Dynamic analysis of machine foundation: when a static force cannot give the full picture.

HKIE-IEM-CIE Tripartite Seminar Recent Developments in Limit State Design for Geotechnical Works
45
Dynamic Analysis of Machine Foundation:
When a Static Force cannot Give the full picture
Gary Yung
AECOM, Hong Kong
E-mail : Gary.WT.Yung@aecom.com
ABSTRACT
Machine foundations provide a robust platform for machinery to operate in a smooth manner with
minimal maintenance requirements. An unexpected vibration can be detrimental to the machine
components, ground settlement, structural integrity of foundation, and other machines or working
personnel adjacent to it. If vibrations become excessive and uncontrollable, a machine can be forced to
shut down or a catastrophic failure can result from fatigue failures of machine components. To avoid
this, the serviceability limit state design takes into account of vibration control. Machine vendors
usually specify different levels of foundation performance and serviceability vibration limits depending
on the machine types for engineers to design the machine foundation.
However, in design office, dynamic design is often considered as a secondary check or sometime even
omitted due to the lack of time and budget allocated to the project. This paper will present various
design approaches from a rule of thumb static design to a comprehensive dynamic analysis and discuss
their limitations. A case study of large machine foundation with stringent vibration limits will be
presented to demonstrate the necessity and complexity of a full dynamic analysis. This study aims to
deliver the message that a static force does not always give the full picture. Project managers and
engineers can achieve a better insight into dynamic analysis for those machine foundations that are
sensitive to vibration, and then be upfront with the clients about the design time and budget required
for the dynamic analysis and evaluation.
1. INTRODUCTION
Machine foundations provides a robust platform for machinery to operate efficiently and reliably.
Vendors continuously improve the productivity and efficiency of their machine by increasing the
machine size and operating speed. On the other hand, the requirements of vibration control can become
more stringent due to higher machinery specifications. Consequently, modern machine foundations are
required to resist larger dynamic forces at higher operating speeds while controlling vibrations.
Unexpected vibrations can be detrimental to the machine components, foundation settlement, structural
integrity of foundation, and other machines or working personnel adjacent to it. If vibration becomes
excessive and uncontrollable, the machine may be forced to shut down, and even lead to fatigue failure
in the machine components resulting in a catastrophic incident. Dynamic analysis of machine
foundation is a crucial justification step to ensure the performance and safety of the machine.
The vibration of foundation subjected to machine load can be assessed by different approaches
including the pseudo-static approach with rules of thumb, natural frequency analysis, modal analysis,
time history analysis and frequency domain analysis. To study the dynamic soil-structure interaction,
an impedance function method is commonly used to study the dynamic behaviours between soil and

46
structure. Dynamic soil stiffness and damping coefficients are frequency dependent, and will be briefly
discussed in this paper.
Site specific geotechnical data and machine dynamic forces are important design parameters for a
success machine foundation design. However, this information is not always available at the time of
design. The difficulty of undertaking the machine foundation design without these parameters will be
highlighted.
Since the interface of soil-structure is often assumed to be rigid, the dynamic effect of flexible behaviour
of machine foundation can be underestimated. A case study of high-speed machinery supported on
shallow foundation will be presented to demonstrate the importance of considering the higher-order
dynamic responses.
2. DYNAMIC EFFECT DUE TO HARMONIC FORCE
The dynamic response of a single degree of freedom system subjected to a harmonic force can be
represented in Figure 1. A dynamic amplification factor (DAF) is given by:
Fig. 1 - Dynamic Amplification Factor of SOF System
frequency of dynamic force
natural frequency of system ( )
k system stiffness
m system mass
If DAF is greater than 1, the machine foundation system is under the influence of resonance. Machine
vendor often requires a frequency separation between the natural frequency of foundation system and
the excitation frequency (e.g. . Under-tuning or over-tuning the foundation system
can be achieved by changing the system stiffness and mass. However, the machine foundation is a
system with multi-degrees of freedom which contains more than one natural frequency. The case study
at the end of this paper will discuss the difficulty of avoiding all the primary resonances.

47
3. DIFFERENT DESIGN APPROACHES
Machine vendors often specify different vibration requirements based on the nature of machine.
Machine such as mobile generators can tolerate the vibrations induced by its own unbalance forces. A
simple rule-of-thumb is sufficient for the type of these machine foundations. In contrast, a critical
machine such as high speed gas turbine in the production line cannot be operated with excessive
vibrations. A dynamic analysis with the consideration of dynamic soil-structure interaction becomes
necessary. Design of machine foundations can be categorised into three approaches::
3.1 Category 1- Static Analysis with Rule of Thumb
Machine vendor sometime is confident that the induced unbalance force of machine is insignificant to
cause any vibration issue based on the empirical results (e.g. mobile generators or small pump stations).
The design can be simplified to rule-of-thumb approach by providing sufficient mass into the foundation
system to control the vibration. ACI 351 recommends the weight of foundation system at least three
times the weight of a rotating machine and at least five times the weight of a reciprocating machine.
The considered machine weight includes both moving and stationary machine parts. No rigorous
dynamic analysis is needed.
3.2 Category 2 – Natural Frequency Analysis
Vendors may require the machine foundation system to be designed such that no system resonance is
occurred during the normal operation. Natural frequency analysis can be used to demonstrate sufficient
frequency separation between the system resonance and the excitation frequency. A sufficient
frequency separation is achieved when any primary natural resonance of the foundation system is less
than 0.5 or greater than 1.5 times the excitation frequency. Unfortunately, in many cases, the frequency
separation cannot be achieved and a vigorous dynamic analysis (Category 3) would then be
recommended.
3.3 Category 3 – Forced Vibration Analysis
Vendor or the client may identify that the machine is crucial and sensitive to vibration. In this case, a
forced vibration response analysis such as a harmonic analysis or time-history dynamic analysis is
required to evaluate the foundation dynamic response. If the soil radiation damping and the dynamic
soil-structure interaction are to be considered in the machine foundation design, a frequency domain
dynamic analysis together with impedance functions would then be recommended. This paper will focus
on Category 3 case. A case study of a large machine foundation subjected to steady-state harmonic
loads will be presented at the end to demonstrate the benefits of this approach.
It should be noted that some key design parameters are considered unnecessary or not provided by the
vendor for the dynamic design, which can limit the analysis options.
4. IMPORTANT DESIGN PARAMETERS
When vibration control is identified as the crucial to the machinery, the design team including
mechanical, geotechnical, and structural engineers will work closely together to provide a foundation

48
solution which satisfies all economic, performance and safety aspects. The design parameters necessary
to perform a detail dynamic analysis include geotechnical investigation, machine weight and centre of
gravity, operating frequency ranges, rotating component weight and balance quality, machine
unbalanced actions and machine vibration tolerances. Amongst these, the geotechnical parameters and
unbalance actions are often the unavailable or unknown at the time of design.
4.1 Dynamic Soil or Rock Properties
Gunter Klein and Dietrich Klein (2003) present typical natural frequencies of machine foundation
system (Figure 2). When the machine such as turbo-generator operates at 30Hz and is supported on a
shallow rock foundation, the foundation system may be difficult to avoid resonance. The dynamic shear
modulus of foundation material is an important parameter to determine the machine dynamic
behaviours. The client and project managers sometime underestimate the importance of in-situ
geotechnical data such as dynamic shear modulus which may require additional site access or
expenditure to obtain.
Fig. 2 - Natural Frequency of Support System
{Source from Gunter Klein and Dietrich Klein (2003)}
The dynamic shear modulus is commonly obtained by measuring shear wave velocity in the field (e.g.
cross-hole method, down-hole method, up-hole method and seismic reflection). The relationship
between dynamic shear modulus and measured-in-field shear wave velocity are as follows:
where
dynamic shear modulus of sub-grade (Pa)
soil density (kg/m3
)
shear wave velocity (m/s)
The case study at the end of this paper demonstrates the difficulty of avoiding resonance without in-situ
dynamic geotechnical data.

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4.2 Unbalanced Forces
Unbalanced forces are essential design parameters to perform the forced dynamic analysis. However,
another common unknown design parameter is the dynamic forces induced by the machine components.
Machine foundation can be excited by dynamic forces due to reciprocating, rotating and impulsive
actions. Reciprocating and rotating machines, such as diesel generators, gas turbines, centrifugal
compressors and electrical motors are subjected to steady-state harmonic dynamic actions. Whereas,
impulsive machines, such as forging hammers and metal forming presses, are subjected to sudden
impulsive force. Discussions in this paper will be limited to the reciprocating and rotating machines.
Salesmen always describe their machines as perfectly balanced and with no significant dynamic force.
The vendor often does not explicitly provide the unbalanced force for the client or designers. In these
cases, the unbalanced force can be estimated by ISO 1940/1 and ACI 351 when the rotating mass and
balance quality grade are given by the manufacturer:
where
dynamic force (zero to peak) (kN)
rotating mass (kg)
mass eccentricity based on the balance quality grade in ISO 1940/1 (mm)
circular operating frequency of the machine (rad/s)
service factor allow for increased unbalance during the machine service life
generally at least equal to 2 or stated by the manufacturer
5 DYNAMIC SOIL-STRUCTURE INTERACTION
The design of machine foundation is complex dynamic soil-structure interaction problem. Academia
has been investigated the soil-structure interaction problem since early 1920s. However, very few
design standards or guidelines have addressed the dynamic soil-structure interaction.
5.1 Review of Dynamic Soil-Structure Interaction
Research activities on the effect of soil-structure interaction in the seismic application had been
considerably increased in the 60s and 70s due to the extensive developments of Nuclear Power Plants
(Roesset 2013). In the early development, linear and frequency independent springs were used to
simulate the stiffness of the foundation. By 70s, the effect of soil-structure interaction was generally
considered as more important for relatively stiff and massive structures such as Nuclear Power Plants.
The radiation damping has been considered as an important factor in the soil-structure interaction
problem. Soil radiation or geometric damping dissipates energy by propagating waves away from the
foundation to the soil mass, increasing the effective damping. However, in soil-structure interaction,
the radiation damping is frequency dependent and is difficult to determine in the time domain. Veletsos
and Wei (1971) and Luco and Wstmann (1971) represented the soil dynamic stiffness and damping
characteristics as frequency dependent, which also called complex valued impedance function. Kausel
(1974) realized that the application of impedance function in seismic engineering can be extend to the
dynamic design of machine foundation. Dynamic response of pile-supported foundation depends on the
dynamic stiffness and damping written by Novak in 1970s. El Naggar and Novak present a model to

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allow for nonlinear soil behaviours in pile foundation with the energy dissipation through radiation
damping and soil hysteresis. Wolf and Deeks (2004) publish a machine foundation textbook to explain
the application of impedance functions together with work examples.
5.2 Impedance Stiffness Approach for Dynamic Soil-structure Interaction
To study the dynamic soil-structure interaction for machine foundation, the soil-foundation interface is
assumed to be rigid. The unbounded soil performs as an energy damper to dissipate energy through
propagating wave energy towards the soil mass, which is called radiation damping or geometric
damping. The impedance stiffness approach has been used to represent the dynamic soil-foundation
behaviours. Practicing engineers often are not familiar with impedance functions involved in dynamic
soil-structure interaction. Indeed, many design standards and regulations are only qualitatively to
describe the general effects and significance of dynamic soil-structure interaction such as in EN 1998-
5:2004. Only few design references such as ACI 351 and Canadian Foundation Engineering Manual
provide guidelines to evaluate the dynamic behaviours of machine foundation by the impedance
stiffness approach. To help understanding of the basic concept of dynamic impedance stiffness, the
equations of motion in time domain and frequency domain are discussed.
The typical equation of motion in the time domain ( is
(1)
where
M mass constant
C damping constant
static spring constant
dynamic action
Frequency domain representation for the harmonic force
(2)
Frequency domain representation for displacement, velocity and acceleration
(3)
(4)
(5)
where
imaginary number,
Euler’s formula,
By substituting (2) to (5) into (1) and cancelling in both sides, an alternative equation of motion
represented in the frequency domain ( ) is:

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Dynamic stiffness is defined as as a function of frequency ( )
thus
At the low frequency, the real part of dynamic stiffness ) is a positive value, the
displacement response is in phase with the driven dynamic force. In contrast, at the high frequency,
the real part of dynamic stiffness becomes a negative value which represents the displacement
response is 180 out of phase to the dynamic force. At the resonance frequency ( , the
real part of dynamic stiffness becomes zero, and only imaginary part ) is remained. Interestingly,
the dynamic response at resonance is only controlled by damping, and the displacement response is
90 out of phase to the dynamic force.
In dynamic design of machine foundation (Wolf and Deeks 2004), a dimensionless frequency ( is
commonly used instead of frequency . The dynamic stiffness is rewritten in terms of spring
coefficient and damping coefficient as a function of dimensionless frequency
where
characteristic length of the foundation
shear wave velocity of the first soil layer
Manual computation of the impedance response functions and dynamic responses can be tedious.
Computer programs such as Matlab and DYNA6 can be used to determine the impedance stiffness and
damping coefficients. DYNA6 has been developed in University of Western Ontario, Canada by El
Naggar and adopted in Canadian Foundation Manual for the machine foundation design.
5.3 Limitations of Impedance Stiffness Approach
The impedance stiffness approach, which assumes the interface of soil-foundation to be rigid, is only
capable of evaluating the dynamic response of a rigid machine foundation with six degrees of freedom.
In practice, however, not all machine foundation can be assumed to be rigid, especially for the high
speed machinery.
In terms of dynamic analysis, rigid and flexible machine foundations can be classified by the natural
frequency of the foundation system and the machine operating frequency. ISO-10816-3 recommends
that the support system can be considered as rigid foundation when the first natural frequency of
foundation system is higher than the operational frequency by at least 25%, otherwise the foundation
system is considered to be flexible.

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For flexible foundations with the consideration of soil-structure interaction, the dynamic response can
be studied using a finite element (FE) model coupling with the frequency dependent soil stiffness and
damping coefficients, determined by the impedance functions with the assumptions of a non-flexible
foundation and a rigid soil-foundation interface (by computer programme such as DYNA6 or Matlab).
6. CASE STUDY OF HIGH SPEED MACHINE FOUNDATION
A foundation with one or more high speed machines can be a complex dynamic soil-structure interaction
problem. While this case can be analysed by assuming the interface of soil-foundation to be rigid with
no flexural deformation of foundation, the effects of flexural deformation can be detrimental to the
vibration control. Some advanced finite element analysis (FEA) packages, such as SAP2000, are
capable of modelling a flexible machine foundation with the dynamic soil stiffness and damping
coefficients in the form of impedance functions..
6.1 Problem Definition
Multiple high speed machines supported on a single block foundation is not uncommon for gas
compression station. As the case study, two high speed centrifugal compressors driven by an electric
motor (E-motor) were supported on a monolithic foundation shown in Figure 3.
Fig. 3 - General Arrangement of Gas Compression Unit
This gas compression station was identified as the crucial facility along the production line. The E-
motor was rated as 25MW with an operational speed of 1800 rpm. High pressure compressor operated
at a speed of 11867 rpm with a gear ratio of 6.59, whereas, low pressure compressor operated at a speed
of 10045 rpm with a gear ratio of 5.58. The total weight of the equipment package including the steel
skids was 126,600 kg. This equipment package was supported on a concrete block foundation. The
weight of concrete foundation without piles was 552,600kg, which provided the foundation to
equipment weight ratio of 4.36. Shallow sandstone was found at the location of the gas compression
station. Shear velocity of sandstone ranging from 600 m/s to 1000 m/s were obtained by geotechnical
field tests. Table 1 specifies a range of allowable peak-to-peak vibration displacements and the
maximum vibration velocity in relative to different operational frequencies.

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Table 1 –Vibration Limits of Machine Foundation
6.2 Machine Foundation Resonated with Shallow Rock
DYNA6 with the consideration of dynamic soil-structure interaction was used to evaluate the natural
frequency of the foundation system. Although the machine foundation was supported on relatively good
shallow sandstone, the vibration response still could not meet the stringent vibration limit given by the
vendor due to the fact that the machine foundation system could not avoid the resonances at the machine
operational speed.
In the early development of the design solution, a concrete block foundation supported directly on the
shallow rock was considered. The natural frequency of this type of foundation, however, appeared to
be fairly closed to the machine operational frequency. Figure 4 illustrates the primary foundation natural
resonances in relative to the machine operational frequencies. Primary natural resonances of the
machine foundation were horizontal (Y), vertical (Z) and rocking (RX) modes. The E-motor operated
at low frequency (24Hz to 32Hz) whereas the compressors operated at high frequency (134Hz to
208Hz). Horizontal natural frequency (Y) appeared to be at the machine operational frequency in this
case. The maximum velocity response thus exceeded the vibration limit of 0.75mm/s shown in Figure
5.
A number of attempts were used to provide sufficient frequency separation between the foundation
natural frequencies and the machine operational frequency, however very little success was achieved.
By enlarging or reducing the foundation footprint, and by increasing or decreasing the mass of
foundation, the horizontal and vertical natural frequencies shifted together in the same manner. Since
the horizontal and vertical natural frequencies were fairly closed to each other, it always one of them
fell within the machine operational frequency. As a result, it was concluded that the block foundation,
in this case, was not a practical solution to avoid resonance.
Fig. 4 - DYNA6 - Primary Natural Frequency (Block Foundation)
Peak-Peak vibration Disp. ( m) Max vibration speed (mm/s)
rpm Hz
Low Pressure Compressor
Low Freq 1800 30 8 0.75
High Freq 10045 167 1.4 0.75
High Pressure Compressor
Low Freq 1800 30 8 0.75
High Freq 11867 198 1.2 0.75
Operational Freq

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Fig. 5 - DYNA6 - Dynamic Response (Block Foundation)
In terms of adjusting individual natural frequency and providing frequency separation, the pile
foundation was a better solution. The concept of pile foundation (Figure 6) is similar to table-top
foundation, which consists of large diameter of piles with sleeves to isolate the machine foundation
from the surrounding rock. The sleeved pile (Figure 7) reduces the horizontal stiffness (reduces
horizontal natural frequency). The large diameter of pile increases the vertical stiffness (increase
vertical and rocking natural frequencies).
Fig. 6 - Pile Foundation isolated from Surrounding Rock
Fig. 7 - Pile Sleeve Detail
Figure 8 presents the dynamic response of pile foundation with the appropriate pile sleeve length to
avoid resonance. The sleeved pile foundation successfully established natural frequency separation. The
primary horizontal resonance occurred before the low machine operational frequency, while the vertical
and rocking natural frequencies occurred in between the low and high machine operational frequencies.

55
The vibration response was controlled under 0.75mm/s within the operational frequency shown in
Figure 9.
Fig. 8 - DYNA6 - Primary Natural Frequency (Pile Foundation)
Fig. 9 - DYNA6 - Dynamic Response (Pile Foundation)
It is noted that DYNA6 did not indicate any vibration response at high operational frequencies. The
flexure behaviour of foundation was not considered in DYNA6, hence, a finite element model coupling
with the frequency dependent pile stiffness and damping coefficient was established to ensure that any
vibrations due to flexure of the foundation were acceptable.
6.3 Finite Element Model with Frequency Dependent Pile Stiffness
A mass foundation with a total number of 12 piles was simulated by using a finite element package
(Strand7) together with frequency dependent pile springs from DYNA6 (Figure 10). The concrete
foundation was modelled using brick elements. Compressor skids were modelled using steel beams.
Masses of compressors, motor and associated components were modelled as translational node masses.
The frequency dependent pile springs were modelled at the bottom of concrete foundation. Frequency
dependent damping ratios were considered as individual modal damping ratios.

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Fig. 10 - Finite Element Model
6.4 Unexpected Vibration due to Flexural Deformation
In general, the primary natural frequencies by Strand7 agreed with DYNA6 results (Table 2). No
primary resonances fell within the machine operational frequency. However, flexural deformation of
foundation appeared to be triggered by the unbalanced force of E-motor shown in Figure 11.
Table 2 – Natural Frequency with Mass Participation Ratio
Force vibration analysis was undertaken separately to evaluate the vibration response due to the E-
motor and compressors. Figure 12 presents the dynamic response of the machine foundation subjected
to the unbalance force due to E-motor or compressors. A clear frequency separation was achieved for
the dynamic response triggered by the unbalanced forces of compressors. The sleeved piles under-tuned
the horizontal resonance; and the relatively large diameter piles over-tuned the vertical resonance.

57
Fig. 11 - Flexural Deformation of Foundation at 39Hz
In contrast, the flexural natural frequency at 39Hz (Mode 4) was triggered by the unbalanced force of
E-motor, which was fairly closed to the machine operational frequency. Fortunately, with a further
adjustment of pile sleeve length, the flexural resonance due to E-motor was shifted towards the higher
frequency, and the total dynamic response satisfied the vendor vibration requirement shown in Figure
13.
Fig. 12 - Dynamic Responses due to E-motor and Compressors
Fig. 13 - Dynamic Responses with Flexible Behaviours of Foundation
0
1
2
3
4
5
6
0 20 40 60 80 100
Velocity(mm/s)
Frequency (Hz)
E motor Compressors
0
1
2
3
4
5
0 20 40 60 80 100 120 140 160 180 200 220 240
MaxVelocity(mm/s)
Frequency (Hz)
0.75mm/s limit

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7. CONCLUSIONS
Different design approaches for vibration control of machine foundations have been presented. Forced
vibration analysis is recommended for all critical machine foundation. Dynamic soil or rock properties
and unbalanced forces induced by the machine components are the important design parameters.
Dynamic soil-foundation interaction written in the form of impedance functions considers the soil
radiation damping and dynamic soil stiffness, however, the soil-foundation interface is assumed to be
rigid, and the dynamic effect of flexible structure might be underestimated.
A case study of using finite element model with dynamic soil stiffness and damping coefficient has
been presented. A sleeved pile foundation was adopted to avoid foundation resonances coincided with
the machine operational frequency. The dynamic effect of flexible machine foundation has been found
that it can be detrimental to the vibration control. Consequently, a finite element model is recommended
to consider the dynamic soil-foundation interaction for the vibration-sensitive machine foundation.
REFERENCES
ACI 351.1R-99 – Grouting between Foundations and Bases for Support of Equipment and Machinery.
Canadian Foundation Engineering Manual 4th
Edition – Canadian Geotechnical Society 2006.
Poulos, Harry G. "Behavior of laterally loaded piles: I-single piles." Journal of the Soil Mechanics and
Foundations Division 97.5 (1971): 711-731.
Günter Klein and Dietrich Klein (2003). Geotechnical Engineering Handbook, Procedures Vol.3 . Ernst
& Sohn.
Wolf John P. (1994) Foundation Vibration Analysis Using Simple Physical Models, PTR Prentice Hall.
Wolf John P. and Deeks Andrew J. (2004) Foundation Vibration Analysis: A Strength-of-materials
Approach, Elsevier.
Roesset Jose M. (2013) Soil Structure Interaction The Early Stages, Journal of Applied Science and
Engineering, Vol. 16, No. 1, pp. 1-8
Veletsos, A. S., & Wei, Y. T. (1971). Lateral and rocking vibration of footings. Journal of Soil
Mechanics & Foundations Div.
Kausel, E.(1974) “Forced Vibration of Circular Foundations on Layered Media” Sc. D. Thesis,
Massachusetts Institute of Technology.
Novak, M. (1974). Dynamic stiffness and damping of piles. Canadian Geotechnical Journal, 11(4),
574-598.
Luco, J. E., & Westmann, R. A. (1971). Dynamic response of circular footings. Journal of the
engineering mechanics division, 97(5), 1381-1395.
Luco, J. E. (1975). Impedance functions for a rigid foundation on a layered medium. Nuclear
Engineering and Design, 31(2), 204-217.
El Naggar, M. H., & Novak, M. (1994). Non-linear model for dynamic axial pile response. Journal of
geotechnical engineering, 120(2), 308-329.

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