k = small spring constant
c = coefficient of viscosity
Rewriting Eq. (2.2) using the relationship for the relative
coordinate Z = Y X we obtain
(2.3)
y
Figure 2.1. Sketch of the Vibrational System


.15
At steady state the linear response will be composed of
sine and cosine functions. Then we may use a complex represen-
tation for Y and Z as
Y = y e1Wft and Z = z e1Wft (2.4)
Substituting Eq. (2.4) into Eqs. (2.1) and (2.3) to solve for the
amplitude at steady state and cancelling common terms, we have
y(-Mwf2 + 4K) + z(iwfC + 4k) = P0 ,
(2.5)
y(+mwf2) + z(-mwf2 + iwfc + 4k) = 0 .
To arrive at a nondimensional form of Eqs. (2.5), we define the
following
w
2
n
4k
m
(2.6)
Dividing both numerator and denominator by the spring constants
and defining the static deflection by P0/4K = Sj, we obtain the
nondimensional solutions of Eqs. (2.5) for Z and Y displacements:
_X_ =
Sd
1 (£f)2 + 10
[i-(jf)2][i-(^)2 + to] Rf?)2 ti+10)
9
(2.7)


16
(2.8)
The magnitudes of Y and X from Eqs. (2.7) and (2.8) are
shown in Figure 2.2 for the X coordinate equal to Y-Z. They are
plotted in centimeters by multiplying the coordinates in the
above equations by the static deflection calculated for the
working model where the forcing input's maximum value was
determined as 8.84 Newtons. These plots are produced by varying
the number of washers placed upon the small mass system. This
plot is not actually a frequency response because by varying the
small m value two terms in Eqs. (2.7) and (2.8) are changed
simultaneously. The mass ratio, R, is changed together with the
natural frequency ratio, wn. That is why this plot of amplitudes
versus small mass is termed a variable natural frequency
response.
2.2 Analysis of Energy Rates
The definition of work is
Work


Amplitude (cm)
17
The F dot V term is a power or energy rate. The energy rate
equation for the two degree of freedom system is obtained by

multiplying Eqs. (2.1) by Y and Eq. (2.2) by X and summing. This
results in
MY Y + mX X + cZ Z + 4KY Y + 4kZ Z = P0 sin(wft) Y . (2.9)
Equivalent expressions for the spring and mass energy rates can
is stated in terms of a coordinate, p, as
m d , ,
m p P = 2 3t ( pp )
k p P = jf ( PP )
(2.10)
Figure 2.2. Variable Natural Frequency Response


18
Using Eq. (2.10) the energy rate equation becomes
v
jf C y2 + f X2 + ^-Y2 + %Z2 ] = P0 s1n(wft)Y c Z2
(2.H)
The energy input and dissipation are written on the Hght
hand side of the energy balance. The storage terms, one for each
spring and mass parameter, are written on the left hand side.
The energy balance requires that the rate of change of energy
storage must equal the difference between input power and
dissipative power.
Clearly the energy rate at the transition times is very
discontinuous. At times t0 and ti the "c" values are changed
discretely while the velocities are the same upon both sub-
intervals at these times. In Position damping the transition
times occur when the velocities are a maximum producing a very
discontinuous dissipation rates. Therefore, it will be
interesting in the analysis to determine just how the energies
are distributed. This will require a graphical method to both
solve the equations of motion and produce the plots of interest.
2.3 Method of Solution
Since the Y displacements are small, a reasonable
assumption would be to ignore any nonlinearities in the Y
coordinate. Thus, the Y displacements would be purely sinusoidal
at steady state. After applying a linearization process one
could arrive at a solution in much the same manner as in


19
Chapter I. This result, in many respects, would be similar to
the solution presented in Chapter I.
The evaluation of the two degree of freedom problem will
be simpler and more accurately approximated by a numerical
computer program. Higher accuracy in the numerical solution can
be obtained than through the linearization process of Chapter I.
In addition, a computerized solution has several advantages
besides accurately solving the two coupled equations of motion
directly. One advantage is that with computer graphics software
the program will be able to produce the many plots required.
After the program is coded, parameters may be changed with the
effects of these changes seen within minutes with graphical
output as discussed in Chapter III.


Chapter III
THE COMPUTERIZED SOLUTION
In this chapter a Runge-Kutta numerical scheme using
small time steps is used to solve the two coupled equations of
motion for a two degree of freedom system. The computerized
solution interfaces computer graphics with the Runge-Kutta
algorithm to obtain the figures presented in this article. The
computerized solution is an accurate and versatile tool to
numerically solve a coupled forced system. Solutions of the
equations in Chapter II are presented for the two discretely
damped cases. Displacement and energy rates are plotted to
illustrate the effectiveness of the computerize numerical tool.
3.1 Input/Output Options
The menu allows the user to interactively change any of
the sixteen system parameters displayed. The displayed items
include: system properties, plotting times, and time step
variables. Lack of space available limited the number of items
that could be displayed. Also described in the menu are the six
different plots that the graphics program will produce. Three of
the plots produced are energy rates versus time and, of course,
displacements versus time are also plotted. The other two plots


21
described in the menu are plots of amplitude versus natural
frequency ratio. One of the plots is for the Y and X
coordinates, while the second is for the Y and Z coordinates.
When the Y and Z coordinates are plotted, measured data from the
model is also graphed. Both of the amplitude plots are graphed
for several values of damping to provide a comparison of damping
value to that of the experimental model.
The user may selectively vary the output of the plots to
a degree. The plotting time interval is displayed in the menu
for the user to interactively change. After the interval has
been selected, the program scales the plot to automatically cover
the width of the display. The displayed graph will cover the
entire screen for any time interval the user has selected in the
menu. This interactive method allows the user a greater
flexibility in achieving the desired graphical output.
3.2 The Runge-Kutta Numerical Scheme
In the numerical Runge-Kutta algorithm the two second
order equations of motion are written as four first order
equations. These four first order equations are-then solved
numerically by advancing the solution through successive small
time steps defined as h. Solving each first order equation
involved for a single time step requires an algebraic approxi-
mation of the first derivative with respect to time.


22
We start with the two equations of motion in terms of X
and Y:

M Y + 4K Y = P0 sin (wft) c (Y-X) 4k (Y-X) ,
m X = c (Y-X) + 4k (Y-X) . (3.1)
Defining the following
X = s
Y = r
variables,
dX
dt= u
air= v
the two second order Eqs. (3.1) can be transformed into the four
first order equations
1)
9
2> &-
3) gjr = Jp [ p0 sin wt 4Kr 4k(r-s) c(v-u) ]
(3.2)
4) 5if = s-C 4k(r-s) + c(v-u) ] ,
using fn ( n = 1,2,3 or 4 ) as an abbreviation for the function
on the right hand side of Eqs. (3.2). The fourth order Runge-
Kutta algorithm for Eqs. (3.2) is obtained by defining


23
ai=h f! (Vi)
bi=h fg (Ui)
cl=h f3 Cti. ri-Si, Vi-Ui]
di=h f4 [ H-si, vn--Ui]
a2=h fi (Vi + .5 h ai)
b2=h f2 (u-j + .5 h bi)
c2=h f3 [ti+.5h ,ri+.5hai, (r-f-s-f )+.5h (ai-bj), (vi~Ui )+.5h (c^-di) ]
d2=h f4 [ (r-f-si )+.5h (ai-bi), (vn--ui )+.5h (ci-dx)]
a3=h (Vi + .5 h a2)
b3=h f2 (u-f + .5 h b2)
c3=h f3 [ti + .5h,ri+.5ha2, (ri-si) + .5h(a2"b2), (v-|-u-j ) + .5h(c2d2)]
d3=h f4 [ (ri-Si)+.5h(a2~b2),(Vi-Ui)+.5h(c2-d2)]
a4=h fi (Vi + h a3)
b4=h f2 (u-f + h b3)
c4=h f3 [ti+h,ri+ha3,(ri-Si)+h(a3-b3),(vi-ui)+h(C3-d3)]
d4=h f4 [ (ri-Si)+h(a3-b3),(vi-Ui)+h(C3-d3)]
and calculating the new variable set at times t.+^ as
r.+j = ri + h(ai + 2a2 + 2a3 + a4)/6.0
si+l = Si + h(bi + ba2 + 2b3 + b4)/6.0
v.+1 = Vi + h(ci + 2c2 + 2c3 + c4)/6.0
ui+1 = Ui + h(di + 2d2 + 2d3 + d4)/6.0


24
This procedure is repeated for small time increments, h, for the
entire simulation
t.+j = t-j + h for i = 0, 1, 2, 3, ...
The typical value of h used was .2 millisecond.
The solution begins with the given initial conditions r0,
s0, v0, and u0. The initial conditions can greatly affect the
time necessary to reach a periodic steady state solution. It is
noted that with the exact initial condition steady state could be
reached immediately. Some difficulty arises in trying to choose
the initial condition that will give this steady state solution
in the least amount of simulation time. The characteristic
response curves are helpful here. They give the correct
amplitude at steady state which is an excellent first choice for
the initial condition. However, refining the initial conditions
to quickly reach steady state can be time consuming and
difficult. The primary difficulty in reaching steady state
occurs as the motion of the large springs, K, is undamped, thus,
transients remain in the springs indefinitely. This appears in
the experimental model also, but in the lab one is free to use
one's hand to damp out this motion. In the numerical solution it
is particularly important to determine initial conditions such
that no transients remain in the larger springs. Four initial
conditions must be known to begin the calculation. They are the
values of initial displacements and velocities as follows:


25
r0 = Y(t0)
vo st-nto)
So = X(to)
u Ht^o)
3.3 Reducing Numerical Algorithm Errors
To handle the periodic changes in damping from one
subinterval to the next is extremely important in the Runge-Kutta
subroutine. If throughout the Runge-Kutta algorithm one randomly
oversteps the transition times at which damping is to be changed
from a ci to a C2 value, these errors are compounded by propa-
gating into the next interval's solution. To reduce overstepping
the transition time, two different time steps are used within the
Runge-Kutta subroutine: a large step size is used on each con-
tinuous interval; and a small step size is used to locate the
transition time. A one microsecond step size is the smallest
step size possible using single precision floating point
variables. This allows only seven decimal digits of accuracy.
If the simulation were carried out to 10 seconds, the addition of
6
one microsecond would be rounded off as the 10 significant
digit is lost. The graphics software does not accept real
numbers declared as double precision. Avoiding mixed mode
arithmetic, this leaves an option of converting all variables of
double precision back to a single precision to be plotted outside
the Runge-Kutta algorithm. Thus, double precision arithmetic is
a cumbersome solution scheme and the decision to use single
precision accuracy was made.


26
Using the two step sizes the maximum error in determin-
ing the transition time is of the order of one microsecond. A
step size of about .2 milliseconds was used to approximate the
continuous subintervals with a high degree of accuracy. The
total precision was remarkably improved by using the two differ-
ent step sizes. The modified Runge-Kutta subroutine provides a
highly accurate fourth order approximation of the time response.
3.4 A Numerical Example
To demonstrate the computerized solution and graphical
output, an example problem is presented. This problem closely
parallels the experimental model two degree of freedom system and
compares the Position and Velocity damped solutions. All of the
parameters used in this example problem are estimated from the
experimental model except the coefficients of damping. The
damping values were chosen to vividly contrast the character of
the two different damping systems. Later, in Chapter IV it will
be explained how the damping values can be estimated from the
experimental model. For this example the damping coefficients
were chosen as follows:
Higher damped subinterval, H(t) : c= 30
Lower damped subinterval, L(t) : c= 2
Constant Average Damping, A(t) : c= 16
The displacement response versus time is plotted in Figure 3.1
for both Position and Velocity damped cases. The discretely


Amplitude (cm) Amplitude (cm)
-0.25 -0.125 ^ 0.0 0.125 0.25 -0.25. -0.125 _0.0 0.125 0.25
Figure 3.1. Comparison of Discretely Damped Displace-
ment Motions


28
damped displacements are compared to a constant damped response
with an averaged "c" value, the A(t) solution. The average A(t)
response curve is sketched in dashed lines for the X coordinate
only. Also shown is the much smaller Y coordinate discretely
damped motion. Since this response only deviates slightly from a
purely sinusoidal motion, no dashed line is presented for
comparison.
The skewing in the period caused by the time duration of
each subinterval being altered is only slightly noticeable in
Figure 3.1. Using the "c" values representative of the experi-
mental model, this effect would hardly be distinguishable. In
the experimental model, as reflected in Figure 3.1, the pre-
dominant feature is an offset from the zero axis in the lower
figure. Looking ahead, this offset feature of the "Velocity"
damped system should be easily discernible in measured experi-
mental data.
3.5 The Energy Rate Plots
In Chapter II section 2 the energy rate equation for a
forced two degree of freedom system was discussed. For the
present example problem the energy rates are plotted to
illustrate the differing energy dissipation rates of the two
systems. As displayed in Figure 3.2, at the transition times the
Position damped system is very discontinuous. This discontinuity
results from the two differing "c" values of damping upon the
H(t) and the L(t) subintervals in the energy calculation. In the


Dissipation (mJ/S) Dissipation (mJ/S)
-38.7 0.0 38.7 77.4 116.1 -54.3 0.0 54.3 108.7 163.0
29
POSITION DAMPED
Figure 3.2 Comparison of Input and Dissipative Powers


30
experimental model this instantaneous change in "c" values is
just not possible. Some elapsed time is required to change the
damping device from the L(t) to the H(t) response. Within the
computerized solution these discontinuities propagate into the
energies of the springs and masses. Figures 3.3 and 3.4 show how
the rates of energy change of the springs and masses vary with
time. As expected in the Position damped solution, Figure 3.3,
the spring and mass terms are affected by the discontinuous
damping. Whereas in Figure 3.4, all the energy rates plotted are
continuous for the Velocity damped case.
With all the energy rate diagrams shown one can see that
the governing energy balance is indeed satisfied. At any
specified time one can verify the energy rate Eq. (2.9) (or 2.10)
using the scale factors provided for each individual plot. These
scale factors are predetermined by the computer program. This
allows each plot to be automatically maximized to fill the entire
height of the display. Without such an algorithm the user would
need to manually rescale each plot for every change in a menu
parameter. Having each plot automatically scaled adds
versatility to the program and provides higher quality output.


31
C\J
SMALL MASS SYSTEM POWERS
Figure 3.3 Position Damped Spring and Mass Powers


mass ^aiu/a; nasb vmu/o;
-94.1 -47.0 0.0 47.0 94.1 -127.5 -63.7 0.0 63.7 127.5
32
SMALL MASS SYSTEM POWERS
Figure 3.4 Velocity Damped Spring and Mass Powers


CHAPTER IV
THE CONSTANT DAMPED EXPERIMENTAL RESULTS
This chapter provides us with some background on the
experimental model. The model was constructed primarily as a
Measurements Laboratory student exercise where it has been used
for the past two semesters. This vibration instrument is an
excellent example of a textbook vibrational problem which is
emphasized from a measurements viewpoint.
This chapter presents data that are obtained from the two
degree of freedom experimental model with constant damping.
First a brief discussion explains the means employed to obtain
the measured data. This discussion includes some statements
about design features of the machine itself. Then in this
chapter the experimental model is shown to provide expected
linear theory results with constant damping. It is important to
first verify the model as a linear system before proceeding to
obtain results for the nonlinearly damped cases.
The experimental model was constructed primarily to be a
practical educational aid. In the M.E. 316 (Measurement II)
laboratory exercise the students must calibrate a displacement
pickup which is a linear variable differential transformer.
After gaining experience on the operation of this displacement


34
pickup during the calibration,- the students measure the
displacement amplitudes versus variable natural frequency for
both the Y and Z coordinates to develop the response curves for
the two degree of freedom system. The forcing frequency is
constant, however the frequency ratio is varied by varying the
/
natural frequency of the small mass system. This is accomplished
by varying the small mass system or more directly stated by the
addition of washers to the small mass platform. The students
develop an amplitude versus variable natural frequency response
which they plot for the two degree of freedom system. In doing
so they achieve hands on experience in the operation of an
experimental vibrational system in addition to developing their
skills in the measurements tasks.
4.1 Design Considerations
The vibrational instrument is designed to be durable yet
the construction provides the students with a clear view of the
instrumentation operation. The students may inspect all the
system components including the damping device, displacement
pickups, and the forcing drive "train to the motors. The damping
device is enclosed within a large plexiglas cylinder so that the
inner components are clearly visible. Knowing the speed of the
two synchronous motors the students determine the forcing
frequency by applying the gear ratios of the driving linkage.
The pickup the students calibrate has been disassembled for their
easy inspection. It is important that all the instrument


35
components be readily available for the students' inspection to
facilitate the laboratory learning experience.
Since students will be handling the vibrational measuring
instrument it is expected that some abuse may occur. To increase
the durability, the apparatus is constructed with the Z
coordinate pickup mounted to a very rigid support. The rigid
support is fastened to the center of a stabilizing spring made
from a thin brass plate four inches in width. The ends of the
stabilizing spring are secured to the foundation of the
instrument support structure. This design helps to constrain the
motion of the Z coordinate bracket to a vertical motion only. In
the vertical displacement direction the stabilizing spring is in
a purely elastic bending mode. This securely fastens the
vibration apparatus to the base and protects the instrument from
a moderate jolt given by a careless student.
Built into the small mass system is a torsional resonance
mode. This is obtained by varying the moment of inertia with the
placement of the washers upon the small mass platform. The
washers are placed at an appropriate radial distance so that a
resonance is observed. This resonance mode occurs with a nearly
zero input in the rotational direction by the constant frequency
forcing function. This demonstrates to students how important it
is to analyze every degree of freedom for a possible resonance
frequency.


36
4.2 Estimating Parameters
The computer program provided an excellent means of fine
tuning the system parameters for the experimental model. System
parameters such as spring constants are particularly difficult to
estimate. The spring constant values were first estimated by
assuming the displacement was a maximum when the natural
frequency was in resonance with the forcing frequency. This
produced only approximate results which were further refined with
the aid of the computer program. The interactive computer
program allows one to vary the spring constants to see the effect
within minutes via the graphic output. This proved to be most
useful in accurately determining the constants for the small
springs.
Mass values were determined directly through accurate
scale readings. These values are subject to some variation due
to changes in the fluid level within the damper. Also, the mass
of the individual washers which are added to the small mass
platform will vary, mainly due to their poor quality. These
errors are small in magnitude and do not affect the results.
4.3 Displacement Measurements
A variable resistance strain gage provides an output
voltage proportional to the displacement motion as a function of
time. A single strain gage senses the Z coordinate's motion as
the elastic stabilizing spring is strained, This is accomplished
by actual sensing the strain in the brass stabilizing spring that


37
secures the small mass system to the rigid Y coordinate's large
mass system. Besides acting as a measurement device, these thin
brass plate springs are good stabilizers. Since the displacement
is proportional to strain it can be dynamically displayed on an
oscilloscope. This pickup was an excellent and inexpensive
secondary measurement method to display the displacement motion
versus time.
A quantitative phase measurement method required less
equipment. In fact, with a stroboscope one can observe the phase
from the forcing function directly. This is accomplished by
adjusting the rate at which the strobe flashes to twice the
forcing frequency. Then one may observe the phase versus
displacement relationship in still frame snapshots. This
fundamental observation of phase readily relates to the students
and conveys the strobe's many uses.
The model is equipped with two linear variable
differential transformer pickups. These pickups provide a more
precise measurement means for sensing the displacements.
However, this displacement signal is encoded in a complex wave
composed of the 60 Hertz excitation signal and the forcing
frequency. This poses some difficulty for the students to grasp
and in all fairness is a bit cumbersome.
These pickups are calibrated to convert a voltage to a
corresponding displacement. The data presented in Table 4.1 were


38
taken from the model by a group of students in the Measurements
Lab. These data are typical of the results an average student
may obtain. The data are from the two degree of freedom system
with constant damping.
Table 4.1
Typical Measured Data with Constant Damping
Number of Frequency Z Coordinate Y Coordinate
Washers Ratio centimeters centimeters
0 .695 .044 .037
2 .740 .0535 .0335
4 .783 .0735 .0335
6 .823 .0975 .027
8 .861 .129 .0255
10 .898 .1955 .021
12 .933 .284 .018
14 .967 .391 .0365
16 1.0 .489 .0605
18 1.031 .391 .0755
20 1.062 .292 .068
22 1.092 .2345 .0575
24 1.121 .185 .0545
26 1.149 .166 .0545
28 1.177 .147 .0515
30 1.204 .137 .0485
A complete description explaining how to operate the
vibrational instrument has been written to assist the students.
These data were obtained by students using this detailed step by
step procedure in Measurements II, M.E. 316.


39
4.4 Plotting Measured Data
The best estimate of the coefficient of damping is
accomplished by measuring displacements over a range of
frequencies to obtain a variable natural frequency response
curve. After measured amplitudes such as the data in Table 4.1
are input into a storage file called "Dat-file" the program will
construct a plot. The subroutine "Actual" plots the data points
while subroutine "Hermite" constructs a cubic Hermite polynomial
to contain each data point. The Hermite polynomial is
interpolated between the data points so the slope is continuous
at each data point.
Superimposed upon the measured data are the theoretical
amplitude curves. These curves are the dimensional amplitudes in
centimeters found by multiplying Eqs. (2.5) and (2.6) by the
static deflection measured from the working model. With both the
theoretical and the actual data plotted, a "c" value of the
damping coefficient can be estimated simply by comparison.
4.5 Analysis of the Data
The data in Table 4.1 are graphed in Figure 4.1 along
with the expected analytic results for these coordinates. From
this graph we may compare the measured data to the analytic
results shown for several different "c" values. As seen in
Figure 4.1, the curve constructed from the measured data is
significantly displaced below the analytic plots. The amount of
this displacement is best qualified at the two end points where


40
AMPLITUDE vs. FREQUENCY
Figure 4.1. Variable Natural Frequency Response
Showing Measured Data in Table 4.1
the analytic curves converge. The lower measured displacements
are explained as an effect of dry friction damping more commonly
known as Coloumb damping.
The entire set of analytic displacement curves displayed
can be multiplied by a constant friction correction factor. From
these corrected response curves a coefficient of damping value
can be estimated for the experimental model which naturally
contains dry friction. Using Figure 4.1 one must first estimate
the amount of friction in the experimental model. The amount of
friction depends greatly upon the alignment of the system,


41
however the data graphed in Figure 4.1 are typical. The
displacement amplitudes are estimated to be decreased to about
70 percent due to frictional losses. The coefficient of damping
for this linear damped case is approximately 4.0 Nm/s for the set
of data plotted in Figure 4.1.


CHAPTER V
DISCRETELY DAMPED EXPERIMENTAL RESULTS
Outlined in this chapter is the design of both damping
devices and the characteristic results measured. A demanding
problem was designing both the required damping devices to meet
the criteria of Velocity and Position regulated damping devices
as described in Chapter I. The construction of these devices was
constrained by very limited funding. An electromagnetic shutter
would have provided a nearly instantaneous change in damping,
however such a device was cost prohibitive.
The design of each damper is presented in detail in this
chapter. Also discussed are the results provided from each
damping device when operated in the vibration instrument. These
results quantitatively describe the characteristic features as
previously observed from the numerical and analytic solution. No
comparison between the two devices is presented since it was not
possible to construct both a Position and Velocity damper device
with equal values of damping upon each subinterval.
5.1 Position Damped Model Results
The position damper device is basically an oil-filled,
plexiglas cylinder containing five circular shafts having two
different diameters and a Teflon sealed 'plunger. These shafts


43
slide through the plunger that has been drilled to pass the
shafts with a small clearance. Thus when the plunger motion
passes back and forth over the discrete diameter change, a
corresponding change in damping occurs. The viscous damping
changes are proportional to the change in the area of the fluid
flow as the plunger travels across the diameter change in the
five shafts.
The vertical position of the plunger is adjustable in
the Position device constructed. Adjustments allow the plunger
to travel solely upon either the small or large diameter section
of the five shafts. This provides a means to determine a "c"
value over each diameter of the shaft individually. The data in
Figure 4.1 came from the smaller diameter shaft section where a
"c" value of 4.0 was approximated. Repeating this procedure to
obtain a "c" value over the larger diameter section of the shaft,
a value of 10 was determined. With the damping values in the
H(t) and L(t) solutions defined, an average "c" value of 7 is
calculated for the A(t) average response.
The displacement motions are not shown for these two "c"
values because the displacement motions would be difficult to
distinguish from the purely sinusoidal curves. However data
measurements are relevant and confirm that the Position damped
system has a peak displacement amplitude nearly equal to the
amplitude for the A(t) response: By referring to Figure 3.1 one
can understand the difficulty in attempting to measure any


44
changes in the periods of each subinterval. Using the dynamic
strain gage, the motion is displayed on an oscilloscope. With
this pickup no discernable difference in the times of the
subinterval lengths can be measured. The losses due to friction
contribute to the measurement problem. Even though the damper
was built to provide the largest difference in damping values
possible and tested with several working fluids, basically the
effect of the skewing of the period is not very prominent. The
only conclusion arrived at from the Position damped case is that
the nonlinear effects cause only a slight deviation from the
averaged A(t) response.
5.2 Velocity Damped Model Results
The velocity damper apparatus provides only limited
results, but in this case the measurement problem is simpler. As
is clearly portrayed in Figure 3.1 the dominant effect of the
Velocity damped motion was the offset of the displacements from
the zero axis. The displacement can be determined from the
linear variable transformer pickups with little error.
The velocity damper is designed with five trapdoor
springs. The trapdoors are opened for motion in one direction
and closed in the other direction allowing the area of fluid flow
through the plunger to be changed as its direction is changed.
However, the design of this device does not permit measurements
to estimate the damping values upon each subinterval.
Nonetheless, this does not interfere with our primary objective


45
to measure the amount of displacement offset that occurs in the
Velocity damped case.
Typical data measured from the working model for a
Velocity damped system are shown in Table 5.1. These data are
from the Velocity damped case where the lesser value of damping
occurs upon the downward motion of the period. Table 5.1 shows
the offset below the axis of reference by the difference in
magnitude of the highest and lowest displacement over a period.
These data are shown for three variable natural frequency ratios
where the displacements are a maximum.
TABLE 5.1
Measured Displacement Offsets of Velocity Damping
Frequency
Ratio
.967
1.00
1.031
Lowest
Displacement
-.513 cm
-.533 cm
-.513 cm
Highest
Displacement
+.349 cm
+.379 cm
+.358 cm
To obtain representative data the distance required to bend the
trap door springs should be only a small percentage of the total
displacements. The transition time, or rather the time for the
springs to open or close, is a smaller percentage at the three
frequency ratios shown in Table 5.1.


46
A similar analogy can be drawn in the Position damped
case where damping changes are proportional to changes in the
diameter of the shaft. The analogy being the plunger's traversal
over the region where the shaft diameter changes should be
relatively small in comparison to the total displacement. The
diameter changes are somewhat fixed to achieve a discretely
damped nonlinear motion effect. But the total displacement must
be large enough that the plunger is operating outside the
transition region a majority of the time. Balancing the damping
transitions while trying to achieve the largest change in "c"
values possible proved difficult. Thus, the results provided
only demonstrate the characteristic features as expected from the
earlier analysis through a prospective limited by physical constraints.


CHAPTER VI
SUMMARY
In summary, nonlinear effects of damping have been
analyzed through three independent means: analytical, numerical,
and experimental. These different methods of analysis all showed
the same characteristic features of nonlinear damping effects.
However, a different but complementary perspective was gained
through each method of analysis.
Results of analysis showed several characteristic
features that correlated each method of analysis. One feature of
nonlinear damping was that the duration of the higher damped
subinterval was slightly longer than the lower damped
subinterval. This skewing feature in the transition times was
observed in both Velocity and Position damped cases. However,
the adjustment in transition times was not as significant as the
amplitudes being varied inversely with the damping factor. The
Position damped case showed the amplitude of the lesser damped
case to be larger than the higher damped subinterval. The
amplitude comparison was made with a constant damped solution
using an averaged coefficient of damping value. Finally, the
most predominant feature was an offset from the axis of reference


48
which occurred in the Velocity damped case and was not observed
in the Position damped case.
The computerized solution was a most useful tool in
providing approximate results to a discretely damped two degree
of freedom system. With this tool energy rates and displacement
motion were plotted for an example problem. These plots
correlated directly to the results measured from the experimental
model. Although the results from the experimental model were
rather limited, the model provided a physical interpretation of
nonlinear damping effects. The effects of the model were
constrained by physical limitations that presented a viewpoint
that was not seen from the analytic results in Chapter I.
A bonus achieved from this project was the addition of a
vibration experiment to the M.E. 316 Measurements Laboratory
experience. The experimental model conveys to the students
several measurement techniques. In applying these techniques the
students gain an understanding of vibrational principles clearly
related through the measurements exercise.
Nonlinear problems were only vaguely discussed in
previous classes. The experience gained in solving this
nonlinear problem is especially valuable since the analysis
conducted correlates different perspectives of engineering
analysis.