Parametric time domain system identification of a mass spring-damper

Session #

Parametric Time Domain System Identification of a Mass-Spring-Damper
System

Bradley T. Burchett

Department of Mechanical Engineering, Rose-Hulman Institute of Technology, Terre
Haute, IN 47803

Abstract

One of the key objectives of any undergraduate system dynamics curriculum is to foster in the
student an understanding of the limitations of linear, lumped parameter models. That is, the
student must come face to face with the fact that models do not perform exactly like the physical
system they are created to emulate. This is best done in the laboratory with a physical system
that has small non-linearities which prevent the student from obtaining an exact match between
model and experiment. This work describes an experiment designed for the sophomore system
dynamics course offered at the Rose-Hulman Institute of Technology. This lab uses a
commercially available hardware system and a digital computer. By a clever combination of
various response data, and using known differences between effective masses, the effective inertia
of motor, pinion, rack and cart are estimated without requiring disassembly of the system.
Typical results are shown.

Introduction

The mechanical engineering and electrical engineering faculty at Rose-Hulman (RHIT) are
currently upgrading the system dynamics and controls laboratory. One of the primary courses
this lab services is Analysis and Design of Engineering Systems (ES 205) which is a sophomore
level system dynamics course taught to all mechanical, electrical, and biomedical engineering
majors. ES 205 focuses on lumped parameter modeling of mechanical, electrical, fluid, and
thermal systems.

The hardware plant used in this lab is the Educational Control Products (ECP) Rectilinear
Control System1, shown in Figure 1. This is a translational mass-spring-damper system driven
by a DC electric motor that provides up to three degrees of freedom of motion. System
stiffnesses may be changed to the user's liking. A variable air damper may be connected to any of
the masses.The plants also provide for varying the system mass by adding or removing 500g
masses. Thus the differences in mass between possible configurations are well known.

Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition
Copyright © 2005, American Society for Engineering Education

Fig. 1. The ECP Rectilinear Control System

Section 2 describes the problem and outlines the theory underlying parametric system
identification, and the data reduction methods needed for this lab. Section 3 outlines the lesson
objectives in detail. Section 4 discusses the data collection procedure. Section 5 presents typical
results.

Problem Statement and Theory

One of the primary misconceptions that students fall into is that lumped parameter models can
exactly predict the behavior of real systems. For example, Figure 2 shows a lumped parameter
schematic of a single mass system.

Fig. 2. Schematic of Lumped Parameter Model

The spring and damper are assumed to contribute no mass to the system. An electric motor is
assumed to provide a force directly to the mass. A parametric differential equation model
matching this schematic is given in Equations 1 and 2.

c k k
˙˙ +
x ˙
x+ x= Kf (t )
m m m (1)

m c
˙˙ +
x x + x = Kf (t ) .
˙
k k (2)


The model and experimental step responses are compared in Figure 3. Although the match is
fairly close, it will never be exact due to small non-linearities in the system.

Fig. 3. Comparison of Expermental and Parametric Model Step Response m = 0.9538kg, k = 216.2 N/m,
c = 14.01 N-s/m

The parametric models in Eqns 1 and 2 can be directly compared to their non parametric
counterparts

˙˙ + 2zw n x + w n x = Kw 2 f (t )
x ˙ 2
n
(3)
and
x
˙˙ 2z
+ x + x = Kf (t ) .
˙
w 2
wn (4)
n

Matching zeroth order coefficients between Eq. 1 and Eq. 3 or second order coefficients between
Eq. 2 and Eq. 4, we obtain the familiar formula for system natural frequency.

2 k
wn =
m (5)

2
Combining this with the definition of damped natural frequency, w d = w n 1 - z , we arrive at
the following equation relating damping ratio, stiffness, mass, and damped natural frequency.

k 2
w di = 1 - zi
mi

The subscript i indicates that we will use several values of mass, but a constant stiffness.
These eqns can be manipulated to the following form which is linear in the unknowns (k and mi)


2
(
mi w di - k 1 - z i = 0
2
) (6)

The values of damped natural frequency ( w di ) are obtained by measuring the response peak time
and applying the relationship

p
tp =
wd . (7)

The values of damping ratio are found by measuring the peak response ( y peak ) and steady state
response ( y ss ) and computing the decimal form of percent overshoot ( M p ).

y peak - y ss
Mp =
y ss

Percent overshoot is related to damping ratio by the formula

Ê - pz ˆ
M p = expÁ 2
˜.
Ë 1-z ¯

Which can be inverted, yielding

2

z=
( ln( M )) p
. (8)
2
( ln( M )) + p
p
2

From their coursework in ES 205, students are familiar with the typical second order response
characteristics such as settling time, percent overshoot, and peak time, and how these
characteristics relate to non-parametric model properties, namely natural frequency and damping
ratio. Homework exercises include determining peak time and percent overshoot from a typical
experimental step response such as that shown in Figure 3, and determining a second order non-
parametric model from these response characteristics. However, without knowing at least one of
the parameters m, c or k of Eqs. 1 and 2 from physical measurements, it is impossible to
uniquely determine the three system physical parameters from the two properties of natural
frequency and damping ratio. This experiment is designed to take the student deeper by
combining information from several step responses.

For the step response of cases four through seven shown in Table 1 below, the student is
required to determine the peak time and percent overshoot, then apply Eqs. 7 and 8 to find the
corresponding damped natural frequency and damping ratio. These values of damped natural
frequency and damping ratio can then be used in Eq. 6 to form four independent equations in the


five unknowns mi , i=1,2,3,4, and k. Three additional eqns are formed from the known difference
between masses

mi +1 - mi = 0.5, i = 1,3
mi +1 - mi = 1.0, i=2 (9)

The seven equations described above form the overdetermined system

Èw 2
d1
0 L (z 2
)
- 1 ˘ Ïm 1 ¸ È 0 ˘
1
Í -1 1 L 0 ˙ Ôm2 Ô Í 0.5˙
Í 0 Ì ˝=Í ˙ (10)
w2 L (z2 - 1 ˙ M
2
) 0
˙Ô k Ô Í M ˙
d2
Í
Î M M M M ˚Ó ˛ Î ˚ .

Solving the system in Eq. 10, results in estimates of the four masses and common stiffness used
for the four step responses.

The damping constant c can then be found by matching coefficients between the expected system
model and standard forms of the model. That is, matching coefficients between the parametric
model of Eq. 1 and the form based on second-order response characterstics in Eq. 3 results in the
following formula for the damping constant c.

wd
c = 2z m 2
1- z

Matching coefficients between the parametric form of Eq. 2 and the Bode form of the second
order ODE given in Eq. 4 results in the formula

2z k 1 - z 2
c=
wd

Lesson Objectives

In this lab, we seek to identify the parameters of a one degree of freedom mass-spring-damper
system. After completing this experiment, the student should be able to identify the effective
mass, stiffness, and damping of a system from measured response data. Although we wish to
ignore motor dynamics, because of the direct connection from motor, to pinion, rack, and
subsequently, mass, we must account for the inertia of these components in our model. Also, the
damper contributes mass which must be lumped in as well. The effective mass includes the
inertial effects of all system elements including, for example, the mass of the damping and spring
elements which are typically neglected in ideal textbook problems.


Data Collection

The student is required to take step response data for the system varying the stiffness, mass and
damping as shown in Table 1. Cases four through seven hold the stiffness and damping constant,
and require incremental mass changes for the solution of Eqn 10 above. The other cases are
intended to illustrate the direct effects of independently varying damping and stiffness.

Fig. 4. ECP Data Export Format

Fig. 5. Data File After Editing
The ECP software exports the data in space delimited columns which are readily imported into
Matlab. Figures 4 and 5 show respectively the typical data file format and how it can be easily
edited to be an executable Matlab script which loads the entire data set.

Table 1. Test Matrix
Case Added Mass Damper Spring
1 1000g, 1/2 turn, Medium
2 1000g, 4 full turns Medium
3 1000g no damper Medium
4 0g 1/2 turn Medium
(Cart Empty)
8 1000g 1/2 turn Stiff
9 1000g 1/2 turn Light

The system is equipped with a removable and adjustable air damper. Case 1 is considered the
benchmark against which all other cases are compared. Cases 1 through 3 illustrate the effect of


varying damping. As implied above, the number of turns which the damper is inserted
determines the relative amount of damping. Case 3 should be attempted in two configurations,
first with the damper disconnected, then with it connected, but the plug removed. In the former
configuration, there is clearly no damping, in the latter, there is negligible damping and added
mass since the internal parts of the damper will move with the system. This demonstrates that
fact that real dampers add mass to the system, so the best way to vary the damping without
varying the system mass is to adjust the plug. The system damping should be relatively
unchanged by connecting the damper and removing the plug, however, connecting the damper
provides a significant increase in system mass which should be obvious from comparing the
frequency of oscillations between the two cases.

Results

Figures 3 and 6-8 show typical experimental step responses compared to the identified linear
models. Table 2 shows the identified parameters. The stiffness common to all four cases is
216.2 N/m. Since Eq. 10 contained more equations than unknowns, the identified parameters do
not satisfy these equations exactly. That is to say, in particular, the differences between
identified mass parameters may vary from the constraints of Eq. 9. This fact also accounts for
the differences in identified damping constants. The damping was not changed at all between
experiments, but the scatter in identified masses ripples through to the identified damping
constants. The mass values shown in Table 1 account for the cart, rack, pinion, damper, and
motor inertias. Thus, the numbers in Table 1 are expected to vary significantly from the
specified added masses in cases 4-7. The static gain values are found by simply dividing the
steady state response value by the step amplitude which was 0.5 volt for all cases.

Fig 6. Comparison of Experimental and Parametric Model Step Response m = 1.4940kg, k = 216.2 N/m,
c = 13.85 N-s/m


For each theoretical response, the identified parameters, m, c, and k were applied to the ODE
model of Eq. 2. the system transfer function is then determined by a LaPlace transform of Eq. 2.

K
G(s) = m
2 c (11)
s + s +1
k k

Table 2. Identified Parameters
Case Mass (kg) Damping (N-s/m) Static Gain K
4 0.9538 14.01 3.618
5 1.4940 13.85 3.633
6 2.4357 15.42 3.698
7 2.9951 16.19 3.876

The identified system transfer function for case 4 (Figure 3) is

3.618 (12)
G4 ( s) = 2
0.004411s + 0.06478s + 1

Fig. 7. Comparison of Experimental and Parametric Model Step Response m = 2.4357kg, k = 216.2 N/m,
c = 15.42 N-s/m

The comparison plot in Figure 5 was made by obtaining the response of G4(s) to a step of
magnitude 0.5.

The identified transfer function for case 5 (Figure 6) is

3.633 (13)
G5 (s) = 2
0.006909s + 0.06407s + 1


Fig. 8. Comparison of Experimental and Parametric Model Step Response m = 2.9951kg, k = 216.2 N/m,
c = 16.19 N-s/m

For case 6 (Figure 7), the transfer function is

3.698 (14)
G6 (s) = 2
0.001126s + 0.07131s + 1

For case 7 (Figure 8), the transfer function is

3.876 (15)
G7 (s) = 2
0.001385s + 0.07489s + 1

By examining the experimental step responses shown, the student should realize that increasing
the system mass decreases the damping ratio and system damped natural frequency. When
stiffness and damping are held constant, increased mass causes the system to oscillate more
slowly, and have a larger initial overshoot. All models match the corresponding experimental data
through the first peak. There is a small amount of unmodeled Coulomb friction in the physical
system which causes the experimental responses to settle out more quickly than the models.
This effect is much more pronounced as the system mass is increased. This is shown by the
progressively worse match in Figures 7 and 8.

Assessment

Seventeen of the students enrolled in the fall term 2004 filled out a confidential self-assessment of
their abilities prior to and after taking the course. The following three questions are regarded as
relevant to this work since only two of the physical labs involve detailed modeling of the spring-
mass-damper system. The second lab, involving frequency response is described in detail in
Burchett and Layton2.


Table 3. Student Self-Assessment Questions.
Question Strongly Agree Disagree Strongly I Don’t
Agree Disagree Know
35% 65% 0% 0% 0%
As a result of this class, I understand
the uses of models in ways that will
help me in future classes.
35% 59% 6% 0% 0%
As a result of this class, I understand
the limitations of models in ways that
will help me in future classes.
35% 41% 18% 0% 6%
This class has helped me better
understand how modeling systems can
be applied in engineering situations.

The students were also asked to rate their knowledge of various course topics and self-confidence
in being able to apply the knowledge gained. They provided self-assessment scores both prior to
and after taking the course. The following scales were provided to guide the students’ self-
assessment:
Knowledge: What you know regarding this concept area.
4= High, I know the concept and I have applied it in this course.
3= Moderate, I know the concept but I still have not applied it.
2= Low, I have only heard about the concept, but do not know it well enough to apply
it.
1= No Clue, I do not know the concept.

Confidence: Level of confidence you have in your ability to solve problems in this area.
4 = High, I am confident that I understand and can apply the concept to problems.
3 = Moderate, I am somewhat confident that I understand the concept and I can apply
it to a new
problem.
2 = Low, I have heard of the concept but I am not sure that I can apply it.
1 = No Clue, I am not confident that I can apply the concept.

The Pre and post course knowledge and confidence scores for the most relevant concept areas are
shown in Table 4. In every category there is approximately 0.7 or better improvement. This
clearly shows that the students’ assessment of their abilities was positively increased through lab
experiences like the one described in this paper.

Concluding Remarks

This paper has shown a method for introducing sophomore engineering students to the use of
step response as a modeling tool. In particular, this lab requires the student to determine the
non-ideal parametric model characteristics that best fit a family of step responses. Along the
way, students are see the effect of varying mass, stiffness and damping in a rectilinear vibrational
system. A post-course survey showed that the students’ knowledge and confidence in the topic
of model determination were significantly increased.


Table 4. Student Self-Assessed Knowledge and Confidence.
Concept Pre Course Pre Course Post Course Post Course
Knowledge Confidence Knowledge Confidence
3.118 2.824` 3.705 3.588
Distinctions between a model
and a real dynamical system
2.705 2.471 3.471 3.235
Various approaches to
modeling dynamical systems
3.176 2.688 3.765 3.471
Comparisons between
predicted response of a
mathematical model and the
response of a physical
system.

Acknowledgements

This material is based on work supported by the National Science Foundation under grant No.
DUE-0310445. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author and do not necessarily reflect the views of the National Science
Foundation.

References
1. Manual for Model 210/210a Rectilinear Control System, Educational Control Products, Bell Canyon,
CA, 1999. http://www.ecpsystems.com
2. Burchett, B. T., and Layton, R. A., “An Undergraduate System Identification Laboratory”,
Proceedings of the 2005 American Control Conference, Portland, OR, June 8-10, 2005.

Author Biography
BRADLEY T BURCHETT is an Assistant Professor of Mechanical Engineering. He teaches courses on the topics
of dynamics, system dynamics, control, intelligent control, and computer applications. His research interests
include non-linear and intelligent control of autonomous vehicles, and numerical methods applied to optimal
control.


Parametric time domain system identification of a mass spring-damper

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