Vibration of hammer foundations *

M. NOVAK and L. EL HIFNAWY

Faculty of Engineering Science, The University of Western Ontario, London, Canada N6A 5B9

Hammer foundations are often deeply embedded and highly damped. Yet, in the prediction of vibration amplitudes, damping is usually neglected. In this paper, two methods are proposed which make it possible to incorporate damping in the analysis. One method, suitable for hand calculations, is based on an energy consideration and the other, suitable for computer analysis, utilizes the notions of the complex eigenvalue problem. It is shown that damping derived from soil may reduce founda- tion amplitudes very significantly, e.g. 50%, particularly for large, embedded foundations.

Methods for the evaluation of foundation stiffness and damping which are needed in the analysis are outlined.

INTRODUCTION

Hammers and other types of shock producing machines generate powerful dynamic effects that are quite short in duration and can be characterized as pulses. Only a part of the shock energy is utilized in the forging process and the rest is dissipated in the foundation causing intense vibra- tion. This may result in settlement and cracking of the foundation and undesirable shaking of the vicinity. Thus, the prediction of the vibration is an important part of hammer foundation design. The basic approaches to the analysis of foundations for shock producing machines were formulated by Rausch, 1 Barkan 2 and a few others but apart from a few contributions, a'4 the area has been rather neg- lected in recent years. The methods of analysis of hammer foundations are well developed but suffer from inconsistent description of soil properties and particularly from the omission or arbitrary definition of damping.

This paper presents two methods of analysis in which damping is accounted for in a rational way: one method is suitable for hand calculation of systems with two degrees of freedom, most often used in practice; the other method can handle any number of degrees of freedom, which is desirable for foundations exposed to eccentric blows, but relies on the use of the computer.

HAMMER FOUNDATION SYSTEMS

There are many types of hammers. The description of the most common types can be found in Refs. 1, 2, 5 and 6. The basic elements of a typical hammer-foundation system are the frame, head (tup), anvil and the foundation block embedded in soil (Fig. 1). When vibration transmission is of particular concern, springs and dampers are used to support the foundation block and/or the anvil and in some more recent designs, the foundation (inertial) block is deleted. In these cases, a reinforced concrete trough is needed to protect the isolation elements from the environment.

The forging action of hammers is generated by the im- pact of the falling head against the anvil. To reduce the stress in the concrete and shock transmission into the frame, viscoelastic mounting of the anvil is usually pro- vided. This may have the form of a pad of hard industrial

* Paper taken from the proceedings of the first International Confer- ence and Exhibition on Soft Dynamics and Earthquake Engineering, Southampton, UK, 13-15 July 1982.

felt, a layer of hardwood or, with very powerful hammers, a set of springs and dampers.

The foundation block or the protective trough are either cast on soil or supported by piles. Examples of the main foundation types are described in Refs. 3, 5 and 6. The various types of foundations can be modeled by lumped mass systems shown in Fig. 2. The one mass model (a) can be used for a foundation with no elastic pad under the anvil; it is also adequate for a foundation with no inertial foundation block in which the anvil rests on springs and dashpots and the protective trough is rigidly supported. In the two mass model (b), the mass m l represents the elastically mounted anvil and me the foundation block sup- ported by soil or piles. With the directly sprung anvil, mass m~ represents the anvil and m2 the protective trough founded on soil or piles. Model (c) comprises the mass of the anvil, m l, mass of the block, m2 and mass of the trough, m3, all elastically supported.

If the blows are centric and the foundation arrangement is symmetrical, only vertical vibrations occur and the foundations have one, two or three degrees of freedom as indicated. With eccentric blows and/or asymmetrical arrangement of the system, horizontal translations and rotations of all masses occur and the number of degrees of freedom grows to three, six or nine for the three models shown in Fig. 2.

D

; l-~ ! - FRA.e i

l/I i I ~ ~ XX'--HEAD ,/,,/r',, ',

/f///11~x,\\xx

PAD ~ FOUNDATION ,~; . . . . ~.;;.-.....g" BLOCK

Fig. 1 Schematic of most common type o f hammer foundation

0261-7277/83/010043-11 $2.00 © 1983 CML Publications Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1 43

Vibration o f hammer foundations: A4. Novak and L. El Hifnawy

I

HEAD ~ mo i

..... ~'P

! Tv BLOCK m

8, A N V I L ..L

I DOF

(a)

Fig. 2.

L I

.EAo me 0 me I

i

ANVIL I mI I Tv, I m, ] T-v,

BLOCK m2 m2

-'Lc 2 k 2 ~ -Lc2~ k2

2 DOF 3 DOF

(b) (c) Basic mathematical models for hammer foundations

STIFFNESS AND DAMPING CONSTANTS OF THE SYSTEM

The prediction of the response of the hammer foundation requires the description of the stiffness and damping of the foundation and the pad under the anvil.

Shallow foundations Stiffness and damping of foundations supported on soil

can be evaluated using elastic halfspace theory. The prin- cipal advantages of this theory are that it accounts for energy dissipation through elastic waves (geometric damp- ing), provides for systematic analysis and describes soil properties by basic constants such as shear modulus or shear wave velocity which can be established by independent experiments. Based on the elastic halfspace theory, a number of approaches were formulated for the determina- tion of foundation stiffness and damping. They were recently reviewed and compared by Roesset. 7 For an em- bedded foundation, stiffness and damping constants can be evaluated using a simple, approximate approach, s'9 How- ever, some assumptions of the halfspace theory differ from real conditions and therefore, some adjustment of the theoretical results is necessary in order to account for the shape of the base, embedment, soil nonhomogeneity and limited thickness of the soil stratum.

The correction for the shape of the base can be made by the introduction of the equivalent radius of a circular base. For a rectangular base and vertical vibration, the equi- valent radius is re = V r d ~ in which a and b are the width and length of the base, respectively.

Experiments ~° indicate that for deep deposits, the half- space theory tends to overestimate the geometric damping of surface foundations in the vertical direction by as much as 100%. The reason for this discrepancy seems to be that soil usually features some layering which reflects elastic waves back to the foundation and reduces geometric damping. To be on the safe side, it appears advisable to divide the theoretical values of the damping due to energy radiation (geometric damping) by a suitable correction factor. Without such a correction, the vertical damping

often reaches very high values which may not materialize. A better way is to account for soft layering in a more rigorous way as shown by Kobori et al., 11 Warburton 12 or more comprehensively by Lueo. 13

Another correction involves embedment effects. The theory indicates that embedment provides a significant source of geometric damping and contributes also to stiff- ness. These theoretical suggestions were, in general, con- firmed by experiments. However, it was also observed that with the heavy vibration, typical of hammers, the soil may separate from the footing sides and a gap may occur. This gap is likely to develop close to the surface where the confining pressure is not sufficient to maintain the bond between the soil and the footing. One way of accounting for footing separation is to assume a slippage zone around the footingJ 4 A rough correction can be made by consider- ing an effective embedment depth smaller than the true one.

Soil material damping. Foundation stiffness and damping are also affected by soil material damping. Material damp- ing of soil is hysteretic and independent of frequency. It is conveniently described using the complex shear modulus

G* = G + iG' = G(1 + / t a n S ) (1)

in which G = the real shear modulus of soil, i = V'~-i, tan6 = G'/G with 6 = the loss angle and G ' = the imaginary part of the complex soil modulus, G*. Another measure of material damping is the damping ratio ~ = ½ tan 6. Material damping can be incorporated using the correspondence principle of viscoelasticity. In the sense of this principle, the shear modulus, G, has to be replaced by the complex shear modulus defined by equation (1). If only the explicitly appearing G is replaced by G*, the stiffness constant, k, and the damping constant, c, including material damping become approximately

kh = k -- 2flc60 (2)

c h = c + 2~k/60 (3)

in which k and c are evaluated without regard to material damping and 60 is circular frequency. With shallow layers, the incorporation of material damping is important because

44 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1

the geometric damping is quite small or may not material- ize at all if the first natural frequency of the hammer foundation is lower than the first natural frequency of the soft layer.

Dynamic properties of soil needed in the above ap- proaches can be established by experiments or est imated using published data. i s, 16

Pile foundations Stiffness and damping of foundations supported by piles

are readily available for endbearing piles in homogeneous soil,~ 7 floating piles in homogeneous soil ~ 8 and floating piles in soil with a parabolic variation of shear modulus with depth. 19 For an arbitrary soil profile they can be calculated using the technique described in Ref. 20. A correction for pile separation can be made as described in Ref. 14. When the piles are closely spaced, an allowance should be made for pile-soil-pile interaction effects (group effects). This factor is treated by Poulos and Davis, 21 Wolf and yon Arx, 22 Waas and Hartmann, 23 Sheta and Novak 24 and others. A review of the various methods available can be found in Roesset. 7

Pads and absorbers When the foundation block or the anvil rest on a pad of

viscoelastic material, the vertical stiffness constant of the pad is

kp = EpAp/h (4)

in which Ep = Young's modulus of the pad, Ap = area of the pad and h = its thickness. The damping constant can be calculated in terms of the complex Young's modulus and is, as in equation (3),

Cp = 2~pkp/t,o (5) where /3p = the damping ratio of the pad material and w = the frequency of the block or anvil vibration.

The stiffness in shear is analogously GpAp/h where Gp = shear modulus of the pad. Rocking stiffness is

kp, ~0 = Ep ]/h (6) in which I = second moment of pad area. Damping con- stants for shear and rocking are obtained as a fraction of stiffness, just as in equation (5).

It is obvious from equations (3) and (5) that frequency independent (hysteretic) material damping results in frequency dependent constants of equivalent viscous damping, c. If material damping is assumed to be viscous, the constant /3 can be replaced by/3'60, in which/3' is the constant of viscous damping and the resultant damping constant becomes frequency independent and equal to

P 2/3pkp.

METHODS OF ANALYSIS

The energy of the impact is determined by the weight of the head and its impact velocity. However, the founda- tion response to the impact also depends on the time history of the force resulting from the impact and trans- ferred to the anvil in the form of a pulse. This pulse is a transient force, P(t), of short duration, Tp. The time history of the pulse and its duration depend on the con- ditions of forging.

A rigorous analysis can be conducted by calculating the footing response for the given time history of the pulse.

Vibration o f hammer foundations: M. Novak and L. El Hifnawy

Methods for this type of analysis were presented for hammers by Rausch ~ and Lysmer and Richart 2s but any general computer oriented method, such as the Fast Fourier Transform or the Wilson 0-method, can be used very effec- tively. For simple pulse shapes, such as half-sine pulse or decaying sine pulse, the calculation can be facilitated by the use of shock spectra available. 4' m The problem is that the exact time history is rarely known and is actually to a high degree random.

Consider the effect of pulse duration. Because the duration of the pulse is very short, it may be possible for the preliminary considerations to replace the real pulse by a rectangular pulse having duration tp < Tp but the same power, i.e.

Po tp = i p P(t) d t

0

For a rectangular pulse, the response of a one mass system can readily be obtained from the Duhamel integral. With damping neglected for simplicity, the peak amplitudes of the response are plotted in Fig. 3. These peak amplitudes are shown for rectangular pulses having different durations, tp, but the same power Potp taken as unity (a unit pulse). The duration of the pulse is expressed as a fraction of the natural period of the system T = 21r/6oo also taken as unity. It can be seen that the peak response decreases as the unit pulse duration increases and that for the ratios tp/T lower than about 0.1, the peak response is practically independent of pulse duration and equal to that obtained with an infinitely short pulse.

The duration of the pulse of hammers is quite short, in the order of 0.01 to 0.02 s. For the most severe impact, which occurs with the blank absent, the pulse duration may be even shorter, about 0.001 or 0.002. 27 Thus, it appears possible to predict the response using the assumption of an infinitely short pulse which is tantamount to the assump- tion that the response is caused by an initial velocity im- parted by the impact. For pulses of longer duration, this assumption is conservative because the response decreases with increasing duration of the pulse.

The approximation which this assumption implies is further justified by a more severe approximation made in almost all hammer response analysis: it is presumed that the anvil remains in full contact with the underlying pad. In reality, the static stress is usually smaller than the dynamic stress and consequently, uplift of the anvil occurs. This aspect is, as a rule, neglected because its inclusion would considerably complicate the analysis.

These two assumptions, i.e. infinitely short pulse and no anvil uplift, are adopted in this paper. The response of the most common two mass model shown in Fig. 2b is investi- gated first. The response of a one mass model in one degree of freedom follows from the elementary theory of vibration. A few examples are given in Ref. 3.

RESPONSE OF TWO MASS FOUNDATIONS IN TWO DEGREES OF FREEDOM

Most hammer foundations are adequately represented by a two mass model shown in Fig. 2b and reproduced in Fig. 4a. With centric blows and a symmetrical arrangement, the model has two degrees of freedom with m~ = mass of the anvil whose displacement is v~ and m2 = mass of the founda-

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1 45

Vibration o f hammer foundations: M. Novak and L. E1 Hifnawy

tion block whose displacement is v2. Denoting the displace- ment vector

{v}= [v, vs] r (7)

the mass matrix is

[ ,m] = [ m , 0 ]

0 ms

and the stiffness and damping matrices are

(8)

=[21 kl + k s ] (9)

[ cl --cl ] (10) [c]=-c~ c~ +c~

Constants ks, C 2 describe stiffness and damping of the foundation. They are available in the references given above and are described in more detail in Ref. 3. Constants k~, cl describe the properties of the anvil pad calculated from equations (4) and (5). With rubber elements or springs with dampers, these constants are determined accordingly.

The governing equations of the system are

['m ~ {9} + [c]{~}+[k]iv}={O} (11)

in which ~ = dSv/dt 2 and (J = d~/dt. The response results from the impact of the hammer

head which issues to the anvil the initial velocity ~)l(0) = d. This velocity follows from the basic formula for collision as

m o e = (1 +kr) ~ Co (12)

mo + m r

where too, Co are the mass and impact velocity of the hammer head respectively and kr is the coefficient of resti- tution, usually taken as 0.5 or so. Applying this initial condition to equation (11), the damped response can be analyzed. It is, however, more convenient to analyze the undamped vibration first.

Undamped vibration The solution for the undamped response is well known

and can be written as

{v l ( t ) l={vH]sin601t+[v12ts in602t (13) vs(t)l Vs~/ tvssl

x x 6 o

W u) 4 Z o o3 ~ 2 Y

W

0 0

~ . 3.

p(t) t P(t)~

I I I I I I I I I I

.2 .4 .6 .8 1.0

PULSE DURATION / PERIOD ( t P / T )

Peak response to rectangular pulse o f unit power vs. pulse duration (Pox tp = 1, Po = l / tp)

HEAD i me MODE 1 MODE 2

' [ ~ 0), ~z ANV,L -Vv, r . . . . . 7, Iv . ~ Tv,z

BLOCK m z

(a) (b) (c)

Fig. 4. Two mass hammer foundation and its vibration modes

in which the two natural frequencies are

: C 6 0 " = - 2 \ m , ms/ ;a / -4[ - -~ l --~2] +'m,m: (14)

The two corresponding ratios of displacements representing the undamped vibration modes are

Vl] --kzs ks2--m260 7 a/= - = f o r / = l o r 2 (15)

v21 kH--m160~ --k21

In the displacement vq, the first subscript identifies the amplitudes of mass ml or ms while the second subscript indicates the frequency and mode with which the ampli- tude vi/ is associated. The two vibration modes are shown in Fig. 4. Denoting the natural frequency of the anvil in case of a rigidly supported foundation block

60a = ~/k, /m, (16)

the vibration amplitudes appearing in equation (13) are 2 2 602 - - 60a

VH = - - (17a) 2 2 601 602 - - 601

s s 60a - - 601 v12 = (17b) 2 2

602 602 - - 601

for the anvil and

7)21 = - -

2 2 2 2 (~= - ~ , ) ( ~ - ~°) ~ 60160~ 6°2: -- 6o21 = vll 1 ----60~ (17c)

2 2 2 2 (60=-60,) (60s-60~) 60, •S2= ~'t~2 ~Oa 2 OJ~ - - ~012 = - - V 2 1 - - COs ( I '7d)

for the foundation. With these amplitudes, the undamped response of both the anvil and the foundation can be calculated from equation (13). Examples of the undamped response are shown in Figs. 5a, 6a and 7a. Figures 5 and 6 were computed for an embedded foundation of the type shown in Fig. 1. The basic dimensions are shown in Fig. 8 and other data are given in Ref. 3. The anvil pad is visco- elastic, 0.5 m and 0.125 m thick respectively. The soil shear wave velocity, Vs, is 152 m/s. Both vibration modes partici- pate in the response but the contribution of the second mode and the general character of the response depend on the stiffness of the anvil pad. The reduction in pad thickness increases the second natural frequency, reduces the ampli- tudes of the anvil and suppresses the contribution of the second mode to the vibration of the footing (Fig. 6a).

Figure 7a shows the undamped response of the same hammer foundation but supported by 15 timber piles. The piles are the same as those described in the example in Ref. 17. The soil shear wave velocity is 100 m/s in this

46 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1

Vibration o f hammer foundations: M. Novak and L. El Hifnawy

case. The pries increase the fundamental natural frequency, w~, despite the lower shear wave velocity as may be expected.

Figures 5a, 6a and 7a show the undamped response. However, the overall character of the response and the

~- / t ~ ANVIL, v= (t)

/ \

, , , ,

~._l ¢o) UNDAMPED ~' I

w ii

l D _

ANVIL, v t ( t ) , vz(t)

. ~ . Yo. ~ ~2o. TIME T• I£-3

(b) DAMPED Fig. 5. Response o f two mass hammer foundation; em- bedded footing, anvil pad 0.Sin thick: {a) undamped, (b) damped (D1 =51%, D2 = Z4%). (Displacements in in., I in. = 2.54 cm, time in seconds}

P 0

LA-

w,;, ~'C

B O

I

S ANVIL, v, (t)

(o) UNDAMPED

o

? w

s ANVIL, v= (t)

~ D A T I O N , v:~ ( 1 )

/V

(b) DAMPED Fig. 6. Response o f two mass hammer foundation; em- bedded footing, anvil pad 0.125 m thick, damping o f soil reduced to ¼: (a) undamped, (b) damped (DI = 12.9%, D 2 = 5.3%). (Displacements in in., 1 in. = 2.54cmj

- - - ~ - ANVIL, v, (f)

~ ?0.. '- ~ • ' •

~ i t (O') UNDAMPED I

o

F ANVIL, v=(t)

euu;-/r~/.~"-/OUNDATION' v2 ( ' )

. . .

(b) DAMPED Fig. 7. Response o f two mass hammer foundation sup- ported by 15 timber piles with anvil pad 0.5 m thick: (a} undamped, (b) damped (1)1 =10.1%, D2 =5.8%}. (Displace- ments in in., I in. = 2.54cm, time in seconds}

magnitude of its amplitudes can be greatly affected by damping. This is shown in the next paragraph in which a simple way of incorporating damping in the analysis is presented.

Damped vibration In the two mass model, damping is defined by the

constants cl and c2 (Fig. 4a). These constants could be considered in the governing equations of the motion, equation (11), but it is more convenient to predict the damped response approximately using the notions of modal analysis and modal damping.

The foundation response comprises the two vibration modes shown in Fig. 4. The damping ratio associated with vibration in each of these modes can be evaluated by means of an energy consideration if it is assumed that the damped mode is approximately the same as the undamped mode.

The work done durin.g a period of vibration T = 21t[t~ I by the damping force P(8) is, in general,

T

W = f P(~) d~(t)

o

( i s )

in which ~ = the relative displacement between the bodies to which the dashpot is attached. For the hammer founda- tion shown in Fig. 4, vibrating in mode / with natural frequency ¢o],

By(t) = 8, I sin ~jt, 82/(t) = 8 2 / ~ ~ / t (19)

in which the relative amplitudes are

6 V = vV--v2i , 621=v2! (20)

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1 47

Vibration o f hammer foundations: M. Novak and L. El Hifnawy

The damping forces generated by the dashpots are

P(~I) = c1~ I = c1~1/601 cos colt (21)

P(~2) = c2~2/= c2~2160i cos COlt

The total work done by these forces follows from equation (18) as

I4/= n60/[cl (vii -- v2/) 2 + c2 v21] (22)

In free vibration, the maximum potential energy of the whole system is the same as the maximum kinetic energy and is

1 2 2 2 - - 1 2 L = 2(mlvq + m2v2/) 60] - 2M/60] (23)

where the generalized mass of mode j

hi~ = rely2~ + m2 v~/ (24)

Then, the damping ratio is deemed as I) /= W/47rL. This yields the damping ratio associated with the foundation vibration in the/ th mode,

1 1)/= ~ [cl(vq--v2/) 2 +c2v~/] (25)

2601Mj where / = 1, 2. In equation (25), cl is the damping constant of the anvil and c2 is the damping constant of the founda- tion. The frequencies appearing in these equations are the natural frequencies 6ol and 602 calculated from equation (14). Thus, for each natural mode, one set o f c , c: may be necessary due to converting a constant hysteretic damping to equivalent viscous damping if material is not considered as viscous. The amplitudes v q and v21 are the undamped amplitudes given by equation (17). Alternatively, arbitrary modal amplitudes complying with equation (15) may be used in equation (25); in this ease, one amplitude can be chosen for each mode, e.g. v~ i = 1, and the other calcu- lated using the ratio a i.

If the frequency 602 ~ 601, which can be the case with a stiff anvil pad, then also vll ~ v21, v22 "~ v12 and

601~- x/k2/(ml +m:), 602 ~ ~

Consequently, equation (25) simplifies to approximate expressions for modal damping ratios

e2 Cl

Dl 2x/k2(ml+m2) D2 2 kx/'k'~lm~ /3t' (26)

Because equations (13) represent the superposition of vibration modes, the damped vibration of the anvil and the foundation block can be written as

{ vl(t) / = [ v l l / exp(--D160xt)sin60'lt v2(t)J ~ v211

+{v12} exp(--O,60:t) sin60'2t (27) ~)22

in which the damped natural frequencies

60; = 6o i ~ (28)

and v~/, v2/are the undamped amplitudes established from equations (17); the damping ratios D i are given by equa- tions (25) or (26). For hand calculations, approximate expressions for the estimate of peak values of the response can easily be deduced from equation (27))

In Figs. 5b, 6b and 7b examples of damped response, calculated using equation (27), are plotted. The founda- tions considered are the same as those assumed for un-

damped vibration. The only difference is the inclusion of soil and anvil pad damping.

In the example shown in Fig. 5, the foundation is em- bedded. The resultant damping is very high for the funda- mental mode, 51%, and 7.4% for the second mode. The response amplitudes are reduced to about one half of the undamped amplitudes. Such a reduction is practically significant because it may suggest a better performance of the foundation than might be expected if damping is neglected. A redesign of the foundation, which might appear necessary, may not be actually needed.

Figure 6b shows the damped response of the embedded foundation with the stiffer anvil pad. The soil damping constant, "c2, was reduced to one fourth of the value used for Fig. 5. This reduction in soil damping, which might be caused by soil layering, results in the reduction of the damping ratio of the fundamental mode to 1 2.9%, about in proportion to the reduction of soil damping. The second mode damping is 5.3%, i.e. only slightly higher than the assumed anvil pad material damping ratio of 5% and less than the 7.4% obtained with full soil damping.

With pile foundations, the character of the response is different. Piles provide greater stiffness but smaller damping as was demonstrated for periodic excitation in Ref. 17. An example of the damped, transient response of a pile sup- ported hammer foundation is shown in Fig. 7b. The foundation analyzed in this example is the same as that assumed in Fig. 7a except for the inclusion of soil and anvil pad damping. The damping of the first mode is only 10.1% which is much less than the 51% predicted for the em- bedded foundation. Accordingly, the reduction of the first peak ampltiudes of the anvil and the foundation due to damping is less significant than for shallow foundations.

More complicated systems. The above approach to damped vibration can readily be extended to include more complicated systems. When the foundation block is sup- ported by absorbers and protected by a trough, a three mass model shown in Fig. 2c can be used to analyze the response.

First, the eigenvalue problem of the system is solved to yield three vibration modes with amplitudes vi/ and fre- quencies 60/. The modal damping is produced by the three dashpots. The modal damping associated with the three vibration modes is obtained by extension of the energy argument, becoming

1 1 ) i - - -

260iM /

in which

[cl (vl/--vz/) z + ez(v2/--v3/) 2 + c3v~/] (29)

3

Mj = E (30) 1=1

and ] = 1, 2 and 3. The damped response of mass mt can be written as

3 V/(t)= ~. vqexp(--Di601t)sin60'lt (31)

1=1

in which vii are the undamped amplitudes obtained from the initial conditions and the modal ratios. (The undamped amplitudes can also be used in equation (29) in lieu of modal displacements.) However, while this procedure is applicable to the more complicated systems it is not com- putat ional ly as advantageous as in the case of two degrees

48 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1

of freedom. For three degrees of freedom .or more, the computer-based procedure described later herein is more efficient.

Complex eigenvalue approach. The damping ratios, 19/, obtained from the energy consideration can be verified by introducing the damping constants, ci, into the governing equations of the motion, equation (11), and solving the complex eigenvalue problem. From the complex eigen- values, mathematically accurate values of modal damping ratios can be established. The complex eigenvalue analysis, also used later herein, is treated in Refs. 28, 29 and 30, and the extraction of modal damping is described in detail in Ref. 31. In this procedure, the solution to equation (11) is written in the form

{v} = A eXt{(I)} (32)

in which A is a complex constant, X a complex eigenvalue (frequency) and {¢b} the complex vector of modal displace- ments. Substitution of equation (32) in to equation (11) leads to the complex eigenvalue problem whose solution yields the complex eigenvalues. These eigenvalues can be written for each vibration mode j as

X/= --D/~j +iCo/ X/1 --D] (33)

in which cb/= IX/I is a frequency close to the undamped natural frequency 60/but generally not equal to it, i = x / ' ~ and D 1 is the damping ratio of mode j. It follows from equation (33) that the modal damping ratio can be calcu- lated as

19] = --ReX//IXil (34)

Modal damping of the foundations used in the examples shown in Figs. 5, 6 and 7 was evaluated by means of equation (34) and compared with the values ascertained by means of the energy consideration, i.e. equation (25). The agreement between the results of the two methods was in all cases excellent even for the highest damping ratio of 51%. Thus, it may be concluded that the more convenient method of evaluating modal damping by means of the energy consideration is sufficiently accurate for hammer foundations. For other types of structures, the energy method may be less accurate as shown in Refs. 31 and 32.

ANALYSIS OF RESPONSE USING COMPLEX EIGENVALUES

When the blows, centres of gravity and elastic centres do not occur on a common vertical line, each mass, m l, under- goes a vertical displacement, vi, a horizontal displaeemant, ui, and a rotation in the vertical plane, ~bt. (Other displace- ments need not be considered because there usually is a vertical plane of symmetry passing through the axis X.) Then, the systems indicated in Fig. 2 have three, six or nine degrees of freedom. An example of such a foundation is shown in Fig. 8. In such situations, the solution of hammer foundation response based on undamped modes loses its advantage of simplicity and an analysis utilizing damped modes and complex eigenvalues becomes preferable. This type of analysis is outlined in the following paragraph.

Governing equations o.f motion The first step of the analysis is to expand the mass,

stiffness and damping matrices of the system to accom- modate horizontal translations and rotations in the vertical

Vibration of hammer foundations: M. Novak and L. El Hifnawy

i

~ ~, o ~ "2

/ / / / / / / / / / \ \ \ \ \ \ \ \ \

1_ 6 .56 _L 6.56 _1_ 3.28 . i (2 .00) - I - (2 .00) - I - ( I .O0)-

FT 16.40 ( m ) ( 5 . 00 ) "

x , x \ \ \ \ \ \ \ \

l

-F

~ ,~:S . . . . . . . . . . .

_ L

o~

Fig. 8. Schematic of hammer .foundation with eccen~- cally mounted anvil

plane (rocking). When the basic characteristics of the shallow or pile foundation are established using the theories referred to above, the determination of the matrices needed is straight forward. For a two mass system with six degrees of freedom, the procedure is outlined in Appendix I. Assume, therefore, that these matrices are known. Then, the governing equation of motion is

[m](6} + [c]{zi) + [k]{u} = {P(t)} (35)

in which the displacement vector

{u} = [ulvt ~klu2v2 ~k2---]r (36)

The force vector {P(t)} describes the impulse,if the response is to be analyzed for a given time history of the load. For a pulse acting with eccentricity e relative to mass m l, the force vector is

{P( t ) }= [O P(t) eP(t) 0 0 . . . ] r

If the response is treated as free vibration triggered by initial velocity {P(t)} = {0}. Both approaches are considered in this section for completeness of the analysis.

Complex eigenvalue method Equation (35) is a set of n differential equations of the second order if n is the number of degrees of freedom. It was shown by Frazer et al. 28 that the solution of these equations can be facilitated by transforming them into 2n differential equations of the first order,

[A]{$} + [B]{z} = {F(t)} (37)

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1 49

Vibration of hammer foundations: M. Novak and L. El Hifnawy

in which

[0] I'm] -EmJ [01 [A]= , [B]= (37a)

EmJ [c] [0] [k] and

(u} (0} {z)= { u } ' {F(t)}= {P(t)} (37b)

The vector {z} comprises the vector of velocities {fi} and the vector of displacements {u} and hence its order is 2n.

Free vibration. For free vibration {F(t)} = {0} and a particular solution to equation (37) can be written as

{z(t)} = exp (~,t) {~} (38)

in which ~ is a complex eigenvalue to be determined and {~} a complex eigenvector of order 2n independent of time. Denoting the identity matrix [I] and substituting equation (38) into equation (37),

(--[A]-' [BI -- )t [II) (q~} = {0} (39)

Equation (39) constitutes the complex eigenvalue prob- lem whose solution by means of a suitable subroutine yields 2n eigenvalues, ?V, and corresponding eigenvectors {~/} for ] = 1, 2, . . . , 2n. The eigenvalues come out in complex conjugate pairs with corresponding pairs of com- plex conjugate modes. All eigenvectors can be assembled in a square modal matrix [~], having the individual eigen- vectors as columns. The eigenvectors are orthogonal which leads to the following orthogonality conditions for damped modes: 29, 33

[~] r [a ] [~] = [',4/~ (40a)

[~] 7" [B] [~] = tBl ] (40b)

in which EAI ] and [`B/,] are diagonal matrices; the elements s~ = x/&.

Forced vibration. The response to a given pulse can be obtained from equation (37) by means of the complex eigelavectors. Introducing the linear transformation

{z(t)) = [~] {q(t)} (41)

in which qi( t )=q/ is a function of time, equation (37) becomes

[A] [~]{q} + [B] [~]{q) = {F(t)} (42)

Premultiplying by [~]T and applying the orthogonality conditions, equations (40), equation (42) reduces to

[A/] {q} -- tB/] {q} = [~]T{F(t)} (43)

Since both [A/,] and [`B/] are diagonal matrices, equa- tions (43) decouple into independent equations

( l / - -Xlq/=fl( t ) /a / ] = 1 , 2 . . . . ,2n (44)

in which

3~(t) = {~/) T{F(t)} (45)

The complete solution to equation (44) is established as a sum of the homogeneous solution, given by the initial

condition q/(0), and a particular integral of the nonhomo- geneous equation. Thus, the complete solution for q/is

t

(t) = q/(0) exp (k/t) + ~ exp [ kl (t -- r)] h (r) q~ dz/A I

0

(46)

The first term of this equation can be used to describe the response of the hammer foundation to initial velocities caused by the impact of the head; the second term can be used to evaluate the response to the pulse given by its time history.

For the application of the initial conditions only the initial velocities of the anvil are nonzero giving

(,~(0)}=[0 vl ~, 0 0 . . . ] r

and {u(0)} = {0}. These initial conditions determine the initial values of {z(t)} by equation (37b) and then, the initial values ofql(0 ) follow from equation (41). The initial velocity I)1 = d is given by equation (12). The initial angular velocity follows from the well known expression

e ~1 = (1 + kr) co (47)

i~(1 + ml/mo) + e 2

in which e is the blow eccentricity relative to mass ml, Co is the impact velocity of the head and i~ = Ii/m~ is the square of the radius of gyration. The mass ml is the mass of the anvil in multi-mass systems or the total mass in one mass systems (Fig. 9).

With ql established from equation (46), {z(t)} follows from equation (41) and this determines the displacements {u} by equation (37b).

When solving the response to a pulse described by its time history the integral part of equation (46) is used. For simple shapes of the pulse, the integral can be evaluated analytically. For more complicated pulses numerical integration is always feasible. The homogeneous solution may be applied to describe the response for time exceeding the duration of the pulse.

The principal advantage of the complex eigenvalue approach is that it incorporates the effect of damping in a rigorous way. Other details on the complex eigenvalue analysis can be found in Refs. 28, 29, 33 and 34. The basics of this approach were already discussed by Rayleigh. 3°

m ~ U

I

/ / / / / / / f / / / / / f / /

3 DOF Fig. 9. One mass system with eccentric hammer blows generating response in three degrees of freedom

50 Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1

Vibration of hammer foundations: M. Novak and L. E! Hifnawy

Examples First, the complex eigenvalue method was used to

analyze the response of symmetrical, two mass hammer foundations with two degrees of freedom. When applied to the cases of shallow and pile foundations shown in Figs. 5, 6 and 7, the complex eigenvalue approach yielded results which are practically identical with those obtained by means of. undamped modes and modal damping. This agree- ment further supports the validity of the modal analyses using modal damping.

In another example, the complex eigenvalue method was used to analyze the response of a hammer foundation with eccentrically mounted anvil shown in Fig. 8. The basic dimensions of the foundation are given in the figure. The foundation differs from that used in Figs. 5, 6 and 7 only by the eccentric position of the anvil and consequently, by the eccentricity of the hammer blows relative to the foundation block. The reference points are the centre of gravity of both the anvil and the foundation block with the frame of the hammer. The system has six degrees of freedom as indicated in Fig. 8. As in the other examples, the initial velocity approach was used.

The response calculated using the first term of equation (46) is shown in Fig. 10. The vertical response shown in Fig. lOa can be compared with Fig. 5b. While the level of the vertical displacements is not changed very much by the eccentricity of the blows, the general character of the response is different. The effective damping of the funda- mental mode is very high (90%) because it is increased by the contribution from rocking and sliding and that is why the response does not oscillate around the equilibrium position. The horizontal response is quite significant for

2-

laJ

M

^ ~ - - v, (t)

( o ) VERTICAL

~';"-J ~ - u ~ ( t ) / / \ /-"2('L_. ~ I I I I , - oL--- r,,E

<~..J (b) HORIZONTAL

"~ ~(t) ( , ,

Z 0 _ . _ _ / ' , , /

.

(c) ROCKING

Fig. ] 1. Vertical, horizontal and rocking components o f response of symmetrical hammer foundation to eccentric blow. (Response in in. and radians; 1 in. = 2.54 cm, time in s)

vl ( t )

0- 0

LD_ (o ) VERTICAL

~- 0 6 I 2d. d. u,E -'1E-3 0d. ld0. 12h.

( b ) HORIZONTAL

20. t0. 60. O0. 100. 120. TlhE TILE-@

(c ) ROCKING

Fig. 10. Vertical, horizontal and rocking components of response of asymmetrical hammer foundation shown in Fig. 8. (Response in in. and radians; 1 in. = 2.54 cm, time in s)

both the anvil and the foundation. The rocking, although not very large, translates into vertical and horizontal dis- placements comparable with the others.

The analysis was repeated for a foundation in which the centre of gravity of the anvil is located on the centreline of the foundation block with all the other data the same as in Fig. 8. However, the hammer blows were considered as eccentric with eccentricity e = 0.5 m. This would be a rare situation for a two mass hammer but it was chosen to illus- trate a case in which initial rotation of the anvil occurs. This rotation follows from equation (47). The response obtained is shown in Fig. 11. In this case, the fundamental vibration mode is less severely damped and the rocking displacements of the anvil are much greater than in the previous case of centric blows against the eccentrically located anvil.

CONCLUSIONS

Two approaches are presented that make it possible to predict damped response of hammer foundations. The first approach, suitable for longhand analysis of two mass systems with two degrees of freedom, is based on the evaluation of modal damping by means of undamped vibration modes.

The second approach is based on complex eigenvalues and damped vibration modes. With the use of the com- puter, it can be applied to complicated hammer founda- tions and particularly to those in which asymmetry of the arrangement or eccentricity of the blows calls for the

Soil Dynamics and Earthquake Engineering, 1983, Voi. 2, No. 1 51

Vibration o f hammer foundations: M. Novak and L. El Hifnawy

consideration of rocking and horizontal translation in addition to vertical vibration.

Both methods were used to analyze a few shallow and pile foundations. This analysis suggests the following observations:

Modal damping evaluated on the basis of an energy con- sideration agrees very well with that obtained from complex eigenvalues.

With eccentric blows, significant horizontal vibration as well as rocking may be generated.

Pile foundations provide less damping than shallow founda- tions.

In the case of shallow foundations, the damping of the anvil is increased by the damping due to soil.

ACKNOWLEDGEMENT

The study was supported by a grant from the Natural Science and Engineering Research Council of Canada.

R E F E R E N C E S

1 Rausch, E. Maschinen Fundamente, VDI-Vedag, Dusseldorf (in German), 1950, Chap. 6, pp. 107-232.

2 Barkan, D. D. Dynamics o f Eases and Foundations, McGraw- Hill Book Co., Inc., 1962, Chap. 5, pp. 185-241.

3 Novak, M. Foundations for shock-producing machines, Can. Geotech. J., 1983, No. 1.

4 Rivin, E. I. Design of vibration isolation systems for forging hammers, Sound and Vibration, April 1978.

5 Major, A. Vibration Analysis and Design o f Foundations for Machines and Turbines, Collet's Holdings Ltd, London, 1962, Chaps. XII and XIII, pp. 221-69.

6 Srinivasulu, P. and Vaidyanathan, C. V. Handbook o f Machine Foundations, Tata McGraw-Hill PubL Co. Ltd, New Delhi, 1976, Chap. 4, pp. 103-34.

7 Roesset, J. M. Stiffness and damping coefficients of founda- tions, Dynamic Response of Pile Foundations: Analytical Aspects, Proc. of a Specialty Session, ASCE National Conven- tion, Florida, 1980, pp. 1-30.

8 Novak, M. Effect of soil on structural response to wind and earthquake, Int. J. Earthquake Engrg and Struc. Dyn., 1974, 3 (1), 79.

9 Novak, M. and Beredugo, Y. O. Vertical vibration of embedded footings, J. Soil Mecl~ and Found. Div., ASCE, 1972, No. SM12, 1291.

10 Novak, M. Prediction of footing vibrations, J. Soil Mech. and Found. Div., ASCE, 1970, 96, No. SM3, 837

11 Kobori, T., Minai, R. and Suzuki, T. The dynamic ground com- pliance of a rectangular foundation on a viscoelastic stratum, Bull. Disaster Prevention Research Inst., Kyoto University, 1971, 20, 289

12 Warburton, G. B. Forced vibration of a body on an elastic stratum, J. Appl. Mech., 1957, 24, 55

13 Luco, J. E. Vibrations of a rigid disc on a Layered viscoelastic medium, Nucl. Eng. and Design, 1976, 36, 325

14 Novak, M. and Sheta, M. Approximate approach to contact problems of piles, Proc. o f Geotechnical Engineering Div., ASCE National Convention 'Dynamic Response o f Pile Foundations: Analytical Aspects', Florida, 30 October 1980, pp. 53-79

15 Kim, T. C. and Novak, M, Dynamic properties of some cohesive soils of Ontario', Cart Geotech. J., 1981, 18 (3), 371

16 Richart, F. E., Hall, J. R. and Woods, R. D. Vibrations o f Soils and Foundations, Prentice-Hall, Inc., 1970, 414 pp.

17 Novak, M. Dynamic stiffness and damping of piles, Can. Geo- tech. J., 1974, II, 574

18 Novak, M. Vertical vibration of floating pries, J. Eng. Mech. Div., ASCE, 1977, 103, No. EMI, 153

19 Novak, M. and Ahoul-Ella, F. Stiffness and damping of piles in Layered media, Proc. o f the Earthquake Engineering and Soil Dynamics, ASCE Specialty Conference, Pasadena, California, 19-21 June 1978, pp. 704-19.

20 Novak, M. and Aboul-Ella, F. Impedance functions of pries in Layered media, J. Engrg. Mech. lh'v., ASCE, 1978, 104, No. EM3, Proceedings Paper 13847, 643

21 Poulos, H. G. and Davis, E. H. Pile Foundation Analysis and Design, John Wiley & Sons, 1980, p. 397

22 Wolf, J. P. and yon Arx, G. A. Impedance functions of a group of vertical pries, Proc. ASCE Specialty Conference on Earth- quake Engineering and Soil Dynamics, Pasadena, 1978, pp. 1024-41

23 Waas, G. and Hartmann, H. G. Analysis o f Pile Foundations Under Dynamic Loads, SMIRT, Paris, 1981, p. 10

24 Sheta, M. and Novak, M. Vertical vibration of pile groups, J. Geotech. Engrg. Zh'v., ASCE, 1982, GT4,570-590

25 Lysmer, J. and Riehart, F. E. Dynamic response of footings to vertical loading, J. Soil Mech. Found. Div., ASCE, 1966, 92, SMP, 65-91

26 Harris, C. M. and Crede, C. E. Shock and Vibration Handbook, McGraw-Hill, Chap. 8, 1976

27 Belyaev, U. V. Peak loads in the co-impacting parts of a ham- mer, Kusnetchno-Shtampovotchnoye Proisvodstvo (Metal Stamping ProductionJ, 1970, No. 8 (in Russian)

28 Frazer, R. A., Duncan, W. J. and Collar, A. R. Elementary Matrices, Cambridge University Press, London, England, 1946, p. 326.

29 Pipes, L. A. and Hovanessian, S. A. Matrix-Computer Methods in Engineering, John Wiley & Sons, Chap. 8, 1969

30 Rayleigh, Lord, The Theory of Sound, Vol. 1, Second Ed., New York, Dover Publications, 1945 (original edition 1877).

31 Novak, M. and El Hifnawy, L. Effect of soil-structure inter- action on damping of structures, Res. Report BLWT-4-1982, Faculty of Engineering Science, University of Western Ontario, p. 33

32 Novak, M. and El Hifnawy, L. Damping of structures due to soil-structure interaction, Proc. 5th Colloquium on Industrial Aerodynamics, Aachen, Germany, June 1982, Part 2, pp. 25-36

33 Foss, K. A. Coordinates which uncouple the equations of motion of damped linear dynamic systems, J. Appl. Mech., ASME, 1958, 25, 361

34 Traill-Nash, R. W. Modal methods in the dynamics of systems with non-classical damping, Int. J. Earthquake Engrg. and Struct. Dyn., 1981, 9, 153

APPENDIX I - S T I F F N E S S A N D DAMPING CONSTANTS OF HAMMER FOUNDATIONS

As an example of the formulation of the stiffness and damping matrices appearing in equation (35) a two mass foundation asymmetrically arranged, such as the one in Fig. 8, is considered.

The numbering of the displacements, the geometry and the reference points are indicated in Fig. 12. The anvil is symmetrical but the foundation is not. There are six degrees of freedom and thus, the stiffness and damping matrices are 6 x 6 with the elements k i /and cii as follows:

kl l = ku~ kl2 = 0 k13 = - k u , y l

k14 = --ku I kls = 0 k16 = - - k u t y 2

k21 = 0 k22 = kv~ k2a = 0

¥

t5 , I ~ I

x

Fig. 12. Notations f o r asymmetrical hammer foundation with six degrees o r freedom

52 Soil D y n a m i c s and E a r t h q u a k e Engineering, 1983, Vol. 2, No . I

Vibration o f hammer foundations: M. Novak and L. El Hifnawy

k24 = 0 k2s = --kv~

k31 = k13 ka2 = k23

ka4 = ku,Yl kas = 0

k41 = k14 k42 = 0

kg4 = k m + ku2 k4s= O

ks i = ki5 k55 = kv, + kv~

k6i = ki6

k26 = --kvt r2

k33 = k~, + ku, y21

k36 = - - k~ + ku l y l y 2

k4a = ka4

k46 = ku~ + ku,Y2

ks6 = k~ir2 + k~r~

k66 = k~2 + k~, + kvlr ~ + ku~r ~ + k m y ~

In these expressions

Yl = d - - e - - h rl =a/2--c

Y2 = e - - f r2 = b--c

and the constants kvl, kul and k~l are the stiffness con- stants of the anvil. They follow from equations (4) to (6) as

kvl = E p A p / h , kul = GpAp/h , k ~ = E p l / h

Constants ku~, kv~, k~o~ and kmo describe the stiffness of the foundation block. For embedded foundations and pile supported foundations, these constants are evaluated using the approaches referred to in the paragraph on stiffness and damping constants of the system.

The damping matrix has the same form as the stiffness matrix and its dements ct! are calculated in the same way as kil except that the constants ku, kv and k~o are replaced by cu, cv and c~. For the anvil, these damping constants follow as a fraction of stiffness from equation (5). The frequency to use is the dominant frequency for the anvil which usually is the second natural frequency and is close to the one given by equation (16).

The mass matrix is diagonal with the diagonal elements being

ml ml I1 m2 m2 12

where ]] and/2 are the mass moments of inertia of the anvil and the block, respectively.

APPENDIX II - NOTATION

A complex constant A. scalar ~p area of anvil or foundation pad a I ratio of modal displacements B i scalar c viscous damping constant or equivalent constant of

viscous damping ch damping constant including material damping Co impact velocity of hammer head

initial velocity of anvil due to head impact D i damping ratio of] th mode E Young's modulus e eccentricity of hammer blow F force J~ generalized force G real soil shear modulus G' imaginary part of complex shear modulus G* complex shear modulus of soil h thickness of pad I second moment of pad area 11, 2 mass moment of inertia kh stiffness constant including material damping kil stiffness constant M/ generalized mass m i mass P force (pulse) q generalized coordinate u horizontal displacement V s shear wave velocity of soft v vertical displacement z variable /3 material damping ratio

loss angle ), complex eigenvalue r variable (dummy time)

complex modal displacement ~b rotation in vertical plane (rocking)

circular frequency ~/ frequency close (or equal) to undamped frequency

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 1 53