Use of frequency-domain concepts in the analysis and design of

Transcription

Use of frequency-domain concepts in the
analysis and design of mechanical vibrating
systems
C. W. de Silva
Industrial Automation Laboratory, Department of Mechanical Engineering, University of
British Columbia, Vancouver, Canada V6T 1Z4
E-mail: desilva@mech.ubc.ca
Abstract This paper investigates the frequency-domain approach for some useful types of analysis
and design related to problems of mechanical vibration. First, some attractive features of the
frequency-domain analysis are noted. The possibility of a unified approach to analysis is indicated
through the existence of analogies between mechanical and non-mechanical systems. The uses of
mechanical impedance and transmissibility functions in the modeling, analysis, and design of
mechanical vibrating systems are illustrated. Applications in the design of vibration isolators and in the
analysis of material damping are given to demonstrate the attractive characteristics of some
frequency-domain techniques in the field of mechanical vibration. The manner in which the material
presented in the paper may be incorporated into an undergraduate core course in mechanical vibration
is indicated.
Keywords mechanical impedance; transmissibility; vibration isolation; material damping
Introduction
The time domain and the frequency domain are related through the Fourier transform [1]. There are significant advantages to working in the frequency domain. For
example, linear differential equations in the time domain are transformed into algebraic equations in the frequency domain. Moreover, this requirement for linearity
may not be highly restrictive in many applications of mechanical vibration, for the
analysis will be valid in the neighborhood of an operating point, and a series of such
points may be used to cover a wide operating region [2]. Furthermore, even though
time (t) is the true independent variable in the evolution and response of a dynamic
system, the fact that frequency is the independent variable in the frequency domain
has particular advantages in the field of mechanical vibration. In many vibration
problems, the primary excitation force typically has a repetitive, periodic nature and
in some cases this periodic forcing function may even be purely sinusoidal. Examples are excitations due to mass eccentricity and misalignments in rotational components, tooth meshing in gears, blade passing in turbines and compressors, and
electromagnetic devices excited by alternating-current or periodic electrical signals.
Frequency-domain considerations are applicable even when the signals are not periodic, in view of the fact that a time signal can be transformed into its frequency
spectrum through the Fourier transform. Hence, a time-domain representation and
analysis has an equivalent frequency-domain representation and analysis, at least for
linear dynamic systems.
International Journal of Mechanical Engineering Education 34/3
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C. W. de Silva
Frequency response functions (frequency transfer functions) are used to model
mechanical vibrating systems in the frequency domain. Impedance functions, receptance functions, and transmissibility functions are particularly useful in this context.
In view of the algebraic nature of the analysis in the frequency domain, these
various transfer functions are algebraically related. This paper highlights some
useful analytical procedures in this regard. The related techniques are illustrated
through applications in the design of vibration isolators and in the analysis of material damping.
Models and analogies
Vibration is a dynamic phenomenon and its analysis, utilization, and effective
control require a good understanding of the vibrating system. A practical way to
achieve this is through the use of a suitable model of the system. A mechanical
vibrating system will typically consist of a set of interacting components or elements, each possessing an input–output (or cause–effect, or causal) relationship.
Its model is some form of representation of the system. An analytical model (or
mathematical model) comprises a set of equations or an equivalent form that approximately represents the system. Mathematical definitions for the system are given
with reference to an analytical model of the system. The analytical properties
that are established and results that are derived are associated with the model rather
than the actual system, whereas the excitations are applied to and the output
responses are measured from the actual system. A model may be employed in the
design of a mechanical system to ascertain its vibration performance. In the context
of vibration testing, for example, analytical models are commonly used to develop
test specifications, to determine the input signal applied to the shaker, and to study
dynamic effects and interactions in the test object, the shaker table, and their interfaces. In product qualification by analysis, a suitable analytical model of the product
replaces the test specimen. In vibration control, a dynamic model of the vibrating
system may be employed to develop the necessary control schemes (e.g. modelbased control).
Analytical models may be developed for mechanical, electrical, fluid, and thermal
systems in a similar manner, because there are clear analogies among these four
types of system [3, 4]. In view of such analogies, a unified approach may be appropriate in the analysis, design, and control of these different systems and mixed
systems. In this context, understanding the analogies between different system types
is quite useful in the development and utilization of models.
Frequency-domain models
Any transfer function is defined as the ratio of output to input. If the output and
input are expressed in the frequency domain, the frequency transfer function (or frequency response function) is given by the ratio of the Fourier transforms of the
output to the input. Frequency-domain representations are particularly useful in the
analysis, design, control, and testing of a vibrating system. The waveforms encountered in process signals can be interpreted and represented as a series of sinusoidal
Use of frequency-domain concepts with vibrating systems
Fig. 1
235
Model of a simple oscillator: (a) Bode diagram; (b) Nyquist plot.
components. Indeed, any waveform can be so represented, and sinusoidal excitation
is often used in the testing of equipment and components. It is usually easier to
obtain frequency-domain models than the associated time-domain models by testing.
By definition, the frequency response function (or frequency transfer function),
with respect to frequency, f, is given by:
G( f ) =
Y( f )
U( f )
(1)
with input u(t) and output y(t). This constitutes a complete model for a linear,
constant-parameter vibrating system. It should be clear that, even though G( f ) is
defined in terms of U( f ) and Y( f ), it is a system model and is independent of the
input (and hence the output): that is, for a physically realizable linear constant-parameter system, G( f ) exists even if U( f ) and Y( f ) do not exist for a particular input.
A frequency-domain model (for the case of one input and one output) may be presented either as a pair of curves, |G( f )| and ∠G( f )versus f (this is the Bode plot),
or on the complex G( f ) plane with the real part plotted on the horizontal axis and
the imaginary part on the vertical axis (this is the Nyquist diagram or argand plot
or polar plot). The shape of these plots for a simple oscillator is shown in Fig. 1.
Analogies
Analogies can be drawn between mechanical, electrical, hydraulic, and thermal
systems. The basic system elements can be divided into two groups: energy storage
elements and energy dissipation elements. The analogy shown in Table 1 between
mechanical and electrical systems is known as the force–current analogy [5]. This
analogy appears more logical than a force–voltage analogy. This follows from the
fact that both force and current are through variables, which are analogous to fluid
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C. W. de Silva
TABLE 1
System type
System-response variables:
Through variables
Across variables
System parameters
Force–current analogy
Mechanical
Electrical
Force, f
Velocity, v
m
k
b
Current, i
Voltage, v
C
1/L
1/R
flow through a pipe, and both velocity and voltage are across variables, which vary
across the flow direction, as in the case of pressure. Note that rotational mechanical
elements possess constitutive relations between torque and angular velocity, which
can be treated as a generalized force and a generalized velocity, respectively.
Proper selection of system variables is crucial in developing an analytical model
for a vibrating system. A general approach may be to use the across variables of Atype (or, across type) energy storage elements and the through variables of T-type
(or, through type) energy storage elements as the system variables (state variables).
Note that if any two elements are not independent (e.g. if two spring elements are
directly connected in series or parallel) then only a single state variable should be
used to represent them both. Independent variables are not needed for D-type (dissipative) elements because their response can be represented in terms of the state
variables of the energy storage elements (A-type and T-type).
For inertia elements, the following statements hold true:
(1) Velocity can represent the state of an inertia element. This is justified because
the velocity, v, at any time, t, can be completely determined with knowledge
of the initial velocity and the applied force, and also because the energy of an
inertia element (kinetic energy) can be represented in terms of v alone.
(2) Velocity across an inertia element cannot change instantaneously unless an infinite force/torque is applied to it.
(3) A finite force cannot cause infinite acceleration. A finite instantaneous change
(step) in velocity will need an infinite force. Hence, v is a natural output (or
state) variable and f is a natural input variable for an inertia element.
For stiffness elements the following statements hold true:
(1) Force, f, can represent the state of a stiffness (spring) element. This is justified
because the force of a spring at any general time, t, may be completely determined with knowledge of the initial force and the applied velocity, and also
because the energy of a spring element (elastic potential energy) can be represented in terms of f alone.
(2) Force through a stiffness element cannot change instantaneously unless an infinite velocity is applied to it.
(3) Force, f, is a natural output (state) variable and velocity, v, is a natural input
variable for a stiffness element.
TABLE 2
237
Definitions of useful mechanical transfer functions
Transfer function
Definition (in the frequency domain)
Dynamic stiffness
Compliance, receptance, or dynamic Flexibility
Impedance (Z)
Mobility (M)
Dynamic inertia
Accelerance
Force transmissibility (Tf)
Motion transmissibility (Tm)
Force/displacement
Displacement/force
Force/velocity
Velocity/force
Force/acceleration
Acceleration/force
Transmitted force/applied force
Transmitted velocity/applied velocity
Mechanical impedance approach
Any type of force or motion variable may be used as an input or output variable in
defining a system transfer function. In vibration studies, through variables (force)
and across variables (velocity) are used in defining the two important frequency
transfer functions: mechanical impedance and mobility. In the case of an impedance
function (Z), velocity is considered the input variable and force the output variable,
whereas in the case of a mobility function (M) the converse applies. Specifically,
M=
1
Z
(2)
One can define several other versions of the frequency transfer function that might
be useful in the modeling and analysis of mechanical vibrating systems [6]. Some
of the more common ones are listed in Table 2. Note that, in the frequency domain,
since acceleration = jw × velocity and displacement = velocity/(jw), the alternative
types of transfer function, as defined in Table 2, are related to mechanical impedance and mobility through the complex frequency factor jw. Specifically:
Dynamic inertia =
Accelerance
=
Dynamic stiffness =
Receptance
=
Force/acceleration =
Acceleration/force =
Force/displacement =
Displacement/force =
Impedance/(jw)
Mobility × jw
Impedance × jw
Mobility/(jw)
The time-domain constitutive relations for the mass, spring and the damper elements are well known. The corresponding transfer relations are obtained by replacing the derivative operator, d/dt, by the Laplace operator, s. The frequency transfer
functions are obtained by substituting jw or j2pf for s. These results are summarized
in Table 3.
Interconnection laws
Any general impedance element or a mobility element may be interpreted as a twoport element in which, under steady conditions, energy (or power) transfer into the
device takes place at the input port and energy (or power) transfer out of the device
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C. W. de Silva
TABLE 3
Impedance and mobility functions of basic mechanical elements
Frequency transfer function (set s = jw = j2pf )
Element
Impedance
Mass, m
Zm = ms
Mobility
1
ms
s
Mk =
k
1
Mb =
b
Mm =
k
s
Spring, k
Zk =
Damper, b
Zb = b
Receptance
1
ms 2
1
Rk =
k
1
Rb =
bs
Rm =
takes place at the output port. Each port of a two-port element has associated with
it a through variable, such as force or current, and an across variable, such as velocity or voltage. Through variables are called flux variables, and across variables are
called potential variables. Through variables are not always the same as flow variables (velocity and current). Similarly, across variables are not always the same as
effort variables (force and voltage). For example, force is an effort variable, but it
is also a through variable. Similarly, velocity is a flow variable and is also an across
variable.
The concept of effort and flow variables is useful in giving unified definitions for
electrical mechanical impedance. But in component connection and circuit analysis,
mechanical impedance is not analogous to electrical impedance. The definition of
mechanical impedance is force/velocity, which is a ratio of (through variable)/
(across variable), whereas electrical impedance, defined as voltage/current in the frequency domain, is a ratio of (across variable)/(through variable). Since both force
and voltage are ‘effort’ variables, and velocity current are ‘flow’ variables, it is convenient to use the definition
Impedance (electrical or mechanical) =
Effort
Flow
(3)
This definition does not particularly help us in analyzing interconnected systems
with mechanical impedance, because mechanical impedance cannot be manipulated
using the rules for electrical impedance. For example, if two electric components
are connected in series, the current (through variable) will be the same for both components, and the voltage (across variable) will be additive. Accordingly, impedance
of a series-connected electrical system is just the sum of the impedances of the individual components. Now consider two mechanical components connected in series.
Here the force (a through variable) will be the same for both components, and velocity (an across variable) will be additive. Hence, it is the mobility, not the impedance,
that is additive in the case of series-connected mechanical components. It can be
concluded that, in circuit analysis, mobility behaves like electrical impedance and
mechanical impedance behaves like electrical admittance. Hence, the ‘generalized
Series Connections
239
Parallel Connections
f1
f
Z1
Z2
M1
v1
M1
f2
M2
Z2
v2
v
Fig. 2
Z1
f
v
M2
v = v1 + v2
f = f 1 + f2
v v1 v2
= +
f
f
f
f
f
f
= 1= 2
v v
v
M = M1 + M2
Z = Z1 + Z2
1
1 1
= +
Z Z1 Z2
1
1 1
= +
M M1 M2
Interconnection laws for impedance and mobility.
series element’ is electrical impedance or mechanical mobility, and the ‘generalized
parallel element’ is electrical admittance or mechanical impedance. The corresponding interconnection laws are summarized in Fig. 2.
Note that it is more convenient to use impedance for a parallel combination and
mobility for a series combination. It should be stated here that the method of receptance analysis [7] is a straightforward variation of the impedance analysis. As a
result, a separate treatment of the topic is not necessary here.
A reciprocity property
To see an interesting reciprocity property, consider the system shown in Fig. 3(a).
In this example the motion of the mass, m, is not associated with an external force.
The support motion, however, is associated with a force, f. The conventional representation, shown in Fig. 3(b), and the corresponding mechanical circuit, shown in
Fig. 3(c), indicate that the spring and damper are connected in parallel and the mass
is connected in series with this pair. By impedance addition for parallel elements,
and mobility addition for series elements, it follows that the mobility function is:
V( f )
ms 2 + bs + k
1
1
1
= Mm +
=
+
=
F( f )
( Zk + Zb ) ms (k s + b) s= j 2 pf ms(bs + k )
(4)
s = j 2 pf
It is noted that when force is the input (i.e. a force source) and the support velocity
is the output, the system characteristic polynomial is ms(bs + k), which is known to
be inherently unstable due to the presence of a free integrator, and has a nonoscillatory transient response.
The impedance function that corresponds to support velocity input (velocity
source) is the reciprocal of the previous mobility function:
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C. W. de Silva
Fig. 3 (a) An oscillatory system in which the mass is not associated with an external
force. (b) Mechanical circuit representation. (c) Impedance representation.
F( f )
ms(bs + k )
=
V ( f ) ms 2 + bs + k
s = j 2p f
and, furthermore,
V1 ( f )
1
=
F( f ) ms s= j 2 p f
The impedance function, F( f )/V1( f ), is not admissible and is physically unrealizable because V1 cannot be an input, for there is no associated force. This is
confirmed by the fact that the corresponding transfer function is a differentiator.
The mobility funtion, V1( f )/F( f ), corresponds to a simple integrator. Physically,
when a force, f, is applied to the support it transmits to the mass, unchanged, through
the parallel spring–damper unit. Accordingly, when f is constant, a constant acceleration is produced at the mass, causing its velocity to increase linearly (an integration behavior). Maxwell’s principle of reciprocity is demonstrated by noting
that the support motion produced by applying a forcing excitation to the mass is
equal to the motion of the mass when the same forcing excitation is applied to the
support, with the same initial conditions. This reciprocity property is valid for
linear, constant-parameter systems in general, and is particularly useful in vibration
testing of multi-degree-of-freedom systems, to determine a transfer function that is
difficult to measure, by measuring its symmetrical counterpart in the transfer function matrix.
Transmissibility function method
Transmissibility functions are transfer functions that are particularly useful in the
analysis of vibration isolation in machinery and other mechanical systems. Two
Fig. 4
Force transmissibility mechanism.
Fig. 5
Motion transmissibility mechanism.
241
types of transmissibility function – force transmissibility and motion transmissibility – can be defined, as given in Table 2. Due to a reciprocity characteristic in linear
systems, it can be shown that these two transfer functions are equal and, consequently, it is sufficient to consider only one of them.
A schematic representation of a force transmissibility mechanism is shown in
Fig. 4. The reason why the suspension force, fs, is not equal to the applied force,
f, is attributed to the inertia paths (broken line in Fig. 4) that are present in the
mechanical system.
A schematic representation of the motion transmissibility mechanism is shown in
Fig. 5. Typically, the motion of the system is taken as the velocity of one of its critical masses. Different transmissibility functions are obtained when different mass
points (or degrees of freedom) of the system are considered.
Example
Consider the two-degree-of-freedom systems shown in Fig. 6. The main system is
represented by two masses linked through a spring and a damper. Mass m1 is conInternational Journal of Mechanical Engineering Education 34/3
242
C. W. de Silva
sidered the critical mass (it is equally acceptable to consider mass m2 as the critical
mass). To determine the force transmissibility using Fig. 6(a) and associated
Fig. 7(a), note that the applied force is divided in the ratio of the impedances between
the two parallel branches. The mobility of the main right-hand branch is:
M=
1
1
+
Zs1 Zm 2 + Zs
and the force through that branch is:
1


1


 M 
F=
F′ = 
F
1 
MZ
1
+
m1


 Zm1 + 

M
The force, Fs, through Zs is given by:
 Zs 
Fs = 
F′
 Zm 2 + Zs 
Consequently, the force transmissibility is given by:
Tf =
Fs 
1
  Zs 
=
F  MZm1 + 1   Zm 2 + Zs 
(5)
where M is as expressed above. To determine the motion transmissibility using
Fig. 6(b) and the associated Fig. 7(b), note that the velocity is distributed in proportion to the mobilities among the series elements. The impedance of the second
composite series unit is:
Fig. 6
Systems with two degrees of freedom: (a) force transmissibility; (b) motion
transmissibility.
Fig. 7
Z=
243
Mechanical impedance representations of the systems in Fig. 6.
1
1
+
Mm 2 Ms1 + Mm1
and the velocity across this unit is:
 1 


 1 
V ′ =  Z V = 
V
1
 Ms Z + 1 
 Ms + 

Z
The velocity, Vm, of mass m1 is given by:
 Mm1 
Vm = 
V ′
 Ms1 + Mm1 
As a result, the motion transmissibility can be expressed as:
Tm =
Vm  1   Mm1 
=
V  Ms Z + 1   Ms1 + Mm1 
(6)
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C. W. de Silva
It remains to show that Tm = Tf. To this end, let us examine the expression for Tm.
Since Zs = 1/Ms, Tm can be written as:
 Zs   Mm1 
Tm = 


 Z + Zs   Ms1 + Mm1 
Note that:
Z=
1
+ Zm 2
Ms1 + Mm1
Hence,





  Mm1  
  Zs 
Zs
Mm1
Tm = 
=





1
1

+ Zm 2 + Zs   Ms1 + Mm1  
+ Ms1 + Mm1   Zm 2 + Zs 
 Ms1 + Mm1

 Zm 2 + Zs



=
 1
M
 m1

 Z

s

1

   Zm 2 + Zs 
 Z + Z + Ms1  + 1 
s
 m2
 


=
 1
Z
 m1

 Z
1

s

1
1    Zm 2 + Zs 

 Z + Z + Z  + 1
s
s1 
 m2

1
(6)*
which is clearly identical to Tf in equation 5. This equivalence of Tf and Tm can be
shown in a similar straightforward manner for systems of higher degrees of freedom
as well.
Resonance considerations
A peak point of a frequency transfer function is termed a resonance. The frequency
at this point is called a resonant frequency. Since a transfer function is the ratio of
a response variable to an excitation variable, it is reasonable to get different peak
frequencies for the same excitation input if the response variable considered is
different. Some results obtained for a damped oscillator model are summarized in
Table 4. The normalized frequency variable is r = w/wn, where w is the excitation
frequency and wn is the undamped natural frequency. The damping ratio is denoted
by z. It is interesting to note that for the same system (simple oscillator) different
resonance values are obtained, depending on the excitation and response variables
that are considered.
Fig. 8
TABLE 4
245
Vibration isolation: (a) force isolation; (b) motion isolation.
Some practical frequency response functions and their resonance points
System
Response/
excitation
Frequency
response function
(normalized)
Normalized
resonant
frequency (rp)
1
Simple oscillator
Displacement/force
(1 − r 2 ) + 2 jzr
Simple oscillator
with velocity
response
Velocity/force
(1 − r 2 ) + 2 jzr
Simple oscillator
with acceleration
response
Acceleration/force
Transmissibility
Force/force or
velocity/velocity
jr
−r 2
(1 − r 2 ) + 2 jzr
1 + 2 jr
(1 − r 2 ) + 2 jzr
Peak magnitude
(normalized)
1 − 2z 2
1
2z 1 − z 2
1
2z
1
1 − 2z 2
1
2z 1 − z 2
1
1 + 8z 2 − 1
2z
1
4z 2
(for small z)
1+
Application in vibration isolation
Vibration isolation involves the design (or selection) of a suitable device to ‘isolate’
the system of interest from vibration excitations. Machine mounts and vehicle suspension systems fall into this category. The former concern force transmissibility
and the latter concern motion transmissibility. The design problem in either case is
to select parameters for the isolator so that the vibrations transmitted to the system
are below specified values within the operating frequency range.
Fig. 8(a) gives a schematic model of force transmissibility through an isolator.
Vibration force at the source is f(t). In view of the isolator, the source system (with
impedance Zm) is made to move at the same speed as the isolator (with impedance
Zs) – a parallel connection of impedances. Hence, the force, f(t), is split so that part
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C. W. de Silva
Fig. 9
(a) Mechanical circuit of Fig. 8(a); (b) mechanical circuit of Fig. 8(b).
of it is taken up by the inertial path (indicated by a broken line) of Zm and only the
remainder (fs) is transmitted through Zs to the supporting structure, which is the
isolated system. It is clear that the force transmissibility is:
Tf =
fs
Zs
=
f
Zm + Zs
(7)
Fig. 8(b) gives a schematic model of motion transmissibility through an isolator.
Vibration motion, v(t), of the source is applied through an isolator (with impedance
Zs and mobility Ms) to the isolated system (with impedance Zm and mobility Mm).
The resulting force is assumed to be transmitted directly from the isolator to the
isolated system and, hence, these two units are connected in series. Consequently,
we have the motion transmissibility:
Tm =
vm
Mm
Zs
=
=
v
Mm + Ms Zs + Z m
(8)
As expected, it is noticed that Tf = Tm = T. It may be easily verified that this transmissibility function is given by:
T=
k + bjw
(k − mw 2 + bjw )
(9)
where w is the frequency of vibration excitation. The associated design problem is
to select the isolator parameters, k and b, to meet the specifications of isolation.
Equation 9 may be written in the form:
T=
w n2 + 2zw nw j
(w − w 2 + 2zw nw j )
2
n
where:
w n = k m = undamped natural frequency of the system
(10)
z=
247
b
= damping ratio of the system.
2 km
Equation 10 may be further converted into the non-dimensional form:
T=
1 + 2zrj
1 − r 2 + 2zrj
(11)
where the non-dimensional excitation frequency is defined as r = w/wn.
The transmissibility function is complex, and has a phase angle as well as a magnitude. In practical applications, however, it is the level of attenuation of the vibration excitation that is of primary importance, rather than the phase difference
between the vibration excitation and the response. Accordingly, it is the transmissibility magnitude that is of interest. It is:
T =
(r
1 + 4z 2 r 2
2
(12)
− 1) + 4z 2 r 2
2
From this expression the following facts can be established:
(1) There is always a non-zero frequency value at which the transmissibility magnitude will peak. This is the resonance.
(2) For small z, the peak transmissibility magnitude is obtained at approximately
r = 1. As z increases, this peak point shifts to the left (i.e. giving a lower value
for the peak frequency).
(3) The peak magnitude decreases as z increases.
(4) All the transmissibility curves pass through the magnitude value 1.0 at the same
frequency ratio, r = 2.
(5) The isolation (i.e. |T| < 1) is given by r > 2. In this region, |T| increases with
z.
(6) The transmissibility magnitude decreases as r increases.
Since r > 2 corresponds to the isolation region, a vibration isolator should be
designed such that the operating frequencies, w, are greater than 2w n . Furthermore,
a threshold value for |T| would be specified, and the parameters k and b of the isolator should be chosen so that |T| is less than the specified threshold in the operating frequency range.
In design problems, what is normally specified is the percentage isolation, I:
I = [1 − T ] × 100%
(13)
For the result in equation 12, this corresponds to:

I = 1 −


 × 100%
(r − 1) + 4z r 
1 + 4z 2 r 2
2
2
(14)
2 2
The isolation curves given by equation 14 are plotted in Fig. 10. These curves are
useful in the design of vibration isolators, as illustrated in the next example.
248
C. W. de Silva
Fig. 10 Curves of vibration isolation.
Example
Consider a milling machine weighing 1000 kg that is rigidly mounted on a concrete
floor. The frequency range of normal operation is 10–60 Hz. It was found (using
load-cell measurements) that the worst-case amplitude of the vibration force transmitted to the floor was 2000 N and the vibrations were nearly sinusoidal. Largeamplitude vibratory motions were noticed during start-up and shut-down procedures.
To reduce floor vibrations that affect adjoining operations and offices, and to maintain machining accuracy during normal operation, vibration isolation was found to
be required. The following specifications are given:
(1) amplitude of vibratory motion at resonance = 1.0 cm or less
(2) level of vibration isolation under normal operation = 80% (approximately)
(3) amplitude of vibratory motion under normal operation = 1.0 mm or less
Let us design a mounting system to achieve these specifications. A schematic
representation of the system is shown in Fig. 11.
We use the force–displacement relation of the model mÿ + b y˙ + ky = f(t), where
the displacement, y, is measured from the static equilibrium position. The amplitude
ratio displacement/force (in the frequency domain) is:
y0
1
1
=
=
2
2
f0 k − w m + bjw k 1 − r + 2 jzr
(15)
Since the resonance is at r ≅ 1.0, for low damping, the corresponding displacement
amplitude is (from equation 15):
Fig. 11
y0 =
249
A milling machine with flexible mounts for vibration isolation.
f0
2 kz
(16)
Using straightforward algebraic manipulation of equation 15, we obtain:
r02 2 y0 mw 02
=
z
f0
(17)
where w 0 is the operating frequency. According to the previous observations, the
vibration isolator has to be designed for the lowest frequency of operation. Accordingly, w 0 = 2p × 10 rad/s. Substituting this and the other parameters into the
r2
right-hand side of equation 17, we get, approximately, 0 = 40. Now, using the
z
isolation curves in Fig. 10 for I = 80%, it can be established that this condition is
satisfied at z = 0.3 where r02 = 3.5. The corresponding value of the mount stiffness
is:
k=
(10 × 2p )2 × 1000
3.5
N m = 11.3 × 10 5 N m
The damping constant of the mount is given by:
b = 2zw n m = 2z km
= 2 × 0.3 11.3 × 10 5 × 1000 = 20 × 10 3 N m s
The operating displacement amplitude (worst case is at the lowest frequency point
of operation) is given by:
250
C. W. de Silva
y0 op =
f0
2000
=
m
5
k 1 − r + 2 jzrop 11.3 × 10 1 − 3.5 + 2 j × 0.3 × 3.5
2
op
= 0.63 × 10 −3 m = 0.63 mm
which is less than 1.0 mm and is acceptable.
Application in damping analysis
Internal damping of materials is crucial in the vibration behavior of devices made
of these materials. Two general types of internal damping can be identified: viscoelastic damping and hysteretic damping [8].
Visco-elastic damping
For a linear visco-elastic material, the Kelvin–Voigt (KV) model for the stress–strain
relationship is given by [8]:
s = Ee + E *
de
dt
(18)
Here, E is the Young’s modulus and E* is a visco-elastic parameter that is assumed
to be time independent. The elastic term, Ee, does not contribute to damping, and
its cyclic integral vanishes.
Two other models of visco-elastic damping are also commonly used. They are the
Maxwell model, given by [8]:
s + cs
ds
de
= E*
dt
dt
(19)
and the standard linear solid (SLS) model, given by [8]:
s + cs
ds
de
= Ee + E *
dt
dt
(20)
It is clear that the SLS model represents a combination of the KV model and the
Maxwell model, and is the most accurate of the three. But, for most practical purposes, the KV model is adequate.
Hysteretic damping
The stress and hence the internal damping force of a visco-elastic damping material depends on the frequency of variation of the strain (and, consequently, the frequency of motion). It has been observed through experiments that for some types
of material the damping force does not significantly depend on the frequency of
oscillation of strain (or frequency of harmonic motion). This type of internal
damping is known as hysteretic damping. A model that satisfies this characteristic
is given by:
s = Ee +
E˜ de
w dt
(21)
251
which is equivalent to using a visco-elastic parameter, E*, that depends on the frequency of motion in equation 18, according to E* = E˜ /w. Consider the case of harmonic motion at frequency w, with the material strain given by the usual complex
form according to:
e = e max e jw t
(22)
equation 21 becomes:
(
)
s = E + jE˜ e
(23)
It follows that this form of simplified hysteretic damping may be represented by
using a complex modulus of elasticity, consisting of a real part, which corresponds
to the usual linear elastic (energy storage) modulus (or Young’s modulus), and an
imaginary part, which corresponds to the hysteric loss (energy dissipation) modulus.
By combining equations 18 and 21, a simple model for combined visco-elastic
and hysteric damping may be given by:

E˜  de
s = Ee +  E * + 
w  dt

(24)
Loss factor
Several parameters are commonly used to represent damping in a mechanical
system. They are summarized here. The damping capacity of a device (damper) is
the energy dissipation in a complete cycle of motion. Specifically,
∆U = ∫ fd dx
(25)
This is given by the area of the hysteresis loop in the displacement–force plane. If
the initial (total) energy of the system is denoted by Umax, the specific damping
capacity, D, is given by the ratio:
D=
∆U
U max
(26)
The loss factor, h, is the specific damping capacity per radian of the damping cycle.
Hence:
h=
∆U
D
=
2pU max 2p
(27)
Note that Umax is approximately equal to the maximum kinetic energy and also to
the maximum potential energy of the device.
Damping capacity per unit volume of a material is given by:
∆u = ∫ sde
(28)
Also, the maximum elastic potential energy per unit volume is given by:
umax = 1 2 s max e max
252
C. W. de Silva
or, in view of the relation smax = Eemax for an elastic material:
2
umax = 1 2 e max
(29)
where emax is the maximum strain in a load cycle. The specific damping capacity can
be expressed using these per-unit-volume definitions as:
∆u
umax
D=
(30)
and in view of equation 27, the loss factor may be expressed as:
h=
∆u
2pumax
(31)
The loss factor is approximately related to the damping ratio by:
h = 2z
(32)
For general types of damping, equation 32 holds, assuming that the equivalent
damping ratio zeq is used in place of z.
Example
For the KV model, given by equation 18, since the elastic term does not contribute
to the cyclic integral in equation 28, we can write:
∆uKV = E * ∫
de
de
dt
(33)
For cyclic motion, we substitute equation 22 into 33 and get:
2
∆uKV = pwE * e max
(34)
Hence, by substituting equations 34 and 29 into 31:
h KV =
2
pwE * e max
1 2
2p × Ee max
2
We have for the KV model:
h KV =
wE *
E
(35)
For the hysteretic KV model we simply substitute E˜ /e for E* to get:
hH =
E˜
E
(36)
For the SLS model given by equation 20, in the frequency domain we simply substitute jw for d/dt:
(1 + jw cs )s = ( E + jwE*)e
253
or,
s=
( E + jwE*)
e
(1 + jw cs )
We multiply the numerator and the denominator of this result by (1 − jwcs) and simplify to obtain:
s=
[( E + w
2
cs E *) + jw ( E * − cs E )]
(1 + w 2 cs )
e
(37)
Compare this with the frequency-domain KV model:
s = ( E + jwE*)e
(38)
It follows that the loss factor for the visco-elastic SLS model may be obtained by
( E* −cs E )
substituting
for E* in equation 35, thus:
(1 + w 2 cs )
hSLS−V =
wE * (1 − cs E E *)
E (1 + w 2 cs )
(39)
Similarly, we get the result for the hysteretic SLS model by substituting
( E˜ w − c E)
s
(1 + w cs )
2
for E* in equation 35, thus:
hSLS− H =
(
˜
E˜ 1 − w cs E E
E (1 + w 2 cs )
)
(40)
These results are summarized in Table 5.
Incorporation in an undergraduate curriculum
A core course in mechanical vibration, in a typical first-degree program in engineering, will cover the following topics [6]:
TABLE 5
Loss factors for various material damping models
Material damping model
Stress–Strain constitutive relation
de
dt
Visco-elastic Kelvin–Voigt
s = Ee + E *
Hysteretic Kelvin–Voigt
s = Ee +
Visco-elastic standard linear solid
s + cs
ds
de
= Ee + E *
dt
dt
Hysteretic standard linear solid
s + cs
E˜ de
ds
= Ee +
w dt
dt
E˜ de
w dt
Loss factor (h)
w E*
E
E˜
E
w E * (1 − cs E E *)
E
(1 + w 2 cs )
E˜ 1 − w cs E E˜
(
E
(1 + w 2 cs )
)
254
Week 1:
Week 2:
Week 3:
Week 4:
Week 5:
Week 6:
Week 7:
Week 8:
Weeks 9 and 10:
Week 11:
Week 12:
C. W. de Silva
Introduction; Vibration in practice
Vibration instrumentation
Time response, free
Time response, forced
Frequency response, general
Frequency response, impedance approach
Multi-degree-of-freedom systems
Modal analysis
Vibration design and control: vibration isolation
Distributed-parameter systems
Energy dissipation and damping
Each week will have three hours of lectures, with a laboratory session and a tutorial session where problems and illustrative examples are discussed and solved. The
material under ‘Models and analogies’, ‘Mechanical impedance approach’ and
‘Transmissibility function method’ of the present paper may be covered in weeks 4
and 5 of the course. The material under ‘Application in vibration isolation’ is appropriate for weeks 9 and 10. The material under ‘Application in damping analysis’
may be included in week 12.
Conclusion
This paper investigated the frequency-domain approach for several types of analysis and design related to problems of mechanical vibration. First, some attractive
features of the frequency domain were noted. The possibility of a unified approach
to analysis in the frequency domain was indicated through the existence of analogies in mechanical and non-mechanical systems. The use of mechanical impedance
and transmissibility functions in the modeling, analysis, and design of mechanical
vibrating systems was illustrated. Applications in the design of vibration isolators
and in the analysis of material damping were given to demonstrate the appealing
characteristics of the frequency-domain techniques in the field of mechanical vibration. The way in which the material presented in the paper might be incorporated
into an undergraduate core course in mechanical vibration was indicated.
Acknowledgement
This work has been supported by a research grant from the National Science and
Engineering Research Council (NSERC) of Canada.
References
[1] J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (WileyInterscience, New York, 1971).
[2] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Willey-Interscience, Now York, 1979).
[3] S. H. Crandall, D. C. Karnopp, E. F. Kurtz and D. C. Pridmore-Brown, Dynamics of Mechanical and
Electromechanical Systems (McGraw-Hill, New York, 1968).
255
[4] J. L. Shearer, A. T. Murphy and H. H. Richardson, Introduction to System Dynamics (AddisonWesley, Reading, MA, 1971).
[5] C. W. de Silva, Control Sensors and Actuators (Prentice Hall, Englewood Cliffs, NJ, 1989).
[6] C. W. de Silva, Vibration: Fundamentals and Practice, 2nd ed. (CRC Press, Taylor & Francis, Boca
Raton, FL, 2007).
[7] R. E. D. Bishop and D. C. Johnson, The Mechanics of Vibration (Cambridge University Press,
Cambridge, 1960).
[8] B. J. Lazan, Damping of Materials and Members in Structural Mechanics (Pergamon, London, 1968).

Use of frequency-domain concepts in the analysis and design of

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