The Fundamentals of
`Modal Testing
`
`Application Note 243 - 3
`
`Η(ω) = Σn
`r = 1
`
`φ φi
`j /m
`ξωω
`ω
`ω
`( n - ) + (2
`n)
`
`2 2 2
`
`2
`
`Page 1 of 56
`
`HAPTIC EX2006
`
`

`

`Preface
`
`Modal analysis is defined as the study
`of the dynamic characteristics of a
`mechanical structure. This applica-
`tion note emphasizes experimental
`modal techniques, specifically the
`method known as frequency response
`function testing. Other areas are
`treated in a general sense to intro-
`duce their elementary concepts and
`relationships to one another.
`
`Although modal techniques are math-
`ematical in nature, the discussion is
`inclined toward practical application.
`Theory is presented as needed to
`enhance the logical development of
`ideas. The reader will gain a sound
`physical understanding of modal
`analysis and be able to carry out
`an effective modal survey with
`confidence.
`
`Chapter 1 provides a brief overview
`of structural dynamics theory.
`Chapter 2 and 3 which is the bulk
`of the note – describes the measure-
`ment process for acquiring frequency
`response data. Chapter 4 describes
`the parameter estimation methods
`for extracting modal properties.
`Chapter 5 provides an overview
`of analytical techniques of structural
`analysis and their relation to
`experimental modal testing.
`
`2
`
`Page 2 of 56
`
`

`

`Table of Contents
`
`Preface
`
`Chapter 1 — Structural Dynamics Background
`Introduction
`Structural Dynamics of a Single Degree of Freedom (SDOF) System
`Presentation and Characteristics of Frequency Response Functions
`Structural Dynamics for a Multiple Degree of Freedom (MDOF) System
`Damping Mechanism and Damping Model
`Frequency Response Function and Transfer Function Relationship
`System Assumptions
`
`Chapter 2 — Frequency Response Measurements
`Introduction
`General Test System Configurations
`Supporting the Structure
`Exciting the Structure
`Shaker Testing
`Impact Testing
`Transduction
`Measurement Interpretation
`
`Chapter 3 — Improving Measurement Accuracy
`Measurement Averaging
`Windowing Time Data
`Increasing Measurement Resolution
`Complete Survey
`
`Chapter 4 — Modal Parameter Estimation
`Introduction
`Modal Parameters
`Curve Fitting Methods
`Single Mode Methods
`Concept of Residual Terms
`Multiple Mode-Methods
`Concept of Real and Complex Modes
`
`Chapter 5 — Structural Analysis Methods
`Introduction
`Structural Modification
`Finite Element Correlation
`Substructure Coupling Analysis
`Forced Response Simulation
`
`Bibliography
`
`2
`
`4
`4
`5
`6
`9
`11
`12
`13
`
`14
`14
`15
`16
`18
`19
`22
`25
`29
`
`30
`30
`31
`32
`34
`
`38
`38
`39
`40
`41
`43
`45
`47
`
`48
`48
`49
`50
`52
`53
`
`54
`
`3
`
`Page 3 of 56
`
`

`

`Chapter 1
`Structural Dynamics Background
`
`Figure 1.1
`Phases of a
`modal test
`
`Test Structure
`
`Frequency Response Measurements
`
`Curve Fit Representation
`
`H (w) = S
`ij
`
`n
`
`r = 1
`
`f fir
`jr
`w w zww
`2
`2
`+ j2
`mr ( r -
`
`r)
`
`Modal Parameters
`
`wz f
`
`— Frequency
`— Damping
`{ } — Mode Shape
`
`the trends and their usefulness in the
`measurement process. Finally, these
`concepts are extended into MDOF
`systems, since this is the type of
`behavior most physical structures
`exhibit. Also, useful concepts
`associated with damping mechanisms
`and linear system assumptions
`are discussed.
`
`Introduction
`
`A basic understanding of structural
`dynamics is necessary for successful
`modal testing. Specifically, it is
`important to have a good grasp of the
`relationships between frequency
`response functions and their individ-
`ual modal parameters illustrated in
`Figure 1.1. This understanding is of
`value in both the measurement and
`analysis phases of the survey. Know-
`ing the various forms and trends of
`frequency response functions will
`lead to more accuracy during the
`measurement phase. During the
`analysis phase, knowing how equa-
`tions relate to frequency responses
`leads to more accurate estimation of
`modal parameters.
`
`The basic equations and their various
`forms will be presented conceptually
`to give insight into the relationships
`between the dynamic characteristics
`of the structure and the correspond-
`ing frequency response function
`measurements. Although practical
`systems are multiple degree of free-
`dom (MDOF) and have some degree
`of nonlinearity, they can generally
`be represented as a superposition
`of single degree of freedom (SDOF)
`linear models and will be developed
`in this manner.
`
`First, the basics of an SDOF linear
`dynamic system are presented to gain
`insight into the single mode concepts
`that are the basis of some parameter
`estimation techniques. Second, the
`presentation and properties of vari-
`ous forms of the frequency response
`function are examined to understand
`
`4
`
`Page 4 of 56
`
`

`

`Spring k
`
`Damper c
`
`Response
`Displacement (x)
`
`Mass m
`
`Excitation
`Force (f)
`
`c
`2km
`
`.
`..
`mx + cx + kx = f(t)
`w 2n = ,k
`or =z
`m
`
`2
`
`n =
`
`cm
`
`s
`1,2
`
`= - + j d
`
`s wswz
`
`— Damping Rate
`— Damped Natural Frequency
`
`t
`
`e-
`
`sw d
`
`5
`
`Structural Dynamics of a Single
`Degree of Freedom (SDOF) System
`
`Although most physical structures are
`continuous, their behavior can usual-
`ly be represented by a discrete
`parameter model as illustrated in
`Figure 1.2. The idealized elements
`are called mass, spring, damper and
`excitation. The first three elements
`describe the physical system. Energy
`is stored by the system in the mass
`and the spring in the form of kinetic
`and potential energy, respectively.
`Energy enters the system through
`excitation and is dissipated through
`damping.
`
`The idealized elements of the physi-
`cal system can be described by the
`equation of motion shown in Figure
`1.3. This equation relates the effects
`of the mass, stiffness and damping in
`a way that leads to the calculation of
`natural frequency and damping factor
`of the system. This computation is
`often facilitated by the use of the def-
`initions shown in Figure 1.3 that lead
`directly to the natural frequency and
`damping factor.
`
`The natural frequency, w
`, is in units
`of radians per second (rad/s). The
`typical units displayed on a digital
`signal analyzer, however, are in Hertz
`(Hz). The damping factor can also be
`represented as a percent of critical
`damping – the damping level at which
`the system experiences no oscillation.
`This is the more common understand-
`ing of modal damping. Although there
`are three distinct damping cases,
`only the underdamped case (z< 1)
`is important for structural dynamics
`applications.
`
`Figure 1.2
`SDOF discrete
`parameter model
`
`Figure 1.3
`Equation of
`motion —
`modal definitions
`
`Figure 1.4
`Complex roots
`of SDOF
`equation
`
`Figure 1.5
`SDOF
`impulse
`response/
`free decay
`
`Page 5 of 56
`
`zw
`

`

`H(
`
`wq w (
`
`w d
`
`w d
`
`Magnitude
`
`Phase
`
`Figure 1.6
`Frequency
`response —
`polar
`coordinates
`
`When there is no excitation, the roots
`of the equation are as shown in
`Figure 1.4. Each root has two parts:
`the real part or decay rate, which
`defines damping in the system and
`the imaginary part, or oscillatory
`rate, which defines the damped
`natural frequency, wd. This free
`vibration response is illustrated
`in Figure 1.5.
`
`When excitation is applied, the equa-
`tion of motion leads to the frequency
`response of the system. The frequen-
`cy response is a complex quantity
`and contains both real and imaginary
`parts (rectangular coordinates). It can
`be presented in polar coordinates as
`magnitude and phase, as well.
`
`Presentation and Characteristics of
`Frequency Response Functions
`
`Because it is a complex quantity, the
`frequency response function cannot
`be fully displayed on a single two-
`dimensional plot. It can, however, be
`presented in several formats, each of
`which has its own uses. Although the
`response variable for the previous
`discussion was displacement, it could
`also be velocity or acceleration.
`Acceleration is currently the accepted
`method of measuring modal
`response.
`
`One method of presenting the data
`is to plot the polar coordinates, mag-
`nitude and phase versus frequency
`as illustrated in Figure 1.6. At reso-
`nance, when w = w n, the magnitude
`is a maximum and is limited only by
`the amount of damping in the system.
`The phase ranges from 0° to 180°
`and the response lags the input by
`90° at resonance.
`
`6
`
`) =
`
`1/m
`w w zww
`( n- ) + (2
`
`2
`
`2 2 2
`
`n)
`
`) = tan-1
`
`2
`
`xwww w2
`
`n-
`
`n
`
`2
`
`Page 6 of 56
`
`

`

`2
`
`n)
`
`2 2 2
`
`H(
`
`) =
`
`H(
`
`) =
`
`Real
`
`Imaginary
`
`Another method of presenting
`the data is to plot the rectangular
`coordinates, the real part and the
`imaginary part versus frequency.
`For a proportionally damped system,
`the imaginary part is maximum at
`resonance and the real part is 0, as
`shown in Figure 1.7.
`
`A third method of presenting the
`frequency response is to plot the real
`part versus the imaginary part. This is
`often called a Nyquist plot or a vector
`response plot. This display empha-
`sizes the area of frequency response
`at resonance and traces out a circle,
`as shown in Figure 1.8.
`
`By plotting the magnitude in decibels
`vs logarithmic (log) frequency, it is
`possible to cover a wider frequency
`range and conveniently display the
`range of amplitude. This type of plot,
`often known as a Bode plot, also
`has some useful parameter character-
`istics which are described in the
`following plots.
`
`When w << wn the frequency
`response is approximately equal to
`the asymptote shown in Figure 1.9.
`This asymptote is called the stiffness
`line and has a slope of 0, 1 or 2 for
`displacement, velocity and accelera-
`tion responses, respectively. When
`w >> wn the frequency response is
`approximately equal to the asymptote
`also shown in Figure 1.9. This asymp-
`tote is called the mass line and has a
`slope of -2, -1 or 0 for displacement
`velocity or acceleration responses,
`respectively.
`
`Figure 1.7
`Frequency
`response —
`rectangular
`coordinates
`
`Figure 1.8
`Nyquist plot
`of frequency
`response
`
`w w ww w zww
`
`2
`
`n -
`( n- ) + (2
`
`2
`
`xwww w zww
`
`-2
`( n- ) + (2
`
`2
`
`2 2 2
`
`n
`
`n)
`
`q w (
`
`)
`
`H(
`
`)
`
`7
`
`Page 7 of 56
`
`w
`w
`

`

`1k
`
`k
`
`w» »» »» -»
`
`Frequency
`
`Frequency
`
`Frequency
`
`1
`mw
`
`2
`
`1m
`
`1m
`
`2
`
`k
`
`Displacement
`
`Velocity
`
`Acceleration
`
`The various forms of frequency
`response function based on the
`type of response variable are also
`defined from a mechanical engineer-
`ing viewpoint. They are somewhat
`intuitive and do not necessarily corre-
`spond to electrical analogies. These
`forms are summarized in Table 1.1.
`
`Figure 1.9
`Different forms
`of frequency
`response
`
`Table 1.1
`Different forms
`of frequency response
`
`Definition
`
`Response
`
`Variable
`
`Compliance
`
`Mobility
`
`Accelerance
`
`X
`F
`
`V
`F
`
`A
`F
`
`Displacement
`Force
`
`Velocity
`Force
`
`Acceleration
`Force
`
`8
`
`Page 8 of 56
`
`w
`w
`

`

`Structural Dynamics for a Multiple
`Degree of Freedom (MDOF) System
`
`The extension of SDOF concepts to
`a more general MDOF system, with
`n degrees of freedom, is a straightfor-
`ward process. The physical system is
`simply comprised of an interconnec-
`tion of idealized SDOF models, as
`illustrated in Figure 1.10, and is
`described by the matrix equations
`of motion as illustrated in Figure 1.11.
`
`The solution of the equation with no
`excitation again leads to the modal
`parameters (roots of the equation)
`of the system. For the MDOF case,
`however, a unique displacement
`vector called the mode shape exists
`for each distinct frequency and damp-
`ing as illustrated in Figure 1.11. The
`free vibration response is illustrated
`in Figure 1.12.
`
`The equations of motion for the
`forced vibration case also lead to
`frequency response of the system.
`It can be written as a weighted
`summation of SDOF systems shown
`in Figure 1.13.
`
`Figure 1.10
`MDOF discrete
`parameter model
`
`k1
`
`k3
`
`m1
`
`c1
`
`c3
`
`k2
`
`k4
`
`m3
`
`m2
`
`c2
`
`c4
`
`Figure 1.11
`Equations of
`motion —
`modal definitions
`
`..
`.
`[m]{x} + [c]{x} + [k]{x} = {f(t)}
`
`{ }r, r = 1, n modes
`
`The weighting, often called the modal
`participation factor, is a function of
`excitation and mode shape coeffi-
`cients at the input and output degrees
`of freedom.
`
`Figure 1.12
`MDOF impulse
`response/
`free decay
`
`6.0
`
`9
`
`0.0
`
`Sec
`
`Amplitude
`
`Page 9 of 56
`
`f
`

`

`H(w) = S
`
`n
`
`r = 1
`
`f fi
`j /m
`w w xww
`( n - ) + (2
`
`2
`
`2 2 2
`
`n)
`
`1
`
`w 2
`
`w 3
`
`Frequency
`
`Mode 1
`
`Mode 2
`
`Mode 3
`
`1
`
`w 2
`
`w 3
`
`Frequency
`
`dB Magnitude
`
`0.0
`
`dB Magnitude
`
`0.0
`
`Figure 1.13
`MDOF frequency
`response
`
`The participation factor identifies the
`amount each mode contributes to the
`total response at a particular point.
`An example with 3 degrees of free-
`dom showing the individual modal
`contributions is shown in Figure 1.14.
`
`The frequency response of an MDOF
`system can be presented in the same
`forms as the SDOF case. There are
`other definitional forms and proper-
`ties of frequency response functions,
`such as a driving point measurement,
`that are presented in the next chap-
`ter. These are related to specific
`locations of frequency response
`measurements and are introduced
`when appropriate.
`
`Figure 1.14
`SDOF modal
`contributions
`
`10
`
`Page 10 of 56
`
`w
`w
`

`

`Damping Mechanism and
`Damping Model
`
`Damping exists in all vibratory
`systems whenever there is energy
`dissipation. This is true for mechani-
`cal structures even though most are
`inherently lightly damped. For free
`vibration, the loss of energy from
`damping in the system results in the
`decay of the amplitude of motion.
`In forced vibration, loss of energy is
`balanced by the energy supplied by
`excitation. In either situation, the
`effect of damping is to remove energy
`from the system.
`
`In previous mathematical formula-
`tions the damping force was called
`viscous, since it was proportional to
`velocity. However, this does not
`imply that the physical damping
`mechanism is viscous in nature. It is
`simply a modeling method and it is
`important to note that the physical
`damping mechanism and the mathe-
`matical model of that mechanism are
`two distinctly different concepts.
`
`Most structures exhibit one or more
`forms of damping mechanisms, such
`as coulomb or structural, which
`result from looseness of joints, inter-
`nal strain and other complex causes.
`However, these mechanisms can be
`modeled by an equivalent viscous
`damping component. It can be
`shown that only the viscous compo-
`nent actually accounts for energy loss
`from the system and the remaining
`portion of the damping is due to non-
`linearities that do not cause energy
`dissipation. Therefore, only the
`viscous term needs to be measured to
`characterize the system when using a
`linear model.
`
`The equivalent viscous damping
`coefficient is obtained from energy
`considerations as illustrated in the
`
`Figure 1.15
`Viscous damping
`energy dissipation
`
`cx.
`
`x
`
`D pwE =
`
`ceq X2
`
`X
`
`Force vs Displacement
`
`Figure 1.16
`System
`block diagram
`
`Input
`Excitation
`
`G(s)
`
`System
`
`Output
`Response
`
`Figure 1.17
`Definition of
`transfer function
`
`Transfer Function =
`
`G(s) =
`
`Output
`Input
`Y(s)
`X(s)
`
`hysteresis loop in Figure 1.15. E is
`the energy dissipated per cycle of
`vibration, ceq is the equivalent vis-
`cous damping coefficient and X is the
`amplitude of vibration. Note that the
`criteria for equivalence are equal
`energy distribution per cycle and the
`same relative amplitude.
`
`11
`
`Page 11 of 56
`
`

`

`Frequency Response Function and
`Transfer Function Relationship
`
`The transfer function is a mathemati-
`cal model defining the input-output
`relationship of a physical system.
`Figure 1.16 shows a block diagram
`of a single input-output system.
`System response (output) is caused
`by system excitation (input). The
`casual relationship is loosely defined
`as shown in Figure 1.17. Mathemati-
`cally, the transfer function is defined
`as the Laplace transform of the out-
`put divided by the Laplace transform
`of the input.
`
`The frequency response function is
`defined in a similar manner and is
`related to the transfer function.
`Mathematically, the frequency
`response function is defined as the
`Fourier transform of the output divid-
`ed by the Fourier transform of the
`input. These terms are often used
`interchangeably and are occasionally
`a source of confusion.
`
`Figure 1.18
`S-plane
`representation
`
`iww n
`
`w d
`
`ss
`
`s-plane
`
`This relationship can be further
`explained by the modal test process.
`The measurements taken during a
`modal test are frequency response
`function measurements. The parame-
`ter estimation routines are, in gener-
`al, curve fits in the Laplace domain
`and result in the transfer functions.
`The curve fit simply infers the loca-
`tion of system poles in the s-plane
`from the frequency response func-
`tions as illustrated in Figure 1.18. The
`frequency response is simply the
`transfer function measured along the
`jw axis as illustrated in Figure 1.19.
`
`12
`
`Page 12 of 56
`
`

`

`Figure 1.19
`3-D Laplace
`representation
`
`Real Part
`
`Imaginary Part
`
`Magnitude
`
`Phase
`
`System Assumptions
`
`The structural dynamics background
`theory and the modal parameter
`estimation theory are based on two
`major assumptions:
`
`l The system is linear.
`l The system is stationary.
`
`There are, of course, a number of
`other system assumptions such as
`observability, stability, and physical
`realizability. However, these assump-
`tions tend to be addressed in the
`inherent properties of mechanical
`systems. As such, they do not pres-
`ent practical limitations when making
`frequency response measurements as
`do the assumptions of linearity
`and stationarity.
`
`jw
`
`jw
`
`jw
`
`s ss s
`
`Transfer Function – surface
`Frequency Response – dashed
`
`13
`
`Page 13 of 56
`
`

`

`Chapter 2
`Frequency Response Measurements
`
`Introduction
`
`This chapter investigates the current
`instrumentation and techniques
`available for acquiring frequency
`response measurements. The discus-
`sion begins with the use of a dynamic
`signal analyzer and associated periph-
`erals for making these measurements.
`The type of modal testing known as
`the frequency response function
`method, which measures the input
`excitation and output response simul-
`taneously, as shown in the block dia-
`gram in Figure 2.1, is examined. The
`focus is on the use of one input force,
`a technique commonly known as sin-
`gle-point excitation, illustrated in
`Figure 2.2. By understanding this
`technique, it is easy to expand to the
`multiple input technique.
`
`With a dynamic signal analyzer,
`which is a Fourier transform-based
`instrument, many types of excitation
`sources can be implemented to meas-
`ure a structure’s frequency response
`function. In fact, virtually any physi-
`cally realizable signal can be input
`or measured. The selection and
`implementation of the more common
`and useful types of signals for modal
`testing are discussed.
`
`Transducer selection and mounting
`methods for measuring these signals
`along with system calibration meth-
`
`Figure 2.1
`System block
`diagram
`
`Excitation
`
`X(
`
`)
`
`H(
`
`)
`
`Y(
`
`)
`
`Response
`
`Figure 2.2
`Structure
`under test
`
`Structure
`
`Force Transducer
`
`Shaker
`
`ods, are also included. Techniques
`for improving the quality and
`accuracy of measurements are then
`explored. These include processes
`such as averaging, windowing and
`zooming, all of which reduce mea-
`surement errors. Finally, a section
`on measurement interpretation is
`included to aid in understanding the
`complete measurement process.
`
`14
`
`Page 14 of 56
`
`w
`w
`w
`

`

`General Test System Configurations
`
`The basic test setup required for
`making frequency response measure-
`ments depends on a few major
`factors. These include the type of
`structure to be tested and the level
`of results desired. Other factors,
`including the support fixture and
`the excitation mechanism, also affect
`the amount of hardware needed to
`perform the test. Figure 2.3 shows
`a diagram of a basic test system
`configuration.
`
`The heart of the test system is the
`controller, or computer, which is the
`operator’s communication link to the
`analyzer. It can be configured with
`various levels of memory, displays
`and data storage. The modal analysis
`software usually resides here, as well
`as any additional analysis capabilities
`such as structural modification and
`forced response.
`
`The analyzer provides the data
`acquisition and signal processing
`operations. It can be configured with
`several input channels, for force and
`response measurements, and with
`one or more excitation sources for
`driving shakers. Measurement func-
`tions such as windowing, averaging
`and Fast Fourier Transforms (FFT)
`computation are usually processed
`within the analyzer.
`
`Figure 2.3
`General test
`configuration
`
`Controller
`
`Analyzer
`
`Structure
`
`Transducers
`
`Exciter
`
`For making measurements on simple
`structures, the exciter mechanism
`can be as basic as an instrumented
`hammer. This mechanism requires
`a minimum amount of hardware.
`An electrodynamic shaker may be
`needed for exciting more complicated
`structures. This shaker system re-
`quires a signal source, a power ampli-
`fier and an attachment device. The
`signal source, as mentioned earlier,
`may be a component of the analyzer.
`
`Transducers, along with a power
`supply for signal conditioning, are
`used to measure the desired force
`and responses. The piezoelectric
`types, which measure force and
`acceleration, are the most widely
`used for modal testing. The power
`supply for signal conditioning may be
`voltage or charge mode and is some-
`times provided as a component of the
`analyzer, so care should be taken in
`setting up and matching this part of
`the test system.
`
`15
`
`Page 15 of 56
`
`

`

`Figure 2.4a
`Example of
`free support
`situation
`
`Figure 2.4b
`Example of
`constrained
`support
`situation
`
`Free
`Boundary
`
`Constrained
`Boundary
`
`system is used, the rigid body fre-
`quencies will be much lower than the
`frequencies of the flexible modes and
`thus have negligible effect. The rule
`of thumb for free supports is that the
`highest rigid body mode frequency
`must be less than one tenth that of
`the first flexible mode. If this criteri-
`on is met, rigid body modes will have
`negligible effect on flexible modes.
`Figure 2.5 shows a typical frequency
`response measurement of this type
`with nonzero rigid body modes.
`
`The implementation of a constrained
`system is much more difficult to
`achieve in a test environment. To
`begin with, the base to which the
`structure is attached will tend to have
`some motion of its own. Therefore, it
`is not going to be purely grounded.
`Also, the attachment points will have
`some degree of flexibility due to the
`bolted, riveted or welded connec-
`tions. One possible remedy for these
`problems is to measure the frequency
`
`Supporting The Structure
`
`The first step in setting up a
`structure for frequency response
`measurements is to consider the fix-
`turing mechanism necessary to obtain
`the desired constraints (boundary
`conditions). This is a key step in the
`process as it affects the overall struc-
`tural characteristics, particularly for
`subsequent analyses such as structur-
`al modification, finite element corre-
`lation and substructure coupling.
`
`Analytically, boundary conditions
`can be specified in a completely free
`or completely constrained sense. In
`testing practice, however, it is gener-
`ally not possible to fully achieve
`these conditions. The free condition
`means that the structure is, in effect,
`floating in space with no attachments
`to ground and exhibits rigid body
`behavior at zero frequency. The
`airplane shown in Figure 2.4a is an
`example of this free condition.
`Physically, this is not realizable,
`so the structure must be supported
`in some manner. The constrained
`condition implies that the motion,
`(displacements/rotations) is set to
`zero. However, in reality most struc-
`tures exhibit some degree of flexibili-
`ty at the grounded connections. The
`satellite dish in Figure 2.4b is an
`example of this condition.
`
`In order to approximate the free sys-
`tem, the structure can be suspended
`from very soft elastic cords or placed
`on a very soft cushion. By doing this,
`the structure will be constrained to
`a degree and the rigid body modes
`will no longer have zero frequency.
`However, if a sufficiently soft support
`
`16
`
`Page 16 of 56
`
`

`

`Figure 2.5
`Frequency
`response
`of freely
`suspended
`system
`
`FREQ RESP
`50.0
`
`10.0
`/Div
`
`Rigid Body Mode
`
`dB
`
`1st Flexible Mode
`
`-30.0
`FxdXY 0
`
`Hz
`
`1.5625k
`
`response of the base at the attach-
`ment points over the frequency range
`of interest. Then, verify that this
`response is significantly lower than
`the corresponding response of the
`structure, in which case it will have
`a negligible effect. However, the
`frequency response may not be mea-
`surable, but can still influence the
`test results.
`
`There is not a best practical or
`appropriate method for supporting
`a structure for frequency response
`testing. Each situation has its own
`characteristics. From a practical
`standpoint, it would not be feasible
`to support a large factory machine
`weighing several tons in a free test
`state. On the other hand, there may
`be no convenient way to ground a
`very small, lightweight device for the
`constrained test state. A situation
`could occur, with a satellite for exam-
`ple, where the results of both tests
`are desired. The free test is required
`to analyze the satellite’s operating
`environment in space. However, the
`constrained test is also needed to
`assess the launch environment
`attached to the boost vehicle.
`Another reason for choosing the
`appropriate boundary conditions is
`for finite element model correlation
`or substructure coupling analyses. At
`any rate, it is certainly important dur-
`ing this phase of the test to ascertain
`all the conditions in which the results
`may be used.
`
`17
`
`Page 17 of 56
`
`

`

`Exciting the Structure
`
`The next step in the measurement
`process involves selecting an
`excitation function (e.g., random
`noise) along with an excitation sys-
`tem (e.g., a shaker) that best suits the
`application. The choice of excitation
`can make the difference between a
`good measurement and a poor one.
`Excitation selection should be
`approached from both the type of
`function desired and the type of exci-
`tation system available because they
`are interrelated. The excitation func-
`tion is the mathematical signal used
`for the input. The excitation system
`is the physical mechanism used to
`prove the signal. Generally, the
`choice of the excitation function
`dictates the choice of the excitation
`system, a true random or burst
`
`random function requires a shaker
`system for implementation. In gener-
`al, the reverse is also true. Choosing
`a hammer for the excitation system
`dictates an impulsive type excitation
`function.
`
`Excitation functions fall into four
`general categories: steady-state,
`random, periodic and transient.
`There are several papers that go into
`great detail examining the applica-
`tions of the most common excitation
`functions. Table 2.1 summarizes the
`basic characteristics of the ones that
`are most useful for modal testing.
`True random, burst random and
`impulse types are considered in the
`context of this note since they are the
`most widely implemented. The best
`choice of excitation function depends
`on several factors: available signal
`
`processing equipment, characteristics
`of the structure, general measure-
`ment considerations and, of course,
`the excitation system.
`
`A full function dynamic signal analyz-
`er will have a signal source with a
`sufficient number of functions for
`exciting the structure. With lower
`quality analyzers, it may be necessary
`to obtain a signal source as a sepa-
`rate part of the signal processing
`equipment. These sources often
`provide fixed sine and true random
`functions as signals; however, these
`may not be acceptable in applications
`where high levels of accuracy are
`desired. The types of functions
`available have a significant influence
`on measurement quality.
`
`Periodic*
`in analyzer window
`Pseudo
`Random Fast
`random
`sine
`
`Transient
`in analyzer window
`Impact
`Burst
`sine
`
`Burst
`random
`
`True
`random
`
`No
`Fair
`
`Fair
`Good
`
`Yes*
`No
`Yes
`No
`
`Yes
`Fair
`
`Fair
`Very
`good
`Yes*
`Yes*
`No
`No
`
`Yes
`Fair
`
`Fair
`Fair
`
`Yes*
`No
`Yes
`No
`
`Yes
`High
`
`High
`Fair
`
`Yes*
`Yes*
`No
`Yes
`
`Yes
`Low
`
`Low
`Very
`good
`No
`No
`No
`No
`
`Yes
`High
`
`High
`Very
`good
`Yes*
`Yes*
`No
`Yes
`
`Yes
`Fair
`
`Fair
`Very
`good
`Yes*
`No
`Yes
`No
`
`Sine
`steady
`state
`No
`Very
`high
`High
`Very
`long
`Yes
`Yes
`No
`Yes
`
`Minimze leakage
`Signal to noise
`
`RMS to peak ratio
`Test measurement time
`
`Controlled frequency content
`Controlled amplitude content
`Removes distortion
`Characterize nonlinearity
`
`* Requires additional equipment or special hardware
`
`Table 2.1
`Excitation
`functions
`
`18
`
`Page 18 of 56
`
`

`

`cable and measuring the transients.
`Self-operating involves exciting the
`structure through an actual operating
`load. This input cannot be measured
`in many cases, thus limiting its useful-
`ness. Shakers and impactors are the
`most common and are discussed in
`more detail in the following sections.
`Another method of excitation mecha-
`nism classification is to divide them
`into attached and nonattached
`devices. A shaker is an attached
`device, while an impactor is not,
`(although it does make contact for a
`short period of time).
`
`The dynamics of the structure
`are also important in choosing the
`excitation function. The level of
`nonlinearities can be measured and
`characterized effectively with sine
`sweeps or chirps, but a random func-
`tion may be needed to estimate the
`best linearized model of a nonlinear
`system. The amount of damping and
`the density of the modes within the
`structure can also dictate the use of
`specific excitation functions. If
`modes are closely coupled and/or
`lightly damped, an excitation function
`that can be implemented in a leakage-
`free manner (burst random for exam-
`ple) is usually the most appropriate.
`
`Excitation mechanisms fall into four
`categories: shaker, impactor, step
`relaxation and self-operating. Step
`relaxation involves preloading the
`structure with a measured force
`through a cable then releasing the
`
`Shaker Testing
`
`The most useful shakers for modal
`testing are the electromagnetic
`shown in Fig. 2.6 (often called
`electrodynamic) and the electro
`hydraulic (or, hydraulic) types. With
`the electromagnetic shaker, (the more
`common of the two), force is generat-
`ed by an alternating current that
`drives a magnetic coil. The maximum
`frequency limit varies from approxi-
`mately 5 kHz to 20 kHz depending
`on the size; the smaller shakers
`having the higher operating range.
`The maximum force rating is also a
`function of the size of the shaker and
`varies from approximately 2 lbf to
`1000 lbf; the smaller the shaker, the
`lower the force rating.
`
`With hydraulic shakers, force
`is generated through the use of
`hydraulics, which can provide much
`higher force levels – some up to
`several thousand pounds. The maxi-
`mum frequency range is much lower
`though – about 1 kHz and below. An
`advantage of the hydraulic shaker is
`its ability to apply a large static pre-
`load to the structure. This is useful
`for massive structures such as grind-
`ing machines that operate under
`relatively high preloads which may
`alter their structural characteristics.
`
`19
`
`Page 19 of 56
`
`

`

`Figure 2.6
`Electrodynamic
`shaker with
`power amplifier
`and signal source
`
`Figure 2.7
`Mass loading
`from shaker
`setup
`
`There are several potential problem
`areas to consider when using a
`shaker system for excitation. To
`begin with, the shaker is physically
`mounted to the structure via the
`force transducer, thus creating the
`possibility of altering the dynamics
`of the structure. With lightweight
`structures, the mechanism used to
`mount the load cell may add appre-
`ciable mass to the structure. This
`causes the force measured by the
`load cell to be greater than the force
`actually applied to the structure.
`Figure 2.7 describes how this mass
`loading alters the input force. Since
`the extra mass is between the load
`cell and the structure the load cell
`senses this extra mass as part of
`the structure.
`
`Since the frequency response is a
`single input function, the shaker
`should transmit only one component
`of force in line with the main axis of
`the load cell. In practical situations,
`when a structure is displaced along
`a linear axis it also tends to rotate
`about the other two axes. To mini-
`mize the problem of forces being
`applied in other directions, the shaker
`should be connected to the load
`cell through a slender rod, called
`a stinger, to allow the structure to
`move freely in the other directions.
`This rod, shown in Figure 2.8, has
`a strong axial stiffness, but weak
`bending and shear stiffnesses. In
`effect, it acts like a truss member,
`carrying only axial loads but no
`moments or shear loads.
`
`20
`
`Power Amplifier
`
`Structure
`
`Ax
`
`Fs
`
`Loading
`Mass
`
`Load
`Cell
`
`Fm
`
`F1
`
`Fs = Fm - MmAx
`
`Page 20 of 56
`
`

`

`The method of supporting the shaker
`is another factor that can affect the
`force imparted to the structure. The
`main body of the shaker must be
`isolated from the structure to prevent
`any reaction forces from being trans-
`mitted through the base of the shaker
`back to the structure. This can be
`accomplished by mounting the shaker
`on a solid floor and suspending