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 (Â£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.