M.R. Svinkin
Consulting Engineer
ABSTRACT: Numerical methods are presented to predict complete vibration response of the soil, buildings or sensitive instruments caused by anticipated, future vibration sources such as construction or heave industry. The suggested methods make use of either Duhamel’s integral or Fourier transforms and experimental soil response.
1 INTRODUCTION
Construction and industrial dynamic sources, such as pile driving and foundations for impact machines, generate elastic waves in soil which may adversely affect surrounding buildings and sensitive instruments (Targets). The effects of these waves range from visible structural damage to serious disturbance of working conditions for sensitive devices and people. Therefore, legitimate concerns frequently arise about possible ground and structure vibrations before the start of construction activities or installation of machine foundations.
Analytical methods (Miller and Pursey, 1954; Broers and Dieterman, 1992; Hanazato and Kishida, 1992; Wolf, 1994) already exist which give accurate results for certain limited cases, but these methods are applicable only to well defined and simple sites like a halfspace or horizontally layered media. Indeed, for the prediction of expected vibrations, it is necessary to have information about the actual soil deposit and to choose a proper soil model to compute vibrations. Computed results from the simple models contain valuable data about general tendencies of wave propagation at a site, but cannot take into account spatial variations of soil properties and produce accurate and complete soil vibration records at any point of interest.
This paper presents numerical methods coupled with experimental soil response measurements to predict ground and structure vibrations before the beginning of construction activities or installation of machine foundations. This approach employs experimental impulse response functions containing real behaviour of soil and structures without the investigation of soil and structure properties. It also provides an opportunity for accurate determination of vibration levels and aids in monitoring of ground, structure and device vibrations prior to start of construction and industrial activities.
2 DEFINITION OF METHODS
The suggested methods for predicting soil and building vibrations are founded on utilization of the impulse response functions technique for predicting complete vibration records on existing soils, buildings and equipment prior to installation of construction and industrial dynamic sources (Svinkin 1973, 1996). The impulse response function is an output signal of the system based on a single instantaneous impulse input (Bendat and Piersol 1993). These functions are applied for studies of complicated linear dynamic systems with unknown internal structures for which mathematical description is difficult or impossible. In the case under consideration, the dynamic system is the soil medium through which waves propagate outward from sources of construction and industrial vibrations. The input signal of the system is the impulse response of the ground at the place of pile driving, dynamic compaction of soil, or installation of a machine foundation; the output signal is the vibratory response of a location of interest situated on the surface or within the soil stratum, or any point at a building receiving vibrations. Output can be obtained, for example, as the vibration traces for displacements at locations of interest. Actually, these records are experimental Green’s functions.
Impulse response functions for the dynamic system being considered are determined by setting up an experiment. Such an approach does not require routine soil boring, sampling, or testing at the site where waves propagate from the vibration source, eliminates the need to use mathematical models of soil bases and structures in practical applications, and provides the flexibility of considering heterogeneity and variety of soil and structural properties. Unlike analytical methods, experimental impulse response functions reflect real behavior of soil and structures without direct investigation of the soil and structure properties. Because of that, the suggested methods have substantially greater capabilities in comparison with other existing methods.
The following is a general outline of the methods for predicting vibrations at a distance from an impact source. It is assumed that the dynamic loads transmitted onto the soil are known or can be found using existing theories. At the place in the field for installation of the wave source, impacts of known magnitude are applied onto the soil. The impact is often created using a rigid steel sphere or pearshaped mass falling from a mobile or bridge crane. The oscillations resulting from the impact on the soil are measured and recorded at the points of interest (target points), for example, at the locations of instruments sensitive to vibration, communication lines and other devices, etc. These oscillations are the impulse response functions (Green’s functions) of the treated dynamic system which automatically take into account complicated soil conditions. Predicted vibrations are computed using Duhamel’s integral or a Fourier transform.
3 APPLICATION OF DUHAMEL’S INTEGRAL
For each single output point, the considered input – soil medium – output system is a one degree of freedom system and predicted displacements can be written as follows
(1)
where
 F(t) = resultant dynamic force transmitted to the ground;
 x,y = coordinates of the output point under consideration at ground or structure;
 h_{z}(x,y,tt) = impulse response function at the output point under consideration;
 t = variable of integration.
Dynamic loads on a machine foundation can be found using existing foundation dynamics theories, for example Barkan (1962) and Richart et al. (1970). It is known that the equation of vertical damped vibrations of foundations for machines with dynamic loads can is given by
(2)
with initial conditions z = z_{0} and t = t_{0} for t = 0. In Equation (2),
 c = viscous damping coefficient;
 k_{z} = spring constant for the vertical mode of foundation vibrations;
 P(t) = exciting force;
 M = mass of foundation and machine.
Parameters of the foundationsoil system M, c and k_{z} are considered known in predicting vibrations.
Figure 1 Dynamic Forces on machine foundations and soil base
Equation (2) can be converted into another form as
(3)
with
(4)
where
 f _{nz} = natural frequency of vertical vibrations of foundation;
 a = effective damping constant.
An expression derived from Equations (3) and (4) for a dynamic load applied to the soil can be written as
(5)
The dynamic force transmitted from the machine foundation to the soil base (Figure 1) depends on the foundation and machine mass, the damping constant, natural frequency of vertical foundation vibrations and vertical foundation displacements as a function of time.
Substitution of Equation (5) to Equation (1) gives
(6)
For an arbitrary dynamic load, P(t), the total foundation displacement is
(7)
where f_{nd} = damped natural frequency of vertical vibrations of the foundation and
(8)
The general solution for determining dynamic displacements of points on the soil or in structures is obtained substituting Equation (7) to Equation (6)
(9)
3.1 Source with impact loads
Impact loads are transmitted to foundations from moulding machines, forge and drop hammers, and many construction operations like pile driving.
Vibration displacements of the source machine foundation can be assigned analytically as a damped sinusoid
(10)
with
(11)
where
 I_{F} = impulse force transmitted from machine to foundation;
 f = modulus of damping;
 k_{z }= coefficient of vertical subgrade reaction;
 A = contact area between foundation and soil.
According to Savinov (1979), the modulus of damping, f, ranges in a relatively narrow range and is slightly dependent on soil conditions. For instance, f values range from 0.004 to 0.008 sec for foundations with contact areas less than 10.0 m^{2}. Coefficient, k_{z¢ }is determined according to Barkan (1962). Also, it is possible to use other approaches for determining values of f and k_{z }.
After substitution of Equation (10) to Equation (6), vibration displacement at a target point is
(12)
3.2 Source with steady state vibration loads
A harmonic dynamic load applied to the foundation can be written as
(13)
where
 P_{0} = load amplitude;
 w = angular frequency.
Such loads are transmitted to foundations under various machines. The most prevalent powerful sources of steady state vibrations are compressors and crushing equipment.
The solution of Equation (3) with the right side of expression (13) is
(14)
with
(15)
After substitution of Equation (14) to Equation (6), vibration displacement in a target point is
(16)
Integration limits were taken (¥ , t) because steady state vibrations are considered.
3.3 Source with transient state vibration loads
Dynamic transient loads are transferred to a foundation from a vibroisolated block for a forge hammer. These dynamic loads can be represented as
(17)
with
(18)
where
 z_{1} = dynamic displacement of the vibroisolated block;
 z^{.}_{1} = dynamic velocity of the vibroisolated block;
 λ = natural frequency of vertical vibrations of the vibroisolated block;
 k_{b} = spring constant for the vertical mode of the vibroisolated block;
 M_{b} = mass of the vibroisolated block and machine;
 c_{b} = viscous damping coefficient of vibroisolation;
 β = damping constant of vibroisolation.
Parameters M_{b}, c_{b} and k_{b} are considered known.
Vibration displacements of the vibroisolated block can be assigned as
(19)
with
(20)
where I_{b} = impulse applied to the vibroisolated block; λ_{1} = damped natural frequency of the vibroisolated block.
Substitution of Equation (19) to Equation (17) gives
(21)
The duration of transient state vibrations is commensurate with the time of attenuation of foundation natural vibrations. For that reason determining dynamic loads transmitted to the soil, F(t), it is necessary to take into account natural foundation vibrations.
Next consider determination of the function F(t) in detail. A general integral of a linear nonhomogeneous Equation (3) with the right side equal to expression (21) is
(22)
where a partial integral will be found in a form
(23)
because the vibroisolated block parameter range eliminates the coincidence of β+iλ with roots of characteristic equation equal α+if_{nd}. The use of the method of indeterminate coefficients gives
(24)
(25)
Differentiation of Equation (22) gives
(26)
Substituting initial conditions to Equation (22) and (26), we obtain arbitrary constants c_{1} and c_{2}. After substitution these constants to Equation (22) and using expression (23), a general solution of Equation (3) is
(27)
The first term of the right side of Equation (27) presents the initial free displacement of a point under consideration determined by initial conditions and independent of the exciting force, the second term is excited free vibrations determined by the exciting force and independent of initial conditions, and third term is forced vibrations.
For zero initial conditions at the time of vibroisolated hammer operations, Equation (27) becomes
(28)
Equation (28) can be transformed as
(29)
with
(30)
(31)
Substituting expression (29) into Equation (6), we obtain vertical or, similarly, horizontal displacements of soil and structures as
(32)
The first integral represents the displacements of a point under consideration excited by free foundation vibrations, and the second one by forced foundation vibrations.
Coefficients in equation (32) are defined as follows
(33)
It is necessary to point out that displacements at target points depends only on parameters observed in experiments.
Duhamel’s integral was applied to compute ground surface vibrations at distance of 8.4 and 14.0 m from the foundation with an area of 80.0 m^{2} under a forge hammer at a site with clay soils. A falling weight of 7.25 tonnes produced the vibration records in Figure 2. The prediction was performed using various frequencies of natural vertical foundation vibrations obtained according to different theoretical approaches. Changes of this frequency affect predicted records only slightly. It can be seen in Figure 2 that measured and predicted records have very close shapes and the difference between maximum amplitudes is 925 % and 210 % at distances 8.4 and 14.0 m, respectively.
Figure 2 Measured (1) and Predicted (25) records of vertical soil vibrations excited by operating large forge hammer with falling mass of 7.25 tonnes
4 APPLICATION OF FOURIER TRANSFORM
Application of the direct Fourier transform to an impulse response function h_{z}(x,y,tτ) gives
(34)
For a real physical system, records can be measured only over some finite time interval T, so that S_{x,y}(iω) is estimated by computing the finite Fourier transform
(35)
The complex Fourier transform can be presented as
(36)
where


Figure 3 Vertical and horizontal amplitudes of soil vibrations at distances 16.6 and 23.3 m from the machine foundation: 1 – Measured vibrations; 2 – Predicted vibrations
In fact, the magnitude spectrum is the transfer function of the considered dynamic system: ground at the place of the dynamic source – soil medium through which waves propagate outward from the source – target point at any location of interest at the soil or in buildings. If the impact applied onto the soil is not instantaneous, the transfer function of the considered dynamic system can be obtained as a ratio of spectrum magnitudes of output to input.
A magnitude spectrum of the dynamic source, P_{s}, transmitted from the foundation onto the soil is
(37)
where


A predicted magnitude displacement spectrum in the target point with coordinates x and y is
(38)
Predicted vertical or, similarly, horizontal displacements of soil and structures as a function of time at the location under consideration may be derived using the inverse Fourier transform of S_{p}(x,y,ω)
(39)
For a source with steady state vibration loads, it is very easy to predict vibration amplitudes in target points. Predicting amplitude Z_{x,y} at target point can be find as
(40)
where


Figure 3 demonstrates examples of predicting amplitudes of soil vibrations from a steady state vibration source. The machine foundation with contact area of 15.1 m^{2} was installed at the site with typeI slumpprone soils (Svinkin & Zhuchkova 1972). The predicted amplitudes of soil vibrations matched well the measured vibration amplitudes excited by the vibration machine installed on the foundation. Comparison was performed at distances 16.6 and 23.2 m from the machine foundation in frequency range of 400800 rpm. Error margins were within 520%.
Comparison of the suggested numerical predicting methods shows a preference for Duhamel’s integral for sources with impact and transient state vibration loads because there are some difficulties in calculation of the inverse Fourier transform for expression with an unknown load function phase spectrum. Besides, Duhamel’s integral is almost insensitive to small changes of original curves.
5 CONCLUSIONS
Numerical methods coupled with experimental soil response measurements are used to predict soil and building vibrations before the installation of construction and industrial vibration sources. Such an approach does not require routine soil boring, sampling, or testing at the site where waves propagate from the vibration source.
Experimental Green’s functions reflect real soil and structure behavior and take into account spatial variations of soil properties. Because of that, the suggested methods have substantially greater capabilities in comparison with other existing methods.
ACKNOWLEDGEMENT
The writer is pleased to acknowledge special contributions to the paper made by Dr. Richard D. Woods, professor of civil engineering at the University of Michigan at Ann Arbor, USA.
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