Proceedings of the Institution of

Civil Engineers
Structures & Buildings 146
November 2001 Issue 4
Pages 391^402

Paper 12467
Received 25/09/2000
Accepted 26/04/2001
V. A. Ivovich M. K. Savovich
Keywords:
Professor of Structural Director of Project,
dynamics/research &
Dynamics, Department of Structural Performance and
development/design methods
Structural Dynamics, Central Technology Division,
& aids
Research Institute for Austru Bau Company,
Building Structures, Moscow Moscow

Isolation of floor machines by lever-type inertial vibration

corrector
V. A. Ivovich and M. K. Savovich

This paper proposes and examines a lever-type vibration o excitation frequency (circular)
isolation system that is used to reduce transmitted om frequency of isolator (circular)
oscillations from machines onto their floor support. os frequency of structure (circular)
This is a system with a lever mechanism for motion c1 coefficient of velocity damping of structure
transformation and an auxiliary mass corrector. The c2 coefficient of velocity damping of isolator
substance of this mechanism is that the corrector fm frequency of isolator
significantly reduces oscillations from low frequency fs frequency of structure
harmonic excitation by virtue of its additional inertial k stiffness of single-mass oscillator
force. It may be used in conjunction with the classical k1 stiffness of structure
passive isolation system when greater performance is k2 stiffness of isolator
required and when a conventional technique cannot L1 L2 length of lever arms
perform adequately. With the addition of a corrector, m mass of single-mass oscillator
the low frequency performance is improved overall. m1 mass of structure
Some analytical results for two-degree-of-freedom m2 mass of machine
(2DOF) machine-floor systems have been presented. M0 equivalent mass of corrector
Thereafter, the numerical analysis of the dynamic M1 equivalent mass of structure
response of this system is carried out using a state M2 equivalent mass of machine
space formulation approach. Extensive parametric pm disturbing force amplitude
studies have shown the effects of various levels of base pT reaction force amplitude
flexibility, viscous damping and mass ratio on the r frequency ratio
efficiency of this isolation. The results obtained are r0 coefficient
useful for the design of lever-type isolators. In fact, the TR transmissibility
highlight of this paper is the design aspect of this novel
isolator. Numerical examples of the steady-state
1. INTRODUCTION
response and the time response of two isolation systems
This research concentrates on medium-sized and large
(with and without the presence of the lever mechanism)
machinery that is placed on the floors of buildings thereby
for floor-supported industrial fans are presented. These
creating a potential source of unwanted machine-induced
examples illustrate the application of the formulation
vibrations. The harmful effects of these vibrations may cause
and the theory. Finally, the proposed isolator has been
serious impairment to the efficiency of workers and the overall
validated by dynamic testing and good agreement
effectiveness of the production machinery. Recently, problems
between theoretical and experimental results has been
of annoying vibrations have increased with the development of
obtained.
floors of lighter construction, longer span and less inherent
damping. In some cases, vibrations may impair production
NOTATION processes. For example, the manufacture of integrated circuits
a coefficient is a very delicate process that can be seriously interrupted by
b lever arm ratio of single-mass oscillator vibrations. Besides the annoyance to the occupants and the
D displacement amplitude impairment of the production process, vibration may also cause
Da displacement amplitude—allowable damage to the floors and the building. In order to suppress
d static deflection machine-induced floor vibrations one alternative is to use
z damping of single-mass oscillator vibration isolation of the source machine. There are two general
zm damping of isolator types of vibration isolation systems that are currently used:
zs damping of structure active and passive isolation. Each type of isolation system
u mass ratio of single-mass oscillator has both benefits and drawbacks. Passive systems tend to be
r mass ratio uncomplicated, easy to implement and of low cost, with zero

Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich 391
 power consumption. In contrast, active systems provide damping. The analytical and experimental results are compared
enhanced capabilities and avoid some limitations of the passive and recommendations are made concerning the acceptability of
systems. Both types of isolation system are generally specified the lever-type isolation concept and possible applications.
by low pass corner frequency and damping ratio. Passive
systems provide isolation through fixed stiffness and This paper is organised as follows: In Section 2 a comparison
damping, and without power consumption. Active systems between conventional passive isolation and isolation using
use a controller to regulate the vibration so they consume a lever-type mechanism is presented. Section 3 presents
power. Active systems can adapt to errors or changes in the modelling and analysis of a 2DOF model using transfer
disturbances and to the response of the isolated machine, while functions and state space formulation. Isolator design and
passive systems cannot. Active systems are also prone to loss experimental results are presented in Section 4 and Section 5
of stability, especially if a loss of power occurs. While passive provides conclusions and discussion on the performance of the
isolation systems use a variety of implementation devices, proposed isolation concept.
including springs, viscoelastic materials, air mounts, sheets of
neoprene, hydraulic dampers as well as combinations of 2. BASIC PRINCIPLES: LEVER-TYPE CORRECTOR
these.1, 2 AGAINST CONVENTIONAL PASSIVE ISOLATION
The simplest model of lever-type isolation is the SDOF system
There have been thousands of publications that discuss the shown in Fig. 1, where w(t) is a base excitation and p(t) is a
use of vibration isolation to mitigate unwanted vibrations. force excitation. The machine and base (floor) are assumed
However, there is little published work on vibration isolation to be rigid and the base mass is considered negligible with
using a lever mechanism. Early investigations3 pointed out respect to the machine mass. For design purposes, the isolator
that lever-type isolation offered only a limited advantage has no mass and is modelled as a spring with stiffness k and a
over conventional passive systems. However, subsequent viscous damper with damping coefficient c. The spring stiffness
investigations4, 5, 6, 7 have shown conclusively that substantial and damping coefficient are assumed to be constant in the
advantages are offered by this isolation if its inherent frequency range of interest. The corrector shown in Fig. 1 is an
capability of focusing on minimising the transmissibility auxiliary mass m0 attached to the machine mass m by some
function at a low frequency excitation are exploited. It has massless lever mechanism with lever arms (L1, L2). The values
been stated2, 8 that the single-degree-of-freedom (SDOF) model for the isolator’s spring, corrector’s mass and lever arms are
is not the ideal design model for vibration isolation using a selected in such a way as to reduce as much vibration in the
lever mechanism. The two-degree-of-freedom (2DOF) model system as possible.
has limitations as well but this model is a step closer to
modelling the flexibility in a system and is better suited for When subjected to steady–state harmonic excitation, the
vibration isolation design. Savovich and Ivovich7 have isolator’s performance can be defined in terms of absolute
discussed the 2DOF model of a lever-type isolation system to transmissibility. The absolute transmissibility measures how
isolate rigid equipment from a flexible floor. They used two well the isolator has reduced the transmitted force to the base
modelling and analysis approaches: frequency response or reduction in the equipment displacement. The absolute
function and state space formulation to calculate the effective- transmissibility is defined as the ratio of the force transmitted
ness of this 2DOF isolation model. It has been shown that the at the base to the excitation force (TR = pT/pm) or the ratio of
use of lever-type isolation may improve substantially the equipment vibration level to the excitation level (TR = a/w0).
isolation of heavy machinery, especially when mounted on a These ratios are equivalent if the system is linear and uniaxial.
floor, in comparison to conventional isolation design. By assuming only base excitation w(t) = w0ejot, the equation of
motion for the SDOF system shown in Fig. 1 is

The objective of this research is to develop a prototype of a M
zðtÞ þ cz_ ðtÞ þ kzðtÞ ¼ M0 w

ðtÞ þ cwðtÞ
_ þ kwðtÞ
1
lever-type isolator designed to be an efficient solution to the
problem of transmitted floor vibrations due to machine-
induced low-frequency excitation. Analytical results were
determined based on vibration theory for 2DOF systems using p(t )
transfer functions and state space formulation of equations of
z(t )
motion. The purpose of the analysis is to provide new isolation
Machine
criteria based on modelling and analysis. One of the objectives
of this paper is to quantify what type of effects the floor m

flexibility will have in the isolation design. Finally, a discussion

of isolation performance for varying parameters, such as Lever mechanism

damping ratio (zm) and mass ratio ( r), will be analysed. w(t ) m0

Performance increases using a lever-type corrector will also be k c Corrector

discussed. To verify the analytical method, a simple run-down

test is conducted in the laboratory. The isolated machine is
initially brought to a speed that is more than its service speed.
L2 L1
The power is then cut off and the machine is allowed to coast
down to zero speed thereby allowing the amplitudes to build up Fig. 1. SDOF isolation model with lever mechanism and
(resonance effect) as the machine passes through the critical auxiliary mass
speed. A free vibration test is performed to determine the

392 Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich
 where M = m + m0(L1/L2)2 and M0 = m0(1 + L1/L2)L1/L2. Define
102 the system transfer function as the ratio of the Laplace
transform of the output (response) to the Laplace transform of
Conventional isolation
Conventional isolation input excitation when the initial conditions are zero
Isolation with lever mechanism
Isolation with lever mechanism
ZðsÞ M0 s2 þ cs þ k
10 2 FðsÞ ¼ ¼
W ðsÞ Ms2 þ cs þ k
Transmissibility, TR

The transmissibility of the system may be obtained by letting

pffiffiffiffiffiffiffi
s = jo, where j = 1, and evaluating jFð joÞj;
1

( )1=2
a ½b2  uð1 þ bÞr 2 2 þ ð2rzb2 Þ2
3 TR ¼ ¼
w0 ½b2  ðu þ b2 Þr 2 2 þ ð2rzb2 Þ2

10–1 pffiffiffiffiffiffiffiffiffi
where, b = L2/L1, u = m0/m, r = o/om, om = k=m and
z = c/2mom. If the machine is excited by force p(t) = pmejot, then
the transmissibility is defined as ratio pT/pm, where pT is the
force transmitted to the base. This transmissibility is equivalent
10–2 to equation (3) if the system is linear. Introducing the zero
0 1 2 3 4 5 value of corrector mass in equation (3) leads to the expression
Frequency ratio, r
for conventional transmissibility. Fig. 2 shows a comparison
between conventional isolation transmissibility and trans-
Fig. 2. Transmissibility comparison for SDOF vibration
isolation models missibility using a lever-type mechanism for two levels of
damping ratios (z): 0 and 0?02. For lever isolation, the mass
ratio and the lever arm ratio
were assumed to be u = 0?02
and b = 0?3, respectively.
p(t ) Several conclusions can be
drawn from Fig. 2.
z2(t )
Machine 2.1. Conventional passive
isolation
m2 The absolute transmissibility is
less than unity, and vibration
pffiffiffi
occurs for o > 2om . The
Lever mechanism
greatest reduction is obtained
m0 at high values of r or at low
k2 c2 Corrector
isolation natural frequency
relative to excitation fre-
L1
quency and minimum damp-
L2 ing. Complete isolation can
z1(t ) theoretically be achieved if
r ! 1 with zero damping.
When the excitation frequency
m1 approaches the natural
frequency of the system, a
resonant condition occurs
and the transmissibility is
unbounded if no damping is
present in the isolator. If
w(t ) damping is present, the trans-
k1
missibility will no longer
c1
approach infinity at the
resonant condition but will
be finite. In the effective
pffiffiffi
isolation range (o > 2om ),
the transmissibility will be
Fig. 3. 2DOF isolation model with lever mechanism and auxiliary mass amplified due to isolation
damping. From Fig. 2 the

Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich 393
 transmissibility level was determined to be TR = 0?407 and damping ratio and lever arm ratio are assumed to be z = 0?01
0?408 at the frequency ratio r = 1?86 for various levels of and b = 0?3.
damping ratios, i.e. z = 0 and z = 0?02. The solid transmissibility
line corresponds to isolation with damping ratio z = 0, while the
dash corresponds to z = 0?02. 3. ISOLATION MODELLING AND ANALYSIS

3.1. Two-degree-of-freedom-model
2.2. Isolation with lever mechanism and auxiliary mass A 2DOF system can be modelled by interconnecting two mass-
For an SDOF system, when a corrector with lever mechanism spring-damper systems, as shown in Fig. 3. This model can
is attached to a system, a transmissibility zero, or anti- simulate a machine or a piece of equipment on a floor where
resonance in the resulting systemffi response, is found at the m2 represents the mass of machine, k2 and c2 represent the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
frequency ratio r0 = b2 =ðu þ ubÞ if no damping is present. stiffness and damping of the isolator, respectively. A mass m0,
If damping is present, the transmissibility will no longer be attached to the machine mass m2 by a lever, can be interpreted
zero at the anti-resonant point but will be very small. The as a lever mechanism. The flexible floor can be modelled as a
corrector mass m0 can be calculated using the above mass-spring-damper system with mass m1, stiffness k1 and
expression in isolation design. The p resonant condition damping c1. Only vertical translations of the masses are to be
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
occurs at the frequency ratio rres = b2 =ðu þ b2 Þ, where the allowed. The system under study, as shown in Fig. 3, can be
transmissibility is unbounded if no damping is present. subjected to a force excitation p(t) = pmejot. This analysis can
If damping is present, the transmissibility will be finite. determine the following quantities for the proposed 2DOF
The frequency ratios defined by the above expressions appear isolation model: transfer functions, frequency response
to be a function of two independent variables: the mass ratio (u) functions and transmissibility. Application of Lagrange
and the lever arm ratio (b). Therefore, the anti-resonant equations of the 2nd kind to the model shown in Fig. 3 results
condition can be controlled by selecting appropriate in the following governing equations of motion
parameters in isolation design. From Fig. 2, the trans-
missibility level was determined to be TR & 0 and TR = 0?023
M1
z1 ðtÞ þ c1 z_ 1 ðtÞ þ c2 z_ 1 ðtÞ þ k1 z1 ðtÞ þ k2 z1 ðtÞ
at the frequency ratio r0 = 1?86, for different values of
damping ratio, i.e. z = 0 and z = 0?02. This is shown by dash–  M0
z2 ðtÞ  c2 z_ 2 ðtÞ  k2 z2 ðtÞ ¼ 0
4
dot and dot transmissibility lines in Fig. 2. This result shows  M0
z1 ðtÞ  c2 z_ 1 ðtÞ  k2 z1 ðtÞ þ M2
z2 ðtÞ
improvement in vibration isolation 18 times greater than
þ c2 z_ 2 ðtÞ þ k2 z2 ðtÞ ¼ pðtÞ
the conventional passive isolation for z = 0?02. The improve-
ment in vibration isolation would be about 34 times for
z = 0?01. A comparison of the effects of different mass ratio where M1 = m1 + m0(1 + L1/L2)2 and M2 = m2 + m0(L1/L2)2. Note
on absolute transmissibility is shown in Fig. 4, where the that linearity of the system has been imposed by assuming that
all stiffness and damping effects can be modelled using linear
constitutive relations. Taking the Laplace transform of each
side of equation (4), under the assumption of zero initial
102
conditions, yields

Mass ratio 1%
Mass ratio 2% ðM1 s2 þ c1 s þ c2 s þ k1 þ k2 ÞZ1 ðsÞ
Mass ratio 3%
Mass ratio 4%  ðM0 s2 þ c2 s þ k2 ÞZ2 ðsÞ ¼ 0
10 5
 ðM0 s2 þ c2 s þ k2 ÞZ1 ðsÞ

þ ðM2 s2 þ c2 s þ k2 ÞZ2 ðsÞ ¼ P2 ðsÞ

Transmissibility, TR

Solving Z1(s) and Z2(s) in matrix form gives

1

 
Z1 ðsÞ
Z2 ðsÞ
" #1
M1 s2 þ c1 s þ c2 s þ k1 þ k2 ðM0 s2 þ c2 s þ k2 Þ
10–1 6 ¼
ðM0 s2 þ c2 s þ k2 Þ M2 s2 þ c2 s þ k2
( )
0
P2 ðsÞ

10–2
0 1 2 3 4 5
Hence, the transfer function from P2(s) to Z1(s) is
Frequency ratio, r

Fig. 4. Effects of different mass ratio on SDOF isolation model Z1 ðsÞ M0 s2 þ c2 s þ k2

with lever mechanism 7 ¼
P2 ðsÞ L

394 Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich
where 2
0 0
6
6 0 0
6
L ¼ b4 s4 þ b3 s3 þ b2 s2 þ b1 s þ b0 6
A ¼6 1
6 fM2 ðk1 þ k2 Þ  M0 k2 g
1
fM0 k2  M2 k2 g
6l l
b4 ¼ M1 M2  M02 6
4
1 1
b3 ¼ 2M0 c2 þ M1 c2 þ M2 c2 þ M2 c1 fM0 ðk1 þ k2 Þ  M1 k2 g fM1 k2  M0 k2 g
l l
3
b2 ¼ M2 ðk1 þ k2 Þ  2M0 k2 þ M1 k2 þ c1 c2 1 0
7
b1 ¼ c1 k2 þ c2 k1 0 1 7
7
7
b0 ¼ k1 k2 1 1 7
11 fM2 ðc1 þ c2 Þ  M0 c2 g fM0 c2  M2 c2 g 7
l l 7
7
5
1 1
fM0 ðc1 þ c2 Þ  M1 c2 g fM1 c2  M0 c2 g
After some manipulation, the following expression for force l l
8 9
transmissibility is obtained from equation (7) > 0 >
>
> >
>
>
> >
>
>
> 0 >
> " # ( )
>
< >
= 1 0 0 0 0
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M
u 2 2 B¼  0 C¼ D¼
u o2s >
u  2s
o2 M0 2
r þ
o2s c2
r >
>
> l >>
>
>
0 1 0 0 0
8 u
pT t om om M22 M2 om 3 >
> >
>
TR ¼ ¼ >
> M >
>
: 1 ;
pm EðoÞ2 þ FðoÞ2 l
l ¼ðM0 Þ2  M1 M2
where
Equations (9) and (10) are solved using a Runge-Kutta fourth
M2 o2s 2M0 c1 c2 order approximation. A mathematical model was implemented
EðoÞ ¼ ð1  aÞr 4  r 2 þ þ1 þ
M1 o2m M1 M1 M2 o2m using a set of M-files in MATLAB (version 4.0, #MathWorks
o2s
Inc.).10 The force excitation or system input, u, was obtained for
þ machine-induced periodic excitations.
o2m

2M0 r2 r2 o2s
FðoÞ ¼ r r2    c2 3.2. Vibration serviceability of building floors
M1 M2 om M2 om M1 om M2 o3m
 Buildings tend to vibrate at well-defined frequencies, with
c1 r 2 1 the largest motions usually at structural resonances. Whole
 
om M1 M1
structure motion frequencies range from approximately 10 Hz
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c1 ¼ 2M1 zm om ; c2 ¼ 2M2 zs os ; om ¼ k2 =M2 ; for low-rise structures to below 1 Hz for tall structures. Mid-
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi floor vibration frequencies are typically between 9 and 20 Hz.
os ¼ k1 =M1 ; a ¼ M02 =M1 M2 ; r ¼ o=om
Some massive large spans could have resonance frequencies
as low as 4 Hz. Mid-wall frequencies are between 10 and
The transmissibility expression given by equation (8) can be 20 Hz for light walls and possibly up to 45 Hz for masonry.
used as an indicator for acceptable isolation design. Machinery can induce vibration of a specific frequency that
could dominate the effects of vibration on occupants.
According to Ivovich and Onischenko,2 a vibration level of
Equation (4) was used to form a state space model, which was 1 mm/s (max. rms) causes substantial disturbances to building
modified in order to account for the fact that the input is force occupants. ISO 2631-211 applies to the human response to
and the output is displacement. The generic form for state vibrations and shocks. In the 8–64 Hz octave band, the
equations is given by equations (9) and (10) threshold of perception is 0?2 mm/s, probable disturbance
occurs at 1 mm/s, and moderate disturbance occurs at
0?4–1 mm/s. The Russian Organisation for Standardisation
9 x_ ðtt Þ ¼ Axx ðtt Þ þ Bu
uðtt Þ
(GOST) have published a standard that provides guidance for
the evaluation of human response to continuous, intermittent
and transient vibrations in buildings.12 Tentative guidance is
10 zðtt Þ ¼ Cxx ðtt Þ þ Du
uðtt Þ
provided in this Standard regarding satisfactory vibration
levels, i.e. levels below which the probability of human reaction
where is low. For industrial buildings of concrete construction and for
x(tt) = (nx1) state vector, where n is the number of states or disturbances at a frequency of 8 Hz, the guide value in the
system order Russian Standard is specified at 0?025 mm. Although vibration
u(tt) = (rx1) input vector, where r is the number of input levels in buildings induced by machinery are rarely high
functions enough to be the direct cause of damage, they could contribute
z(tt) = (px1) output vector, where p is the number of outputs to the process of deterioration from other causes, for example,
A = (nxn) square matrix, called the system matrix vibration-induced building damage such as cracks in walls and
B = (nxr) matrix, called the input matrix ceilings, separation of masonry blocks, or cracks in floors. The
C = (pxn) matrix, called the output matrix German Standard DIN 415013 provides guidance values in
D = ( pxr) matrix, which represents any direct connection terms of frequency dependent peak-particle velocity for upper
between the input and output. storeys. The probability of building damage due to vibration

Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich 395
 below the guidance value is very small. For industrial buildings
of concrete construction and for vibration at a frequency up to 102
10 Hz, the guidance value in this Standard is specified at
40 mm/s. Finally, with regard to the user, it can be concluded
that disturbance is likely when vibration velocity levels
exceed 1 mm/s. Damage problems can occur when vibration
levels exceed about 40 mm/s and are attributed to the lower 10
frequency of floor vibration. The velocity could be transformed
into displacement amplitudes by the simple harmonic equation

Transmissibility, TR
of motion z_ = oz, where z is the displacement and o is the
frequency in Hz multiplied by 2p.
1
3.3. Numerical example
As an example, consider an industrial fan, which is trans-
mitting vibration into an elevated concrete floor-slab. The
vibration, while not severe, is annoying to the personnel and
fs = 9·416 Hz
high-vibration sensitivity exists. The fan rotating at 480 rpm 10–1 fs = 13·076 Hz
( f0 = 8 Hz) gives a harmonic excitation force of 5000 N fs = 17·160 Hz
amplitude in the vertical direction. For numerical results the fs = 21·054 Hz
ANSYS
parameter values are listed in Table 1. In order to isolate
machine-induced floor vibrations, one solution is flexible
mounting of the fan. This involves a rigid sub-base (steel-work) 10–2
10–1 1 10
under the fan that is supported on six identical springs. The
Frequency ratio, r
design requirement is to have no more than Da = 0?025 mm of
induced vibration on the floor, with a goal of no more than Fig. 5. Transmissibility plots of 2DOF conventional passive
0?001 mm.12 A 2DOF lumped parameter model, shown in Fig. 3, isolation system for different floor flexibility
was used to analyse the vibration of the complete system. The
undamped natural frequencies of the floorpand isolator in its
ffiffiffiffiffiffiffiffiffiffiffiffiffi
uncoupled modes
pffiffiffiffiffiffiffiffiffiffiffiffiffi are denoted by fs = 1/2p k 1 =M 1 and configuration (solid line), obtained isolation efficiency of
fm = 1/2p k2 =M2 . For the parameter values listed in Table 1, 77?6% is inadequate in isolation design. In this case, an
fm = 2?011 Hz and fs = 9?147 Hz. The lever arm ratio was chosen isolation efficiency of 91?6% is required. Also, from these
to be L1/L2 = 2. By solving equation (10) for u = 0, eigenvalues results it appears that the floor flexibility tends to decrease
and corresponding eigenvectors are obtained. Vectors of isolation performance. A finite-element model (FEM) of two
undamped and damped natural frequencies are mass systems was analysed using a software package called
8 9
>
> 3523 þ 58886j >
>
  >
< >
12714 3523  58886j = 102
12 fog ¼ rad=s fo g ¼
58992 > >
> 0250 þ 12711j >
>
: >
;
0250  12711j

Figure 5 shows the transmissibility plotted against the

frequency ratio plot for the fan mounted on a spring isolation 10
system. From the plot, a system resonant frequency of 2?029 Hz
was determined. In addition, other resonant peaks are observed
Transmissibility, TR

at four specific floor frequencies: 9?03 Hz, 12?727 Hz,

16?897 Hz and 20?581 Hz.
1
From Fig. 5, the transmissibility levels were determined to be
TR = 0?224 (D = 0?04 mm > Da = 0?025 mm), 0?102, 0?087 and
0?08. This is shown by the lines in Fig.5. For baseline

fs = 9·147 Hz
10–1 fs = 12·703 Hz
Property Value
fs = 16·670 Hz
fs = 20·453 Hz
Floor mass m1 = 8000 kg
Machine mass m2 = 4800 kg
Corrector mass m0 = 53 kg
Floor stiffness k1 = 28000 kN/m
10–2
Isolation stiffness k2 = 800 kN/m 10–1 1 10
Floor damping coefficient c1 = 53?7 kNs/m Frequency ratio, r
Isolation damping coefficient c2 = 2?5 kNs/m
Fig. 6. Transmissibility plots of 2DOF isolation system with
Table 1. Lumped parameter model values lever mechanism for different floor flexibility

396 Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich
ANSYS (version 5.0 #Swanson Analysis Systems). The mass the relative velocity of two connecting nodes. More detail can
elements m1 and m2 used in the analysis are type MASS21, be found in the ANSYS manuals.14 Comparison of the FEM
which are lumped masses with no rotational inertia and have model and the theoretical model is shown in Fig. 5. The results
one degree of freedom. The spring and damper element used is show that theoretical predictions match the FEM model, which
type COMBIN14, which has a linear spring rate and a viscous confirms the theoretical model. Fig. 6 shows a typical
damping coefficient producing a restoring force proportional to transmissibility plot of the composite 2DOF system including
lever mechanism. The
curves in Fig. 6 show that
102
besides two resonance
peaks there is the corrector-
induced anti-resonance
pffiffiffiffiffiffiffiffiffiffiffiffiffi
o = k2 =M0 = 50?24 rad/s
(8 Hz). This basic physical
behaviour has great practical
10
potential, which means that if
excitation is provided by
machinery operating at
Transmissibility, TR

frequency o, vibration can

be reduced by attaching the
1
lever mechanism so that the
anti-resonance is created near
that frequency. The different
lines show the effect of floor
Isolation damping 0·5%
Isolation damping 1·0% flexibility on this isolation.
10–1 Isolation damping 2·0% With lower floor stiffness, the
Isolation damping 3·0% level of attenuation is less at
the anti-resonant point. From
Fig. 6 the transmissibility
level was determined to be
10–2 TR = 0?035 (D = 0?0062 mm <
10–1 1 10
Da = 0?025 mm), 0?018, 0?014
Frequency ratio, r
and 0?012. The obtained
Fig. 7. Transmissibility plots for varying isolation damping isolation efficiency of 96?5%
for baseline configuration is
satisfactory in isolation
design. As can be expected,
the inclusion of the floor
flexibility has degraded the
isolator performance that was
obtained for the SDOF system.
10 In this case, results show
an improvement of six to
seven times in the vibration
isolation compared to con-
ventional passive isolation.
Transmissibility, TR

The isolation damping value

1
used in all calculations thus
far was 2%, which is con-
servative. This value is based
on material damping levels
Mass ratio 1% and is likely to be high for
10–1 Mass ratio 2% the lever mechanism. The
Mass ratio 3%
results of varying damping
Mass ratio 4%
from 0?5% to 3%, while keep-
ing the other parameters from
Table 1 fixed, are shown in
10–2 Fig. 7. The next step is to vary
the mass ratio r = m0/m2.
10–1 1 10
Frequency ratio, r Fig. 8 shows a comparison of
transmissibility functions for
Fig. 8. Transmissibility plots for varying mass ratio different values of mass ratio.
Fig. 9 shows the contour

Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich 397
 1·0 TR
45·00
0·9
26·96
16·15
0·8 9·679
5·800
0·7 3·475
Isolation damping: ζm

2·082
0·6 1·247
0·7474
0·5 0·4478
0·2683
0·4 0·1608
0·09633
0·05772
0·3 0·03458
0·02072
0·2 0·01241
0·007438
0·1 0·004457
0·002670
0 0·001600
(a)
1·0
TR
0·9 44·1396
26·4724
15·8766
0·8
9·5219
5·7107
0·7 3·4249
Isolation damping: ζm

2·0541
0·6 1·2319
0·7388
0·5 0·4431
0·2658
0·4 0·1594
0·0956
0·0573
0·3
0·0344
0·0206
0·2
0·0124
0·0074
0·1 0·0044
0·0027
0 0·0016
0 10 20 30 40 50 60 70 80 90 100
Excitation frequency, ω: rad/s
(b)

Fig. 9. Contour plots for 2DOF model with lever mechanism: (a) floor frequency fs = 9?147 Hz; (b) floor frequency fs = 20?453 Hz

plots for the 2DOF system with lever mechanism for two diagrams. It can be seen that the floor vibration amplitude
different levels of floor flexibility: floor frequencies decreases to about 16% of the original conventional isolation.
fs = 9?147 Hz and fs = 20?453 Hz. Note the spreading of the This result is in accordance with the theoretical solution. The
light grey zone around the anti-resonant point for the stiffer MATLAB model of the conventional isolation case (m0 = 0)
case in Fig. 9(b). The free vibration response of the system agrees with the ANSYS model. This study has demonstrated
for the given initial conditions is shown in Fig. 10. It that the lever-type corrector is effective in controlling single
is shown for undamped (c1 = c2 = 0), proportionally damped frequency vibration of a floor mass (m1). The level of
(k1/k2 = c1/c2 = 35) and non-proportionally damped (k1/k2 = 35, attenuation achieved using this system is in the order of
c1/c2 = 40) cases. It should be mentioned that results obtained 96?5%, with 2% damping present in the mechanism. It
showed that mode 2 for the system with lever mechanism has should be mentioned that it gave a higher attenuation for
no node point (a point that always remains stationary). Fig. 11 lower isolation damping, e.g. for isolation damping of 1?0%
shows the response of the 2DOF system with and without the the transmissibility level is 0?02 and for 0?5% it is 0?01
lever mechanism mounted, and the corresponding phase (see Fig. 7).

398 Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich
 0·0010

0·0005
z1(t ): m
0

–0·0005

–0·0010

0 2 4 6 8 10

0·00004

0·00002
z2(t ): m

–0·00002

–0·00004
0 2 4 6 8 10
Time: s
(a)
0·0010

Proportionally damped
0·0005 Non-proportionally damped
z1(t ): m

–0·0005

–0·0010
0 1 2

0·00004
Proportionally damped
0·00002 Non-proportionally damped
z2(t ): m

–0·00002

–0·00004
0 2 4 6 8 10
Time: s
(b)

Fig. 10. (a) Undamped free vibration of 2DOF model with lever mechanism, initial conditions = [0?001, 0, 0, 0]; (b) comparison of
proportionally and non-proportionally damped free vibration of 2DOF model with lever mechanism

4. ISOLATOR DESIGN AND EXPERIMENTAL RESULTS The system uses the angular motion mechanism given by the
The design of a lever-type isolator is shown in Figs 12 and 13. corrector mass movement. The workable frequency span of the
The impetus behind the concept is that for some applications isolator can be estimated from 3 Hz to approximately 13 Hz.
(building floor protection applications), the isolator must be The method under discussion was tested on a small experi-
as simple as possible. To be of practical use, the design must mental setup located at the Structural Dynamics Laboratory of
incorporate a means of adjusting the absorbing frequency, since the Building Research Institute (CNIISK), Moscow. This consists
the corrector mass m0 is often difficult to adjust. The damping of a variable-speed D.C. motor, a V-belt driving a shaft with an
in the lever mechanism must be low. Referring to Fig. 12, unbalanced rotating disc and a basic framework (0?561 m)
the prototype isolator consists of a spring unit, lever mech- supported by four compressive springs with and without lever
anism with auxiliary mass unit and bearings (see Table 2).15 mechanism and auxiliary mass (see Fig. 14). The motor is

Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich 399
 1 X 10–4 0·003
Theory
ANSYS
0·002

5 X 10–5

0·001
Displacement, z1(t ): m

Z1(t ): m/s
0 0

–0·001

–5 X 10–5

–0·002

–1 X 10–4 –0·003
0 5 10 15 –1 X 10–4 –5 X 10–5 0 5 X 10–5 1 X 10–4
Time: s (a) Z1(t ): m
1X 10–4 0·0015
Theory

0·0010

5 X 10–5

0·0005
Displacement, z1(t ): m

Z1(t ): m/s
0 0

–0·0005

–5 X 10–5

–0·0010

–1 X 10–4 –0·0015
0 5 10 15 20 –0·00006 –0·00004 –0·00002 0 0·00002 0·00004 0·00006
Time: s Z1(t ): m
(b)

Fig. 11. Dynamic response and phase diagram of 2DOF model: (a) model without lever mechanism; (b) model with lever mechanism

initially brought to a speed

Section A-A Section B-B that is more than the service
70 70 speed (1200 rpm). The power
is then cut off and the speed
of the motor is allowed to
coast down to zero due to
75

damping in the system. Since

75
40

the damping is small, the

coasting period is
large enough (~120 s) to
allow the amplitudes to build
400
up (resonance effect) as the
7 60 120 motor passes through its
A critical speeds. A wide range
4 of masses (15–400 kg) can be
3
clamped beneath the motor
B 1 2
assembly to increase the
8
3 inertia of the system. Table 3
summarises the isolator pro-
150

totype characteristics.
4
6
A
10 The mass (m2) is the approx-
B 4
imate mass of motor, steel
framework and added mass
which is assumed to be
9 200 5 96?164 = 384?4 kg. The ver-
tical amplitude of harmonic
Fig. 12. Section view of the lever-type isolator (all dimensions in mm) disturbing force is 98?1 N.
The velocity response of the

400 Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich
 Position Description

1 Lever
2 Tunable auxiliary mass corrector
3 High resistance steel spring, as per GOST
Standard12
4 Bearing
5 Support
6 Rectangular base plate
7 Rectangular top plate
8 Linking tie
9 Slip-proof rubber mat (synthetic rubber)
10 Attachment hole with bolt

Table 2. Isolator with lever mechanism and auxiliary mass

Property Value
Fig. 13. Lever-type isolator design
Maximum displacement d = 57?2 mm
Isolation frequency fm = 1?98 Hz
Damping ratio zm = 0?02
framework (V2) and floor slab (V1) was measured with Corrector mass m0 = 4?8 kg
Messgeratewerk Zwantiz (Germany) velocity transducers type Lever arms L1 = 120 mm, L2 = 60 mm
IOO-1 in conjunction with signal amplifiers and vibrograph of Stiffness k = 16481 N/m
type TSS-101 (see Fig. 14). The two transducers were used to
measure the amplitude of the vertical vibrational velocity as a Table 3. Isolation system characteristics
function of frequency in steps of 1 Hz from 1 to 20 Hz. The
magnitudes of the displacement responses after transformation
are given in Fig. 15, which show that there are two resonant transmissibility function. Numerical analysis of the dynamic
frequencies and one anti-resonant frequency in the frequency response of this system has been performed using a Runge-
range under consideration. The corresponding theoretical Kutta approximation and state space formulation of the
frequency responses are overlaid in the same figure. equations of motion. Comparison of the finite element analysis
and the theoretical model has been made and results showed
5. CONCLUSIONS that theoretical predictions match the FEM model, which
In this research, an improved vibration isolation system with confirms the theoretical solution. The damping associated with
lever mechanism has been designed and tested, using the the lever-type corrector has been reduced by using bearings,
simple test setup. Theoretical analysis has been undertaken to thus improving the maximum level of attenuation possible. The
predict the behaviour of the isolator using a 2DOF lumped lever mechanism with auxiliary mass has been shown to be
parameter model in conjunction with frequency response or able to reduce the vibration effectively. The level of attenuation
achieved showed an improve-
ment of up to seven times in
D.C. motor comparison to conventional
Added mass isolation. The operation
Velocity transducer
Framework range of frequencies for the
isolator has been improved by
V2
tunable corrector mass. The
proposed isolator has been
Floor
validated by dynamic testing
Lever isolator V1
and good agreement between
theoretical and experimental
results have been obtained.
Vibrograph Amplifier Results showed that isolation
Amplifier
with the lever mechanism
might provide adequate
isolation of low frequency
harmonic excitation but
other issues, such as random
excitation, must also be
considered. Coloured plots
and more results can be
viewed and downloaded
Fig. 14. Experimental setup for isolation with lever mechanism and auxiliary mass at the following website:
http://vibrodesign.ru.

Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich 401
 6. KUBOTA M., ISHIMARY S.,
TAKAHITO N. and IPPEI H.
Theory–with lever mechanism Dynamic response-
Theory–with lever mechanism
Theory–without lever mechanism
controlled structures
10–1
Theory–without lever mechanism with lever mechanisms.
Experiment, V1–with lever mechanism
Proceedings of the
Experiment, V2–with lever mechanism
Experiment, V1–without lever mechanism Smart Structures and
10–2 Materials Conference,
Displacement amplitude: m

Experiment, V2–without lever mechanism

San Diego, 1998, 3325,
35–44.
10–3 7. SAVOVICH M. K. and
IVOVICH V. A. Improve-
ment in safety and per-
formance of isolation
10–4
systems under industrial
vibrations using a lever-
type corrector. Journal of
10–5 Earthquake Engineering
& Structural Safety,
2000, No. 4, 39–43
10–6
(in Russian).
8. NELSON F. C. Shock
0 5 10 15 20
vibration isolation:
Excitation frequency: Hz
breaking the academic
Fig. 15. Frequency response amplitudesFexperimental and model-predicted paradigm. ASME PVP,
Seismic and Vibration
Isolation, 1991, No. 5,
1–6.
REFERENCES 9. SCIULI D. and INMAN D. Comparison of single and two-
1. FROLOV F. V. and FURMAN F. A. Applied Theory of Vibration degree-of-freedom models for passive and active vibration
Isolation Systems. Hampshire Publications. New York, isolation design. Proceedings of the Smart Structures and
1990. Material Conference, San Diego, 1996, 2720, 293–304.
2. IVOVICH V. A. and ONISCHENKO V. Y. Vibration Protection in 10. MATLAB. User’s manual. The Math Works Inc., Natick,
Engineering. Mashinostroenie, Moscow, 1990 (in Russian). 1995.
3. GLAZIRIN V. S. Methods of lowering the dynamic loads 11. ISO. Evaluation of human exposure to whole-body
transmitted to the supporting structures. Stroitelnaya vibration–Part 2: Continuous and shock-induced vibration
Mekhanika i Raschet Soruzhenii, 1971, No. 3, 43–47 in buildings (1–80 Hz), 1998, Geneva, Switzerland 2631-2.
(in Russian). 12. RUSSIAN STANDARD INSTITUTION. Occupational Safety Stan-
4. IVOVICH V. A. Guidelines for vibration protection of dards Systems. Vibrational Safety. General Requirements.
industrial buildings. Technical Report No. 649. The Central GOST 12.1.012-90. Moscow, 1991 (in Russian).
Research Institute for Building Structures (CNIISK), 13. GERMAN STANDARD INSTITUTION. Structural Vibration in
Moscow, 1988 (in Russian). Buildings. Part 3: Effects on Structures. DIN 4150. Berlin,
5. IVOVICH V. A. and ARUTJUNYAN M. V. Dynamic analysis of 1984.
vibration isolation systems with lever-type corrector. 14. ANSYS. User’s manual. Swanson Analysis Systems.
Proceedings of the Central Research Institute of Building 15. SAVOVICH M. K. and IVOVICH V. A. Antivibrador con Palanca
Structures—CNIISK, 1991, No. 624, 115–127 (in Russian). de Inercia. Spanish patent license, 2000 (applied for).

Please email, fax or post your discussion contributions to the secretary: email: lyn.richards@ice.org.uk; fax: +44 (0)20 7799 1325;
or post to Lyn Richards, Journals Department, Institution of Civil Engineers, 1^7 Great George Street, London SW1P 3AA.

402 Structures & Buildings 146 Issue 4 Isolation of floor machines Ivovich / Savovich