Download - Vibration Control of a Beam
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Nodal Control of Vibrating Structures: Beam
A Thesis
Submitted to the Graduate Faculty of the Louisiana State University and
Agricultural and Mechanical College in partial fulfillment of the
requirements for the degree of Master of Science in Mechanical Engineering
in The Department of Mechanical Engineering
by
Akshay Nareshraj Singh
B.E. in Mechanical Engineering Maharaja Sayajirao University, 1999
December 2001
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To my parents Alka and Nareshraj Singh and my younger brother Abhishek
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Cataloging Abstract
Vibration control is an important engineering problem and many methods for both active and passive vibration absorption have been developed. This thesis deals with developing a method to achieve nodal control at the point of excitation in a Bernoulli-Euler beam. It is established that, for a uniform Bernoulli-Euler beam, the steady state motion at the point of excitation can be absorbed by means of a control force determined from displacement information at the point of application. A closed form solution for the control gain is presented and a criterion for implementing the control by active and passive means is developed. The result for the control gain is generalized for the case of a non-uniform beam. Chapter 4 shows through some examples that the theory can be also applied to eliminate the steady state motion at any desired location other than the point of excitation. Analysis is also performed to determine the optimal control force and investigate the stability of the overall system. Several controllability graphs are shown and meaningful conclusions are drawn from these graphs. An experiment is designed to validate the proposed theory and display its practicality. A uniform steel beam supported at two locations is tested. Modal testing is performed to extract natural frequencies in order to characterize the system and assist in formulation of an appropriate mathematical model. The steel beam is then excited by a known harmonic force supplied by a vibration exciter and a spring, with suitable spring constant obtained by performing the control gain calculations on the model, is used to absorb the motion at the free end. It is confirmed, that the theory developed in this thesis produces accurate results, and that it can serve as a vital tool in developing practical solutions to structural control problems. Akshay Nareshraj Singh, B.E., Maharaja Sayajirao University, 1999 Master of Science, Fall Commencement, 2001 Major: Mechanical Engineering Nodal Control of Vibrating Structures: Beam Thesis directed by Associate Professor Yitshak Ram Pages in thesis, 105. Words in abstract, 297
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Acknowledgements
The guidance and support provided by my major professor Dr. Yitshak Ram is
acknowledged. Thanks go to Dr. Michael Khonsari and Dr. Su-Seng Pang for serving
on my graduate committee and evaluating my thesis.
The intellectual input of my colleagues Kumar Vikram Singh, Jaeho Shim, Sumit
Singhal and Madhulika Sathe, as well as the assistance of Mr. Ed Martin in constructing
the experiment is also acknowledged.
Last but not the least, my gratitude goes to my aunty Mrs. Anita Singh and uncle Dr.
Vijay Singh who have always been there for me, and to my parents Alka and Nareshraj
Singh who have made me what I am today.
The research was supported in part by a National Science Foundation research grant
CMS-9978786.
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Table of Contents
Dedication..ii Cataloging Abstract.iii Acknowledgements ........................................................................................................ iv Table of Contents............................................................................................................ v List of Tables................................................................................................................. vii List of Figures .............................................................................................................. viii List of symbols ................................................................................................................ x Abstract ......................................................................................................................... xii Chapter 1: Introduction................................................................................................. 1 Chapter 2: Literature Survey and Background .......................................................... 6
2.1 Introduction ...................................................................................................... 6 2.2 Passive vibration control .................................................................................. 6 2.3 Active vibration control.................................................................................... 9
Chapter 3: Beam: Theory and Background .............................................................. 11 3.1 Introduction .................................................................................................... 11 3.2 Equation of motion for a non-uniform beam.................................................. 12 3.3 Natural frequencies and modeshapes ............................................................. 15 3.4 Steady state response and natural frequencies for a uniform clamped cantilever beam............................................................................................... 17 3.5 Summary......................................................................................................... 23
Chapter 4: The Control Gain ...................................................................................... 24 4.1 Introduction .................................................................................................... 24 4.2 Dynamic absorption in a uniform beam and a formula for the control gain ............................................................................................... 24 4.3 Analysis of results .......................................................................................... 29 4.4 Some illustrations ........................................................................................... 36 4.5 Summary......................................................................................................... 44
Chapter 5: Stability and Optimality ........................................................................... 45 5.1 Introduction .................................................................................................... 45 5.2 Stability analysis............................................................................................. 45 5.3 Optimality....................................................................................................... 48 5.4 Summary......................................................................................................... 50
Chapter 6: Experimental Verification........................................................................ 52 6.1 Introduction .................................................................................................... 52 6.2 Proposed model for the experiment................................................................ 52
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6.3 Determination of natural frequencies ............................................................. 53 6.4 Determination of the control gain................................................................... 64 6.5 Control system design .................................................................................... 66 6.6 Procedure........................................................................................................ 70 6.7 Experimental result......................................................................................... 70 6.8 Validation of the proposed theory .................................................................. 70 6.9 Summary......................................................................................................... 72
Chapter 7: Conclusions and Recommendations........................................................ 73 7.1 Conclusions .................................................................................................... 73 7.2 Recommendations for future research............................................................ 75
References ..................................................................................................................... 77 Appendix A.................................................................................................................... 79
Matlab Programs ........................................................................................................ 79
Appendix B.................................................................................................................. 100 Eigenvalues............................................................................................................... 100
Vita............................................................................................................................... 105
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List of Tables
TABLE 4.1: Comparison of control gain formulae........................................................ 35
TABLE 4.2: Comparison of control gain formula for static case .................................. 35
TABLE 6.1: Comparison of natural frequencies............................................................ 63
TABLE 1 (Appendix B): Eigenvalues ..................................................................... 100 TABLE 2 (Appendix B): Eigenvalues for a=0.25 .................................................. 101 TABLE 3 (Appendix B): Eigenvalues for a=0.5..................................................... 102 TABLE 4 (Appendix B): Eigenvalues for a=
21 ................................................... 103
TABLE 5 (Appendix B): Eigenvalues for a=2/3..................................................... 104
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List of Figures
FIGURE 1.1: A schematic for active control ................................................................... 2
FIGURE 1.2: Vibration control of a harmonically excited beam .................................... 4
FIGURE 2.1: Single-degree-of-freedom dynamic absorber ............................................ 7
FIGURE 3.1: Equation of motion for a beam ................................................................ 13
FIGURE 3.2: Uniform cantilever beam subject to harmonic excitation ........................ 17
FIGURE 3.3: Steady state amplitude for a uniform cantilever beam............................. 22
FIGURE 4.1: Vibration control of a harmonically excited beam .................................. 25
FIGURE 4.2: Plot of control gain against ............................................................. 29 FIGURE 4.3: Plot of control gain against - (superimposed) ................................. 30 FIGURE 4.4: Clamped and clamped-double-hinged uniform beam.............................. 31
FIGURE 4.5: Static deflection of a clamped-hinged beam............................................ 32
FIGURE 4.6: Illustration demonstrating control gain calculations................................ 37
FIGURE 4.7: Controlled uniform beam......................................................................... 39
FIGURE 4.8: Uncontrolled uniform beam..................................................................... 41
FIGURE 4.9: Controlled uniform beam......................................................................... 42
FIGURE 4.10: Implementation of nodal control............................................................ 43
FIGURE 5.1: Passively controlled uniform beam.......................................................... 46
FIGURE 5.2: Stability analysis and equivalent stiffness ............................................... 47
FIGURE 5.3: Plots of control gain, control force and inverse of static deflection against the beam span x for cases with excitation frequency of (a) 10= , (b) 20= , (c) 30= and (d) 80= ................................ 49 FIGURE 6.1: Mathematical model of the test beam used in the experiment................. 53
FIGURE 6.2: Dimensions of the test beam used in the experiment............................... 57
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FIGURE 6.3: Clamping details ...................................................................................... 58
FIGURE 6.4: Impact Hammer ....................................................................................... 59
FIGURE 6.5: Accelerometer .......................................................................................... 59
FIGURE 6.6: Modal Analysis ........................................................................................ 61
FIGURE 6.7: VirtualBench DSA display ...................................................................... 63
FIGURE 6.8: Controlled test beam ................................................................................ 64
FIGURE 6.9: Control gain and control force variation along the beam span ................ 65
FIGURE 6.10: Test beam modeshape before and after control ..................................... 66
FIGURE 6.11: Spring housing configuration................................................................. 67
FIGURE 6.12: Attaching the vibration exciter to the beam........................................... 67
FIGURE 6.13: Schematic of the experimental setup ..................................................... 68
FIGURE 6.14: Attaching the springs and the shaker ..................................................... 69
FIGURE 6.15: Experimental setup................................................................................. 69
FIGURE 6.16: Active Vibration Control ....................................................................... 71
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List of symbols
)(, xAA area of cross-section
iiii DCBA and ,,, constant coefficients for each i
A square matrix
b force vector
control gain static deflection
)(, xEE the Youngs modulus of elasticity
pe pth unit vector
)(tf harmonic force
non-dimensional parameter )(, xII moment of inertia
pk spring stiffnes of the primary system
sk spring stiffnes of the secondary system
L length of the beam
eigenvalues of clamped-hinged beam pm mass of the primary system
sm mass of the secondary system
),( txM bending moment
eigenvalues of clamped-double-hinged beam )(, x density
x
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BS bending stiffness
)(tu control force
),( txV shear force
)(xv shape function
),( txw deflection
frequency of excitation n natural frequencies
z displacement vector
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Abstract
Vibration control is an important engineering problem and many methods for both
active and passive vibration absorption have been developed. This thesis deals with
developing a method to achieve nodal control at the point of excitation in a Bernoulli-
Euler beam. It is established that, for a uniform Bernoulli-Euler beam, the steady state
motion at the point of excitation can be absorbed by means of a control force
determined from displacement information at the point of application. A closed form
solution for the control gain is presented and a criterion for implementing the control by
active and passive means is developed. The result for the control gain is generalized for
the case of a non-uniform beam. Chapter 4 shows through some examples that the
theory can be also applied to eliminate the steady state motion at any desired location
other than the point of excitation. Analysis is also performed to determine the optimal
control force and investigate the stability of the overall system. Several controllability
graphs are shown and meaningful conclusions are drawn from these graphs.
An experiment is designed to validate the proposed theory and display its practicality.
A uniform steel beam supported at two locations is tested. Modal testing is performed
to extract natural frequencies in order to characterize the system and assist in
formulation of an appropriate mathematical model. The steel beam is then excited by a
known harmonic force supplied by a vibration exciter and a spring, with suitable spring
constant obtained by performing the control gain calculations on the model, is used to
absorb the motion at the free end. It is confirmed, that the theory developed in this
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thesis produces accurate results, and that it can serve as a vital tool in developing
practical solutions to structural control problems.
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Chapter 1: Introduction
Most mechanical components are subject to vibrations, which, depending on
circumstances may be desirable or undesirable. On one hand, vibrations of guitar
strings produce wonderful music and on the other hand, vibrations in an automobile
may cause excessive discomfort and fatigue to the driver.
This thesis is concerned with the case of undesirable vibrations and in particular in
developing methods for elimination of steady state response. Components are designed
to withstand definite levels of vibrations. Design, in vibrations, is used to denote an
educated method of choosing and adjusting the physical parameters of a vibrating
system in order to obtain a more favorable response [1]. Modifications of physical
parameters namely mass, damping, and stiffness, in order to improve the vibrational
response of the system fall in the category of passive control. The most common
passive control device is a vibration absorber, which manifests in the form of layers of
damping material added to vibrating structures. Passive control may also involve
changing values of mass and stiffness and hence is also referred as redesign.
Chapter 2 describes a passive control device called single-degree-of-freedom dynamic
absorber. A single-degree-of-freedom dynamic absorber is made up of single mass and
spring and may or may not have a damper. Essentially, the dynamic vibration absorber
introduces additional degree of freedom in the original dynamic system, which results
in a different steady state response. The values of mass, stiffness and damping can be
modified to tune the response of the resulting system to desired levels.
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However, altering the physical parameters of the system may not always yield a desired
response. In these situations, one has to try implementing active control. Active
control uses external active device, called an actuator, which assists in shaping the
system response. The actuator (e.g. a piezoelectric device, a hydraulic piston, or rack
and pinion) is capable of applying control force to the system under consideration. The
control force is determined based on a mathematical rule, which operates on the system
response measured in realtime by a sensor. The mathematical rule used to apply the
force from the sensor measurement is called the control law.
Dynamic System
Actuator
Sensor
ProcessorDynamic System
Actuator
Sensor
Processor
FIGURE 1.1: A schematic for active control
The system comprising both, the actuator and the sensor together with the electronic
circuitry that reads the sensor output and calculates corresponding input to the actuator
is called the control system [5]. Figure 1.1 shows a schematic for implementing active
control.
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Much work has been done on tuning vibrational response of multi degree of freedom
systems by applying results obtained from the study of dynamic absorbers. A single-
degree-of-freedom dynamic absorber is attached to continuous system like a beam to
control single a mode of vibration under the influence of harmonic excitation. Aida,
Toda, Ogawa and Imada in [11] and Kawazoe, Kono, Aida, Aso and Ebisuda in [12]
discuss a beam type vibration absorber capable of suppressing several vibration modes
of beams.
The subject investigated in this thesis is elimination of steady state vibration at a desired
location in beams by means of nodal control. The developed theory will provide a
strong foundation for realizing realistic and convenient methodologies in control
applications in cases like surgical procedures, drilling and turning operations etc.
However, one of the many direct applications of this method is structural vibration
control in an aircraft wing. Several measurements such as vibrational response, air
temperature, wind velocity etc are required in order to monitor flight conditions in an
aircraft. These data also assist the pilot in flying the aircraft. Sensors and data
collection circuitry form an entire network of the electrical wiring all in and around the
airplane body. Data acquisition devices are also located on the wing of the airplane.
Shielding of these devices from undesirable vibration of the wing is critical in order to
avoid noise in the gathered data and prevent damage to electrical wiring. Exclusion of
steady state vibration at the locations of these devices provides the motivation for this
investigation.
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Consider a non-uniform Bernoulli-Euler beam of length . Suppose that the beam is
excited by a harmonic force
L
( ) ttf cos= , as shown in Figure 1.2(a).
x
( ) ttf cos=
( )txw ,
L
(a) Uncontrolled beam
x
( ) ttf cos=
( )txw ,
L
(b) Controlled beam
( ) ( )tawtu ,=a
x
( ) ttf cos=
( )txw ,
L
(a) Uncontrolled beam
x
( ) ttf cos=
( )txw ,
L
(b) Controlled beam
( ) ( )tawtu ,=a
FIGURE 1.2: Vibration control of a harmonically excited beam
The steady state motion of a prescribed point of the beam may be vanished by applying
a concentrated control force u at ),( ta ax = as shown in Figure 1.2(b).
The work here focuses on determining a closed form solution for the control gain that absorbs the motion of the beam at Lx = . Chapter 4 describes the method in detail and also provides criterion to determine the type of the control i.e. active nodal control
or passive nodal control. It also provides an illustration to show that motion can be
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absorbed not only at the point of excitation but also at any other point along the beam
span. Chapter 5 discusses the criterion for stability of the controlled system and
optimality in relation to the control force. In order to validate the findings, an
experiment is designed exhibiting nodal control by means of a passive element, a
spring, in a steel beam under the influence of harmonic excitation. It is thus shown that
the approach presented in this thesis is highly practical.
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Chapter 2: Literature Survey and Background
2.1 Introduction
Some fundamental knowledge and some key results found in the literature survey
related to vibration control are presented here. The survey covers topics related to the
design of both active and passive vibration absorbers.
2.2 Passive vibration control
In 1911 Frahm invented a device for stabilization of rocking oscillations of ships [4].
This device is now known as a dynamic absorber. The dynamic absorber is extremely
simple in principle and has large practical applications. For example Lee, Nian, and
Tarng in [6], Al-Bedoor, Moustafa, and Al-Hussain in [9], Yamashita, Seto, and Hara in
[10] describe design of a dynamic vibration absorber for vibration control in turning
operations, synchronous motor-driven compressors, and piping systems, respectively.
Theory of single-degree-of-freedom dynamic absorber
The dynamic vibration absorber is an additional mass-spring system, which is
appropriately chosen to neutralize the steady state force acting on a particular degree of
freedom. Consider the single-degree-of-freedom system shown in Figure 2.1(a), under
the harmonic excitation of ( ) tFtf sin0= . Let this system be called the primary system. Upon a harmonic excitation, the system vibrates with two frequencies, the
frequency of excitation , and the natural frequency of the system,
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,ppn mk= (2.2.1) where subscript p denotes the parameters associated with the primary system. The
objective is to eliminate the forced component of vibrations. This is implemented by
attaching additional single-degree-of-freedom mass-spring system, which is called the
secondary system, with the mass and the spring stiffness , to the primary system.
The global system is shown in Figure 2.1(b).
sm sk
(a) Primary system
(b) Global system
( )tx p
( ) tFtf sin0=
pmpk
( )txs
sksm
( )tx p
( ) tFtf sin0=
pmpk
(a) Primary system
(b) Global system
( )tx p
( ) tFtf sin0=
pmpk
( )tx p
( ) tFtf sin0=
pmpk
( )txs
sksm
( )tx p
( ) tFtf sin0=
pmpk
FIGURE 2.1: Single-degree-of-freedom-dynamic absorber
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The mathematical model of the global system has the form
,sin00
0 0 tF
xx
kkkkk
xx
mm
s
p
ss
ssp
s
p
s
p
=
++
&&&&
(2.2.2)
which can also be written as
( )
=+=++
.0sin0
sspsss
sspsppp
xkxkxmtFxkxkkxm
&&&&
(2.2.3)
Both masses vibrate with forced harmonic vibrations of the form
( )( )
==
,sinsin
tXtxtXtx
ss
pp (2.2.4)
where the constants , are the amplitudes of the forced component of vibration.
Substituting (2.2.4) in (2.2.3) gives
sp XX ,
( )
=+=++
.020
2
sspsss
sspsppp
XkXkXmFXkXkkXm
(2.2.5)
Since is to be eliminated, substituting pX 0=pX in the above set of equations yields
=+=
.020
ssss
ss
XkXmFXk
(2.2.6)
The second equation in (2.2.6), gives
,2s
s
mk= (2.2.7)
and the first equation in (2.2.6) provides the amplitude of vibration of , sm
.0s
s kF
X = (2.2.8)
Hence, it is concluded that the vibratory motion of the primary mass can be eliminated
provided that the stiffness and the mass values for the secondary mass are chosen such
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that they satisfy (2.2.7). In this case the secondary mass vibrates out of phase to the
external harmonic excitation and the spring force exactly contradicts the harmonic
force, causing the forced motion of to vanish. pm
This idea can also be extended to a multi-degree-of-freedom system. Ram and Elhay in
[8], show that a multi-degree-of-freedom dynamic absorber may absorb the steady state
motion associated with the harmonic excitation of several frequencies. There are,
however, some limitations associated with practical implementation of the dynamic
absorber. Firstly, it is not always feasible to attach the absorber to the specific degree of
freedom of which the motion is to be absorbed. Secondly, application of dynamic
absorber increases the dimension of the system, and hence introduces new natural
frequencies that may interfere with other excitations. Thirdly, the theory of dynamic
absorbers for damped systems is not fully developed. It is, therefore, not clear how the
dynamic absorption phenomenon may be used in eliminating the steady state motion of
a damped system that is excited by a harmonic force [7]. Herzog in [16] investigates
the topic of performance degradations of dynamic systems implementing passivity-
based control. He has analyzed the topic in terms of flatness of response of the
controlled system in the vicinity of the natural frequency of the dynamic absorber.
2.3 Active vibration control
As described earlier in Chapter 1, in some cases implementing the active vibration
absorber is imperative. Nishimura, Yoshida and Shimogo in [17] have studied optimal
design method of the active dynamic vibration absorber for multi-degree-of-freedom
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systems. The method was also validated by performing numerical simulations and an
experiment on a 3-degree-of-freedom building like structure. Aida, Toda, Ogawa and
Imada in [11] and Kawazoe, Kono, Aida, Aso and Ebisuda in [12] demonstrate control
of several mode shapes of a beam by using a beam type vibration absorber with
boundary conditions same as the main beam. It is shown that for specific vibration
modes, mode equations of a beam with beam-type dynamic absorber are approximate
equivalents of the motion of two-degree-of-freedom system. Hence, the dynamic
absorber system can be tuned by Den Hartog method.
Gaudreault, Liebst, Bagley in [18] present four techniques for combining active
vibrational control and passive viscous damping. The motivation behind the work is
some findings, which reveal that the passive damping can reduce the amount of active
damping needed to control structural vibrations. However, inappropriate design of the
passive damping can produce contrary results in that it may increase system reaction
times, reducing control effectiveness.
The partial-pole assignment problem is addressed by Ram in [15]. The paper
determines the force required to place a few poles of the spectrum while leaving the rest
unchanged and the conditions under which the solution is unique. The work of Ram in
[3] lays the foundation for this thesis. The paper provides a closed form control gain
solution for absorbing the harmonic response at a desired location in an axially
vibrating rod and analyzes the stability of the controlled system.
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Chapter 3: Beam: Theory and Background
3.1 Introduction
In order to determine the dynamic behavior of mechanical systems, one needs to
develop an appropriate mathematical model. Mechanical systems can be modeled as
lumped-parameter systems, where it is assumed that the motion of the system is
governed by the mass of the system concentrated at a particular point. The modeled
system is known as a lumped parameter system, which has finite number of lumped
masses connected to each other by means of springs and dampers. Even though a
discrete model provides an acceptable solution to the system, it is not capable of
accounting for the flexibility of various structures. Engineering problems such as
swaying of tall buildings, torsional and bending vibrations of shafts and vibrations in
wings of aircraft demand an insight into elastic behavior of structural members. These
elastic systems consist of continuous mass and elasticity throughout their span. Hence,
these systems are modeled assuming that the mass of the system is distributed in the
entire system as infinitesimally small elements. Such a model for a mechanical system
is known as a distributed parameter model. There are only few distributed parameter
systems such as beams, bars, strings and plates that have closed form solutions.
However, study of these systems provides understanding of behavior and modeling of
most complex systems, which cannot be solved in a closed form manner.
This chapter deals with study of vibration of continuous beams. The equation of motion
for a beam is described and a closed form solution is provided. Investigation is done in
terms of natural frequency and steady state response by considering the case of a
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cantilever beam subjected to dynamic excitations. This chapter forms the foundation to
the problem investigated in the thesis. The source of most of the material presented in
Sections 3.2 and 3.3 is [2].
3.2 Equation of motion for a non-uniform beam
Consider a non-uniform Bernoulli-Euler beam of length as shown in Figure 3.1(a).
The transverse vibrations are denoted as
L
( )txw , . The cross-sectional area is ( )xA , modulus of elasticity is ( )xE , density is ( )x , and moment of inertia is ( )xI . The external force applied to the beam per unit length is denoted by . ( )txf ,
From strength of materials, the bending moment ( )txM , is related to the beam deflection by ( txw , )
( ) ( ) ( ) ( )22 ,,
xtxwxIxEtxM
= . (3.2.1) One can look on an infinitesimal element of the beam, shown in the Figure 3.1(b), and
determine the model of flexural vibrations. It is assumed that the deformation is small
enough such that the shear deformation is much smaller than . From Newtons
second law in the - direction,
( txw , )y
( ) ( ) ( ) ( ) ( ) ( ) ( )22 ,,,,,
ttxwdxxAxdxtxftxVdx
xtxVtxV
=+
+ . (3.2.2)
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( )txf ,
( )txw ,
dxxL
x
y
( )txw ,
( )txf ,
Undeformed x-axis
x dxx +dx
( )txM ,
( ) ( ) dxx
txMtxM + ,,
( )txV ,
( ) ( ) dxx
txVtxV + ,,
(a)
(b)
( )txf ,
( )txw ,
dxxL
x
y ( )txf ,
( )txw ,
dxxL
x
y
( )txw ,
( )txf ,
Undeformed x-axis
x dxx +dx
( )txM ,
( ) ( ) dxx
txMtxM + ,,
( )txV ,
( ) ( ) dxx
txVtxV + ,,
(a)
(b)
FIGURE 3.1: Equation of motion for a beam
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Here is the shear force at the left end of the element , ( txV , ) dx ( ) ( )dxtxVtxV x ,, + is the shear force at the right end of the element . The term on the right hand side of
the equality sign is the inertia force of the element. The sum of the moments on the
element yields
dx
( ) ( ) ( ) ( ) ( ) ( )[ ] 02
,,,,,, =+
++
+ dxdxtxfdxdxx
txVtxVtxMdxx
txMtxM . (3.2.3)
Here the right hand side in the equation vanishes, since it is assumed that the rotary
inertia of the element is negligible. Simplification of this expression yields, dx
( ) ( ) ( ) ( ) ( ) 02
,,,, 2 =
++
+ dxtxfdx
xtxVdxtxVdx
xtxM . (3.2.4)
Since is small, and hence dx ( )2dx is negligible. The above expression takes the form
( ) ( )x
txMtxV = ,, . (3.2.5)
This expression relates shear force and the bending moment. Substituting (3.2.5) in
(3.2.2) gives
( )[ ] ( ) ( ) ( ) ( )22
2
2 ,,,t
txwdxxAxdxtxfdxtxMx
=+ . (3.2.6)
Substituting (3.2.1) in (3.2.6) and dividing by yields dx
( ) ( ) ( ) ( ) ( ) ( ) ( )txfx
txwxIxExt
txwxAx ,,, 22
2
2
2
2
=
+
. (3.2.7) If no external force is applied ( )txf , is zero. The equation of motion of beam for
is then given as 0 ,0 >
-
The above expression (3.2.8) is a fourth order partial differential equation, which
governs the vibration of a non-uniform Bernoulli-Euler beam. If the parameters ,
, and
)(xE
)(xA )(xI )(x are constant then (3.2.8) can be further simplified to give
( ) ( ) ,0,, 44
22
2
=+
x
txwct
txw (3.2.9)
where
.A
EIc = (3.2.10)
3.3 Natural frequencies and modeshapes
The equation of motion (3.2.9) contains four spatial derivatives and two time
derivatives. Hence, in order to determine a unique solution for , four boundary
conditions and two initial conditions are needed. Usually, the values of displacement
and velocity are specified at time
),( txw
0=t , and so the initial conditions can be given as, 0)0,( and ,0)0,( == xwxw & , (3.3.1)
where dots denote derivates with respect to time. The method of separation of variables
is used to determine the free vibration solution,
)()(),( tTxVtxw = . (3.3.2) Substituting (3.3.2) in (3.2.9) and rearranging yields
( ) ( ) .)(
1)(
22
2
4
42
==dt
tTdtTdx
xVdxV
c (3.3.3)
Here, is a positive constant. The above equation can now be written as two
ordinary differential equations as shown below.
2
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( ) ,0)(444
= xVdx
xVd (3.3.4)
( ) ,0)(222
= tTdt
tTd (3.3.5) where,
EIA
c
2
2
24 == . (3.3.6)
The solution to (3.3.5) can be given as
tBtAtT sincos)( += . (3.3.7) The constants A and B can be evaluated using two initial conditions given by (3.3.1),
and the solution to (3.3.4) is assumed as,
sxexV =)( . (3.3.8) Substituting (3.3.8) in (3.3.4) and simplifying furnishes,
044 = s . (3.3.9) The roots of (3.3.9) are
iss == 4,32,1 , , (3.3.10) hence, the solution to (3.3.4) can be given as
xexexeexV xixixx +++= 4321)( . (3.3.11) Equation (3.3.11) can also be expressed alternatively as
xxxxxV coshsinhcossin)( 4321 +++= , (3.3.12) where are different constants, which can be evaluated using the four
boundary conditions. The natural frequency of the beam can be therefore computed
from (3.3.6) as
4321 and ,,
16
-
( ) 42422 ALEI
AlEIL
AEI
=== , (3.3.13) where, is a non-dimensional parameter.
3.4 Steady state response and natural frequencies for a uniform clamped cantilever beam The dynamic behavior of a beam can be determined by analyzing the case of a uniform
clamped cantilever beam shown in Figure 3.2.
x
( ) ttf sin=( )tbw ,
L
a
x
( ) ttf sin=( )tbw ,
L
a
FIGURE 3.2: Uniform cantilever beam subject to harmonic excitation
The boundary conditions at the clamped end are no displacement and no slope, which
can be given as,
( ) 0,0 =tw , (3.4.1) and,
( ) 0,0 =
xtw , (3.4.2)
respectively. The boundary conditions at the free end are no bending moment and no
shear force, represented by
17
-
( ) ( ) ( ) 0,22
=
xtLwxIxE , (3.4.3)
and,
( ) ( ) ( ) 0,22
=
xtLwxIxE
x. (3.4.4)
The steady state response ( )tbw , , measured at bx = , of the beam under the influence of harmonic excitation of ( )t tf sin= at some other position ax = , is described by a Green function. The Green function is a function of
Frequency of excitation Position of excitation ax = , and Position of interest where the response is to be measured . bx =
The beam is separated in two parts, ax 0 , and a Lx < , and denoted by
( ) ( )( )
-
At ax = , the deflection, the slope and the moment are the same for both parts of the beam. The shear force differs by EI1 . These four conditions represent the matching
conditions at ax = and can be described by ( ) ( )tawtaw ,, 21 = , (3.4.9)
( ) ( )x
tawx
taw
= ,, 21 , (3.4.10)
( ) ( )2
22
21
2 ,,x
tawx
taw
= , (3.4.11)
and
( ) ( )EIx
tawx
taw 1,,3
23
31
3
=
. (3.4.12) Separation of variables gives
( ) ( ) txvtxw ii sin, = , 2,1=i . (3.4.13) Substitution of (3.4.13) in (3.4.6) and (3.4.7) yields
0 12
1 = vAvEI , ax
-
The equations (3.4.14) and (3.4.15) can be written as
0 14
1 = vv , ax
-
The determinant of is determined for different values of A . The values of which make singular are designated as A n . Then the natural frequencies of the beam n are
( ) 42422 ALEI
ALEIL
AEI
nnn === , (3.4.24) where is a dimensionless parameter. The fundamental natural frequency of a cantilever beam leads to 875.11 = . Denoting EI , the bending stiffness of the beam, by , the first natural frequency for the cantilever beam can be expressed as, BS
41 5156.3 ALSB
= . (3.4.25) Now,
zAb 1= (3.4.26) allows determination of the values for constants for i . iiii DCBA and ,,, , 2,1=
The steady state amplitude at any point other than the point of excitation is given by the
Green function as below
( ) ( )( )
-
where a is the point of harmonic excitation and b is the point of measurement of the
steady state amplitude. The system parameters and ,,, LIAE can be chosen arbitrarily. The figure shows 875.11 = , 694.42 = , 854.73 = , and 995.104 = . Hence, for known values of and ,,, LIAE the first four natural frequencies of the cantilever beam can be given by (3.4.24).
(e)
7.85 7.855 7.86
0
)(bw
0 2 4 6 8 10 12
0
)(bw
0 2 4 6 8 10 12
0
)(bw
(b)
(d)
0 2 4 6 8 10 12
0
)(bw
0 2 4 6 8 10 12
0
)(bw
(a)
(c)
(e)
7.85 7.855 7.86
0
)(bw
0 2 4 6 8 10 12
0
)(bw
0 2 4 6 8 10 12
0
)(bw
0 2 4 6 8 10 12
0
)(bw
(b)
(d)
0 2 4 6 8 10 12
0
)(bw
0 2 4 6 8 10 12
0
)(bw
(a)
(c)
FIGURE 3.3: Steady state amplitude for a uniform cantilever beam
22
-
However, the cantilever beam, being a distributed parameter system, has infinite
number of natural frequencies, which can be approximated by the formula ( )2
12 n for
[5]. 5>n
Cases (a) and (c) represent the response at a collocated point, i.e., the point of excitation
and the point of measurement are the same. In cases (b) and (d) the response is at a
non-collocated point, i.e., the point of excitation and the point of measurement are
different.
In Figure 3.3(c) the pole in the neighborhood of 854.73 = is not observed. This is because the pole and the zero are very close to each other. A magnified view of the
region marked by circle in Figure 3.3(c) is shown in Figure 3.3(e). Here, it is clear that
the pole and the zero are extremely close to each other.
3.5 Summary
One popular approach in controls is pole-zero cancellation. Here, the idea is to place
some zeros on some poles to reduce vibration in the rod. However, a major drawback
of the method is that one has to be extremely careful in placing the zeros, because the
slightest error in positioning of a zero can lead to a non-vanishing pole, which will
make the system unstable.
23
-
Chapter 4: The Control Gain
4.1 Introduction
This chapter deals with development of the theory for elimination of steady state
response at a prescribed location for the case of a harmonically excited Bernoulli-Euler
cantilever beam. A closed form expression for the control gain is established for the
uniform cantilever beam and the results are then generalized to the case of a non-
uniform beam. Several graphs indicating the control gain requirement with the change
in excitation frequency are shown. Investigation is performed on the obtained results to
draw meaningful conclusions and provide groundwork for further development of the
proposed theory.
4.2 Dynamic absorption in a uniform beam and a formula for the control gain Consider a uniform beam of length , modulus of elasticity L E , density , cross-sectional area , and moment of inertia A I . The axial distance from the supported end
of the beam is x . Suppose that the beam is excited by a harmonic force ( ) ttf cos= , as shown in Figure 4.1(a).
24
-
FIGURE 4.1: Vibration control of a harmonically excited beam
x
( ) ttf cos=
( )txw ,
L
(a) Uncontrolled beam
x
( ) ttf cos=
( )txw ,
L
(b) Controlled beam
( ) ( )tawtu ,=a
x
( ) ttf cos=
( )txw ,
L
(a) Uncontrolled beam
x
( ) ttf cos=
( )txw ,
L
(b) Controlled beam
( ) ( )tawtu ,=a
Then as shown in Chapter 3, the steady state transverse vibrations of the beam are
governed by the differential equation
022
4
4
=+
twA
xwEI , Lx
-
and
( ) tx
tLwEI cos,33
= . (4.2.5)
The conditions (4.2.2) and (4.2.3) describe no-displacement and no-slope at 0=x , condition (4.2.4) imposes no bending moment at Lx = and condition (4.2.5) shows that shear induced by the exciting force is tcos . The steady state motion of a prescribed point of the beam may be vanished by applying a concentrated control force at ax = ,
( ) ( )tawtau ,, = , (4.2.6) where is a constant. The partial differential equation for the controlled system is then
( ) waxtwA
xwEI =
+
2
2
4
4
, Lx
-
( ) ( )( )
-
( ) 02 =Lv . (4.2.18) Hence, the four conditions in (4.2.14), the first three conditions of (4.2.15), and the
condition (4.2.18), can be written in matrix form
bAz = , (4.2.19) where
=A
LshLchLsLcLchLshLcLs
LchLshLcLsachashacasachashacas
ashachasacashachasacachashacasachashacas
3333
2222
22222222
000000000000
00000000001010
,
[ ]TDCBADCBA 22221111=z , (4.2.20) 8
1 ebEI
= , (4.2.21)
where is the ppeth unit vector, and s, c, sh, and ch represent sine, cosine, hyperbolic
sine and hyperbolic cosine, respectively. Once is found the shape functions z ( )xiv , , are determined by (4.2.16). The fourth condition expressed by (4.2.15) can then
be resolved for
2,1=i , i.e.
( ) ( ) ( )( )
=
avavavEIa
1
21, . (4.2.22)
The values of the control gain for various points of excitation a , as function of the exciting frequency
8.0 ,6.0 ,4.0 ,2.0= are shown in Figures 4.2(a)-4.2(d).
LAIE and ,,, can be chosen arbitrarily.
28
-
(b)
(a)
0 2 4 6 8 10 12-8
-4
0
4
8
0 2 4 6 8 10 12-8
-4
0
4
8
(c) (d)
0 2 4 6 8 10 12-8
-4
0
4
8
0 2 4 6 8 10 12-8
-4
0
4
8
310
310
310
310
(b)
(a)
0 2 4 6 8 10 12-8
-4
0
4
8
0 2 4 6 8 10 12-8
-4
0
4
8
(c) (d)
0 2 4 6 8 10 12-8
-4
0
4
8
0 2 4 6 8 10 12-8
-4
0
4
8
(b)
(a)
0 2 4 6 8 10 12-8
-4
0
4
8
0 2 4 6 8 10 12-8
-4
0
4
8
(c) (d)
0 2 4 6 8 10 12-8
-4
0
4
8
0 2 4 6 8 10 12-8
-4
0
4
8
310
310
310
310
FIGURE 4.2: Plot of control gain against
4.3 Analysis of results
Large amount of information can be derived from the above graphical results. The plots
for are singular at certain values of . Also, the positions of these singularities change with the change in control location ax = . However, Figure 4.3 shows the above four graphs superimposed on each other. It can be seen from these graphs that
the zeros of ( ) ,a are invariant of the control location. Further analysis is performed in order to understand this behavior.
29
-
0 2 4 6 8 10 12
-8000
-4000
0
4000
8000
0 2 4 6 8 10 12
-8000
-4000
0
4000
8000
0 2 4 6 8 10 12
-8000
-4000
0
4000
8000
FIGURE 4.3: Plot of control gain against - (superimposed) Consider the clamped-hinged uniform beam shown in Figure 4.4(a). The natural
frequencies of this beam are indeed the zeros of ( ) ,a . Also, the singularities in ( ) ,a are the natural frequencies of the beam shown in Figure 4.4(b) which is
clamped in the left end and hinged both at ax = and Lx = .
From the above arguments and from [19] and [3], the control gain ),( a can also be determined from the following formula
( )
=
=
=1
2
1
2
1
1,
i i
i ica
, (4.3.1)
30
-
(a) Clamped-hinged uniform beam
L
(b) Clamped-double-hinged uniform beam
L
a
(a) Clamped-hinged uniform beam
L
(b) Clamped-double-hinged uniform beam
L
a
L
a
FIGURE 4.4: Clamped and clamped-double-hinged uniform beam
where is a constant, c i denotes the eigenvalues of the clamped-hinged beam shown in Figure 4.4(a) and i denotes the eigenvalues of the clamped-double-hinged beam in Figure 4.4(b). Constant c can be evaluated by considering the static case i.e. when
0= , applied to the case shown in Figure 4.4(a). Substituting 0= in (4.3.1) gives ( ) ca =0, . (4.3.2)
Denote by the static deflection at ax = of the clamped-hinged beam shown in Figure 4.5(a) due to a collocated unit static load 1=U . This is a static problem where the deflection of any point on the beam is no longer a function of time , and
can be denoted by . Hence the mathematical model of the beam shown in Figure
4.5(a) can be described by
),( txw
(w
t
)x
31
-
LxUaxdx
wd
-
0)(22
=dx
Lwd . (4.3.7)
The beam is separated in two parts as described below.
-
[ ]TDCBADCBA 22221111=z , (4.3.14) 6
1 ebEI
= . (4.3.15)
For each value of the constants can be evaluated from . Then the static
deflection
a bAz 1= can be given as
2,1 ,23 =+++= iDaCaBaA iiii . (4.3.16) By the linearity of the problem (4.3.3), the static deflection due to a force applied to
the beam shown in Figure 4.5(b) is
F
F . Hence, invoking the control law (4.2.6) gives ( ) FaF 0,= , (4.3.17)
which yields
1=c , (4.3.18)
by virtue of (4.3.2). The control gain is therefore
( )
=
=
=1
21
2
1
1,
i i
i ia
. (4.3.19)
Note that the control gain given by (4.3.19) is identical to that expressed in (4.2.22).
Table 4.1 shows the comparison between the values of control gain as determined from (4.3.19), using different numbers of eigenvalues and , and that determined from (4.2.22), for .10=
34
-
TABLE 4.1: Comparison of control gain formulae
64.7608271.9743164.52690245.735447
67.2449174.8187566.70656254.081601
64.9247872.1497964.68074246.365423
64.7489071.9736864.52439245.69129as determined from (4.2.22)
64.7803572.0130564.55730245.821325
a=2/3a=a=0.5a=0.25
as determined from (4.3.1)No. of eigenvalues
64.7608271.9743164.52690245.735447
67.2449174.8187566.70656254.081601
64.9247872.1497964.68074246.365423
64.7489071.9736864.52439245.69129as determined from (4.2.22)
64.7803572.0130564.55730245.821325
a=2/3a=a=0.5a=0.25
as determined from (4.3.1)No. of eigenvalues
( ) 5.05.0
It is evident that with increase in number of eigenvalues used to calculate from (4.3.19) the comparison converges to the exact value determined from (4.2.22). Again,
the accuracy of the numerical value of from (4.3.19) is largely influenced by the accuracy in the numerical value of eigenvalues and . In calculations for in Table 4.1 the eigenvalues and have been determined by solving the transcendental eigenvalue problem as shown in [13]. The eigenvalues are listed in Appendix B. This
argument can be further strengthened by comparing values of for the static case calculated at . 25.0=a
TABLE 4.2: Comparison of control gain formula for static case
364.0889364.0927
as determined from (4.3.1)as determined from (4.2.22)
364.0889364.0927
as determined from (4.3.1)as determined from (4.2.22)
35
-
Here, 364.0889 is the exact value for . The inaccuracy in the value determined from (4.2.22) is due to singularity of in (4.2.19) for A 0= . Hence, 364.0927 is the value of control gain for 00001.0= . Also, it is important to note that the boundary condition for the beam shown in Figure 4.5(a) is clamped-hinged. This is because the
desired objective is to achieve no steady state displacement at the free end in the beam
shown in Figure 4.1(a).
4.4 Some illustrations
Some simple examples depicting the calculation procedure are presented for better
understanding of the theory developed in the previous sections. Example 1 shows the
stepwise numerical calculations of the control gain, the control force and the steady
state displacement at some positions of interest. Example 2 demonstrates that control
can be achieved in order to eliminate the steady state motion at a point other than the
free end by making a small modification to the theory. Simulated mode shapes are
shown in example 3, which display beam mode shapes before and after implementation
of the nodal control.
Example 1:
A uniform Bernoulli-Euler beam is described by
035 22
4
4
=+
tw
xw , ,20 (4.4.1)
( ) 0,0 =tw , ( ) 0,0 =
xtw , ( ) 0,22
2
=
xtw , and ( ) t
xtw 10sin2,23
3
=5 (4.4.2)
Hence, EI = bending stiffness = 5, and A = 3. Amplitude of harmonic force = 2.
36
-
x
( ) ttf 10sin2=
( )txw ,
L
x
( ) ttf 10sin2=
( )txw ,
L
FIGURE 4.6: Illustration demonstrating control gain calculations
Determine control gain and control force needed to eliminate steady state displacement
at the free end.
From separation of variables
( ) ( ) txvtxw 10sin, = , (4.4.3) and,
605/310/ 224 === EIA . (4.4.4) Hence, (4.4.1) gives
060 = vv , (4.4.5) and, (4.4.2) gives
( ) ( ) ( ) 0200 === vvv , ( ) 522 =v . (4.4.6) The beam is separated into two parts, 0 75.0 x , and 275.0 < x , denoted as
( ) ( )( )
-
Boundary condition are
( ) ( ) ( ) 0200 211 === vvv , ( ) 5222 =v , (4.4.10) and
( ) ( )75.075.0 21 vv = , ( ) ( )75.075.0 21 vv = , ( ) ( )75.075.0 21 vv = , (4.4.11) and
( ) ( )( ) ( )75.075.075.05 121 vvv = . (4.4.12) The conditions in (4.4.11) and (4.4.12) are the matching conditions at . The
general solution of (4.4.8) and (4.4.9) is given as
75.0=x
( ) ( ) ( ) ( ) ( ) xDxCxBxAxv iiiii 41414141 60cosh60sinh60cos60sin +++= , (4.4.13) where i . The desired objective is to achieve no displacement at the free end, i.e., 2,1=
( ) 0,2 =tw , (4.4.14) or equivalently,
( ) 022 =v . (4.4.15) The following problem is solved to determine the coefficients .2.1for , ,,, =iDCBA iiii
bAz = , (4.4.16) where
=A
364.2818447.2818164.14252.160000680.1012650.1012839.5089.50000
736.130732.130754.0657.00000710.31750.30826.3735.6710.31750.30826.3735.6049.11394.11420.2375.1049.11394.11420.2375.1094.4970.3494.0 0.870-4.094 3.970 0.494-0.870
00000783.202.78300001010
,
[ ]TDCBADCBA 22221111=z , (4.4.17)
38
-
852 eb = . (4.4.18)
,0144.0 ,0032.0 ,0054.0 ,0032.0 ,0054.0 21111 ===== ADCBA 0686.0 and ,0687.0,0125.0 222 === DCB ,
(4.4.19)
( ) ( ) ( )75.060cos0032.075.060sin0054.075.0 4/14/11 +=v ( ) ( )75.060cosh0032.075.060sinh0054.0 4/14/1 + , (4.4.20)
( ) 0.0022 75.01 =v . (4.4.21) The control gain is determined as
( ) ( ) ( )( ) .5.178175.075.075.0510,75.0
1
21 ==v
vv (4.4.22)
Since is negative, the control can be implemented by means of spring and the system is stable.
The control force is given by
( ) ( ) ( ) tvtwtu 10sin75.05.1781,75.05.1781,75.0 1== . (4.4.23) ( ) ttu 10sin-3.9018,75.0 = . (4.4.24)
x
( ) ttf 10sin2=( )txw ,
L
5.1781=Kx
( ) ttf 10sin2=( )txw ,
L
5.1781=K
FIGURE 4.7: Controlled uniform beam
The steady state displacement at the free end is as required and is given as
39
-
( ) ( ) ( )260cos0125.0260sin0144.02 4/14/12 =v ( ) ( )260cosh0686.0260sinh0687.0 4/14/1 + . (4.4.25)
( ) ( ) 010sin101.7764 10sin2),2( -152 == ttvtw . (4.4.26)
Example 2:
The control can also be implemented so as to eliminate steady state motion at any other
prescribed point, for example at 5.0=x . This can be mathematically expressed as ( ) 0,5.0 =tw , (4.4.27)
or equivalently,
( ) 05.01 =v . (4.4.28) The sixth row in matrix , in A bAz = changes to accommodate for the condition in (4.4.26). The new matrix is given by
=A
364.2818447.2818164.14252.160000680.1012650.1012839.5089.50000
0000135.2886.1178.0984.0710.31750.30826.3735.6710.31750.30826.3735.6049.11394.11420.2375.1049.11394.11420.2375.1094.4970.3494.0 0.870-4.094 3.970 0.494-0.87000000783.202.78300001010
,
( ) 0.0046 75.01 =v . (4.4.29) The control gain is
( ) .-958.111010,75.0 = (4.4.30) The control force is given by
( ) ( ) ( ) tvtwtu 10sin75.0-958.111,75.0-958.111,75.0 1== (4.4.31)
40
-
( ) ttu 10sin-4.3685,75.0 = (4.4.32) The steady state displacement at 5.0=x is, as desired
( ) ( ) 010sin101.4728 10sin5.0),5.0( -151 == ttvtw . (4.4.33)
Example 3:
Consider a uniform Bernoulli-Euler beam
022
4
4
=+
tw
xw , ,10 (4.4.34)
The position of excitation is at 8.0== bx . The desired objective is to eliminate steady state motion at the free end. The mode shapes of the beam before and after
implementing nodal control at position 3.0== a , ,,
x
,
are plotted. For the case with the
uncontrolled beam, the coefficients 2,1for =iDC iiBA ii are determined and the
mode shapes can be determined from (4.4.35) given below.
x
( ) ttf sin=( )txw ,
L
b
x
( ) ttf sin=( )txw ,
L
b
FIGURE 4.8: Uncontrolled uniform beam
( ) xDxCxBxAxv iiiii coshsinhcossin +++= , i . 2,1= (4.4.35)
41
-
x
( ) ttf sin=( )txw ,
L
( ) ( )tawtu ,=
b
a
x
( ) ttf sin=( )txw ,
L
( ) ( )tawtu ,=
b
a
FIGURE 4.9: Controlled uniform beam
For the case with the controlled beam, the coefficients are
determined and the mode shapes are again determined from (4.4.35). The mode shapes
before and after implementing nodal control are shown in Figure 4.10. It is clear from
the figures that the condition of no motion at the free end is achieved as desired.
3.2,1for , ,,, =iDCBA iiii
42
-
Before Control After Control
0 0.2 0.4 0.6 0.8 1-0.008
-0.004
0
0.004
0.008
Mode 2
0 0.2 0.4 0.6 0.8 1-0.008
-0.004
0
0.004
0.008
Mode 1
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 3
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 2
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 4
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 2
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.08
-0.04
0
0.04
0.08
Mode 1
0 0.2 0.4 0.6 0.8 1-0.08
-0.04
0
0.04
0.08
Mode 1
x x
)(xv)(xv
Before Control After Control
0 0.2 0.4 0.6 0.8 1-0.008
-0.004
0
0.004
0.008
Mode 2
0 0.2 0.4 0.6 0.8 1-0.008
-0.004
0
0.004
0.008
Mode 1
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.008
-0.004
0
0.004
0.008
Mode 2
0 0.2 0.4 0.6 0.8 1-0.008
-0.004
0
0.004
0.008
Mode 1
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 3
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 2
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 3
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 2
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 4
0 0.2 0.4 0.6 0.8 1-0.004
-0.002
0
0.002
0.004
Mode 2
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.08
-0.04
0
0.04
0.08
Mode 1
0 0.2 0.4 0.6 0.8 1-0.08
-0.04
0
0.04
0.08
Mode 1
x x
)(xv)(xv
0 0.2 0.4 0.6 0.8 1-0.08
-0.04
0
0.04
0.08
Mode 1
0 0.2 0.4 0.6 0.8 1-0.08
-0.04
0
0.04
0.08
Mode 1
x x
)(xv)(xv
FIGURE 4.10: Implementation of nodal control
43
-
4.5 Summary
The problem of absorbing steady state motion at the free end of a cantilever beam under
the influence of harmonic excitation is investigated. It has been shown that for a fixed
excitation frequency and point of application of control force there exists a unique
control gain that absorbs the harmonic motion at the free end. A closed form solution for the control gain is given. This result is better than the one given by Ram in
[3] because it does not require infinite product of eigenvalues.
44
-
Chapter 5: Stability and Optimality
5.1 Introduction
The concept of stability explains the limitations on motion of a vibrating system. A
stable system vibrates in specific bounds while an unstable system has an unbounded
motion. The study of stability also assists in tuning the response of a dynamic system
so that it remains in desired limits. Control systems design involves determining
various parameters like and . Desired overshoot, settling time, response levels,
system robustness, cost of implementing control and simplicity of implementing and
monitoring the control are some of many important factors governing the choice of
these parameters. Study of optimality provides an insight into smart selection of these
physical parameters. Conditions ensuring the stability of the controlled system are
determined in this chapter. Graphs demonstrating regions of active and passive control,
stability and optimal control locations are shown and discussed.
cm, k
5.2 Stability analysis
If the control gain is negative then the control can be implemented by means of a spring of constant = , which should be attached between point and the ground as shown in Figure 5.1.
a
45
-
x( ) ttf cos=( )txw ,
L
FIGURE 5.1: Passively controlled uniform beam
In this case the system is stable. Note that there are many frequency intervals for which
the control gain is negative. Since the beam in Figure 4.4(b) is obtained by imposing a
single constraint to the beam shown in Figure 4.4(a), the eigenvalues i interlace the eigenvalues i in the sense that
L32211 if kk
-
Corollary 2
The controlled system is stable if and only if
1< (5.2.4)
(a) Controlled beam
F
x ( )txw ,
L
x
1
xF
(b) Equivalent lumped parameter model
for displacement at x = a
FIGURE 5.2: Stability analysis and equivalent stiffness
Proof: By definition from (4.3.16), static deflection is positive. Hence, inequality (5.2.4) holds for negative . This is the passively controlled case where the control can be implemented by attaching a spring as discussed above, for which the system is
obviously stable. Instability may arise only for the case where is positive. Consider
47
-
now the case when is positive. From the linearity of the differential operator in (4.2.1) it follows that there is a linear relation between force and deflection in the beam
i.e. without the control, ),( tauF = . Hence the deflection at ax = in the controlled beam can be considered as governed by the sum of two springs, one negative with
constant applied by the control, the other positive with constant 1 applied by the flexibility of the beam, as shown in Figure 5.2(a) and Figure 5.2(b), where x is an infinitesimal element of the beam. The system is stable if and only if the equivalent
spring is positive, i.e., 0>1 , and otherwise unstable. This is precisely the inequality (5.2.4).
1
5.3 Optimality
The topic of optimality discussed here is based on the minimal control force
requirement. Several graphs are shown in Figure 5.3. Considerable information can be
drawn from these graphs in terms of understanding stability, optimality, and type of
control. The graphs are a plot of control force u , control gain ),( ta and inverse of
static deflection i.e. over the span of beam described by (4.2.1) through (4.2.5) for
four different values of excitation . Here, LIE , A and ,, can be chosen arbitrarily.
48
-
x0 0.2 0.4 0.6 0.8 1
-2
0
4
8
-10
-5
0
5
10
0 0.2 0.4 0.6 0.8 1x
0 0.2 0.4 0.6 0.8 1
-2
0
2
4
x0 0.2 0.4 0.6 0.8 1
-5
0
5
x
(a) (b)
(d)(c)
control force control gain /1
310
310 310
310
x0 0.2 0.4 0.6 0.8 1
-2
0
4
8
-10
-5
0
5
10
0 0.2 0.4 0.6 0.8 1x
0 0.2 0.4 0.6 0.8 1
-2
0
2
4
x0 0.2 0.4 0.6 0.8 1
-5
0
5
x
(a) (b)
(d)(c)
control force control force control gain control gain /1 /1
310
310 310
310
FIGURE 5.3: Plots of control gain, control force and inverse of static deflection against the beam span x for cases with excitation frequency of (a) 10= ,
(b) 20= , (c) 30= and (d) 80= The following observations can be made from the graphs.
1. The inverse of static deflection is always positive (consistent with the
definition).
Static deflection is positive by definition and hence the inverse of static deflection is
clearly positive.
49
-
2. Inverse of static deflection is unbounded at 0=x and . 1=xStatic deflection is 0 at (clamped end) and at 0=x 1=x (desired objective). Hence, the inverse is .
3. Control force is unbounded near 0=x . At location just next to 0=x the control gain requirement is very large and the displacement is almost zero. Hence the control force is large. At the control
force should be zero because the displacement is zero.
0=x
4. The control force is -1 at 1=x . Since the excitation amplitude at 1=x is unity, the control force at is 1. 1=x
5. The control gain is positive in some regions and negative in some others.
The control gain follows Corollary 1. Again if 0 , active control has to be implemented.
6. The control gain is unbounded at 0=x and 1=x . It is not possible to determine control gain at 0=x numerically because in (4.2.19) becomes singular. However, using pseudo inverse one can calculate the control gain. It
is obvious that infinite amount of control gain is needed to achieve the state of no steady
state motion at the free end. Also, if control is implemented at the end
A
1=x , the control gain required is infinity because the displacement is zero and the control force
which is the product of control gain and steady state amplitude at the end has to be 1.
5.4 Summary
The sign of the control gain i.e., positive or negative, determines the type of control to
be implemented i.e., active or passive. Equations (5.2.2) and (5.2.3) in Corollary 1
50
-
establish the conditions for the sign of the control gain . If < 0, the control can be realized by a passive element i.e., spring of constant =k . However, if > 0, then the control has to be implemented by active means. Corollary 2, through (5.2.4),
establishes the conditions for the controlled system to be stable. The optimal control
force requirements may be obtained from the graphs shown in Figure 5.2.
The issue of robustness is more critical than the issue of optimality of control force,
because at the free end the control force requirement is minimum, but the control gain
required is infinity. A passively controlled system is more robust in comparison to a
system controlled by active means. It is obvious from the graphs that the flatness of the
control gain graph over the beam span is a direct indication of robustness. Also with
increase in the frequency of excitation the flatness of the control gain graphs goes on
decreasing and also there are few locations where control can be implemented by
passive means and active control is impending.
51
-
Chapter 6: Experimental Verification
6.1 Introduction
The analysis so far has been purely theoretical and has shown that the steady state
motion at a prescribed location on a beam can be eliminated by describing the problem
appropriately and then performing simple mathematical manipulations. The analysis
also helps in determining the value of necessary the control gain, the magnitude of the
control force, the type of control i.e. active or passive and the location of control. The
ultimate objective however, is to be able to implement the theory in practical
applications. A small experiment is carried out in order to validate the method
developed and results derived. This chapter describes the mathematical model, the
experimental setup and the results obtained from the experiment.
6.2 Proposed model for the experiment
In order to validate the theory a simple experiment is constructed so as to mimic the
configuration shown in Figure 6.1. The beam is under the influence of a harmonic
excitation at cx = . The objective is to implement nodal control by using a spring so as to eliminate the steady state displacement at the free end i.e. at . This is a simple
configuration and can be easily modeled from the information available from previous
analysis and illustrations.
Lx =
52
-
( ) ttf sin=x
L
( )txw ,
c
b
( ) ttf sin=x
L
( )txw ,( )txw ,
c
b
FIGURE 6.1: Mathematical model of the test beam used in the experiment
It is imperative that there is significant resemblance in terms of dynamics between the
experimental model constructed and the schematic model shown in Figure 6.1. For this
reason modal analysis is performed on a model of a double-simply-supported beam, as
shown in Figure 6.6. First few natural frequencies of the beam are extracted and
compared to natural frequencies of the beam in Figure 6.1 obtained analytically.
6.3 Determination of natural frequencies
(A) Analytical determination of natural frequencies.
The natural frequencies for the double-simple-supported beam can be extracted
analytically by solving transcendental eigenvalue problem from [13]. However, more
simply, one can write the equation to the beam and impose the boundary conditions (in
this case double-simple-supports and free end) and solve for natural frequencies by
solving a linear algebraic problem as demonstrated in Chapter 3, section 3.4. The
results are not as accurate as one obtained by solving the transcendental eigenvalue
53
-
problem, but provide significantly close approximation. The governing differential
equation for the beam is given by
022
4
4
=+
twA
xwEI , Lx
-
( ) 0,1 =taw , ,( ) 0,2 =taw ( ) ( )xtaw
xtaw
=
,, 21 , ( ) ( )222
21
2 ,,x
tawx
taw
= . (6.3.9)
From separation of variables
( ) ( ) txvtxw ii sin, = , 2,1=i . (6.3.10) Direct substitution of (6.3.10) in (6.3.6) and (6.3.7) gives,
0 2 = ii AvvEI , 2,1=i . (6.3.11) Again,
EIA 24 = . (6.3.12)
Hence,
0 4 = ii vv , 2,1=i . (6.3.13) Accordingly, the boundary conditions are
( ) ( ) ( ) ( ) ,0 ,0,00,00 2211 ==== LvLvvv (6.3.14) and the matching conditions are
( ) ( ) ( ) ( ) ).( ),(,0,0 212121 avavavavavav ==== (6.3.15) The solution to (6.3.13) is
( ) xDxCxBxAxv iiiii coshsinhcossin +++= , 2,1=i (6.3.16) where, for ,,, iii CBA ,iD 2,1=i are constants. Equation in (6.3.16) and the conditions
in (6.3.14) and (6.3.15) together give a set of algebraic equations, which are represented
here in matrix form
bAz = (6.3.17) where,
55
-
=
aaaaaaaaaaaaaaaa
aaaaaaaa
LLLLLLLL
coshsinhcossincoshsinhcossinsinhcoshsincossinhcoshsincos
coshsinhcossin00000000coshsinhcossin
sinhcoshsincos0000coshsinhcossin0000
0000101000001010
A ,
, [ ]TDCBADCBA 22221111=z (6.3.18) and
[ ]T00000000=b . (6.3.19) Clearly there are infinite numbers of solutions for for i for every
position
,,, iii CBA ,iD 2,1=
ax = of the simple support. The values of , which make the matrix singular, give the eigenvalues
Z
for the Bernoulli-Euler beam by the following relationship
AEI
4= . (6.3.20)
The natural frequencies of the beam are then given by taking the square root of the
eigenvalues and dividing them by 2 , i.e.,
2=nf . (6.3.21)
The physical parameters namely, modulus of elasticity E , the cross-sectional area ,
density
A
and moment of inertia I are assumed to be invariant along the beam span. Parameters and A I are determined from the physical dimensions of the beam.
56
-
h
w
AISI 1005Steel
L
h
w
AISI 1005Steel
L
FIGURE 6.2: Dimensions of the test beam used in the experiment
Beam Dimensions:
mm 0.1727=L , mm 8.50=w ,
mm 0.6=h . Hence,
2mm 8.30468.50 === hwA , (6.3.22) and
433
mm 914.40 12
68.5012
=== hwI . (6.3.23)
The beam is made up of AISI 1005 Steel.
Beam properties:
GPa 200=E . -3m-Kg 7870= .
57
-
Simulation is performed in MATLAB (Appendix A Program 11) for the double-simple-
supported case discussed above and also for the clamped-simple-supported case.
(B) Determination of natural frequencies from modal test.
A test where the structure or component is vibrated with a known excitation, often out
of its normal service environment, which includes both the data acquisition, and its
subsequent analysis, is called modal testing [14]. It is performed in order to extract the
natural frequencies of the test beam. The complete experimental setup is shown in
Figure 6.6. The following material gives a brief description of the setup, necessary
equipment, procedure of performing the test and some necessary precautions.
(i) Test Beam
The test beam is fastened to a heavy metal frame as shown in Figure 6.4 using clamps
Bolt
Nut
Test Beam
Clamps
Bolt
Nut
Test Beam
Clamps
FIGURE 6.3: Clamping details
at two locations. The details of the clamping are shown in Figure 6.3.
(ii) Impact Hammer
Model 291M78-086C05 from PCB Piezotronics is used to cause an impact. It consists
of an integral ICP quartz force sensor mounted on the striking end of the hammerhead.
58
-
The frequency range is approximately 5 kHz and hammer range is approximately
22000N.
FIGURE 6.4: Impact Hammer
Its resonant frequency is near 28 KHz. Figure 6.4 shows a picture of the hammer and
the beam.
(iii) Accelerometer
FIGURE 6.5: Accelerometer
Quartz shear ICP accelerometer, Model 353B33, from piezoelectric is used in the test.
The range of frequencies is from 2 Hz to 4000 Hz with %5 and voltage sensitivity is 100mV per g .
59
-
(iv) Data Acquisition
NI-4551 from National Instrument is used for data collection. BNC 2140 also from
National Instruments is used to provide ICP power to the accelerometer and the hammer
and connect them to the computer via NI 4551. VirtualBench DSA (Dynamic Signal
Analyzer) is used for signal processing and monitoring.
(v) Connecting Cables
The connecting wires for hammer and accelerometer are recommended to have
impedance in the range of 50~75 .
Modal analysis technique
The natural frequencies of a system are a function of the physical parameters, material
properties and boundary conditions of the system. Hence, the frequency response
function of the test beam should remain invariant under ideal conditions irrespective of
the point of measurement and the point of excitation. The accelerometer is mounted on
the beam as shown in Figure 6.5. An impact is made at several locations along the span
of the beam. Again, frequency response is also obtained for several different locations
of accelerometer for same position of impact. The schematic shown in Figure 6.6 gives
a clear picture of the setup and data acquisition for modal analysis. The clamping of the
beam is very critical. Repeated impacts on the beam may cause loosening of the bolts
in the clamps. Also the heavy metal frame should be as rigid as possible and should not
move. These factors change the characteristics of the system and influence the
frequency response function.
60
-
Test
Bea
mIm
pact
Ham
mer
Acc
eler
omet
er
ICP
Powe
r U
nit
PC w
ith
Dyn
amic
Sig
nal
Ana
lyze
r
Hea
vyM
etal
Fram
e
Test
Bea
mIm
pact
Ham
mer
Acc
eler
omet
er
ICP
Powe
r U
nit
PC w
ith
Dyn
amic
Sig
nal
Ana
lyze
r
Hea
vyM
etal
Fram
e
Test
Bea
mIm
pact
Ham
mer
Acc
eler
omet
er
ICP
Powe
r U
nit
PC w
ith
Dyn
amic
Sig
nal
Ana
lyze
r
Hea
vyM
etal
Fram
e
FIGURE 6.6: Modal Analysis
61
-
Some important considerations for the Modal Test
It is essential to mount accelerometer securely on the structure to avoid relative motion between the accelerometer and the structure. Insecure mounting of
accelerometer can cause noise in the collected data for the system. In this test,
an accelerometer is first attached on the beam using a double sided scotch tape
as seen in Figure 6.5 and it is wrapped using a tape around the beam and the
accelerometer for secure mounting.
The location of accelerometer is decided after observation of the dynamic motion of beam from the side. The mode shape of slow natural frequency can
be observed looking at the motion from the side. The mounting position is then
decided avoiding the nodal points between free end and clamped support.
The impact of the hammer should be quick and sharp. Also for low frequency range analysis a soft hammer tip should be used. It should be insured that the
amplitude spectrum for the hammer is approximately flat in the frequency range
to be analyzed. This is shown in Display 2 of Figure 6.7.
Figure 6.7 shows the display of the VirtualBench DSA. Sharp peaks in the frequency
response shown in Display 3 are the natural frequencies of the test beam. Time
waveform of the accelerometer and the hammer is shown in Display 1 and the amplitude
spectrum is shown in Display 2.
62
-
FIGURE 6.7: VirtualBench DSA display
Table 6.1 shows a comparison of the natural frequencies for the two cases with natural
frequencies derived from modal analysis on the test beam.
TABLE 6.1: Comparison of natural frequencies
11.26
73.8570.9568.804
44.13
23.17
3.6
Modal Analysis of Test Beam
(Hz)
55.16
22.28
3.31
AnalyticalClamped Simple support
(Hz)
3.131
21.142
40.763
AnalyticalDouble Simple support
(Hz)
Natural frequency
11.26
73.8570.9568.804
44.13
23.17
3.6
Modal Analysis of Test Beam
(Hz)
55.16
22.28
3.31
AnalyticalClamped Simple support
(Hz)
3.131
21.142
40.763
AnalyticalDouble Simple support
(Hz)
Natural frequency
63
-
The double-simple-supported case is chosen because the approximation of natural
frequencies is closer to the experimental result. The frequency of 11.26 Hz is believed
to be coming from the overall system comprising the test beam and the heavy metal
frame.
6.4 Determination of the control gain
The control gain calculations are performed as discussed in Chapter 4, Section 4.2.
Actual test beam parameters and dimensions are used. A harmonic force of 45N and
frequency of 3.75 Hz acts exactly in the middle of the test beam span between the two
supports.
( ) ttf 56.23sin45=
x
m7272.1
0.327m ( ) ( )tawtu ,=0.654m
a
( ) ttf 56.23sin45=
x
m7272.1
0.327m ( ) ( )tawtu ,=0.654m
a
FIGURE 6.8: Controlled test beam
Graph of control gain variation with changing location of control along the beam span is
shown in Figure 6.9. It is clear from the graphs that is negative between the two supports (shown by triangles in the graph). Hence the control can be implemented by
means of a spring.
64
-
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-100
-80
-60
-40
-20
0
20
40
60
80
100
control force in N control gain in N/mmx
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-100
-80
-60
-40
-20
0
20
40
60
80
100
control force in N control gain in N/mmx
FIGURE 6.9: Control gain and control force variation along the beam span
In order to verify the accuracy of the calculations, simulation is done to plot the
modeshape of the test beam (Appendix A Program 14 and Program 15) before and after
implementing control. The Figure 6.10 shows that the steady state motion at the free
end is eliminated after implementing control. However, Figure 6.10(b) shows that there
is discontinuity in the shape of the beam. This helps in understanding the huge control
force requirement in the range 0.2m 0.3m and also 0.6m 0.7m.
65
-
0 0.4 0.8 1.2 1.6-2.5
-1.5
-0.50
0.5
x
)(xv
210
0 0.4 0.8 1.2 1.6-0.8
-0.4
0
0.4
0.8
x
)(xv
210
(a) Before Control (b) After Control
0 0.4 0.8 1.2 1.6-2.5
-1.5
-0.50
0.5
x
)(xv
0 0.4 0.8 1.2 1.6-2.5
-1.5
-0.50
0.5
0 0.4 0.8 1.2 1.6-2.5
-1.5
-0.50
0.5
x
)(xv
210
0 0.4 0.8 1.2 1.6-0.8
-0.4
0
0.4
0.8
x
)(xv
210
0 0.4 0.8 1.2 1.6-0.8
-0.4
0
0.4
0.8
0 0.4 0.8 1.2 1.6-0.8
-0.4
0
0.4
0.8
x
)(xv
210
(a) Before Control (b) After Control
FIGURE 6.10: Test beam modeshape before and after control
6.5 Control system design
The graph in Figure 6.9 shows that the most optimal place for placing the spring is at
0.4138m from the first support. This is because before 0.4138m the control force
requirement increases sharply. Also after 0.4138m the control gain requirement
increases sharply even though the control force requirement decreases. The necessary
controlling spring should have a spring constant of 59.34N/mm.
In practical situation a spring of 59.34N/mm positioned at 0.4138m from the first
support may not necessarily eliminate the motion of the free end. However, this
simulation serves as a guideline in choosing both the spring and the location of the
spring on the test beam. A very simple configuration is designed which is extremely
flexible, both in terms of housing springs of variable length and diameter, and also, in
66
-
positioning the springs at different locations along the span of the test beam between the
two supports. The configuration is shown in Figure 6.11.
BeamBeam
FIGURE 6.11: Spring housing configuration
The photograph shown in Figure 6.14(a) shows how the device can be easily clamped
on the heavy metal frame. The vibration exciter is attached to the test beam by the
configuration shown in Figure 6.12. This is also shown in Figure 6.14(b).
Beam
VibrationExciter
Beam
VibrationExciter
FIGURE 6.12: Attaching the vibration exciter to the beam
The complete experimental setup is shown in Figure 6.13. Photograph of the same is
shown in Figure 6.15.
67
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3.6
50
H
Z
Spri
ng S
uppo
rt
Func
tion
Gen
erat
or
Cha
rge
Am
plifi
er
Bea
m S
uppo
rts
Spri
ngs
Vib
ratio
nEx
cite
r
Test
Bea
m
Hea
vyM
etal
Fram
e3
.65
0
HZ
3.6
50
H
Z
Spri
ng S
uppo
rt
Func
tion
Gen
erat
or
Cha
rge
Am
plifi
er
Bea
m S
uppo